Essays on actor models in exchange networks and social dilemmas

ESSAYS ON ACTOR MODELS IN EXCHANGE NETWORKS AND SOCIAL DILEMMAS

RIJKSUNIVERSITEIT GRONINGEN

ESSAYS ON ACTOR MODELS IN EXCHANGE NETWORKS AND SOCIAL DILEMMAS

Proefschrift ter verkrijging van het doctoraat in de Psychologische, Pedagogische en Sociologische Wetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. D.F.J. Bosscher, in het openbaar te verdedigen op donderdag 28 juni 2001 om 14.15 uur door

Marcus Adrianus Leonardus Maria van Assen geboren op 1 april 1972 te Waalwijk

Promotores: Prof. dr. F.N. Stokman Prof. dr. W. Raub Prof. dr. T.A.B. Snijders

Leescommissie: Prof. dr. W.B.G. Liebrand Prof. dr. S.L. Lindenberg Prof. D. Willer

ISBN 90-367-1446-x

Preface It proved difficult to write a preface that was, at least to a certain degree, satisfying to me. Initially I thought that a short and to the point preface, such as “Folks, thanks for everything”, would suffice. This would not be satisfactory though, because it does not answer the question about which folks I thank and for what. The other extreme would have been a preface in which I provide a long list of folks who have made my life enjoyable over the first 348 months of my life general, and in particular over the last 15.8% of it. However, this preface would then have consisted of an enumeration of at least five pages. Moreover, a similar enumeration would be more appropriate in an autobiography than in a preface of a dissertation. Therefore, I have chosen to thank in the preface only those people who have helped me in constructing this dissertation. A consequence of this choice is that I do not mention some folks here who are very much more important to me than the present dissertation. I owe many thanks to Frans Stokman and Tom Snijders, two of my promoters. They formulated the project “Parameter estimation in models of collective decision making” for which I applied and was accepted. Although in my dissertation I estimated zero parameters in models of collective decision making and although I mentioned collective decision making only twice, Frans and Tom have had a big impact on this dissertation. I especially thank Tom for his suggestions regarding the statistical tests of some of my hypotheses. I thank Frans in particular for involving me in his discussions with Dave Willer, for being my guide at conferences, and for introducing me to the research field of network exchange (chapter 6 and 7 of this dissertation) 42 months ago. I also owe many thanks to my promoter Werner Raub and to Chris Snijders. Firstly, Werner stimulated me to work out the relation between individuals’ risk aversion and cooperation in social dilemmas. Later Werner, Chris, and I jointly prepared an experiment and the articles that are the basis of chapters 3 to 5 of this dissertation. The joint discussions with Werner and Chris were not only motivating, but secretly I even enjoyed them. I want to thank Werner in particular for his involvement with the last phase of my writing and his suggestions with respect to the introductory chapter. I appreciated and still appreciate the cooperation with Chris very much. If everybody were like Chris then I would never again work on an article alone. Chapter 2 was the result of a traineeship during my study of mathematical psychology at the University of Nijmegen. However, without Hein Fennema and Peter Wakker this chapter would never have been accomplished. For inspiring and instructive discussions with respect to the subjects in this dissertation I thank Phil Bonacich, Dudley Girard, Károly Takács, Jeroen Weesie, and Dave Willer. I owe many thanks to my wife Marieke van Onna for editing the dissertation, for which she sacrificed days of her time. I thank Jan Kratzer and Ruud Brekelmans for their help with constructing figures, Tamás Bartus for his help with running analyses in Stata, Graeme Blake for correcting my English, and Rita Smaniotto for checking and improving parts of the text.

Contents 1 Introduction 1.1 The importance of the micro-level in predicting and understanding macro-level outcomes in the social sciences 1.2 The importance of studying the micro-level in order to understand macro-level outcomes when macro-level predictions are accurate 1.3 The importance of studying the micro-level in order to predict macro-level outcomes: the case of social dilemmas

1

15

2 Measuring the Utility of Losses by Means of the Tradeoff Method 2.1 Introduction 2.2 History 2.3 The tradeoff method 2.4 The experiment 2.5 Results 2.6 Discussion

19 21 22 24 27 29 34

3 8

3 Effects of Risk Preferences in Repeated Social Dilemmas: A Game-theoretical Analysis and Evidence from Two Experiments 3.1 Introduction 3.2 Effects of risk preferences on cooperation in repeated social dilemmas: theoretical background 3.3 Experiment 1 (Raub and Snijders, 1997) 3.4 Utility measurement 3.5 Experiment 2 3.6 Conclusion and discussion Appendix 3.1: Simulation of other actor's strategy in repeated PDs

41 45 47 51 61 64

4 Effects of Individual Decision Theory Assumptions on Predictions of Cooperation in Social Dilemmas 4.1 Introduction 4.2 The effects of loss aversion on cooperation in social dilemmas 4.3 Probability weighing 4.4 Discussion Appendix 4.1: Proof of Theorem

67 69 69 72 73 75

5 The Effect of Nonlinear Utility on Behavior in Repeated Prisoner’s Dilemmas 5.1 Introduction 5.2 Theoretical background and previous empirical results 5.3 Hypotheses 5.4 Experiment 5.5 Results 5.6 Conclusion and discussion Appendix 5.1: Estimation of utility function Appendix 5.2: Simulation of other agent's strategy in repeated PDs

77 79 79 84 85 89 94 98 99

37 39

Contents 6 Bargaining in Exchange Networks 6.1 Introduction 6.2 Background 6.3 Models and hypotheses 6.4 Hypotheses 6.5 Data 6.6 Results 6.7 Conclusion and discussion

101 103 105 114 116 119 120 131

7 Two Representations of Negotiated Exchange: A Review and Comparison 7.1 Introduction 7.2 Exchange representations in economics and sociology 7.3 Relations between representations of negotiated exchanges 7.4 Consequences of representation inequivalence 7.5 Advantages of the non-reduced exchange representation Appendix 7.1: Equivalence of equiresistance formulations Appendix 7.2: Nash solution in bilateral monopoly situation (II)

137 139 143 152 159 175 179 180

8 Epilogue 8.1 Introduction 8.2 Measuring utility by means of the tradeoff method 8.3 The relation between actors’ utility and their behavior in PDs 8.4 Bargaining in exchange networks 8.5 Two representations of negotiated exchange: a review and comparison 8.6 Conclusions

183 185 186 187 189 191 192

References

195

Samenvatting (Summary in Dutch)

211

1 Introduction

2

Chapter 1

Introduction 1.1

3

The importance of the micro-level in predicting and understanding macro-level outcomes in the social sciences

1.1.1 The importance of the micro-level A central problem in sociology and economics is that of accounting for collective phenomena. Collective phenomena that are addressed in sociology are numerous and diverse, examples of which include suicide rates (Durkheim, 1951 [1897]), group solidarity (Hechter, 1987), revolutions (Coleman, 1990, Ch. 18), and the existence of norms (Coleman, 1990, Ch. 10-11). In economics, a central problem is to predict market prices. The present dissertation is an attempt to contribute to our understanding of two other collective phenomena that receive a lot of attention in social science research in general, and sociological research in particular: outcomes of social dilemmas (chapters 3 to 5) and outcomes in exchange networks (chapters 6 and 7). How do economists and sociologists attempt to explain collective phenomena? One possible mode of explanation of collective phenomena entails the examination of processes inherent to the system, involving its component parts or units at a level below that of the system, known as the micro-level. Coleman (1990, p. 2) refers to this mode of explanation as the internal analysis of system behavior. The internal analysis of the system can be represented in a diagram containing three kinds of relations. The diagram is shown in Figure 1.1 (also in Coleman, 1994, pp. 167-168; Raub, 1984, 1.2; Lindenberg, 1992). Following Coleman, in Figure 1.1 and in the remainder of the text the terminology macro-level is used instead of the collective level.1 The three relations are: (1) Effects of macro- or systemic level phenomena on orientations, preferences, information, and the set of feasible actions of actors (which can be either individuals or organizations) on the micro-level. (2) Actor behavior on the micro-level. (3) The combination or aggregation of these actions, in some institutional structure, to bring about outcomes at the systemic or macro-level. Internal analysis of the system is practised by both economists, in particular microeconomists (Kreps, 1990, p. 3), and some sociologists, in particular those adhering to rational choice theory (Coleman, 1987, p. 167; Wippler and Lindenberg, 1987; Lindenberg, 1992; Raub and Voss, 1981, 1.4). Why do microeconomists and rational choice sociologists ascribe importance to the micro-level in order to predict and understand collective or macro-level phenomena? In the first chapter of his standard work Foundations of social theory Coleman (1990, pp. 3-5) offers five theoretical arguments in favor of the internal analysis of system behavior. Most relevant for the present discussion is his argument that “… an explanation based on internal analysis of system behavior in terms of actions and orientations of lower-level units is likely to be more stable and general than an explanation which remains at the system level” (Coleman, 1990, p. 3)2. Coleman’s argument can also be reformulated more forcefully. A relation between variables on the macro-level is mediated by actors’ behavior on the microlevel. Therefore, if we do not understand processes on the micro-level (relation (2) in Figure 1.1) and the links between micro- and macro-level (relations (2) and (3)), then we have an 1 Although collective phenomena mainly occur at the macro-level, they can also occur at the level of

individuals. Therefore, Figure 1.1 and the terminology macro-level are somewhat misleading. However, I have chosen to remain consistent with Coleman’s work and terminology because of its familiarity. 2 Coleman’s arguments in favor of the internal analysis of system behavior are discussed in more detail in section 6.1 of this dissertation.

4

Chapter 1 System or macro-level 1

3 2

Figure 1.1:

Actor or micro-level

Internal analysis of system behavior represented by three kinds of relations.

incomplete understanding of why and how the relation between the macro variables originated. 1.1.2. How much emphasis on the micro-level? After Max Weber, Homans (1958, 1974) was among the first sociologists to propose the basing of theoretical analyses of macro-level phenomena on a micro-level model of individual behavior. According to Homans there are no general laws that are specifically social, and only psychological general propositions are needed to explain human activity (Willer, 1999, p. 6). Homans adopted these general propositions from behaviorism in learning psychology, according to which the actions of an individual must be understood in terms of the learning history of that individual. Homans’ approach can therefore be regarded as an attempt to reduce sociology to psychology (Wippler and Lindenberg, 1987, p. 139). Homans’ approach has been criticized for two reasons. Firstly, his approach fails in explaining system behavior and its relation to actor behavior in the micro-level. Homans asserts that social structures are a consequence of characteristics of individuals, and that they are an incidental by-product of the everyday activities of individuals (Willer, 1999, p. 6). However, much research, for example the network exchange research as discussed in Chapter 6 and Chapter 7 of this dissertation, demonstrates that structures, independent of characteristics of individuals, can have a large effect on both actor behavior at the micro-level and outcomes at the macro-level. Secondly, the principal task of sociology lies in the explanation of macro-level phenomena, not of the behavior of single actors. Focusing on the behavior of actors at the micro-level can divert attention from the functioning of the system at the macro-level (Coleman, 1990, p. 2). Therefore, Coleman and other rational choice sociologists propose to keep the model of actor behavior at the micro-level as simple as possible.3 They argue that psychological theories of action, such as behaviorism, are too complex for the analysis of collective phenomena (Coleman, 1990, pp. 13-21). Rational choice theory in sociology distinguishes itself from Homans’ approach, which places both analytical and explanatory primacy at the micro-level, by placing analytical primacy at the 3 The phenomenon to be explained at the macro-level is not independent of the specifications and

simplifications of the micro-level model (Lindenberg, 1998). A consequence of simplifying the micro-level model is that less aspects of the macro-level phenomenon can be explained. The simplest model assumptions should therefore always be realistic enough to allow a description of the phenomenon to be explained. Lindenberg (2001) calls this principle sufficient complexity. In order to avoid complicating the discussion in the present chapter, I do not attend to this dependency and principle.

Introduction

5

macro-level but explanatory primacy at the micro-level (Coleman, 1990; Lindenberg, 1985; Wippler and Lindenberg, 1987). The standard rational choice model as used by both sociologists and microeconomists assumes that actors on the micro-level maximize their utility subject to constraints (von Neumann and Morgenstern, 1944; Becker, 1976). In the standard rational choice model actors are assumed to have stable preferences and accurately foresee the consequences of their actions. The strength of the rational choice model lies in its simplicity and analytical tractability. According to its critics, its weakness lies in the doubtful psychological reality of its behavioral assumptions (Flache, 1996, p. 25). Experimental research on individual decision making, mainly carried out by psychologists (see Camerer, 1995, for an overview), demonstrated that individuals systematically violate the rational choice model in a variety of circumstances. However, some rational choice theorists are not concerned with the descriptive (in)accuracy of the standard model. They consider the model as a normative model whose purpose is to prescribe actor behavior at the micro-level, rather than to describe actor behavior (Tversky, 1975). In spite of its failures to describe individual decision making behavior at the micro-level, the majority of rational choice theorists in economics and sociology believe that the standard rational choice model is also adequate enough to provide descriptive theories for a wide range of phenomena at the macro-level (Coleman, 1987). For a justification of this perspective of rational choice theorists, see for example Friedman (1953), Kreps (1990, Ch. 1), Russell and Thaler (1985, p. 1074), Lindenberg (1985), and Thaler (1987). The most common justification is that using the rational choice model does not mean that actors are assumed to be rational, but that actors are assumed to behave as if they were seeking rationally to maximize their utility (Friedman, 1953, p. 21; Becker, 1976, Ch. 5). The above discussion reveals two perspectives on how much emphasis should be placed on the micro-level in our attempt to predict and understand macro-level outcomes. The majority of rational choice theorists choose for a simple and an often descriptively inadequate model of rational action on the micro-level. On the other hand, in similar fashion to Homans, one can attempt to model and describe actor behavior as accurately as possible. In my opinion, the answer to the question of which perspective to take depends on whether one wants to predict or understand the outcomes at the macro-level.4 For accurately predicting the outcomes at the macro-level, the micro-level model need not be realistic, as is explained by Friedman in his book The methodology of positive economics (Friedman, 1953). In his view, economics is a positive science in which theory is only to be judged by its predictive power towards the class of macro-level phenomena to which the theory is applied. According to Friedman (1953, p. 15), “… the relevant question to ask about the ‘assumptions’ of a theory is not whether they are descriptively ‘realistic’, for they never are, but whether they are sufficiently good approximations for the purpose in hand.” Hence Friedman has no problems with adopting the as if rational choice model if it yields accurate predictions of outcomes at the macro-level. It seems that in his view, and in the view of the majority of rational choice theorists, it is primarily important to predict what happens, rather than to understand why things happen (Kreps, 1990, p. 4).

4 The method of decreasing abstraction (see Lindenberg, 1992) is also relevant to the present discussion.

Briefly, according to this method a richer explanation of a macro-level phenomenon requires a more detailed (less abstract) explanatory theory.

6

Chapter 1

Coleman (1990, p. 4) argues that an internal analysis of the system based on actions and orientations of actors at the micro-level provides an understanding of the system behavior that a purely macro-level explanation does not. However, internal analysis of the system is not a sufficient condition for understanding system behavior. Outcomes at the macro-level can only be truly understood when their causes can be identified,5 that is, when the micromodel concerning all three relations in Figure 1.1 is at least an approximately accurate description of what actually ‘happens’ at the micro-level. Only then is the explanation more general and stable in the sense that one can be more confident that the explanatory mechanism including its micro-model can be extrapolated to explain other similar macrolevel phenomena. So what does Coleman’s statement mean? Does it mean that he strives for an understanding of macro-level outcomes through an accurate micro-level model? In my opinion, the answer is no. Suppose that an internal analysis of a system based on a simple rational choice model yields an inaccurate prediction of a macro-level outcome. There are two ways open to a rational choice theorist to enrich the standard model. The first way is uncontroversial in the fields of economics and rational choice sociology. It is to adopt a more complex view of the environment and of the social organization or interactions among actors (Kahneman, Knetsch, and Thaler, 1987, pp. 114-115; Coleman, 1987, 1990), that is, relations (1) and (3) in Figure 1.1. A more controversial way is to complicate the model of the actor at the microlevel, that is, relation (2) in Figure 1.1. In the rational choice tradition the model can be complicated by adding arguments to the actor’s utility function or by abandoning, weakening, or modifying the standard assumptions of rational expectations. Rational choice theorists, including Coleman (1987; 1990, p. 19), prefer the first way to the second. At least three reasons have been put forward to justify their preference. The first, already mentioned reason, is that the functioning of the system is the primary interest of sociologists (Durkheim, 1964 [1895]), including rational choice theorists. A second reason is that adding complexity to the model of the actor makes it more difficult to derive unequivocal predictions of actor behavior from a specification of the environment, and consequently, predictions of the resulting macro-level outcomes. Finally, it appears all too easy to lengthen the list of non-economic motives or cognitive errors that might affect actor behavior (Kahneman et al., 1987, p. 115). Although these reasons are important considerations, they should not always be decisive and lead to heuristics like “… it is especially important that the individual-action component remain simple” (Coleman, 1990, p. 19). Sometimes it is useful to add complexity to the model of the actor at the micro-level. Tradeoffs between the model’s parsimony and accuracy, rather than heuristics, should determine whether to add complexity to the model of the actor or to the links between the micro- and macro-level. On the basis of the discussion in this subsection two guidelines can be derived in order to determine when adding complexities to the model of the actor at the micro-level should be considered. The first guideline is involved with the prediction of macro-level outcomes, and is similar to Kahneman’s (1988, p. 17) guideline for economists of when to be concerned with the empirical validity of assumptions in the model of the actor. The second 5 Of course, the understanding of system behavior is not a simple binary yes-no issue. The argument here is that

(i) the depth of understanding of relationships on the macro-level is increased if their explanation incorporates a micro-level model, and (ii) “truly” understanding corresponds to a depth of understanding that necessitates an approximately accurate description of micro-level processes.

Introduction

7

guideline is involved with understanding macro-level outcomes. (I)

(II)

If the prediction of a macro-level outcome is insufficiently accurate, adding complexity at the micro-level should be considered if (a)

it consistently has a substantial effect on the accuracy of predictions of the macro-level outcome in comparison to the effect of adding complexity to the micro-macro links;

(b)

the boundary conditions for the occurrence of effects on the macro-level outcome are specified with reasonable precision.

A necessary condition for understanding a macro-level outcome is to have at least an approximately accurate model of the actor at the micro-level. Therefore, even when the macro-level outcome can be predicted with reasonable accuracy, the model at the micro-level should be tested and adding complexity to the model should be considered if it is insufficiently accurate.

Of course the guidelines are rather imprecise. It is the researcher who has to judge whether effects are ‘substantial’, and whether a prediction or a model is ‘approximately’, ‘reasonably’, or ‘sufficiently’ accurate. 1.1.3. Preview In this dissertation, complexity is added to the model of the actor at the micro-level in order both to predict and understand better two macro-level phenomena: outcomes in exchange networks and outcomes of social dilemmas. It will be shown that outcomes in exchange networks can often be predicted relatively easily using different actor models. These accurate predictions can be the result of different theories involving different actor models on the micro-leve, and hence accurate predictions do not imply an accurate micro-level model. Therefore, following guideline (II) above, the behavior of actors at the micro-level must be studied in order to understand outcomes in exchange networks. In Section 1.2, examples of macro-level phenomena, including exchange networks, are provided which can be explained using different actor models. Because a considerable number of economic macro-level phenomena can be predicted reasonably accurately using a rational choice or rational actor model, the examples in Section 1.2 are selected from economic research. The section is concluded with a short introduction to the research of exchange networks as reported in Chapter 6 and Chapter 7. Other macro-level phenomena are more difficult to predict reasonably accurately. One example is an actor’s cooperation in social dilemmas. In this field of research, changes in assumptions of the actor model can result in substantial changes to the predictions of macrolevel outcomes. Therefore, following guideline (I) above, we considered adding complexity to the actor model to be able to better predict the outcomes of social dilemmas. Section 1.3 briefly explains how variations in the actor model can result in different predictions. In the research described in this dissertation an attempt is made to explain outcomes of social dilemmas by relating actors’ behavior in social dilemmas to their utilities in a rational actor model. The accompanying hypotheses that are derived and tested in Chapter 3 to Chapter 5 are summarized in Section 1.3.

8 1.2

Chapter 1 The importance of studying the micro-level in order to understand macro-level outcomes when macro-level predictions are accurate

The primary purpose of this section is to demonstrate the importance of studying micro-level processes in order to understand macro-level phenomena. However, this section is also relevant to social scientists, for example Friedman (1953), who are not concerned with the realism of the micro-level model and are satisfied with accurate macro-level predictions. The second purpose of this section is to contradict the belief of a subset of these scientists, not shared by Friedman, that accurate prediction of macro-level outcomes implies that the microlevel (rational actor) model is also correct. 1.2.1 The N* game: Rational equilibrium without rationality This subsection is entirely based on Kahneman (1988). Kahneman (1988) demonstrated empirically that a macro-level outcome resulting from actors who do not behave rationally can be predicted by an economic model assuming rational actors. He and his colleagues studied actor behavior in a game they called the N* game. Fifteen individuals participated in one game. The game, which consisted of 20 periods. In each period a number was announced (3 < N* < 15), with N* varying randomly from period to period. Participants in the game decided independently to each other, in each period, whether to ‘enter’ the market or not. If the number of entrants is denoted by E, the payoff to entrants was equal to $[0.25 + 0.50(N*E)], while the non-entrants received $0.25. Therefore, entrants received more than nonentrants when N* > E, equal sums when N* = E, and less when N* < E. The equilibrium prediction is N* = E because only then does no actor have an incentive to change his behavior. The results of the study are that in the vast majority of trials E-1 ≤ N* ≤ E+1, which is therefore very close to the equilibrium prediction. Conversations with the participants after the experiment showed, however, that most of the participants’ strategies were completely unfounded. Kahneman (1988, p. 12) concludes that “The equilibrium outcome (which could be generated by the optimal policies of rational policies of rational players) was produced in this case by a group of excited and confused people, who simply did not know what they were doing.” He argues that the fact that an equilibrium was achieved is not surprising because there is no stable alternative, and only a little intelligence is required from the participants in order to recognize apparent regularities in the group response to variations of N*. Not only is the assumption of rational actors unnecessary to explain the results of the study, but in addition the results do not present evidence in favor of the assumption. However, theorists of macro-level phenomena do sometimes seem to make the incorrect inference that an accurate prediction implies a valid model of the (rational) actor. In the next subsection an attempt is made to understand why this incorrect inference is made and why it is so persistent. 1.2.2 The fallacy “IF P THEN Q implies IF Q THEN P” and its persistence Empirical psychological research has demonstrated that most people have great difficulties in understanding the logic of the “IF P THEN Q” rule.6 Wason’s selection task was especially devised to test individuals’ understanding of this rule (see, e.g., Eysenck and Keane, 1990 for an overview of research carried out using Wason’s selection task). In this task four cards, with one symbol one each side, are laid down in front of the subjects. The visible side of the 6 Relevant to the present discussion is the work of Popper (1959), who proposes the falsification of scientific

theories on the basis of the if ... then rule.

Introduction

9

cards show respectively P, not-P, Q, not-Q. The subjects are required to select the cards that need to be turned over to test the rule “IF P (on one side) THEN Q (on the other side)”. Of 128 subjects only 5 (4%) gave the correct answer (the P and not-Q cards). The most common error was that subjects selected the P and Q cards (59 out of 128, 46%). This observation should be interpreted with caution, because it is not known, at least not to me, what the subjects actually thought. However, the observation that many people also selected the Q card suggests that they did not realize that if the rule “IF P THEN Q” is true, Q can also be the result of an event other than P. It was as if they believed that “IF P THEN Q” implies “IF Q THEN P”. The relevance of the results of Wason’s selection task becomes clear when substituting for P and Q respectively “actors at the micro-level are rational” and “accurate prediction of macro-level outcomes”. According to the results above the most likely error is the inference that if “accurate prediction of macro-level outcomes” can be the result of “actors at the micro-level are rational” (IF P THEN Q) then these accurate predictions are evidence that actors at the micro-level are indeed rational (IF Q THEN P). Naturally, the inference is false because accurate predictions might also be explained by irrational actors (IF not-P THEN Q). However, there are three reasons for the belief of a large number of economists and rational choice sociologists that P is true. Firstly, the rational choice model has worked reasonably well in some fields of research, for example market research and exchange network research. That is, the rule “IF P THEN Q” has very often been confirmed. The result of repeated confirmation of the rule is that economists (and perhaps also some exchange network theorists and other rational choice sociologists) “attach a very large prior probability (0.9944?) to the proposition that people have consistent utility functions and in fact maximize utilities in an objective sense” (Simon, 1987, p. 38).7 Secondly, because of the strengthening of the belief that P is true, one becomes less inclined to test whether it is indeed true, hence rejections of P have not been observed. Thirdly, because of the researchers’ large prior probability that P is true, even direct tests of P that falsified P have not been taken seriously. Initially the only rejections of P came from research on individual decision making carried out by experimental psychologists in laboraties with usually hypothetical payoffs. This research was considered illegitimate or irrelevant by economists (Russell and Thaler, 1985, 1074; Thaler, 1987). They instead “… tend to look for evidence that the theory makes correct predictions and resist advice that they should look instead directly at the decision mechanisms and processes” (Simon, 1987, p. 38). Later P was also tested by economists themselves, but even then falsifications of P were sometimes disregarded. An illustration of economists disregarding direct micro-level evidence against P comes from research on bidding in first-price auctions (for an elaborated discussion of what follows, see Kagel, 1995, pp. 503-536). In a first-price auction the actor with the highest bid gets the item he is bidding for and pays his bid. It can be demonstrated that a rational actor bids less than his reservation value of the item in the auction, but that the difference between his bid and his reservation value decreases in the concavity of his utility function, or in other

7 A considerable amount of both theoretical and empirical research carried out in the last twenty years, briefly

mentioned in the next subsection “Markets”, has demonstrated that (1) actors are often irrational, and (2) irrational actors can produce macro-level outcomes consistent with the model of rational actors. Therefore, I believe that the prior probability is an overestimation of the “present” prior probability.

10

Chapter 1

words, when he becomes more risk averse (Cox, Smith, and Walker, 1988).8 A typical finding in first-price auctions is that bidding is above the risk-neutral Nash equilibrium. Cox et al. (1988) rationalized this observation by assuming that actors are risk averse. Therefore, they modified the model of the rational actor (P) such that the bids and the macro-level outcome were predicted more accurately (Q). Their approach is criticized for a number of different reasons. I will focus on the criticism that is most relevant to the discussion here, which is the empirical relation on the micro-level between the actor’s risk aversion and his bid (P). Walker, Smith, and Cox (1990) tested this relation on the micro-level, taking into account actors’ utilities by measuring them using the binary lottery procedure (Roth and Malouf, 1979). They found that even after taking into account actors’ utilities, actors bid more than could be expected from risk-neutral bidders. Instead of drawing the conclusion that risk aversion is not the sole, or even the primary factor underlying bidding above the riskneutral Nash equilibrium, they concluded that the binary lottery procedure can not be used reliably to induce risk neutral bidding (Kagel, 1995, pp. 533-534). Therefore, they disregarded direct evidence that conflicted with their model of a rational actor (P). Additional evidence against Walker et al.’s (1990) hypothesis comes from Isaac and James (2000). They found no relation between actors’ risk aversion directly measured using the well-known Becker-DeGroot-Marschak (1964) procedure and risk aversion indirectly measured by the same actors’ bidding in a first-price auction.9 The illustration above confirms three observations. Firstly, a phenomenon could be predicted reasonably accurately using a model of a rational actor, which led researchers to believe that this model is true. Secondly, empirical evidence that contradicts the model was disregarded. Thirdly, presuming that risk aversion is not the factor underlying the phenomenon, bidding above the risk-neutral Nash equilibrium can be predicted but is still not understood. Only a further examination of processes and relations at the micro-level can lead to a better understanding of the phenomenon. The above illustration is also relevant to our research into explaining the outcomes of social dilemmas, described in Chapter 3 to Chapter 5. In these chapters, as in the illustration, actors’ behavior, their degree of cooperation, is related to their utilities in order to obtain better prediction and understanding of outcomes at the macro-level. 1.2.3 Markets The double auction is the most commonly used trading mechanism in experimental markets. In a double auction both buyers and sellers actively post and accept prices publicly. There is typically an “improvement rule” which specifies that bids of buyers (offers from sellers) must be successively higher (lower). An acceptance concludes the deal between the buyer with the highest bid and the seller with the lowest offer, and commonly cancels all outstanding bids and offers. This procedure is repeated until the trading period ends or until the remaining buyers and sellers cannot reach an agreement (see for an elaborate description of the double auction, Holt, 1995, pp. 368-369). A large number of market experiments using the double auction trading mechanism demonstrated that the market price quickly converges to competitive equilibrium outcomes even when (i) no trader knows anything about the 8 It is assumed that actors have constant relative risk aversion. 9 The Spearman rank order correlation between the two measures was even negative (-0.33), although not

significantly so.

Introduction

11

valuation conditions of other traders, (ii) the traders do not have any understanding or knowledge of market supply and demand conditions, or (iii) traders do not have any trading experience (Holt, 1995, 370; Smith, 1962, p. 116; 1982, pp. 945-946) With the example of the N* game still clear in mind, one wonders whether in order to predict prices in markets, assumptions about the actors at the micro-level can also be largely irrelevant. If the latter is true, accurate predictions naturally do not imply that the behavior of actors at the micro-level conforms to rules of rational choice, and therefore an understanding of the macro-level outcomes is lacking. A number of economic studies have been carried out to examine the effects of violations of the assumption of rational actors on the obtained and predicted macro-level outcomes. Results of some of these studies are discussed briefly in the remainder of this subsection to demonstrate that the rational actor model does not always yield accurate predictions of market outcomes. The fact that in some but not all markets the rational actor model yields accurate predictions stresses the need to study the micro-level and the micro-macro link in order to understand when and why the rational actor model can produce accurate macro-level outcomes. Some studies demonstrate that only very little intelligence is required to attain the predicted equilibrium prices in procedures similar to the double auction. Gode and Sunder (1993, see also Roth, 1995, pp. 52-55), for example, compare experimentally observed human behavior with the simulated behavior of what they call “zero-intelligence traders”. Under some general conditions a tendency towards equilibrium prices is observed in markets containing these zero-intelligence traders, who know just enough to avoid buying or selling at a loss. A difference observed on introducing human subjects is that human subjects attain equilibrium prices much faster over repetitions of the market, because human subjects are able to learn while the zero-intelligence traders were not. Other studies focussed on finding cases where deviations from rationality are not corrected by competition in the market and as a result can have consequences on market prices. Three different approaches can be identified in these studies. One approach begins with anomalies at the market level (not-Q) and searches for explanations based on behavioral models of individual choice (P or not-P). Among other studies, Camerer (1995, p. 675) classifies the illustration in Section 1.2.2. under this approach. A second, more theoretical approach starts with models of individual behavior and derives its implications for the macrolevel outcome. For example, Russell and Thaler (1985) demonstrated that the existence of a market is not sufficient to eliminate the effect of quasi-rational behavior on the outcome. Haltiwanger and Waldman (1985) made a distinction between situations where “rational” actors have a disproportionately large effect on the predicted outcome, and situations where “quasi-rational” actors have a disproportionately large effect. They illustrate this distinction with some examples. Akerlof and Yellen (1985) constructed four examples in which small deviations from rationality result in only second-order (small) effects on individual outcomes but in first-order (large) effects on the resultant equilibrium. A third, experimental approach begins with individual “errors” and constructs a market in which they might persist, in order to search for domains where predictions obtained using rational actors might fail. Research on experimental markets demonstrates that rational choice predictions of outcomes of market institutions other than the double auction are often less accurate (Smith, Williams, Bratton, and Vannoni, 1982; Smith, 1982). A more dramatic demonstration that deviations from rationality can have a substantial effect on market

12

Chapter 1

phenomena comes from research on the endowment effect. While Kahneman (1988) presented the N* game to demonstrate that assumptions about actors can be largely irrelevant in some economic situations, he employs the endowment effect to illustrate that deviations from rationality can also have large effects on macro-level outcomes. Endowment effects received considerable attention in psychological and economical research (see e.g. Camerer, 1995, pp. 665-670). The endowment effect refers to people’s preference for the things they own, leading to much higher selling than buying prices for the same object. Note that this is a violation of economic theory, which predicts that the prices involved when a person buys and sells an object should be about the same. Kahneman (1988) and other studies (see Camerer, 1995) demonstrated that these buying-selling price gaps persist in market settings and can have a large effect on the market price. The results of the three approaches briefly outlined above demonstrate that in general the market cannot be relied upon to correct the effect of individual irrationalities in the absence of supporting evidence. According to Tversky and Kahneman (1987, p. 91), the burden of specifying a plausible corrective mechanism should rest with those who make this claim. Russell and Thaler (1985, pp. 1080-1081) continue to remark that, because irrationalities will not disappear in the aggregate, research on individual decision making is highly relevant to economics (and of course to rational choice sociology), even when the goal is to predict behavior. On similar grounds, Camerer (1995, p. 676) concludes that the aggregation question should therefore be studied theoretically as well as experimentally. Additionally, other examples in this section and in Section 1.2.1 demonstrated that in the cases where the effects of actors’ behavior on macro-level outcomes are small, processes at the micro-level should also be studied in order to understand these outcomes. Despite these recommendations from some economists, mainstream economics is not concerned with elaborating on and testing micro-level assumptions. In the next section we will see how some rational choice sociologists deal with the micro-level when examining exchange networks. 1.2.4 Exchange networks Exchange as a typical example of bargaining is intensively studied in economics, and also in social psychology and sociology. The issue under investigation in Chapter 6 and Chapter 7 is a particular type of exchange, called negotiated exchange. Its relation to other types of exchange is discussed in Chapter 7. Negotiated exchange involves a joint decision process, such as bargaining, to determine the terms of exchange (Cook, 1995; Molm, 1997; Molm, Takahashi, & Peterson (2000). The terms are agreed on at the same time, even if the transactions do not occur simultaneously. In general and also in our investigation it is assumed that the agreements are strictly binding, that is, the distribution of outcomes follows directly and automatically from an agreement. Negotiated exchanges where the agreements are strictly binding can be considered as cooperative games in game theory. There are two representations of negotiated and other types of exchange that are most common in economic and sociological research. The non-reduced exchange representation characterizes exchange situations by actors having different distributions of endowments and different interests or utilities in these endowments. In Chapter 7 it is explained that the nonreduced exchange representation, borrowed from pure exchange in economics, is more valid and more general than the other representation, known as reduced exchange representation, which characterizes exchange by a division of a fixed-prize or common resource pool. A

Introduction

Figure 1.2:

13

Reduced exchange representation of a simple exchange network.

review and comparison of both representations is also provided in Chapter 7. The reduced exchange representation is much more common in exchange network research than the non-reduced representation. The example in this subsection and our research on actor bargaining in exchange networks reported in Chapter 6 is also based on the reduced exchange representation. The reduced exchange representation of a very simple exchange network is presented in Figure 1.2. Two actors, B1 and B2 have the opportunity to exchange with actor A, that is, to divide a common resource pool or price of 24 valuable units, e.g. units of money. A description of the common ‘rules of the game’ or bargaining protocol is provided in Section 6.2.1. An exchange network can be considered as a market with additional restrictions determining who can exchange with whom. If exchange networks are markets the question arises whether outcomes in exchange networks can be predicted as accurately as outcomes in markets like the double auction. A large number of empirical studies have been carried out on simple exchange networks like the one in Figure 1.2 under the so-called 1-exchange rule. This rule specifies that an actor at each position can make a maximum of one exchange during a bargaining period. For the network in Figure 1.2 this implies that one of the B actors is necessarily excluded from exchange. A number of theories of network exchange have been developed to predict the outcomes in exchange networks operating under the 1-exchange rule. Theories have been developed by, for example, Berg and Panther (1998), Bienenstock and Bonacich (1992), Braun and Gautschi (2000), Cook and Emerson (1978), Friedkin (1992), Markovsky, Willer, and Patton (1988), and Yamaguchi (1996). Although there is some debate about which theory yields the best predictions, the predictions of all seven theories are often surprisingly accurate. Therefore theorists conclude that the theories and their experimental tests show how “accounting for decisions at the levels of individuals and relations can improve our understanding of structural effects, and that only by accounting for structural contingencies can lower level processes and outcomes be fully comprehended.” (Markovsky, Skvoretz, Willer, Lovaglia, and Erger, 1993, p. 208). The conclusion that market outcomes were the result of actors behaving rationally at the micro-level turned out to be a fallacy. Could the conclusion above with respect to the predictive success and understanding of exchange network outcomes also be a fallacy? In my opinion the answer is yes, for two reasons. Firstly, although nearly one hundred studies have been carried out on exchange networks, only a handful focussed on actor bargaining at the micro-level. Secondly, the seven exchange network theories mentioned above are very different from each other with respect to their assumptions about micro-level processes (see Skvoretz and Willer, 1993). Two of the theories (Berg and Panther, 1998; Braun and Gautschi, 2000) are even based on a bargaining protocol that is entirely different to that used in the network exchange experiments. Still they provide accurate predictions. Both reasons suggest that a large number of models of the actor at the micro-level can produce accurate predictions of macro-level outcomes. They also suggest that the conclusion cited above is too premature. Following guideline (II) in Section 1.1.2, not accurate predictions but rather

14

Chapter 1

studies of micro-level processes can improve our understanding of how actor bargaining at the micro-level determines exchange outcomes at the macro-level. To illustrate that a large number of models of the actor at the micro-level can produce accurate predictions of macro-level outcomes, consider the exchange network in Figure 1.2. In experiments the B actors obtain outcomes close to the minimum value of 1 while A obtains the remaining points in the exchange. Markovksy (1987) simulated different combinations of actor strategies and found that not every combination yielded an extreme resource division. However, the variations of actor strategies that he used were unrealistic and are therefore cannot be regarded as evidence against the hypothesis that a large number of actor strategies can produce the extreme resource division. On the basis of his simulations, Markovsky (1987) suggested that actors’ reactions to exclusion and inclusion in the previous round were important in generating the extreme resource distributions. Reactions to exclusion and inclusion can indeed be sufficient to generate extreme distributions in the long term. For example, consider the following family of strategies for the Bi actors xi ,t = xi ,t −1 − ci ei ,t −1 + d i (1 − ei ,t −1)

(1)

where a strategy is characterized by three integer-valued parameters (1 ≤ xi,0 ≤ 23; 0 ≤ ci ≤ 5; 0 ≤ di ≤ 5). The offer of Bi to A at round t is denoted by 1 ≤ xi,t ≤ 23, and ci and di denote the increase in the offer to A after exclusion (ei,t-1 = 1) and inclusion (ei,t-1 = 0) respectively in the previous round. The offer xi,t obtains a value equal to 1 when the right side of (1) produces a value smaller than 1, and 23 when the right side of (1) produces a value larger than 23. If it is assumed that A accepts the best offer of x1,t and x2,t then it can be easily shown that all strategy combinations with c1 + c2 > d1 + d2 yield extreme resource distributions after a large number of rounds. However, actor bargaining is certainly not described accurately by any strategy from this family. The study in Chapter 6 demonstrates that actors usually make a large number of offers to each other in one round, not zero (as A does in the simple model here) or one (the B’s in this model). The preceding discussion emphasizes that actor bargaining should be studied in order to understand how it mediates the relation between network structure and network exchange outcomes at the macro-level. Simulation studies and the handful of experimental studies of actor bargaining, perhaps partly because of Markovsky’s (1987) study, mainly focussed on the effects of exclusion and inclusion in combination with actor position in the network. Unfortunately, however, these studies neglected other main variables involved in actor bargaining in exchange networks. That is, they do not provide a model or explanation of the whole bargaining process, which consists of actors making initial offers to each other followed by concessions when the initial offers do not complete an agreement. Chapter 6 summarizes the studies carried out on actor bargaining in exchange networks. More importantly, a micro-level model of actors’ initial offers and concessions is constructed and tested using experimental data. Although the model is based on insights from the (mainly bargaining) literature, it is not derived from a rational actor model. Because accurate macrolevel predictions can be produced with either a rational actor model or a different micro-level model, the rational actor model is abandoned in order to discover how actors actually bargain. The main aim of the study reported in Chapter 6 was to construct a model with which it is possible to find out empirically what variables determine actor bargaining behavior, and to test and estimate the effects of these variables. The study focuses not only on the effects of

Introduction Table 1.1:

15 An example of a PD game. Actor 2

Actor 1

Cooperate Defect

Cooperate (R1 = 0, R2 = 0) (T1 = 10, S2 = -15)

Defect (S1 = -15, T2 = 10) (P1 = -10, P2 = -10)

exclusion and inclusion in the previous round, and the position of the actor in the network, but also on the effects of bargaining time and differences between actors. Therefore, the study can be considered as an exploratory investigation of actor bargaining in exchange networks. On the basis of this exploratory study, a model of boundedly rational actors that can predict both exchange outcomes and actor bargaining at the micro-level, can perhaps be developed. 1.3

The importance of studying the micro-level in order to predict macro-level outcomes: the case of social dilemmas

1.3.1 Introduction Naturally, macro-level phenomena whose outcomes are sensitive to actor behavior or to changes in actor behavior also exist. While in sections 1.2.1 and 1.2.3 situations were discussed where actor behavior had only small effects on macro-level outcomes, a few market studies were also discussed in Section 1.2.3 demonstrating that in certain situations actor behavior can have large effects on macro-level outcomes. Unsurprisingly, particularly in these situations it is more difficult to construct a model of the actor that allows the generation of accurate predictions of macro-level outcomes. It is also unsurprising that rational choice theorists are more willing, or less resistant, to examine micro-level processes when predictions of macro-level outcomes are inaccurate than if they are accurate, as for the phenomena discussed in Section 1.2. Because the examination of micro-level processes is less controversial in the case of inaccurate macro-level predictions, we will directly proceed with a discussion of issues relevant to this topic, which is the focus of chapters 3 to 5 of this dissertation; that is, cooperation in social dilemmas, or more specifically, cooperation in repeated prisoner’s dilemma (PD) games. 1.3.2 Cooperation in the PD: failure of the standard rational choice model Social dilemmas are characterized as situations in which individual or actor interests are opposed to the interests of the group to which the actor belongs. That is, in these situations each actor is best off when he acts in his own private interest, however, when all actors in these situations act in their own interests they all obtain an outcome that is worse than if they had not acted in their own interests. This paradoxical situation can be illustrated with a famous example of a social dilemma with only two actors, the PD game. An example of a PD game, which is also used in the experiment analyzed in Chapter 3 and Chapter 5, is presented in Table 1.1. Assume that actor 1 and actor 2 play the PD game once, also called a one-shot PD game, with outcomes T = 10 > R = 0 > P = -10 > S = -15. Furthermore, assume that as in the standard rational choice model, actors attempt to maximize their own utility and that the outcomes represent utilities. Then the dominating strategy of both actors is to defect (T > R and P > S). Therefore, if they act in their own interests both actors will defect, but this leads

16

Chapter 1

to them both obtaining a worse outcome (P) than if both had chosen to cooperate (R). The standard rational choice model predicts that actors in a one-shot PD game defect, that is, it predicts a macro-level outcome that in principle could have been better for both actors. Consider a PD game that is repeated a fixed number of times, for example 100, with the number of repeats known to both actors. The standard rational choice model predicts that actors will also defect on each repetition of the game (Luce and Raiffa, 1957, pp. 97-102). However, when actors in experiments are confronted with either a one-shot PD or a repeated PD game, a large number of actors choose to cooperate instead of to defect (e.g., Cain, 1998; Roth, 1995, pp. 9-10; Ledyard, 1995; van Lange, Liebrand, Messick, and Wilke, 1992). Therefore, the standard rational choice model, based on the assumption that actors maximize their own outcomes, yields inaccurate predictions. Naturally, an inaccurate model is unsatisfactory and one would like to complicate his model in order to improve its predictive accuracy. In this case it is not obvious how complexity can be added at the macro-level. The macro-level only consists of the structure of the PD (Table 1.1) and the result of the choices of the two actors (that is, the outcomes of one of the four cells of Table 1.1). Therefore, adding complexity to the model of the actor seems to be the most appropriate way to proceed. This can be done in one of two ways. Firstly, one can abandon the rational choice approach and attempt to find characteristics both of the dilemma situation and of individuals that have an effect on the individuals’ cooperation in social dilemmas in general, and in prisoner’s dilemmas in general. Many studies have succeeded in this attempt in the sense that factors that influence cooperation and were seemingly irrelevant according to the standard rational choice model, have been identified (van Lange et al., 1992).10 A second approach is to enrich the standard rational choice model by adding extra arguments to the utility function of the actor. Note that guideline (I) in Section 1.1.2 states that this procedure should only be considered if the effects of these extra arguments on the macro-level outcome are substantial. In the next subsection it is argued that this is indeed true in the case of predicting cooperation in the PD game. 1.3.3 Predicting cooperation in the PD game: adding complexity to the rational choice model Cooperation in one-shot PD games or in PD games that are repeated a fixed number of times can be explained by a rational choice model when actors’ utilities are made dependent on variables other than their own outcomes. For example, cooperation can be rational when actors possess a sufficient degree of altruism, or when actors are motivated by considerations of fairness, or when preferences are influenced by certain kinds of moral judgments (Cain, 1998, p. 136). These factors may transform the payoff matrix of the PD game such that defection is no longer the dominating strategy. For example, consider the following social orientation or social value model of an actor i’s utilities (see for example also Cain, 1998; Taylor, 1987; Liebrand and McClintock, 1988; Weesie, 1994; Snijders, 1996): ui ( x i , x j ) = x i + θ x j

(2)

10 Factors can be considered to be irrelevant for the standard rational model when they are not related to the

strategic aspects of the situation. Two factors seemingly irrelevant for the standard rational choice model that are found to affect cooperation in social dilemmas, are: communication without commitment or “cheap talk”, and perceptions of other’s characteristics (Van Lange et al., 1992).

Introduction

17

The model asserts that i’s utility is a function of his outcome xi and the other actor’s outcome xj times a constant θ, which can be considered as a social orientation parameter. For positive values of social orientation defection no longer becomes the dominating strategy. More specifically, if 1/4 < θ < 2/3 in our example the structure of the PD game changes such that actor 1 prefers outcome (S1 = -15, T2 = 10) over outcome (P1 = -10, P2 = -10). For even larger values of social orientation, in our example 2/3 ≤ θ, cooperation even becomes the dominating strategy; that is, an additional change in the preferred order of outcomes of actor 1 is that he now also prefers outcome (R1 = 0, R2 = 0) over outcome (T1 = 10, S2 = -15). A rational choice theorist might argue that he has found an explanation of cooperation in PD games; actors cooperate because they also take the outcome of the other player into account. However, it should be noted that this would be a mistake, like that in the illustration of Section 1.2.2. It is not sufficient to add theoretical postulates about the arguments or shape of the utility function (Simon, 1987, p. 28). One should explicitly test at the micro-level whether there is indeed a relation between an actor’s social orientation and the likelihood that he cooperates in a PD game. Some studies indeed report a correlation between these two variables (Cain, 1998; van Lange, 1999), demonstrating the potential success of the approach to complicate the rational choice model. 1.3.4 The effect of nonlinear utility on behavior in repeated prisoner’s dilemmas In chapters 3 to 5, actor behavior is examined in another version of the PD game, the indefinitely repeated PD game. In each round of this game there is a known constant probability 0 < p < 1 that the game will continue into the next round. In this game, as opposed to the one-shot PD game and the PD game with a fixed number of repetitions, cooperation can be rational behavior of a self-interested egoistic player. A well-known result derived from game theory, a popular branch of rational choice theory, is that conditions for cooperation become more favorable when either of two conditions apply (e.g., Friedman, 1986, pp. 88-89, or chapters 3 to 5 of this dissertation): Firstly, when the shadow of the future (value of p) is increased, and secondly, when the utility function of both actors for the range of outcomes in the PD game becomes more concave. Concavity of utility corresponds to diminishing marginal utility; each additional unit of the outcome yields less additional utility than the previous unit of the outcome. The relation between the concavity of an actor’s utility function and his propensity to cooperate in an indefinitely repeated PD is derived and tested in chapters 3 to 5. In Chapter 3 the relation is tested for PD games with only negative or only positive outcomes. In Chapter 4 it is derived that loss aversion should improve conditions for cooperation in mixed PD games with both positive and negative outcomes, as in our example in Table 1.1. In Chapter 5 the relation is tested for four PD games, where one has only negative outcomes, one has only positive outcomes, and two are mixed PD games. In the experiments reported the actor’s utility was assessed using two methods. One method assumes that actors are rational and make decisions based on a standard theory of rational choice in economics, known as Expected Utility theory. The other method, known as the tradeoff method, assumes that actors are only boundedly rational and do not treat probabilities objectively. The tradeoff method is explained and related to other utility assessment methods in Chapter 2.

2

Measuring the Utility of Losses by Means of the Tradeoff Method

Fennema, H. (2000). Decision making with transformed probabilities. Unpublished doctoral dissertation, NICI, University of Nijmegen, The Netherlands, pp. 57-76. Fennema, H. & Van Assen, M.A.L.M. (1999). Measuring the utility of losses by means of the tradeoff method. Journal of Risk and Uncertainty, 17, 277-295.

20

Chapter 2

Abstract This paper investigates the shape of the utility function for losses. From a rational point of view it can be argued that utility should be concave. Empirically, measurements of the utility for losses show mixed results but most evidence supports convex rather than concave utilities. However, these measurements use methods that are either biased by the certainty effect or require complex parametrical estimations. This paper re-examines utility for losses, avoiding the mentioned pitfalls by using the tradeoff method. We find that utility for losses is convex. This is contrary to common assumption in the economics literature. Also, we investigate properties of the tradeoff method showing a new violation of procedure invariance. Our findings demonstrate that diminishing sensitivity is an important phenomenon for utility elicitation.

Key words utility elicitation, prospect theory, tradeoff, risk aversion, diminishing sensitivity, procedure invariance.

Acknowledgments Peter Wakker improved the present paper with comments and discussions. Mathieu Koppen gave many helpful comments. Mark Rijpkema conducted part of the experiment.

Measuring the Utility of Losses by Means of the Tradeoff Method 2.1

21

Introduction

For economists it is a truism that the utility of money is marginally decreasing, and therefore that utility is concave (Bentham, 1789; Marshall, 1920; Samuelson, 1937; Pratt, 1964). Empirically, this has indeed been well established for gains (Friend and Blume, 1975; Cohn, Lewellen, Lease, and Schlarbaum, 1975; Wolf and Pohlman, 1983; Szpiro, 1986). For losses the situation is less clear. Prospect Theory (Kahneman and Tversky, 1979; Tversky and Kahneman, 1992) has suggested the very opposite of classical economics for losses, that is, utility is convex rather than concave. We first review the experimental literature concerning the shape of utility for losses. As will be shown, most measurements done so far indicate a convex utility for losses but the evidence is not conclusive. Hence for losses one of the most basic aspects of utility, that is, whether marginal utility is increasing or decreasing, is as yet an unsettled question. A major difficulty lies in the very measurement of utility. Experimental research in decision making shows that responses to choice problems are affected by many factors not covered by Expected Utility theory. One major deviation from Expected Utility was shown as early as Allais (1953): people are very attracted to sure gains compared to risky prospects (certainty effect). This implies that the most widely used methods for utility elicitation, the certainty equivalence and probability equivalence methods (Farquhar, 1984), will yield biased utilities because these methods are based on Expected Utility (Hershey, Kunreuther and Schoemaker, 1982; Hershey and Schoemaker, 1985; Johnson and Schkade, 1989). McCord and de Neufville (1986) proposed a utility elicitation method that avoids the certainty effect. Their method is, however, still prone to probability distortion, another deviation from Expected Utility. To side-step the described biases we measure the utility of losses assuming Cumulative Prospect Theory (CPT), introduced by Tversky and Kahneman (1992); a similar theory was developed by Luce and Fishburn (1991) and Starmer and Sugden (1989). CPT retains the familiar reference point of original prospect theory thus permitting losses to be treated differently than gains. CPT generalizes Expected Utility by allowing (rank-dependent) probability distortions, thus accommodating major empirical deviations from Expected Utility such as the certainty effect, the common consequence effect and the common ratio effect. CPT also allows distortion of probabilities to depend on the sign of the outcome (signdependence). Because of the above characteristics, CPT allows for a considerable improvement of utility elicitation. Assuming more complex models like CPT does not solve problems concerning utility elicitation by itself. To derive a utility from a certainty equivalence or probability equivalence method assuming CPT avoids biases due to the certainty effect, but introduces the problem that a weighting function must be known to measure utility (and vice versa). By simultaneously estimating both functions, a utility can be derived but this involves statistically complicated estimations under parametrical assumptions concerning both functions. To avoid problems concerning parametrical assumptions and estimation procedures, we adopted the tradeoff method (TO method; Wakker and Deneffe, 1996) to elicit utility. This method allows utility elicitation without knowledge or special assumptions about the weighting function. This method is also valid for other theories that transform probabilities,

22

Chapter 2

such as original prospect theory (Kahneman and Tversky, 1979), transformation of fixed outcome probabilities (Edwards, 1962), rank-dependent transformation (Quiggin, 1982), and prospective reference theory (Viscusi, 1989). Since the TO method has been developed only recently, we give a self-contained presentation in Section 2.3. Section 2.3 also describes two different TO procedures that were employed to further study the robustness of the TO method. Section 2.4 reports results, showing convexity for losses and a new violation of procedure invariance. First, we turn to a discussion of evidence and arguments concerning the utility of losses presently available in the literature. 2.2

History

Describing the origins of the concept of utility, Stigler (1950, p.63) notes that 'the principle that equal increments of utility-producing means (such as income or bread) yield diminishing increments of utility is a commonplace'. In other words, an amount of money (say $100) is of more value to the pauper than to the millionaire. This principle of diminishing marginal utility provides a rational argument for concave utility. The concept of concave utility was introduced as early as Cramer (1728) and Bernoulli (1738) to explain commonly found behavior in the St. Petersburg paradox. In Expected Utility theory a concave utility explains commonly observed risk-averse behavior. Concave utility is also required in Debreu's (1959) derivation of equilibria for economic transactions. Since the introduction of prospect theory by Kahneman and Tversky (1979) it is also a commonplace that people evaluate decisions with respect to a reference point (usually the status quo), allowing them to frame decisions in terms of gains and losses (see also Markowitz, 1952; Edwards, 1954; Laughhunn, Payne, and Crum, 1980). With respect to the value of outcomes, the distinction between gains and losses is central in prospect theory. First, it yields the means to model the well-established phenomenon of loss aversion ('losses loom larger than gains'). Second, but more important for our purpose, it permits to conjecture that the utility (or 'value') of losses is convex rather than concave. This contradiction between the hypothesis of prospect theory and the economic prediction based on diminishing marginal utility calls for an empirical investigation. Although Expected Utility theory with a concave utility1 has been the leading theory in decision making since von Neumann and Morgenstern (1944), utility functions have not been measured extensively (Stigler, 1950; Farquhar, 1984). In particular the utility of losses has received little attention. Moreover, the few experiments investigating the shape of the utility function for losses show diverse results, to which we turn next. Using small amounts of money, Davidson, Suppes, and Siegel (1957) found 3 concave, 7 convex, and 5 linear utility functions for losses. Officer and Halter (1968) tested the utility of the cost of fodder reserves for five farmers. Two farmers showed a convex utility for losses, two farmers showed a concave utility function, and one farmer had a utility that was nearly linear. Green (1963) elicited utility for four managers in large chemical companies. All four showed concave utilities for losses. Swalm (1966) elicited the utilities 1 It is controversial whether marginal utility used in risky decisions is equivalent to marginal utility used in

riskless situations. Our discussion is therefore restricted to marginal utility for decision under risk. For a discussion on this issue see Dyer & Sarin (1982), Wakker (1994), and Smidts (1997). For some results showing concave utilities for gains see Parker and Schneider (1988). For a review of experiments on utility in riskless situations see Galanter (1980).

Measuring the Utility of Losses by Means of the Tradeoff Method

23

for gains and losses for 'about 100' executives. The paper reports only a non-random selection of 13 cases, the majority (10) of which showed convex utilities for losses. Fishburn and Kochenberger (1979) reanalyzed 26 cases from various previous papers, including the 13 reported cases of Swalm (1966). Their conclusion that the majority of functions are convex for losses is therefore heavily based on the selection made by Swalm. Besides that, still 10 cases (38%) reviewed in their paper showed a concave utility function for losses. Kahneman and Tversky (1979) refer to these results when they present their well-known value function; no new experimental evidence is provided. Besides being far from conclusive, the above reports are biased in favor of convexity. Measurements were done using the certainty equivalence and probability equivalence methods that are biased because Expected Utility was assumed (cf. Farquhar, 1984). These methods require a gamble on the one hand and a sure outcome on the other such that the subject is indifferent between both. For example, a subject may be indifferent between a prospect that yields a 50% probability of losing $100 or else nothing and a sure loss of $40. In these situations it is well known that the attractiveness of the gamble is enhanced because it gives at least a chance of breaking even, instead of a choice for a sure loss.2 For losses, the certainty effect therefore enhances risk seeking.3 If we correct for this certainty effect, preferences that can be explained by Expected Utility with a convex utility may also be compatible with a linear or even concave utility for losses, depending on the magnitude of the certainty effect. Therefore, many experiments that find risk seeking for losses (e.g. Libby and Fishburn, 1977; Laughhunn, Payne and Crum, 1980; Currim and Sarin, 1989; Erev and Wallsten, 1993) do not necessarily imply that subjects show convex utilities for losses. Recently, Tversky and Kahneman (1992) found evidence for convexity of the utility for losses. They corrected for the certainty effect by assuming CPT and simultaneously estimating utility and weighting functions. Their procedure has the disadvantage that parametric assumptions concerning both functions are needed. They report no individual, but only aggregated results. Based on the median parametric estimate for the utility they conclude that the utility of losses is convex for most subjects tested. Gonzalez and Wu (1999) simultaneously fitted utility and weighting functions in a way that is not restricted to specific parametric families. Their study also reports mainly convex utilities. Shortly after we performed our study, Abdellaoui (2000) measured the utility of losses using the TO method, also reporting mainly convex utility for losses. If a convex utility for losses is not considered rational but still is accepted as the common utility in prospect theory, an explanation is needed. The explanation offered by Kahneman and Tversky is based on diminishing sensitivity. This phenomenon is well known in psychophysics: a (physically) constant difference between stimuli is more easily detected near the natural reference point than farther away from it. It causes psychological sensitivity functions to, for example, auditory stimuli to be concave. A compelling example of diminishing sensitivity when applied to the perception of monetary values can be found in Savage (1954). Savage describes the event of a man buying a car. The man wants to buy a car at $2,134.56 and is tempted to buy the car with a radio installed at $2,228.41, 'feeling that the 2 Note that also a 'breaking-even' effect may exist, further enhancing risk seeking. 3 The certainty effect has been studied mostly for gains, where it enhances the attractiveness of the sure gain

and thus enhances risk aversion.

24

Chapter 2

$1500

$2000

1/3

1/3

2/3

2/3 - $25

Figure 2.1:

?

A stimulus used in the outward TO procedure for losses.

difference is trifling'. However, reflecting that if he already had the car, he would certainly not pay $93.85 for the radio, he decides not to buy the radio. This change of preference can obviously not be rationally explained in terms of final wealth. Rather does it show that the perceived difference between $0 and $93.85 is larger than the perceived difference between $2,134.56 and $2,228.41. For losses, a similar argument can be given: the larger the loss, the more trivial a small increase of the loss appears. So it is plausible that the perception of numbers, rather than intrinsic value, causes measurements of utility for losses to be convex. 2.3

The tradeoff method

First CPT is briefly described, restricted to the types of prospects that will be considered in this paper. In all cases probabilities are given and there are at most two outcomes. A prospect is denoted by (x1,p1; x2,p2) yielding $xi with probability pi. Because there are only two outcomes, p2 equals 1−p1 but for consistency of notation we maintain p2. The CPT value of a prospect is the sum of separate evaluations of gains and losses, that are both rank-dependent (Quiggin, 1982). That is, weighting functions w+ and w− are defined for probabilities associated with gains and losses, respectively. In the case of a mixed prospect, where x1>0>x2, the CPT value is given by w+(p1) u(x1) + w−(p2) u(x2)

(1)

The CPT value of a positive prospect, where x1 ≥ x2 ≥ 0, is given by w+(p1) u(x1) + [w+(p1 + p2) − w+(p1)] u(x2)

(2)

and the CPT value of a negative prospect, where x1 ≤ x2 ≤ 0, is given by w−(p1) u(x1) + [w−(p1 + p2) − w−(p1)] u(x2)

(3)

Note that these formulas all agree with original Prospect Theory (Kahneman and Tversky, 1979), so that our analysis and experiment apply to this theory as well. As usual, weighting functions are increasing with w−(0) = w+ (0) = 0 and w−(1) = w+(1) = 1. Next we show how,

Measuring the Utility of Losses by Means of the Tradeoff Method

$1500

$2000

1/3

1/3

2/3

2/3 - $60

Figure 2.2:

25

?

A second example from the outward TO procedure for losses.

assuming CPT, the TO method elicits utility without requiring any assumptions about the form of the weighting functions (details and proofs are offered by Wakker and Deneffe, 1996). The subject's task is to choose between two prospects (see Figure 2.1) that are both defined on the same events E1 (upper branch) and E2 (lower branch) having probabilities p1 (1/3) and p2 (2/3) respectively. If event E1 occurs the right prospect yields more money than the left (an additional $500 in the example). If E2 occurs, the left prospect yields a smaller loss than the right. So, the task for the subject is essentially to make a tradeoff: does the extra money that the right prospect yields in the case of E1 outweigh the smaller loss that the left prospect yields in the case of E2? The free value (labeled '?' in Figure 2.1) is varied in order to find the preference switching point. For example, if the free value is −$26, the possibility of losing one additional dollar (if one chooses the right prospect) in the case E2 occurs can be easily outweighed by the chance of receiving an additional $500 if E1 occurs. If the additional loss of the right prospect in event E2 is increased, at some point the tradeoff will turn in favor of the left prospect. Let us assume, for example, that the preference switching point where the subject is indifferent between the left and right prospect lies at −$60. The first indifference point by itself does not give any clear information about the utility function, because the choices made depend crucially on how the subject transforms the probabilities of both events. It tells us that the CPT value of the left prospect equals the CPT value of the right, which, by applying (1), results in the following equation w+(p1)[u($2000)−u($1500)] = w−(p2) [u(−$25) − u(−$60)]

(4)

Exact information about the utility function becomes available with the responses to the subsequent questions. For the second question the experimenter changes the −$25 value of the left prospect, for example to the amount of money that was the preference switching value in the first series, −$60 (see Figure 2.2). The preference switching value of the new question may then be at −$100, leading to the following equality based on CPT:

26

Chapter 2

$1500

$2000

1/3

1/3

2/3

2/3 ?

Figure 2.3:

-$150

An example of a stimulus from the inward TO procedure for losses.

w+(p1)[u($2000)−u($1500)] = w−(p2) [u(−$60) − u(−$100)]

(5)

Combining (4) and (5) and dropping the common factor w−(p2), it can be seen that u(−$25) − u(−$60) = u(−$60) − u(−$100)

(6)

The procedure proceeds by replacing the value −$60 by −$100 in the left prospect in Figure 2.2. The subject may now be indifferent between the left and the right prospect if, for example, the value −$150 is substituted for the question mark in the right gamble. This would reveal that for the subject the utility difference between −$100 and −$150 is equal to both utility differences in (6). A series of such questions can be asked, each time replacing the value set by the experimenter with the preference switching value given by the subject in the previous question. In this way, we can measure a series of monetary values that are equally spaced in utility units. Note that the TO procedure is chained, in the sense that an answer given by the subject is used as input in the next question. Because the intervals thus generated are increasingly farther away from the reference point, this procedure is called the outward TO procedure. The TO method can be modified such that a series of intervals is generated that goes towards zero as the procedure progresses. In this case, a tradeoff has to be made where the right prospect is fixed and the left prospect has a free value that is varied to find the preference switching point (see Figure 2.3). This is called an inward TO procedure. The equalities that are derived are independent of w+ and w−. The only assumption to be made is that subjects do not change the 'weighting' of the probabilities p1 and p2 during the experiment, an assumption that must be made for any elicitation procedure. Other requirements of the application of the TO method assuming CPT can be satisfied by a proper choice of stimuli. First, the outcomes associated with E1 of both prospects must have the same sign, as well as the outcomes associated with E2. (The inward TO method therefore stops if the next monetary interval passes zero.) Also, the rank-order of the outcomes of both

Measuring the Utility of Losses by Means of the Tradeoff Method

- $75

- $275

2/3

2/3

1/3

1/3 $200

Figure 2.4:

27

?

The stimulus used in the outward TO procedure for gains.

prospects may not change during the process. This is satisfied in the example because the outcomes associated with E1 are positive ($1500 and $2000) and therefore are always the preferred outcomes. 2.4

The experiment

The experiment was conducted with 64 undergraduate students at the University of Nijmegen, most of whom were majoring in psychology. The actual number of subjects was 68, but the data of 4 subjects were discarded. The data of three subjects clearly indicated that they had not understood the instructions. One subject did not finish the experiment. The experiment was ran individually in two separate sessions (a single session would be strenuous).4 The first session took about 50 minutes. The second session took less time (about 40 minutes), because the instruction was shortened and the subjects were more experienced at the task. Subjects were paid a fee of Hfl 15 for participation (about $8). Some subjects preferred course credit instead of money. We used hypothetical payoffs; the responses of the subjects did not affect the reward subjects received for participation. We would prefer using real incentives. The obvious problem is that the experiment involves losses. We could first give subjects money such that losses would be covered, but then a considerable part of the subjects would probably treat this money as 'house' money which distorts the responses (Thaler and Johnson, 1990). Other solutions also introduce income effects. In addition, studies using real incentives yield results very similar to studies using hypothetical payoffs (Tversky and Kahneman, 1992; Camerer, 1995; Beattie and Loomes, 1997). Given the many problems for real incentives for losses, the virtually general finding of no difference between real and hypothetical payment in individual choice under risk, and the importance of settling the nature of marginal utility for losses, we have decided to carry out our experiment without using real incentives. Each subject completed a utility measurement procedure for gains and losses, in separate sessions, with the order counterbalanced. Each session consisted of three parts, all combined with an introduction. The first part was an outward TO procedure consisting of four chained questions. The starting prospects that were used are illustrated in Figures 2.1 4 A computer program was employed that can be obtained from the second author upon request.

28

Chapter 2

Table 2.1:

Percentage of convex, concave and linear parts for losses and gains. Losses

Convex Concave Linear

Outward 47% 34% 19%

Inward 65% 10% 25%

Gains CE 66% 17% 17%

Outward 14% 79% 7%

Inward 10% 85% 5%

CE 20% 73% 7%

(losses) and 2.4 (gains), except that the values in the experiment were Dutch guilders instead of dollars ($1 equals about Hfl 2). The starting value of the series (−Hfl 25 for losses, Hfl 200 for gains) is labeled TO0; revealed preference switching values are labeled TO1 to TO4, respectively. In the second part of a session a biserial certainty equivalence (CE) method was applied. The subject had to find a preference switching value (i.e. the certainty equivalent) such that he or she was indifferent between obtaining this value for sure or obtaining the prospect (TO4 , 0.5; TO0 , 0.5). This certainty equivalent (labeled CE2) was then used in a fifty-fifty prospect together with TO0 to obtain a certainty equivalent labeled CE1, and together with TO4 to obtain a certainty equivalent labeled CE3. The third and final part of each session consisted of an inward TO procedure (see Figure 2.3) that used the most extreme preference switching value generated in the first part (i.e. TO4) as the starting point for a trial of four questions. Note that if a subject maximizes Expected Utility (EU), then all three methods yield the same preference switching values (Wakker and Deneffe, 1996). During the instruction the subject was told to imagine that each question involved a one-time gambling situation involving real money that he or she was allowed to play only once. To help subjects imagine the hypothetical situation, one of the introductory questions was illustrated using a casino-like setting with real banknotes and dice. Also, the experimenter was present during all instructions to answer any questions. A bounding, choice-only method was used to elicit the preference switching values during the instruction of all three parts: the computer generated a series of choices narrowing an interval containing the preference switching value. During the experiment, subjects were encouraged to use the choice-only method, but they were allowed to state their preference switching value at any time by typing a value. Subjects were allowed to type a value because the use of a bounding method (that we do consider superior to the matching task) in a task that is very familiar is time consuming and may easily tire the subject. For example, when the computer program has not yet converged, a subject may already have a preference switching value in mind. If that is the case, further choices will not change the mind of the subject, but will take up valuable attention span. Values that violated stochastic dominance were not accepted. In that case the computer program indicated that the value given was not 'logical' and continued by again presenting the question. This procedure was mainly adopted to prevent typing errors.

Measuring the Utility of Losses by Means of the Tradeoff Method

29

Money (Hfl) -150

-100

-50

0 0

Outward TO procedure -1

CE Method

-2

Utility

Inward TO procedure

-3

-4

-5

Figure 2.5:

2.5

Median values of the tradeoffs and certainty equivalents for losses, converted to utility functions.

Results

The order in which the gain and loss sessions were held did not have a significant effect in any of the comparisons; therefore the data of the two conditions are pooled. The main hypothesis of the experiment concerns the shape of the value function for losses. First, we used a nonparametric test. As explained before, the chained TO method yields a series of monetary intervals that yield the same difference on the utility scale. A linear utility implies that all intervals are equal. A convex utility implies that when the amounts to be lost get larger, the intervals need to be larger to create an equal utility difference (this can easily be inferred from Figure 2.5). Therefore, if an interval exceeds the previous interval at some indifference point (i.e. TO1, TO2, or TO3), this reveals a convex part of the utility. More 1

precise, a part surrounding TOi is called convex if TOi < 2 (TOi−1 + TOi+1) . A part surrounding TOi is called concave if the opposite inequality holds, and is called linear when both sides are equal. For the CE method the classification is made likewise. Table 2.1 shows the percentage of convex, concave and linear parts for all subjects. For all methods, subjects provided three parts that can be classified as concave, convex or linear. Subjects with two or three concave parts were classified as revealing a concave utility. Subjects were similarly classified as revealing a convex or linear utility. If a subject revealed one convex, one concave and one linear part, the subject was not classified. Table 2.2 shows that for losses, there were significantly more subjects classified convex than concave for the CE method and inward TO procedure (bin(56,.5) p R > P > S. We assume that the PD is repeated in a following round with continuation probability w.4 Therefore, w is the constant probability that round t+1 will be played after round t has been played, while 1-w is the probability that the game terminates after round t has been played. It is assumed that both actors know the value of w. 4 The meaning of w as the continuation probability of the repeated PD is equivalent to interpreting the repeated

game as an infinitely repeated PD with exponential discounting of constituent game payoffs with discount parameter w. Both interpretations are used in the literature. The continuation probability interpretation is chosen here because it corresponds to the experimental setting in both experiments.

42

Chapter 3

In round t, each actor has been informed about the behavior of the other actor in all previous rounds 1,2, … t-1. We assume that actors evaluate outcomes using their utility function U. As is well known, analyzing the one-shot PD results in the conclusion that a rational selfinterested actor will choose to defect, because both T > R and P > S. However, the situation changes in the repeated PD. Assume that actors either choose defection in all rounds or play a “trigger” strategy. An actor playing a trigger strategy cooperates until the other actor defects, and as soon as that happens, the actor defects in all rounds thereafter. Hence a trigger strategy represents conditional cooperation to the extreme. Mutual cooperation, resulting in R for both actors instead of the worse P, is feasible for rational self-interested actors if it is supported by a subgame perfect equilibrium. If we assume that actors either play defection in all rounds or play a trigger strategy, the repeated two-person PD has a (subgame perfect) equilibrium such that both actors cooperate on the equilibrium path if, and only if w≥

U (T ) − U ( R ) U (T ) − U ( P )

(1)

Equation (1) reveals that the shadow of the future as reflected by the continuation probability w, must be larger than a certain value to make cooperation possible in equilibrium. If w is smaller than the ratio in (1) there is no equilibrium that yields only cooperative choices, because the trigger strategy, the strongest possible punishment for a defection, does not provide a strong enough punishment (P instead of R as long as the game lasts) for a first defection in comparison to the gain from this defection (T - R). Note that the ratio in the equation does not contain the PD outcomes itself but their utilities. In the derivation it is assumed that both actors evaluate outcomes similarly, that is, they have equal utility functions.5 A derivation of the above result and an elaborate discussion of related results can be found in Chapter 4 of Taylor (1987). To see the effect of the utility function on the conditions for conditional cooperation, consider Figure 3.1. Figure 3.1 depicts a utility function, known as an S-shaped utility, commonly found and hypothesized in the individual decision making literature (Tversky and Kahneman, 1992; Fennema and van Assen, 1999). The x-axis depicts the outcome, for example a monetary value, and the y-axis depicts its evaluation or utility. It is assumed that the function is strictly increasing, that is, the larger the outcome the more desirable it is for the actor. If we consider linear utility functions then the ratio in (1) is equal to the ratio of payoffs (T-R)/(T-P). This is not the case if we consider a part of a utility function with decreasing marginal utility (negative second derivative), as in the gain domain of the function in Figure 3.1. In the case of decreasing marginal or concave utility the middle outcome R is evaluated more highly relative to outcome T than in the case of linear utility, as can be seen by a higher positioning of U(R) on the y-axis then R’s position in the case of linear utility. The fact that R is evaluated more highly leads to a smaller ratio in (1) because the numerator contains the reciprocal difference U(T)–U(R). Therefore, conditions for cooperation in the 5 In fact, to derive (1) we only require that the actors’ ratios of utilities in (1) are equal and that this is common

knowledge (they know it, and both know that the other actor also knows it, etc.). This does not imply equal utility functions, because utilities for outcomes different from T, R, and P might be different for both actors. We do need to assume that in calculating the equilibria, subjects do not under- or overestimate probabilities. However, in our theoretical argumentation we explicitly take the possibility into account that actors do transform probabilities nonlinearly. We return to this issue in the discussion section.

Effects of Risk Preferences in Repeated Social Dilemmas

Figure 3.1:

43

S-shaped utility function. Ratio in (1) is highest if utility is concave (in gain domain with outcomes P, R, and T), intermediate if utility is linear, and lowest if utility is convex (in loss domain with outcomes P', R', and T').

repeated PD would be better for decreasing marginal utility functions because a smaller continuation probability would suffice for mutual cooperation to be in equilibrium. On the other hand, we can also imagine that utility functions have increasing marginal or convex utility, as in the loss domain of the function in Figure 3.1. Here we have the opposite relation. The middle outcome R is evaluated less (more negatively) in comparison to outcome T than in the case of linear utility, resulting in less favorable conditions for mutual cooperation; a larger continuation probability is needed for mutual cooperation to be in equilibrium. Therefore, our analysis suggests that if actors are rational and comply with behavioral assumptions of game theory then conditions for cooperation are better for concave utility functions and worse for convex utility functions. Our main hypothesis is then H1:

The cooperation rate in the repeated PD is larger for people with concave utility functions than for people with convex utility functions.

We already mentioned that people are considered to have S-shaped utility and that this would imply that actors’ cooperation rates are larger when outcomes represent gains than when outcomes represent losses. Note that this prediction is exactly opposite to the predictions of Walder and Berejikian discussed earlier. We believe that the main reason for the difference in predictions is the interpretation of the concept “risk”. Note that in our introduction we discuss our hypotheses in terms of “risk preferences”, but in this paragraph we do not use the concept “risk” in deriving our hypotheses. The reason is that the risk concept is ambiguous in that it loses its obvious interpretation in interdependent situations, as we will explain below. “Risk”, and the related concept “risk preferences”, are commonly used in the field of individual decision making. If an actor makes a decision under conditions of risk, he chooses between alternatives of which at least one consists of probabilistic outcomes, and where the

44

Chapter 3

values of the probabilities are known to him and are not equal to 0 or 1 (see Luce and Raiffa, 1957, for a discussion on terminology). The concepts risk aversion and risk seeking refer to the behavior of actors under conditions of risk. An actor’s behavior in a situation is called (i) risk seeking when he prefers to gamble over obtaining a certain outcome equal to the expected value of the gamble, (ii) risk neutral when he is indifferent between the gamble and its expected value, and (iii) risk averse when he prefers the expected value over the gamble. For example, if an actor prefers $10 to a gamble ($20, ½, $0), with a 50% chance of winning $20 and a 50% chance of playing even, then this behavior is known as risk averse. Risk preferences are directly related to the shape of the utility function if we assume that actors are rational in the sense of Expected Utility theory as formulated by von Neumann and Morgenstern (1944).6 Risk aversion then implies and is implied by a concave utility function, risk neutral behavior implies and is implied by a linear utility function, and risk seeking behavior implies and is implied by a convex utility function. The utility of a gamble is then equal to a linear combination of the utilities of the outcomes of the gamble, with weights equal to the probabilities. For example, in the loss domain of Figure 3.1, the utility of a gamble ($T, p, $P) is equal to the point pU($T) + (1-p)U($P), which is on the line connecting U(T) and U(P). The utility U[p$T+(1-p)$P] of the expected value of the gamble is equal to the point on the utility curve at this value on the x-axis. Note that the whole utility curve in the loss domain is below the straight line, implying that an actor with a convex utility function would prefer the gamble above its expected value, which is risk seeking behavior. Therefore, under Expected Utility theory we could substitute our hypothesis H1 with H1’

The cooperation rate in the repeated PD is larger for people who are risk averse in gambles containing the outcomes of the PD than for people who are risk seeking in these gambles.

It is important to emphasize that because there are reasons not to believe in Expected Utility as a good descriptive theory of human behavior, as we will argue in our section on utility measurement, we will refer to H1 as our main hypothesis. A natural question to ask is whether actors’ risk preferences depend on the range of the outcomes used in gambles. For economists it was a truism that the utility of money is marginally decreasing, and therefore that utility is concave (e.g. Bentham 1789). Prospect theory (Kahneman and Tversky, 1979; Tversky and Kahneman, 1992) suggested that utility is indeed concave for gains, but convex for losses. Empirically it is well established that utility for gains is concave for most actors (see e.g. Fennema and van Assen, 1999). However, the evidence for the shape of the utility function for losses was inconclusive because of variability between individuals and particular bias in the methods used to assess utilities. Fennema and van Assen (1999) elicited utility for a pool of subjects using a utility assessment method that avoids this biases. On the basis of this method, they found that utility for losses was convex for most subjects, supporting Prospect Theory and not the traditional economic literature. Using the additional assumption of S-shaped utility, hypothesis H1’ can be restated as H1’’

The cooperation rate in the repeated PD is larger when outcomes in the PD are

6 In the section on utility measurement we will see that risk preferences and utility are not directly related

because actors tend to transform probabilities.

Effects of Risk Preferences in Repeated Social Dilemmas

45

gains then when the outcomes in the PD are losses. Walder (1994) and Berejikian (1992) proposed a hypothesis that is the opposite of H1’’. We think that the reason for this difference is the ambiguity of the concept “risk”. In the reasoning of Walder and Berejikian the concept of risk is not limited to choice behavior in gambles in comparison to the expected value of the gamble. In their work, “risk” refers to the fact that some outcomes are uncertain even after having chosen a particular course of action (e.g., defection or cooperation). However, in this interpretation there are multiple risks involved in the decision. In a repeated PD, if an actor cooperates (s)he runs the risk of being exploited by the other actor and hence obtaining the sucker's payoff S. If the actor chooses not to cooperate (s)he runs the risk that his or her choice will lead to future retaliation by the other actor. Because both cooperative and non-cooperative behavior is in some sense risky, according to the latter use of the concept risk, it is unclear how a risk averse actor might proceed. We believe that Walder and Berejikian relate their reasoning only to the risk of being exploited by the other actor when cooperating, and if this is done, it leads to the complementary hypothesis of H1’’. To avoid ambiguities and to be consistent with the individual decision making literature, we use the term “risk averse” for an actor who prefers the expected value of a gamble over the gamble. We now briefly restate the results of a first experimental test of hypotheses H1’ and H1’’, as reported by Raub and Snijders (1997). 3.3

Experiment 1 (Raub and Snijders, 1997)

3.3.1 Method The experiment was run over eight sessions using different subjects. In total, 190 subjects participated. To test our hypotheses each subject played two repeated PDs, one PD with outcomes in the loss domain and one PD with outcomes in the gain domain. Outcomes T+, P+, and R+ were equal to 16, 7 and 4 Dutch guilders respectively.7 Outcome S+ was varied over the sessions between -1, 0, and 1 guilders. Outcomes T-, R-, P-, and S- in the loss domain were obtained by subtracting 16 guilders from the outcomes of the PD in the gain domain. The value of the continuation probability w in the experiment was 0.75. All participants completed three tasks. The assessment of risk preferences of subjects for both value domains was conducted in the first part of the experiment. To test the relation between subjects' behavior in repeated PDs and their risk preferences, the outcomes of the gambles used to elicit risk preferences were equal to those used in the repeated PDs. Subjects were required to make a choice between (T+, 1-w, P+) and the certain outcome R+. Probability w was equal to the continuation probability also used in the repeated PDs. Additionally, subjects made a choice between (T-, 1-w, P-) and the certain outcome R-. The order in which both choices had to be made was varied across different sessions. In the second task subjects completed a comprehension check and questionnaire, which contained several questions about possible relevant background variables and in which their comprehension of the repeated PD was tested. The most time consuming third and last task of the experiment consisted of subjects playing the repeated PD in both the loss and gain domain. Sessions were run in groups with a maximum of 30 subjects. Participants were 7 At the time we conducted the first experiment, 1 US dollar was equivalent to approximately 1.7 Dutch

guilders.

46

Chapter 3

evenly divided over two semi-separated sides of a room such that they could not see each other. Pairs of players playing a repeated PD together were separately determined at random for each repeated PD. The order in which both repeated PDs were played was counterbalanced between sessions. Subjects were not allowed to communicate during the experiment and were told that they would not find out the identity of the players they were connected with. The procedure of the repeated PD was roughly in line with the procedure used in Murnighan and Roth (1983). Choices of subjects were written down on paper and the experimenters passed the choices on to their partners on the other side of the room. A roulette wheel with 36 slots was then spun. If the ball fell in one of 27 slots (with probability equal to 0.75) mentioned on the card of a pair of players then this pair of players played another round of the repeated PD. At the start of the experiment subjects received 20 Dutch guilders. We hoped to minimize the gambling with the house money effect (Thaler and Johnson, 1990) by explicitly mentioning that the money was meant to cover their time and effort, and by letting participants put the money in their wallet immediately. The gambling with the house money effect might make participants more risk seeking than they would have been if they had not received 20 guilders at the start of the experiment. 3.3.2 Results As expected, the majority of subjects were risk seeking in the loss domain (66.3%). Unexpectedly, only 44.7% of the subjects were risk averse in the gain domain, and across risk patterns for both domains, of 190 subjects only 43 (23%) had risk preferences in agreement with Prospect Theory. One explanation for the absence of risk aversion in the gain domain is that outcomes of the gambles are low in comparison to the outcomes used in other experiments. It is known from other studies that subjects in general are more risk averse when the range of outcomes of the gambles is larger (Krzysztofowicz and Duckstein, 1980). Another explanation is provided in the section on utility measurement. The main hypothesis to be tested was whether there is a positive relation between risk aversion and cooperation in the repeated PD. Cooperation was measured by the first response in the repeated PD. An important reason for neglecting the responses in round 2 and beyond is that the responses in later rounds are very likely be influenced by the responses of their opponent in previous rounds. Comparing the proportions of cooperation for subjects who were risk seeking with those subjects who were risk averse confirmed our hypotheses. In the gains domain, 38% of the risk averse subjects cooperated, as opposed to 26% of the risk seeking subjects (p = 0.054, Fischer exact test). In the losses domain the observed difference was larger; of the risk averse subjects 48% cooperated, as against only 26% of the risk seeking subjects (p = 0.002, Fischer exact test). The connection between risk preferences and cooperation was also tested on the basis of a probit analysis of the probability of cooperation, using as independent variables a subject’s risk aversion, whether the game was one involving losses or gains, and some other background or control variables such as age and sex. The effect of risk preferences remained significant (p < 0.05) in the probit analyses including all variables. The effects of all other variables except one were not significant. The one variable for which the effect was significant indicated that subjects who participated in the experiment as a course requirement were more inclined to cooperate than others (p < 0.01). Summarizing the results, our

Effects of Risk Preferences in Repeated Social Dilemmas

47

hypothesis that risk preferences have an effect on cooperation in repeated social dilemmas was confirmed. 3.4

Utility measurement

Although other experiments suggest that most subjects have S-shaped utility, in our Experiment 1 we found that only 23% of all subjects were risk seeking for losses and risk averse for gains. Although risk seeking behavior was observed for losses, the majority of subjects were also risk seeking for gains. One explanation of this unexpected finding of a majority of risk seekers for gains is that our design did not sufficiently account for the gambling with the house money effect. There are two other complementary explanations for this unexpected finding. Firstly, there can be an experimental bias. The subject’s payoff in the experiment was not fixed but depended on the subject’s responses, the responses of other subjects, and on chance. In the worst case scenario a subject would earn 3 Dutch guilders and in the best possible scenario a subject would earn 36 guilders. The expected utility of participating in the gamble for a person who is risk averse for gains would be lower than for a person who is risk seeking for gains. Therefore, the experiment might attract a relatively large proportion of subjects who are risk seeking for gains. Other experiments investigating risk preferences used either fixed payoffs (e.g., Tversky and Kahneman, 1992; Fennema and van Assen, 1999) or a variance in the payoffs that was substantially lower than in our experiment. Therefore Experiment 1 might indeed have had a (larger) bias in selecting subjects who are risk seeking for gains. Another explanation for the relative abundance of risk seeking behavior involves probability weighting by subjects (this argument was made earlier by van Assen, 1998). In Section 3.2 we explained that, assuming Expected Utility theory, subjects with concave utility prefer the expected value of a gamble over the gamble itself because only then pU[T] + (1-p)U[P] < U[pT + (1-p)P] = U[R]. However, if subjects transform not only utilities but also probabilities then Expected Utility no longer holds and risk aversion is no longer equivalent to behaving in accordance with concave utility, just as risk seeking behavior is no longer equivalent to behaving in accordance with convex utility. We will show below that the abundance of risk seeking for gains can at least partially be explained by concave utility for gains together with probability weighting as assumed in (Cumulative) Prospect Theory of Tversky and Kahneman (1992). For example, assume that a subject’s utility function is concave and can be 0.88 represented by the power function u[x] = (x) . Furthermore, assume that the subject overweighs a probability of 0.25 relative to a probability of 0.75, such that the weight of probability 0.25 is π[0.25] = 0.27 and π[0.75] = 1 - π[0.25] = 0.73. Then, although utility is concave, the subject prefers the gamble (T+, 0.25, P+) over its expected value R+: 0.27 * (16)0.88 + 0.73 * (4)0.88 > (7)0.88. Evidence for subjects transforming probabilities in this way – overweighing small probabilities and underweighing moderate to large probabilities – is overwhelming in individual decision making research. Most researchers in the field of individual decision making accept Expected Utility theory as an adequate normative theory or as a way of making the underlying mechanisms of decision situations apparent. Most of them also agree that the theory is not an adequate descriptive theory of how people actually make decisions. Kahneman and Tversky (1979) describe sets of choice situations involving gambles that elicit responses from subjects that cannot be accommodated by Expected Utility theory. They could convincingly explain the deviations from Expected Utility theory by

48

Chapter 3 Probability weight 1 .8 .6 .4 .2 0 0

Figure 3.2:

.2

.4 .6 Objective probability

.8

1

A typical probability weighting function, reflecting overweighing of a small probability and underweighing of moderate to large probabilities.

assuming that subjects transform probabilities in a particular way. Their theory of how actors transform probabilities, known as Prospect Theory, was developed in 1979, and was extended by Tversky and Kahneman (1992), being relabeled Cumulative Prospect Theory. In Cumulative Prospect Theory different weight functions π for probabilities belonging to losses and gains are assumed. A property of the functions π is that for gambles with two outcomes in the same domain (losses or gains) the functions assign weight π(p) to the probability p corresponding to the largest absolute outcome. The smallest absolute outcome is assigned decision weight 1- π(p). The decision weight functions π can be different for losses and gains. If the outcomes of the two-outcome gamble are from different value domains with probability q that the positive outcome occurs and with probability 1-q that the negative outcome occurs, then the decision weights assigned to the outcomes are π+(q) and π-(1-q) respectively. Note that, as opposed to the decision weights of two-outcome gambles with outcomes from the same value domain, the sum of decision weights does not need to be unity. Intuitively, the probability transformations can be understood as a higher sensitivity for changes in probability near certainty than in the middle range of probabilities. This tendency is corroborated by several empirical studies (Kahneman and Tversky, 1979; Tversky and Kahneman, 1992; Camerer, 1995; Camerer and Ho, 1994; Gonzalez and Wu, 1999). A typical probability weighting function found in these studies is depicted in Figure 3.2. The argument from Cumulative Prospect Theory used to explain the relative abundance of risk seeking behavior in the gains domain can be summarized as follows. A typical subject has concave utility for gains and overweighs the probability of 0.25 for the largest outcome T+. The overweighing of the largest outcome makes the gamble more attractive. This increase in attractiveness overcomes the effect of the concavity of the utility function that makes the gamble less attractive. Both effects taken together result in a preference for the gamble over the certain outcome R+. Allowing for probability weighting has considerable consequences for our reasoning outlined above and for our experimental test. We stated that under Expected Utility the

Effects of Risk Preferences in Repeated Social Dilemmas

49

?

X 1/3

1/3

2/3

2/3 -5

Figure 3.3:

-10

Structure of the gambles used in the tradeoff method to assess utilities in the gain domain. The gain X in the left gamble is either X1 = 0 , X2 = 5, X3 = 10, or X4 = 20. The subject specifies the amount ? such that (s)he is indifferent between the two gambles.

20 1/3

1/3

2/3

2/3 X

Figure 3.4:

50

?

Structure of the gambles used in the tradeoff method to assess utilities in the loss domain. The loss X in the left gamble is either X1 = 0 , X2 = 5, X3 = -10, or X4 = -20. The subject specifies the amount ? such that (s)he is indifferent between the two gambles:

hypotheses H1 and H1’are equivalent. If probability weighting is taken into account, this is no longer true. With our first experiment we tested assumption H1’, but we were mainly interested in hypothesis H1 which describes the relation between the shape of the utility function and cooperation, as derived from game theory. Moreover, the utility assessment method used in Experiment 1 is not robust with respect to probability distortions and therefore provides only indirect information on subjects’ utility functions. For a stricter test of hypothesis H1 we need a utility assessment method which is robust with respect to probability distortions. However, all traditional utility assessment methods (see Farquhar, 1984, for a general overview) assume that subjects do not transform probabilities. Wakker

50

Chapter 3

and Deneffe (1996) developed a method that can measure utility functions independently to utility transformations. Their method, called the “tradeoff method”, will be used to assess utilities in Experiment 2. The tradeoff utility assessment method is robust with respect to probability transformations. It can be used to assess utilities under different theories of individual decision making and under risk assuming probability transformations, including Expected Utility theory, Prospect Theory, Rank Dependent Utility theory, and Cumulative Prospect Theory.8 We will give a self-contained explanation of the tradeoff method here, assuming that subjects have different decision weight functions for gains and losses, as in Cumulative Prospect Theory. The subject’s task in the tradeoff method is to choose between two gambles with two outcomes. The lowest outcomes, and therefore also the largest outcomes, occur with the same probability for both gambles. Moreover, the outcomes with equal probabilities are of the same value domain, that is, both are losses or both are gains. In the gambles used in the tradeoff method here, the lowest outcome is from the loss domain and occurs with probability 2/3, and the largest outcome is from the gain domain and occurs with probability 1/3. The structures of the choice situations used in the second experiment to elicit utilities in the gain and loss domains, are shown in Figure 3.3 and Figure 3.4 respectively. Consider first the structure of the choice situation in the gain domain. The losses from the gambles are fixed at a loss of 5 guilders in the left hand gamble and a loss of 10 guilders in the right hand gamble.9 The gamble on the left hand side contains a gain X. The subject’s task in the long run is to specify the value of the gain such that (s)he is indifferent between choosing the gambles on the left and right hand sides. Consider two choice situations that differ in gain X in the left hand gamble, e.g. X3 = 10 and X4 = 20. The responses of the subject in these situations are denoted by ?3 and ?4 respectively. Assuming Cumulative Prospect Theory, the response ?3 to the first situation implies that π-(2/3)u(-5) + π+(1/3)u(10) = π(2/3)u(-10) + π+(1/3)u(?3). Rearranging the terms in the equality leads to U (? 3 ) − U (10) = [U ( −5) − U ( −10)]

π − (2 / 3) π + (1 / 3)

(2)

Repeating this exercise for the second choice situation leads to the same equation but with the left hand side equal to u(?4) - u(20). Thus, the two responses in the choice situations together imply that the subject’s utility differences u(?4) - u(20) and u(?3) - u(10) are equal. Note that possible probability transformations are canceled out because they have identical effects in both choice situations. The only assumption we make using the tradeoff method is that both probability transformations and utility for losses of 5 guilders and 10 guilders, are invariant over the choice situations. In a similar fashion, utilities can be assessed for the losses domain. Wakker and Deneffe (1996) and also Fennema and van Assen (1999) used a so-called “chained outward” tradeoff method to elicit the utilities. The term “outward” is used because 8 Another merit of the tradeoff method as opposed to traditional utility assessment methods is that it can be used

under conditions of uncertainty, that is, when outcomes are obtained from events of which the probabilities are unknown. Explanations of how the tradeoff method works under conditions of uncertainty and what considerations should be taken into account when using the tradeoff method assuming one of the different theories of individual decision making under risk, can be found in Wakker and Deneffe (1996) and Fennema and van Assen (1999). 9 At the time we conducted the second experiment 1 US dollar was worth approximately 2 Dutch guilders.

Effects of Risk Preferences in Repeated Social Dilemmas

51

the absolute value of the subject’s response ? is larger than the absolute value of the outcome X. Note that likewise the outward tradeoff method is used in the choice situations of Figure 3.3 and Figure 3.4. The method is said to be “chained” because the subject’s response in a given situation is substituted for the X outcome in the next choice situation. As a result, a sequence of connected equal utility intervals is obtained for each subject. In our experiment we are interested in the shape of utility for a narrow domain of outcomes used in the repeated PDs. The outcomes used in the PDs are in the interval ranging from a loss of 25 guilders to a gain of 25 guilders. For our purposes an unchained tradeoff method seemed more convenient, yielding a more precise measurement of the interval of relevant outcomes. In our new experiment we substituted for X in Figure 3.3 the values of 0, 5, 10, and 20 guilders, and 0, 5, -10, and -20 guilders for the losses in Figure 3.4. In our Experiment 1, risk preferences were assessed using single questions relating the outcome R to gambles (T,1-w,P) that have an expected value equal to R. We also chose to use this simple method in Experiment 2, although elaborated with two additional questions, alongside the tradeoff method. By including both measurements in the experiment we are able to compare the results of both experiments, and are able to compare the results of the tradeoff method with the results of the traditional method. Empirical studies (Wakker and Deneffe, 1996; Fennema and van Assen, 1999) have found that the tradeoff method results in less concave utility than a method comparable to that used in Experiment 1. However, it was not evident that this result would also be found in the second experiment. In the two experimental studies cited, utility was assessed for the whole gain domain. Previous research has shown that when utility is assessed using the simple method in gambles with outcomes lying far apart utility is more biased to concavity (Krzysztofowicz and Duckstein, 1980). More details of the precise utility assessment methods used in the second experiment are provided below, where the design and results of the second experiment are discussed in detail. 3.5

Experiment 2

To test the hypothesis that there is a relation between cooperation in a repeated PD and the shape of utility in the domain of the payoffs used in the PDs, an experiment consisting of two parts was performed. In the first part, utility of subjects was assessed, while in the second part subjects played a number of repeated PDs. The method and results of the experiment are reported below. 3.5.1 Method 3.5.1.1 Subjects Subjects were first recruited by flyers and by advertising in the university newspaper. Because only a small number of students was reached using these methods we sent an email to approximately 13,000 students of the University of Groningen in which we advertised our experiment. A sufficient number of potential subjects reacted positively to our email. In total 227 subjects participated in the experiment, of whom 216 finished both sessions. Most subjects were undergraduates from the University of Groningen. The age of subjects was between 17 and 29, and 50.5% of them were male.

52

Chapter 3

3.5.1.2 Procedure A computer program was written for each of the two parts of the experiment, which were run on different days.10 We attempted to ensure that the amount of time between participating in the first and second part of the experiment was at least 24 hours and at most a week. The experimental sessions took place in a large room with 15 computers distributed in five rows of three. In the experimental sessions both subjects participating in the first part and subjects participating in the second could be present. Although the configuration of the room was such that it was barely possible to observe other subjects' responses, the subjects participating in the second part were always positioned in the back of the room to prevent the other subjects suspecting what might be involved in the second part. Subjects were explicitly asked to be silent during the experiment. One experimenter was always present in the room to answer any questions of the subjects. In the first part of the experiment, subjects' utility functions were assessed in different ways (see below). In the second part, subjects played five different repeated PDs. Outcomes P for mutual defection in these PDs were 5, 0, -5, -10, -20 Dutch guilders respectively. The continuation probability w of the PDs was equal to (T-R)/(T-P) = 0.5. In the PDs, T = P + 20, R = P + 10, and S = P - 5, resulting in a domain of PD outcomes ranging from -25 to 25. In the first part of the experiment utility was obviously assessed for precisely this domain. 3.5.1.3 First part of the experiment The first part consisted of five tasks. The subjects started with an extension of the utility assessment procedure used by Raub and Snijders (1997). For the second task, subjects were required to indicate their indifference point in comparisons of a gamble with a certain outcome. Subsequently, the TO method was used to elicit subjects' utilities. The fourth task was a questionnaire. All subjects had to complete the first four tasks, and if they were completed within 45 minutes the subject continued with a test-retest reliability task. Here the subject was given exactly the same choice situations as in the first three tasks. A more detailed description of each of the tasks is provided below. Task 1: Extension of utility assessment procedure as used by Raub and Snijders Raub and Snijders assessed risk preferences for the domain [P,T] in a PD with these outcomes and a continuation probability w equal to 0.75, by requiring subjects to make a preference comparison between gamble (T, 1-w, P) and the expected value R of the gamble. Subjects preferring the gamble are "risk seeking", while subjects preferring the certain outcome R are "risk averse". Subjects also made made preference comparisons between (T, 1w, P) and R, but this time with w equal to 0.5. We refer to this way of measuring the risk preference as the "traditional method". To extend the measurement two additional preference comparisons were constructed: a comparison between (T, 1/3, P) and R, and one between (T, 2/3, P) and R. Using the answers to these three preference comparisons, four different types of risk preferences can be distinguished: "very risk averse" if a subject does not prefer any of the gambles above the certain outcome, "risk averse" if the subject only prefers the gamble when the probability of obtaining the best outcome T from the gamble is equal to 2/3, "risk seeking" if the subject prefers the gambles for probabilities 2/3 and 0.5, and "very risk seeking" if the subject prefers all gambles above R. In the analyses, we refer to this way of 10 The computer programs are written in Turbo Pascal 7.0 and are available from the first author upon request.

Effects of Risk Preferences in Repeated Social Dilemmas

53

measurement as the "extended traditional method." A brief instruction explained the task to the subjects, who were told to imagine that they would choose between the gamble and the certain outcome only once and that the outcomes of the choice situation were their own money. To practice the procedure, subjects made three preference comparisons between (60, p, -20) and 20 with p equal to 2/3, 0.5, and 1/3 respectively. The task contained four sets of three preference comparisons, with P equal to 0, -5, 10, or -20. The order of the sets was varied among subjects with a Latin Square design (Edwards, 1968) consisting of four orderings. Within a set, subjects always first stated their preference for the comparison with p equal to 0.5. After subjects had stated their preference they were asked to confirm their preference. They were also given the possibility of changing answers they gave earlier. Task 2: Extra choice situations Five additional choice situations involving a gamble and a certain outcome were put before to the subjects. In theory, the answers of the subjects in these situations allow us to estimate parametric utility functions. Because we do not report on parametric utility functions here, we also do not report on the details of the second task. Task 3: The tradeoff method Utility was assessed for gains and losses using the unchained tradeoff method as described above. Four choice situations were used in each value domain. The choice situations for assessing utility for gains and losses are shown in Figure 3.3 and Figure 3.4 respectively. For gains, outcomes 0, 5, 10 and 20 are substituted for X, and for losses outcomes 0, -5, -10, and –20 are substituted. To obtain the subject's indifference point "?" in the right hand gamble of a particular choice situation, the computer program first substitutes outcome X + Y for "?". The value of Y is 5, 10, 15, or 20 in the case of gains, and -5, -10, -15, or -20 in the case of losses. The probability of each of the values of Y is 0.25. In general the subject then has four options: he can either be indifferent between the two gambles, he can prefer the right hand gamble or the left hand gamble, or he can recognize that he has made a “mistake” earlier in the procedure and choose to return to an earlier choice situation to change a response. If the subject is indifferent between the gambles, the indifference point is determined and the computer program provides a choice situation with another value of X (and "?"). If the subject prefers one of the gambles to the other, the computer program generates another value of "?" in the right hand gamble of the same choice situation. Consider, for example, that the subject prefers the left hand gamble to the right hand gamble. In the case of losses, this implies that the indifference point of the subject is between X and X + Y. The computer program then generates the following choice situation where the right hand gamble is made more attractive than in the first situation. The new value of "?" will then be equal to a loss of X + 0.5Y, bisecting the interval which contains the indifference point. If the subject had preferred the right hand gamble, the loss of the right hand gamble in the previous choice situation is increased by an additional loss Y, again chosen randomly from the distribution specified above. This procedure, choosing "?" with bisection or an additional random loss, is continued

54

Chapter 3

until the subject indicates he is indifferent between the two gambles.11 In the case of gains the procedure is identical, choosing "?" with bisection or an additional random gain. Half of the subjects started with the tradeoff method for gains, while the other half started with the TO method for losses. The order of choice situations within each of the two value domains was such that a subject first determined his or her indifference point for the two most extreme values of X (0 and 20, or 0 and -20), followed by that for the two other choice situations (X equal to 5 and 10, or -5 and -10). The 24 possible orderings were varied among subjects using a Latin Square design. The instructions for the tradeoff method given to subjects were extensive because previous research with the method indicated that at least initially, subjects have some problems in grasping the complexity of the choice situation. The procedure described above was explained and applied to find the subject's indifference point "?" such that he is indifferent between (40, 1/3, -10) and (?, 1/3, -20). To assist the subject in finding the value of "?", he was asked to think of how much more the gain "?" (in the right hand gamble) should be than the 40 guilders (in the left hand gamble) in order to compensate for the additional loss of 10 guilders (compared to the left hand gamble), which is twice as likely as the additional gain. The instruction also contained a second choice situation in which subjects had to specify a negative indifference point. The subjects were again told to imagine that they would choose between the gambles only once and that the outcomes of the choice situation involved their own money. Task 4: Questionnaire The participants were asked about their age, sex, number of siblings, whether they studied or not, whether they had a (part-time) job, subjects they followed in secondary school, the political party they voted for at the last elections, and whether they had knowledge of game theory. Subjects' responses to the questions were used as controls in our analyses. The questionnaire was concluded with a measure of the social orientation of the participant (Messick and McClintock, 1968; McClintock and Liebrand, 1988). Social orientation acknowledges that for one's utility in a choice situation, not only one’s own payoff is important but also the payoff of the other actors involved in the choice situation. That is, in the PD, ego's utility of the payoff can also depend on the payoff obtained by alter. The higher ego's social orientation, the more important alter's payoff is for one’s utility relative to one’s own payoff. A high social orientation can transform a PD game into another game in which defection is no longer the dominant choice. Therefore, social orientation is a relevant background variable for our analyses in the results section. Social orientation is measured in the questionnaire by requiring the subjects to order pairs of outcomes (53,53), (64,39), (74,30), and (58,104). The first outcome is that of ego and the second outcome is that outcome for alter. Assuming that the actors' utility of outcome (x, y) is equal to x + θy, where θ is the social orientation parameter, then six possible preference orders of the four outcome pairs are obtained from different values of θ, and hence six different classes of participants with varying social orientation.

11 The procedure was also stopped when the length of the interval, which contained the indifference point, was

equal to 1. The subject was then required to state the value which was closest to his "real" indifference point. This resulted in integers as indifference points.

Effects of Risk Preferences in Repeated Social Dilemmas

55

Task 5: Test-retest reliability The choice situations in tasks 1, 2, and 3 were repeated once until 45 minutes had passed since the start of the experimental session. The order of choice situations in Task 1 and Task 2 was randomized. The choice situations of Task 3 with extreme values of X were repeated first, followed by the four other choice situations. 3.5.1.4 Second part of the experiment The second part of the experiment started with extensive instruction towards the subjects. The PDs were presented to the subjects using 2x2 tables. The relation between the choices made by the players and their outcomes was explained by letting participants play a repeated PD twice, with outcomes T = 60, R = 20, P = -20, S = -40, and w equal to 0.5 as in the games of interest. Subjects were told that they did not play against a player present in the lab but against a randomly selected player from a previous experiment, drawn with replacement before each repeated PD. The experiment from which players were drawn was actually that of Raub and Snijders (1997). In Appendix 3.1 we explain how we were able to model the strategies of human players accurately and realistically by making use of the responses of subjects in the 1997 experiment. Advantages of the procedure we used are also enumerated in Appendix 3.1. A lot of effort was invested in the instruction to make participants in our second experiment realize that (i) they played against a human player, although the computer generated the responses, and (ii) that they were not deceived in any way (which was true). After participants had played the two repeated PDs it was explained to them how their payoff would be determined at the end of the experiment. The procedure for rewarding the subjects is described later. Finally, before playing the repeated PDs of interest, subjects were told to imagine that outcomes of the PDs involved their own money. After the instruction the subject played four repeated PDs in a random order. The value of P in the PD was -20, -10, -5, or 0. The subjects were told that their responses in these games could have an effect on their reward at the end of the session. If 40 minutes had not yet passed since the start of the experimental session, the subject continued to play these four repeated PDs in a random order. The responses in these additional games had no effect on the subject's reward. The subjects were, however, told that their behavior in these games was important for the experiment and they were asked to behave as if they would experience the outcomes as real payoffs. When either the subject had played each repeated PD twenty times or 40 minutes had passed, the subject played a fifth repeated PD with P equal to 5. This game was also important with regard to the subject's reward. The computer program then showed the possible rewards for participating in the experiment. 3.5.1.5 Reward for participation in the experiment After completion of the first part of the experiment, subjects were given a reward of 35 guilders (about US$ 17.5). Rewards were not dependent on the subject's responses in the preference comparisons. We were not concerned that using hypothetical payoffs in the choice situations would bias our results because previous studies on individual decision making under risk using real incentives yielded results similar to studies using hypothetical payoffs (e.g., Tversky and Kahneman, 1992; Camerer, 1995; Beattie and Loomes, 1997).

56

Chapter 3 Table 3.1:

Risk preferences for the gains and losses domain using the traditional method with a single choice per subject.

Risk averse for gains

No Yes

Risk averse for losses No Yes 114 37 45 20

Total

159

57

Total 151 65 216

To make the repeated PDs in the second part of the experiment as realistic as possible and because payoffs dependent on behavior were used in Experiment 1, we chose to make the subject's reward dependent on his or her behavior (and on the behavior of the other player). Since it was possible for the subject to money in some of the repeated PDs, and hence in the second part of the experiment, the reward for participation in the first part was high. Separating the two parts had the advantage that the "gambling with the house money effect" (Thaler and Johnson, 1990) was, in all likelihood, minimized. The subject's reward in the second part of the experiment was determined by the random selection method. The outcome of the last round of each of the five different repeated PDs played for the first time was selected as a possible payoff. This was explained to the subjects during the instruction at the start of the second part of the experiment. As shown by Raub and Snijders (1997), selecting only the last round excludes wealth effects and does not affect the strategic structure of the repeated PD. We were mainly interested in the subjects' behavior in the four repeated PDs with P = 0, -5, -10, and -20. In the present study, only the subjects’ response to the two repeated PDs with P equal to 0 (positive PD) and -20 (negative PD) are analyzed; responses to the two other (mixed) PDs are analyzed in a future study. Because most outcomes in the last round of a repeated PD were determined by mutual defection, the expected value of the subjects' reward with the random selection method would have been negative. Therefore, the fifth repeated PD with P = 5 was included at the end of the experiment. At the end of the experiment the subjects drew one card from a pile of six. To further increase the expected value of the procedure, two jesters were included which were associated with the fifth repeated PD; the other repeated PDs were associated with clubs, diamonds, hearts, or spades. By the simulation of both actor strategies we predicted that the expected value of the second part of the experiment was around zero guilders. Our prediction was approximately correct, the average reward given to subjects in the second part being -0.4 guilders. The minimum reward for participation in both parts of the experiment was 10 guilders (35-25) and the maximum reward was equal to 60 guilders (35+25). Because participation in the second part was not lucrative for the subjects, we were concerned that a number of subjects would not turn up. Therefore, we asked them to sign a contract with their name, address, telephone number, and a promise that they would arrive for the second part. Only 8 subjects did not turn up for the second part of the experiment.

Effects of Risk Preferences in Repeated Social Dilemmas Table 3.2:

Risk averse for gains

57

Risk preferences for the gains and losses domains using the traditional method with three choices per subject. The values ‘0’, ‘1’, ‘2’, ‘3’, represent the number of gambles out of three that the subject prefers above the certain outcome. Therefore, a larger value corresponds to greater risk aversion.

0 1 2 3 Total

0 11 17 7 1 36

Risk averse for losses 1 2 19 8 67 25 34 15 3 2 123 50

3 2 2 1 2 7

Total 40 111 57 8 216

3.5.2 Results 3.5.2.1 Distribution of risk preferences We first report on the data obtained on subjects’ risk preferences based on the traditional method employed by Raub and Snijders (1997). That is, to assess risk preferences for gains, we only used the subject’s preference for the gamble (T+, 1-w, P+) and the certain outcome R+. The choice between (T-, 1-w, P-) and the certain outcome R- determines the risk preference of the subject for losses. We found that only 30.1% (65/216) of the subjects are risk averse for gains, and that 73.6% (159/216) are risk seeking for losses. Compared to Raub and Snijders (1997) we find a relatively large percentage of risk seeking choices, especially in the gains domain. Only 20.8% (45/216) of the subjects are categorized as risk averse for gains and risk seeking for losses. Table 3.1 summarizes these results. In Table 3.2 we report the frequencies of the degree of risk aversion based on three choices per domain. One crucial difference between the experiment reported by Raub and Snijders (1997) and this one is the use of the more sophisticated measurement of risk preferences, the tradeoff method. Based on this method, subjects can be categorized in the same way as with the traditional method. As outlined previously in the section on utility measurement, the tradeoff method generates four responses ?1, ?2, ?3, ?4 per value domain (gain or loss). These four responses provide three independent pieces of evidence to determine the shape of utility in this domain (see also Fennema and van Assen, 1999). For example, in the case of gains, if the difference ?1 – 0 is smaller than ?2 – 5 then this is evidence for concavity of the utility function (more money needs to be added to 5 than to 0 to bridge the same utility difference). Similarly, subsequent differences can be compared and evidence can be listed. If two or more pieces of evidence point to concave, linear, or convex utility, then the subject’s pattern of responses is coded in Table 3.3 as concave, linear, or convex respectively. If a response pattern reveals exactly one piece of evidence for each of convex, linear, and concave utility then the subject is categorized as “missing”. We have 45 “missings” in the losses domain and 34 in the gains domain. We observe that of the 142 subjects who are assigned a risk preference, the largest proportion (54 subjects) have concave utility for gains and convex utility for losses, in accordance with S-shaped utility. Therefore, the tradeoff method does generate response patterns that are more in line with standard results on risk preferences.

58 Table 3.3:

Utility for gains

Chapter 3 Utility for the gains and losses domain using the tradeoff method. “Missing” is denoted by ‘*’.

Convex Linear Concave * Total

Convex 9 4 54 6 73

Utility for losses Linear Concave 0 11 18 5 7 34 7 13 32 63

* 4 3 30 8 45

Total 24 30 125 34 213

The tradeoff method is also more stable than the traditional method. The average testretest reliability correlation between the responses before and after the questionnaire in a given choice situation equals 0.67 for the tradeoff method, whereas this average correlation equals 0.50 for the traditional method. Surprisingly, the correlation between the results from the extended traditional method and the tradeoff method is low (0.10, p=0.16). An explanation and discussion of this weak correlation is postponed until the conclusion and discussion section. 3.5.2.2 Risk preferences and cooperation Hypothesis H1’’ stipulated that the cooperation rate is higher for the gains domain and our results support this hypothesis. The percentage of cooperation in the first repeated PD for gains is 36.5, whereas this percentage is only 26.9 for the losses domain (paired t-test, t=2.16, p=0.02). The finding of a higher cooperation rate for gains together with evidence of Sshaped utility as assessed by the tradeoff method suggests that our hypotheses H1’ and H1 will also be supported. However, this is not the case, as we demonstrate below. Hypothesis H1’ stated that there is a relation between risk preferences, as measured by the traditional method, and the cooperation rate in the repeated PD, such that a higher cooperation rate corresponds to greater risk aversion. If we compare the choices of subjects in the repeated PDs, we find small differences between risk averse and risk seeking subjects according to the traditional method. In the first round of the first gains dilemma that subjects play, 38.5% of the risk averse subjects cooperate, as opposed to 35.1% of the risk seeking subjects (1-sided Fisher’s exact=0.37). If we use the extended version of the traditional method, we find that the percentage of cooperation decreases for only the most risk seeking subjects: the most risk averse subjects score 37.5%, and the more risk seeking categories of subjects score 38.6%, 36.9%, and 30%. In the first round of the first losses dilemma that subjects play, 31.6% of the risk averse subjects cooperate, as opposed to 25.2% of the risk seeking subjects (1-sided Fisher’s exact=0.22). The extended version of the traditional method yields a percentage of cooperation that decreases as subjects become more risk seeking, from 42.9% to 30.0% and 26.8% to 19.4% for the most risk seeking subjects. In summary, the differences in cooperation rates between risk averse and risk seeking subjects are small, but show the correct trends for the traditional measurements. We next analyze whether a relation between risk preferences and behavior in repeated PD’s can be established. For this aim we use subjects’ risk preferences as mentioned above,

Effects of Risk Preferences in Repeated Social Dilemmas Table 3.4:

Logistic regression on the choice in round 1 of the repeated PD excluding the control variables. To account for the multiple cases per subject, standard errors are calculated on the basis of Huber (1967). Absolute t-values are given in brackets. ‘*’ denotes significance at the 5% level, ‘**’ denotes significance at the 1% level.

Risk seeking Gains dilemma Constant Adj. R2 Observations Table 3.5:

59

Risk preference assessment Traditional (1) Traditional (3) Tradeoff -0.22 (0.98) -0.21 (1.49) 0.12 (0.86) 0.42* (2.03) 0.43* (2.05) 0.51* (2.04) -0.84** (3.75) -0.62* (2.06) -1.20** (3.47) 0.01 0.01 0.01 432 432 347

Logistic regression on the choice in round 1 of the repeated PD including the control variables. To account for the multiple cases per subject, standard errors are calculated on the basis of Huber (1967). Additional independent variables are included, absolute t-values are given in brackets. ‘*’ denotes significance at the 5% level, ‘**’ denotes significance at the 1% level.

Risk seeking Gains dilemma Male Know game theory Social orientation Constant Adj. R2 Observations

Risk preference assessment Traditional (1) Traditional (3) Tradeoff -0.25 (1.09) -0.22 (1.49) 0.12 (0.84) 0.46* (2.17) 0.46* (2.18) 0.53* (2.08) 0.34 (1.52) 0.33 (1.52) 0.36 (1.49) 0.59** (2.59) 0.58** (2.60) 0.65** (2.65) -0.03 (0.36) -0.02 (0.42) -0.03 (0.32) -1.15** (4.18) -0.93* (2.68) -1.57** (4.02) 0.01 0.03 0.03 432 432 347

and we correlate these with the choices of subjects in the first round of the first gains PD (P = 0) that they played, and in the first round of the first losses PD (P = -20) that they played. Therefore, we consider two cases per subject simultaneously. Only the choices in the first rounds were used in the analysis, because choices in subsequent rounds are affected by the choices of the other player. Table 3.4 reports the results using a repeated measures design (logistic regression with two cases per subject). Table 3.5 reports the same results, including several extra independent variables. In contrast to our first experiment we find no significant relation between risk aversion as measured by the traditional method and the probability of cooperation. We observe that only the traditional method in its extended form (using three choices instead of a single choice) comes close to being significant at the p=0.10 (tvalue=1.65) level. The size of the effect on the probability of cooperation of a single difference in category equals 0.05 for both traditional methods of utility measurement. Separate analyses performed on only the gains or only the losses dilemma give similar results. The added independent variables in Table 3.5 show some interesting results. Men are

60

Chapter 3

Table 3.6:

Logistic regression on the choice in rounds 1 and 2 of the repeated PDs for gains, and for rounds 1 and 2 of the repeated PDs for losses, excluding the control variables. To account for the multiple cases per subject, standard errors are calculated on the basis of Huber (1967). Absolute t-values are given in brackets. ‘*’ denotes significance at the 5% level, ‘**’ denotes significance at the 1% level.

Cooperation in the gains dilemma Cooperation in the losses dilemma

Risk seeking Adj. R2 Observations Risk seeking Adj. R2 Observations

Risk preference assessment Traditional (1) Traditional (3) Tradeoff 0.07 (0.26) -0.07 (0.47) 0.10 (0.49) 0.00 0.00 0.00 431 425 358 -0.20 (0.80) -0.32* (2.07) 0.38* (2.51) 0.01 0.01 0.02 431 431 347

not more likely to cooperate, and those who claim to have heard of “game theory” do cooperate more often (an effect size of about 0.12 on the probability of cooperation). Additionally, contrary to most previous results on the effects of social orientation in social dilemma situations, subjects with high social orientation values are also not more likely to cooperate in the repeated PD. Moreover, we find that cooperation occurs more often in round 1 of the gains dilemma (an effect size of about 0.10 on the probability to cooperate). Therefore, the argument that “losses loom larger than gains and/or people are more risk seeking for losses, thus subjects are more willing to risk being the sucker in a repeated PD for losses, and are hence more likely to cooperate in a repeated PD for losses” is not only a mistake theoretically, as we outlined in the previous sections, but also is not corroborated empirically. Hypotheses H1 stated that there is a relation between the shape of the utility function as measured by the tradeoff method and the cooperation rate in the repeated PD, such that a larger cooperation rate corresponds to greater concavity of the utility function. In the case of the first round of the first gains dilemma, the tradeoff method yields no difference in cooperation rates. Subjects with concave utility cooperate in 37.6% of the cases, and the percentages of those who have linear utility or convex utility are 36.7% and 37.5% respectively. For the first round of the losses dilemma, the trend of the percentages of cooperation given by the tradeoff method is in the opposite direction. Subjects with concave utility cooperate in 23.8% of the cases, which increases to 25.0% and 32.9% for those who have linear and convex utility respectively. The tradeoff method thus gives results that clearly contradict our hypothesis: we find no difference for the gains dilemma, but a slight increase in cooperation for subjects who have more convex utility. In similar fashion to our analyses of the results of the traditional methods, we analyze whether there is a relation between concavity of the utility function and subjects' behavior in repeated PDs. Results of the analyses are reported in Table 3.4 and Table 3.5. We observe that in both the analysis with control variables and without, the relation between shape of the utility function and cooperation is not significant. The effect size of concavity of utility on the probability of cooperation of a single difference in category equals 0.017.

Effects of Risk Preferences in Repeated Social Dilemmas

61

Evidently, the effects of both risk aversion and the shape of the utility function on cooperation as put forward in H1 and H1' receive little support. However, if we extend our analyses and include the choice in the second round of the repeated PD, we do find some significant effects. Table 3.6 reports these results. Table 3.6 shows that risk preferences may have an effect in the losses dilemma, more sothan in the gains dilemma. For the losses dilemma, the extended traditional method shows a significant effect in the hypothesized direction (effect size = 0.07), whereas the tradeoff method reveals an effect of the same size in the opposite direction. If we run additional analyses, including extra rounds of the losses dilemma, we find that the negative effect of the extended traditional method remains significant at p=0.10 with an average effect size of approximately 0.10 for a difference of two categories. The positive effect of the tradeoff method disappears immediately when the third rounds of the losses dilemma are included, and does not return when subsequent rounds are included. 3.6

Conclusion and discussion

In Raub and Snijders (1997) a theoretical analysis suggested that a connection exists between the shape of the utility function and cooperation rates in a repeated PD. The theoretical argument there is that concavity should favor cooperation. Raub and Snijders (1997) assumed Expected Utility theory and in that case risk preferences and the shape of the utility function coincide. An experimental test supported their argument that risk aversion (concavity) favors cooperation. Moreover, cooperation was on average less prevalent in situations where it was necessary to avoid a loss. Van Assen (1998) argued, among other things, that the utility assessment method used by Raub and Snijders (1997) needed to be improved. We reported here on an experiment meant to replicate, combine, and extend the work in these two papers. We conducted an experiment consisting of two separate sessions, one to measure a subject’s risk preferences and utility function, and a second to measure cooperation in different repeated PDs. Three hypotheses were put to the test. One hypothesis (H1’’) stated that cooperation should be more easily achieved in a situation where mutual cooperation is necessary to achieve a gain than in a situation where mutual cooperation is necessary to avoid a loss. Our results clearly support this hypothesis. Once again, the intuitively appealing argument that people are more likely to contribute to the collective good in a situation where cooperation is necessary to avoid a loss, as argued, for instance, in Berejikian (1992), is refuted. Moreover, the acceptance of hypothesis H’’ together with the fact that framing effects exist leads to the conclusion that a party interested in the development of cooperation can influence the probability that cooperation will emerge by framing the situation as a gains dilemma. The second hypothesis (H1’) stated that the cooperation rate in the repeated PD is greater for people who are risk averse in gambles containing the outcomes of the PD than for people who are risk seeking in these gambles. Although significant (but small) positive effects of risk aversion were found by Raub and Snijders (1997), the evidence in our additional experiment is less convincing. Using the same method of measuring risk preferences as in the earlier experiment, we find significant effects of risk aversion only when we extend the measurement from a single gamble to three gambles. Risk aversion favors cooperation only in the repeated PD for losses, with an effect size on the probability of cooperation of about 0.10. The third hypothesis (H1) stated that the cooperation rate in the

62

Chapter 3

repeated PD is greater for people with concave utility functions than for people with convex utility functions. To test this hypothesis, we measured utility functions using the tradeoff method (Wakker and Deneffe, 1996; Fennema and van Assen, 1999), a method that does not depend on the assumption that subjects evaluate probabilities linearly. The tradeoff method does elicit risk preferences that are more in line with earlier results in the literature on risk preferences. Whereas the traditional method as used by Raub and Snijders (1997), and replicated here, resulted in a relatively large number of risk seeking choices in the gains domain, the tradeoff method shows more convex utility for losses and much more concave utility for gains. However, there is no relation between utility as measured using the tradeoff method and cooperation in the repeated PDs Surprisingly, the shape of utility as measured using the tradeoff method hardly correlates with risk preferences as measured by the extensive traditional method. Several complementary reasons can be provided for the low correlation between the results of using the two methods. Firstly, the bias in the traditional method resulting from probability weighting can vary between individuals, which decreases the correlation. Secondly, it is likely that the correlation is also obscured because we make use of rather crude nonparametric measures of utility and risk preferences. We expect the correlation between the two measures to be higher when estimating risk preferences and utility parametrically. In principle it is possible to use the data to estimate the utility function in a parametric way, but we did not do so because our main goal was to test the relation between the shape of the utility function and cooperation in the repeated PDs. Other disturbing factors that could have been involved in reducing the correlation are random and independent errors of subjects in both methods. The moderate test-retest reliability correlations indicate that within-subject variation is indeed substantial. Although there are reasons to expect only a low to moderate correlation between the two methods, the finding that it is near zero is worrying and implies that the question should be asked of whether utility as a subject characteristic exists at all (Shoemaker, 1982). A recent study by Isaac and James (2000) also questions the view that individuals have a stable and consistent risk attitude. They measured utility of individuals using two methods and found a negative correlation between the two methods, although the correlation was not significantly different to zero. Weber and Milliman (1997) hypothesize that risk preference might be a stable personality trait after all, but that differences in the choices made in different situations or methods may be the result of changes in risk perception. That is, different choices made by the same individual in different circumstances can be the result of different perceptions of risk, while risk preferences remain stable. Nevertheless, in the context of the present study, the low reliability and small correlations between methods imply that if a correlation is expected between assessed utility and cooperation in the PD games, then it is very likely to be small. While the low reliability and small correlations between methods at least partly explain the lack of empirical support for hypotheses H1’ and H1’’, several other disturbing factors may be involved as well. Firstly, it is theoretically possible that subjects in our study were trying to reach an equilibrium, where behavior on the equilibrium path alternated between cooperation and defection, yielding payoffs (S, T) and (T, S). However, this is unlikely for two reasons. Firstly, outcome R was larger than the average of outcomes S and T. Subjects could therefore observe that alternating defection and cooperation does not lead to a higher payoff than adhering to mutual cooperation. Secondly, alternating response patterns

Effects of Risk Preferences in Repeated Social Dilemmas

63

were observed only infrequently observed in both experiments. Somewhat related to this argument is the argument that (T-R)/(T-P) may not be the correct index to study in the repeated PD (although it is the right index to study if game theoretical rationality is assumed). However, Murnighan and Roth (1983) found that this index was one of the best predictors of cooperation rates in repeated PDs with different payoffs. The experiment that we conducted was different to the original experiment of Raub and Snijders (1997) in an important way other than just using a more extensive measurement of risk preferences and utility. Instead of letting subjects play repeated PDs against each other, we let them play against the computer, telling them so. The computer was programmed to mimic the behavior of a subject in the earlier experiment. This allowed subjects to wok through the experiment at their own pace, without delaying others, thereby speeding up matters dramatically. Although subjects were explicitly and repeatedly told during the instruction that the computer program played like a human player randomly selected from the Raub and Snijders (1997) experiment, this procedure may have resulted in a certain lack of reality for the subjects (“there was no real person on the other side”). However, after completing the experiment, only a few (less than five) subjects asked for a detailed typed explanation of the computer program they played against. The explanation was an elaborated version of Appendix 3.1. The fact that only a few subjects asked for the explanation suggests that participants believed that they had not been tricked, and that they played against a computer that faithfully mimicked a human player. The data from the experiment allow other tests of the effects of risk preferences on cooperation in repeated PDs to be studied. As van Assen (1998) showed, one can expect more dramatic effects of risk preferences for repeated PDs where the payoffs are mixed (that is, partly positive and partly negative). Although we do not report it here, we did also let subjects play such mixed dilemmas. We plan a future study on the effect of loss aversion on cooperation in repeated mixed dilemmas. Theoretically, two improvements of our current models seem particularly useful. Thus far, we have assumed that ego considers the utility function of alter to be the same as his own. A logical extension would be to include incomplete information about the utility function of alter, reflecting in one way the uncertainty of ego when considering the move of alter. Additionally, we could improve the theory by looking more closely at the effects of evaluating probabilities nonlinearly on the properties of the equilibrium condition. In measuring the utility functions we took care of this, but likewise, there are arguments that the equilibrium condition will be influenced if we base our theory on Cumulative Prospect Theory. For example, part of the derivation of the equilibrium condition involves calculating (infinite) discounted sums. These discounting weights are typically probabilities, calling for adaptations of the derivation that accurately reflect the nonlinear evaluation of probabilities.

64 Appendix 3.1

Chapter 3 Simulation of other actor's strategy in repeated PDs

The optimal social dilemma experiment would allow the experimenters to gather a maximum number of responses per subject per time unit, requiring as little time of as few experimenters as possible, without losing the reality of actual interaction. We use a way of letting subjects interact that is not very common: they interact with a computer program that pretends to behave like a human and we inform the subjects about this. In Experiment 1 a procedure was used where subjects played repeated PDs against other players in the same room. This procedure has several disadvantages. First of all, it is rather slow and quite demanding for the experimenters. The experimenters - sometimes five in number per session - passed the responses of subjects to their opponents. Only after all subjects had made their choice and had received the choice of their opponent was a roulette wheel spun to determine the pairs that could continue to the next round of the repeated PD. This procedure is quite timeconsuming and in fact a lot of the subjects’ time is spent waiting (until other subjects had made their choice, or until the experimenters had delivered the choices of the opponents, or until the roulette wheel was spun). This is not efficient and moreover, it tempts subjects to start looking around or talking to others. We considered speeding up the procedure by using a computer network connecting pairs of players, which also requires less effort on the part of the experimenters during the experiment. However, the slower one of the two subjects still determines the speed of the procedure, and subjects may not believe that they are actually playing against another subject. Therefore, we chose to try a procedure that has the advantages of being fast and not very demanding, and avoids the disadvantage that subjects are not sure whether they are being tricked into believing that they are not interacting with someone else. We told subjects that they were interacting with a computer program that mimics the behavior of a person in previous experiments. This implies that subjects play at their own pace, any number of subjects can play at the same time, and the experimenters' efforts during the experiment are minimized. Naturally, it might be the case that "playing against the computer" is not as realistic to the subjects as playing against a real opponent would be. A statistical model of actors' strategies was constructed and fitted to an actor's behavior in Experiment 1. The model is a logistic regression model P (C i ) =

1 1 + exp( − X i )

where P(Ci) denotes the probability that the actor cooperates in round i. Function Xi, denoting the propensity of actor z to cooperate in round i, is dependent on four parameters:  i −1   i −1      X i = β 0 + β D  ∑ 2( j − i + 1)(1 − y jA)  + β C  ∑ 2( j − i + 1) y jA +  j =1   j =1    i −1   i −1  β M  β D  ∑ 2( j − i + 1)(1 − y jZ)  + β C  ∑ 2( j − i + 1) y jZ        j =1   j =1  

Variable YjZ denotes actor z's choice in round j, YjA denotes the other actor's choice in round j, where y is 1 if the choice is cooperation and is 0 if the choice is defection. The sums in the equation imply that the responses in round i-1-k are 2-k times as relevant for the propensity to

Effects of Risk Preferences in Repeated Social Dilemmas

65

cooperate than the responses in round i-1. Moreover, using a discount factor equal to 0.5 implies that the responses in the previous round are never less relevant than the whole history of the game before the previous round. The parameters have the following interpretation:

β0

Propensity to cooperate in the first round of the repeated PD

βD

Effect of defection in the previous round on the propensity to cooperate in the present round

βC

Effect of cooperation in the previous round on the propensity to cooperate in the present round

βM

Relative importance of the opponent's choice in comparison to the own choice in the previous round for the propensity to cooperate in the present round

The statistical model was fitted to the data obtained from Experiment 1. More precisely, the observed frequencies of histories in round 1, round 2, round 3, and the proportions of mutual defection or mutual cooperation up to round 5 were predicted by the logistic model. The model was able to predict the 2 ('C' and 'D') + 4 ('DD', 'DC', 'CD', 'CC') + 8 ('DDD', 'DDC', etc.) + 2 ('DDDDD' and 'CCCCC') proportions very accurately. Estimates of the parameters β0, βD, βC, and βM were equal to –0.7369, -1.1, 1.3, and 0.85 respectively. These estimates indicate that cooperation in the previous round had a larger effect on the present response than defection, and that the subject’s own response had a larger effect on the present response than the response of the opponent. The chi-square statistic comparing expected and observed frequencies was equal to 3.24. It is difficult to devise a statistical test because the frequencies are clearly not independent. However, even for 1 degree of freedom the obtained chi-square value is not significant, which indeed indicates that our statistical model accurately describes actor's behavior in the first experiment. To use the statistical model in Experiment 2 with the fitted values of the four parameters, the computer program generates random numbers in the unit interval [0,1]. The program cooperates if the number is in interval [0,P], where the value of P is determined by the history of the game up to the previous round, otherwise it defects.

4 Effects of Individual Decision Theory Assumptions on Predictions of Cooperation in Social Dilemmas

Van Assen, M.A.L.M. (1998). Effects of individual decision theory assumptions on predictions of cooperation in social dilemmas. Journal of Mathematical Sociology, 23, 143-153.

68

Chapter 4

Abstract Raub and Snijders (1997) show that, under the assumption of S-shaped utility, conditions for cooperation in social dilemmas are more restrictive if outcomes represent gains than if outcomes represent losses. They neglected two interesting issues in their paper: conditions for cooperation in social dilemmas with both losses and gains as outcomes, and the effect of probability weighing on these conditions. In this comment it is shown that, under assumptions of Prospect Theory, conditions for cooperation are best if dilemmas include both positive and negative outcomes, and that these conditions improve with increasing loss aversion. Furthermore, it is shown that probability weighing can effect conditions to cooperate as well.

Key Words Social dilemma, prospect theory, loss aversion, risk preferences, decision weights

Acknowledgements I thank Alexander Gatig for providing me with an earlier draft of Raub and Snijders (1997), and I thank Werner Raub, Chris Snijders, and Peter Wakker for making comments on an earlier draft of this paper. Finally, I thank Sasa Bistrovic for pointing out an inaccuracy in the description of the proof in the appendix.

Effects of Individual Decision Theory Assumptions on Predictions of Cooperation 4.1

69

Introduction

In their first article on Prospect Theory, Kahneman and Tversky (1979) reported substantial empirical evidence for three regularities in individual decision making: S-shaped utility, loss aversion, and probability weighing. S-shaped utility refers to convex and concave utility functions for, respectively, losses and gains. Loss aversion refers to the relative steepness of the utility function for losses in comparison to gains. Probability weighing refers to the observation that individuals overweigh small and underweigh large probabilities. It is argued in this note that S-shaped utility, loss aversion, and probability weighing have important implications for cooperation in social dilemmas. Raub and Snijders (1997) compared conditions for cooperation in strictly positive and strictly negative repeated PDs. They derived that, under the assumption of S-shaped utility, conditions for cooperation are more restrictive if outcomes represent losses than if outcomes represent gains, i.e., that a gain among smaller gains is a more powerful motivator to cooperate than a loss motivates actors to cooperate to obtain a smaller loss. Raub and Snijders however neglected two interesting issues in their paper: conditions for cooperation in (mixed) PDs with both losses and gains, and the effect of probability weighing on these conditions. Both issues are addressed in this paper. More specifically, it is shown in Section 4.2 that: (1)

conditions for cooperation in mixed PDs improve with increasing loss aversion;

(2)

in well-defined circumstances, assuming S-shaped utility, conditions for cooperation in mixed PDs are less restrictive than for strictly positive (and strictly negative) PDs.

Hence in mixed social dilemmas a loss can be a more powerful motivator to cooperate than a gain and loss aversion increases the motivating power of a loss in these dilemmas. In Section 4.3 it is demonstrated how probability weighing can effect the willingness to cooperate: (3)

In strictly positive and strictly negative repeated PDs, assuming probability weighing, subjects are less (more) inclined to cooperate if the PD's continuation probability is large (small) than can be expected from the value of the probability alone.

The latter result can explain the absence of high cooperation rates in Raub and Snijders' experiment. It is also demonstrated that Raub and Snijders' use of a utility assessment procedure not taking probability weighing into account can explain why only a minority of subjects seemed to choose in accordance with S-shaped utility. 4.2

The effects of loss aversion on cooperation in social dilemmas

Following Raub and Snijders an indefinitely repeated standard 2-person PD1 (Γ:∆) is considered as an example of a social dilemma. Using their notation, the outcomes of the PD are denoted by T for unilateral defection, R for mutual cooperation, P for mutual defection, and S for unilateral cooperation. By definition of the PD, S < P < R < T. The set (Γ:∆) contains all PDs with outcomes T+∆, R+∆, P+∆ > S', where ∆ and S' are realvalued constants with S'< P+∆. The PD (Γ:∆) is indefinitely repeated in rounds 1,2,...,t with a constant probability of ω 1 The results of their derivations and the derivations in this text can be generalized to n-person dilemma games of the Schelling-type (1978).

70

Chapter 4

to play the game at round t+1 given that round t has been played, reflecting exponential discounting of game pay-offs. It is assumed that both actors have complete information and common knowledge with respect to the structure of the repeated PD, and that they are informed on the behavior of the other actor in all previous rounds 1,2,...,t-1. Furthermore, they are assumed to behave in accordance with the same utility function u. A well-known result in game-theory (e.g., Friedman, 1986, pp. 88-89) is that a (subgame perfect) equilibrium exists such that actors choose to cooperate in all rounds if and only if

ω ≥ ω u ( Γ : ∆ )=

u(T + ∆ ) - u(R + ∆ ) u(T + ∆ ) - u(P + ∆ )

(1)

In the sequel, the behavior of ωu(Γ:∆) is studied as a function of ∆ without Raub and Snijders' restriction to strictly positive or strictly negative PDs (outcomes T+∆, R+∆, and P+∆ having identical sign). Without loss of generality, outcome P is fixed at value 0. S-shaped utility is assumed together with loss aversion. Loss aversion is modelled by assuming that u(-x) = λu(x) for all x ≥ 0, with λ > 1. The higher the value of λ the more loss aversion. To acknowledge that ωu is dependent on the value of the loss aversion parameter, ωu is denoted by ωu(Γ:∆,λ). To examine the behavior of ωu(Γ:∆,λ) four domains of ∆ are distinguished: ∆1 ≥ 0, 0 > ∆2 ≥ -R, -R > ∆3 ≥ -T, and -T > ∆4. Raub and Snijders remark that ωu(Γ:∆1,λ) and ωu(Γ:∆4,λ) are independent from the value of λ. Our first result, however, states that conditions for cooperation in mixed PDs improve with increasing loss aversion.

ω u ( Γ : ∆ 2 ,λ ) =

u(T + ∆2 ) - u(R + ∆ 2 ) u(T + ∆ 2 ) + λu(- ∆ 2 )

(2)

ω u ( Γ : ∆3 , λ ) =

u(T + ∆ 3 ) + λu(-R - ∆ 3 ) u(T + ∆ 3 ) + λu(- ∆ 3 )

(3)

The result is easily proved: both ωu(Γ:∆2,λ) and ωu(Γ:∆3,λ) are strictly decreasing in λ, which demonstrates that loss aversion indeed promotes cooperative behavior in mixed PDs. More interesting is a comparison of ωu(Γ:∆2,λ) and ωu(Γ:∆3,λ) to ωu(Γ:∆1,λ) to see whether conditions for cooperative actions are less restrictive in a mixed PD than in a strictly positive PD. Although some results can be derived without assuming a parametric form of the utility function, more can be said about the behavior of ωu(Γ:∆,λ) as a function of its parameters if additional assumptions are made on the utility function. Tversky and Kahneman (1992) found empirical support for a utility function from the power family. Following Tversky and Kahneman (1992) the power family is assumed, but with the restriction that the power parameters for losses and gains are identical: 2 u(x) = xα

(x ≥ 0), and

u(x) = -λ(-x)α

(x < 0).

(4)

Parameter α is in the interval (0,1) because S-shaped utility is assumed. Since ωu is dependent on α as well, it is a function of three parameters and is from now on denoted by ωu(Γ:∆,λ,α). 2 In their 1992 experiment Tversky and Kahneman found identical estimates for the power parameter for losses

and gains.

Effects of Individual Decision Theory Assumptions on Predictions of Cooperation

Figure 4.1:

71

ωu(Γ:∆,λ,0.88) as a function of ∆ and λ, for ωL = 0.75 (T = 16, R =12), and ωL = 0.25 (T = 16, R =4).

The second result of this section concerns the behavior of ωu(Γ:∆2,λ,α) in relation to ωu(Γ:∆1,λ,α) and is stated in the following theorem. The proof of the theorem and an extension are presented in Appendix 4.1. Theorem

ωu(Γ:∆2,λ,α) < ωu(Γ:∆1,λ,α) for all ∆2 in the interval (-

RT ,0), and for all ∆1, 0 < α < 1 and λ = 1. R +T

Because ωu(Γ:∆2,λ,α) and ωu(Γ:∆3,λ,α) decrease in λ (see (2) and (3)), both the left border of the interval and the numbers [ωu(Γ:∆2,λ,α) - ωu(Γ:∆1,λ,α)] and [ωu(Γ:∆3,λ,α) - ωu(Γ:∆1,λ,α)] decrease if λ is increased. Together with the results on the effect of loss aversion the Theorem RT ,0) if both do not cooperate, implies that if both actors receive a loss in the interval ( R +T they are more prone to cooperate than in strictly positive PDs (Γ:∆1) and that they are even more so for increasing loss aversion. Thus, in social dilemmas with both negative and positive outcomes a loss can be a more powerful motivator to cooperate than a gain and loss aversion increases the motivating power of a loss in these dilemmas. Unfortunately, no general results of ωu(Γ:∆3,λ,α) in comparison to ωu(Γ:∆1,λ,α) could be derived independently from the values of α, λ, and ∆3. For some values of these parameters ωu(Γ:∆3,λ,α) is less and for others greater than ωu(Γ:∆1,λ,α) for all ∆1. To clarify the results derived in this section, the behavior of ωu(Γ:∆,λ,α) as a function of its parameters is illustrated by examples, all depicted in Figure 4.1. The two horizontal curves represent distinct values for ωL = ωu(Γ:∆,1,1); ωL = 0.25 (T = 16, R = 12), and ωL = 0.75 (T = 16, R = 4). Three curves for each of the two values are shown in the figure, all with

72

Chapter 4

α = 0.88, the value found in the Tversky and Kahneman (1992) experiment for both losses and gains. For each ωL, the upper curve, middle, and lower curve represent the behavior of ωu(Γ:∆,λ,0.88) for respectively λ = 1 (no loss aversion), λ = 2.25, and λ = 5. The value λ = 2.25 was found by Tversky and Kahneman (1992) as well. If the Theorem is applied to the examples it is found that ωu(Γ:∆2,λ,α) < ωu(Γ:∆1,λ,α) 64 192 if λ = 1 and ∆2 ε ( ,0) for ωL = 0.75 and ∆2 ε ( ,0) for ωL = 0.25. If it is assumed that 20 28 the values for α and λ found by Tversky and Kahneman are correct, then the intervals have more negative boundaries; by calculation it is obtained that the intervals are (-13.519,0) and (-9.150,0) for respectively ωL = 0.25 and ωL = 0.75. Note that these intervals contain the outcome -R, which means that conditions for cooperation in PDs with three negative outcomes (S, P, and R) can be less restrictive than in strictly positive dilemmas. The results of Raub and Snijders and the results derived in this section can be summarized in other words as follows: People are more inclined to cooperate when they are more risk averse. Since it is assumed that people are risk averse for gains and risk seeking for losses, more cooperation is expected for strictly positive than for strictly negative PDs. Loss aversion causes people to be more risk averse for some mixed PDs, and therefore people are more inclined to cooperate in the latter PDs. 4.3

Probability weighing

In investigating the effects of probability weighing on both the results derived in Section 4.2 and the results of Raub and Snijders, it is assumed that decision makers weigh probabilities as formulated in Cumulative Prospect Theory (CPT: Tversky and Kahneman, 1992, pp.300301). In CPT different weight functions π for probabilities belonging to losses and gains are assumed. A property of the functions π is that for strictly positive or strictly negative twooutcome gambles the decision weight functions assign the decision weight π(p) to the probability p belonging to the largest absolute outcome, and 1-π(p) to the probability assigned to the smallest absolute outcome. Note that the property implies that both decision weights add to one. It is assumed, in agreement with empirical results from numerous studies (e.g., Kahneman and Tversky, 1979; Tversky and Kahneman, 1992; Camerer, 1995; Camerer and Ho, 1994) that subjects overweigh rather small probabilities (say, smaller than 0.30) and underweigh moderate to large probabilities (say, larger than 0.50) belonging to the largest absolute outcome. Subjects in Raub and Snijders' experiment participated in strictly positive and strictly negative dilemmas which had a probability of 0.75 of continuing. If the PD is continued subjects gain (loose) an additional amount of money since all outcomes of a single trial of the PD are positive (negative). Therefore, it is assumed that the decision weight for ω is equal to π(ω), the decision weight for the largest absolute outcome. Assuming that subjects have the same decision weight function π, it can be derived that cooperation is feasible, if

π ( ω ) ≥ ωu( Γ )=

u(T) - u(R) . u(T) - u(P)

(5)

The formula above is (1) with ∆ = 0 and ω replaced by π(ω). From the formula above our third result can be derived: Because of overweighing (underweighing) of a small (large) probability ω, subjects are more (less) inclined to cooperate in strictly positive and strictly

Effects of Individual Decision Theory Assumptions on Predictions of Cooperation

73

negative PDs than can be expected from probability ω alone. In the next paragraph this result is used to explain one of Raub and Snijders' findings that cannot be explained without probability weighing. Raub and Snijders choose the outcomes T, R and P in their experiment such T -R . Raub and Snijders (Table 1) found cooperation rates smaller than 0.50 for that ω = T -P three out of four groups with risk averse response patterns for losses or gains. Their results can be explained if π(0.75) < ωu(Γ) < 0.75. Assuming the value function of Tversky and Kahneman (1992) with α = 0.88 and the outcomes used in Raub and Snijders (T = 16, R = 7, P = 4), it can be calculated that ωu(Γ) = 0.733. Hence a small underweighing of ω is sufficient to explain the absence of high cooperation rates in the experiment of Raub and Snijders. In the analysis above it was assumed that the utility function was correctly measured. However, some critical comments can be made on the measurement of utility function characteristics by Raub and Snijders. They assessed whether the functions for gains (losses) were concave by asking the subjects to choose between the two options; (a) receiving R+ (R-), (b) receiving P+ (P-) with probability ω and T+ (T-) with probability 1-ω. They argue that subjects with concave utility functions will choose the certain option. Raub and Snijders (text under Table 1) however found "... that - contrary to Kahneman and Tversky's conjecture only 23% of all participants chose in accordance with S-shaped utility". Their argument and statement is not correct if probability weighing is assumed. According to CPT the probability 1-ω (0.25) assigned to outcomes T+ and T- is overweighed3 which implies that subjects can prefer the risky option above the certain option, i.e., that they are risk seeking, even if they have a concave utility function. The latter can both explain the unexpected high number of risk seeking response patterns for both losses and gains (44%, Table 1) and the rather small percentage (23%) of expected response patterns. Thus the results of Raub and Snijders in Table 1 need not be in conflict with the assumption of S-shaped utility and in principal but can be explained by probability weighing as in CPT. If the method of Raub and Snijders is not valid to measure characteristics of the utility function, then what method is? All traditional utility assessment methods (see Farquhar, 1984, for a general overview) assume that subjects do not transform probabilities. Only recently, Wakker and Deneffe (1996) developed a method that can measure utility functions independently from probability transformations. Applications of the method and a discussion of its pros and cons in comparison with traditional utility assessment methods can be found in Wakker and Deneffe (1996) and van Assen (1996). 4.4

Discussion

Results from the field of individual decision making are seldomly used in deriving predictions on both subjects' and collective behavior in social dilemma-like situations (Raub and Snijders, 1997). That is surprising, since this and Raub and Snijders' paper clearly show that results from research in the field of individual decision making, like S-shaped utility, loss aversion, and probability weighing, have substantial implications for predictions of cooperation in social dilemmas. 3 The probabilities assigned to T and T- are overweighed since, respectively, both π(0.25) > 0.25 and 1-π(0.75) +

> 0.25.

74

Chapter 4

The results derived in both papers suggest applications as well. Actors evaluate outcomes in terms of gains and losses relative to a (subjective) reference point rather than in terms of final outcomes. The reference point, however, is dependent on the description of the choice situation (e.g., Kahneman and Tversky, 1979). Thus, if a group of actors is better off when its members cooperate, their leader(s) can enhance conditions for cooperation by framing the situation as a situation in which defection leads to a loss, and cooperation to a larger gain or a smaller loss. Since the application of individual decision theory to repeated PDs yielded new insights in conditions for cooperative behavior, it can be fruitful to apply individual decision theory to other social dilemmas as well. The predictions resulting from these analyses can then be tested in empirical studies like in Raub and Snijders (1997).

Effects of Individual Decision Theory Assumptions on Predictions of Cooperation

75

Appendix 4.1: Proof of Theorem The proof of the Theorem consists of several steps: (i)

calculate the minimum of ωu(Γ:∆1,λ,α),

(ii)

proof that ωu(Γ:∆2,λ,α) = min[ωu(Γ:∆1,λ,α)] if ∆2 = -

(iii)

proof that ωu(Γ:∆2,λ,α) < min[ωu(Γ:∆1,λ,α)] for all ∆2 ε ( -

RT for λ = 1 and all α ε (0,1), R +T RT ,0) for λ = 1 and all α R +T

ε (0,1). (i)

Since ωu(Γ:∆1,λ,α) is increasing in ∆1 (see Theorem 3 of Raub and Snijders) the minimum of ωu(Γ:∆1,λ,α) is obtained for ∆1 = 0. Hence the minimum is equal to T α - Rα Tα

(ii)

(6)

Modify (2) by substituting λ = 1 and using (4). Then equate the modified equation (2) to (6) to obtain after some manipulations: α α α α Rα (- ∆ 2 ) - T α (R + ∆ 2 ) - T α (- ∆ 2 ) + Rα (T + ∆ 2 ) = 0

Substituting ∆2 = (iii)

(7)

RT in (7) results in the equality 0 = 0. R +T

It is sufficient to show that the second derivative of (the left hand side of) (7) with respect to ∆2, which is the numerator of the difference ωu(Γ:∆2,λ,α) - min[ωu(ΓRT :∆1,λ,α)], is positive for all ∆2 ε ( ,0). Differentiating (7) twice, dividing by (1R +T α) and rearranging terms yields α

-

α

α

α

R T + - R 2 -α + T 2 -α 2 -α 2 -α (T + ∆ ) (R + ∆ ) (- ∆ ) (- ∆ )

(8)

The value of (8) is clearly larger than zero for all ∆2 and thus also for all ∆2 ε RT (,0), because the positive terms are larger than the negative terms. R +T A result more general than the Theorem can be derived by extending the set of PDs * (Γ:∆1) to all strictly positive PDs (Γ :∆1) with the same ωL, i.e., with outcomes T' = aT, R' = aR, and a > 0. Because the minimum of ωu(Γ*:∆1,λ,α) (obtained in ∆1 = 0) is equal to the minimum of ωu(Γ:∆1,λ,α), the Theorem applies to set (Γ*:∆1) as well.

5 The Effect of Nonlinear Utility on Behavior in Repeated Prisoner’s Dilemmas

Van Assen, M.A.L.M., & Snijders, C. (2001). The effect of nonlinear utility on behavior in repeated prisoner’s dilemmas. Submitted to Journal of Economic Behavior and Organization.

78

Chapter 5

Abstract The present study focuses on the effect of agents’ utility on their cooperation in repeated twoperson prisoner’s dilemma games (PDs). A game-theoretical analysis suggests that conditions for cooperation in the PDs improve with concavity of utility, increasing risk aversion, and increasing loss aversion in the case of PDs with both negative and positive outcomes. The hypotheses were tested in an experiment in which participants played a number of different repeated PDs and their utility was elicited. The results provide no evidence that concave utility promotes cooperation in repeated PDs, but evidence that it is promoted by risk aversion and possibly loss aversion is provided. JEL classification:

C72, C90, D81

Keywords utility elicitation, prospect theory, repeated two-person prisoner’s dilemma, cooperation, risk preferences

Acknowledgements We would like to thank Werner Raub and Jeroen Weesie for many helpful discussions concerning the research presented in this paper, and Werner Raub and Tom Snijders for their suggestions in improving the manuscript.

The Effect of Nonlinear Utility on Behavior in Repeated Prisoner’s Dilemmas 5.1

79

Introduction

Studies of individual decision making provide substantial empirical evidence that utility is not linear with respect to money. Although the variability in decision makers’ utility is substantial (e.g., Gonzalez and Wu, 1999), utility is often S-shaped, that is, concave for gains and convex for losses (e.g., Kahneman and Tversky, 1979; Tversky and Kahneman, 1992; Fennema and van Assen, 1999)1. In addition, people seem to exhibit loss aversion (Kahneman et al., 1979; Kahneman and Tversky, 1984; Tversky and Kahneman, 1991, 1992), that is, they dislike a loss more than they like a comparable gain (“losses loom larger than gains”). The implications of these deviations from linear utility on behavior in situations where agents’ outcomes are interdependent are not well understood. In the present study we focus on the effect of agents’ evaluation of outcomes on cooperation in repeated prisoner’s dilemmas. Such an analysis has until recently been widely neglected (Ledyard, 1995, p.143), despite theoretical and empirical arguments in favor of a closer analysis of this effect. Raub and Snijders (1997) derived that concavity of utility is positively related to favorable conditions for cooperation in infinitely or indefinitely repeated prisoner’s dilemma games (PDs). Moreover, they tested and confirmed this hypothesis for PDs where all outcomes are gains and for PDs where all outcomes are losses (“positive PDs” and “negative PDs”). Van Assen (1998)2 derived that, in theory, loss aversion has a substantial effect on conditions for cooperation in PDs with both positive and negative outcomes (“mixed PDs”). He also argued that Raub and Snijders’ (1997) assessment of utility might have been biased by nonlinear probability weighing because it assumed that subjects made choices in accordance with Expected Utility theory. Continuing our common research program, van Assen and Snijders (2001)3 carried out a second experiment employing both the traditional utility assessment method of Raub and Snijders (1997) and a utility assessment method that is robust with respect to probability distortions. The present study again examines positive and negative PDs, and hence serves as a replication and extension of the results of Raub and Snijders (1997). The present study is distinguished from previous studies in our research program in two respects. Utilizing both the traditional and a theoretically unbiased assessment, cooperation in negative, positive and mixed PDs can be related to agents’ choice behavior in positive, negative, and mixed gambles. In addition, our study enables a conclusion to be made with respect to the effects of loss aversion on cooperation as hypothesized by van Assen (1998). We first outline the theoretical background in more detail. 5.2

Theoretical background and previous empirical results

5.2.1 Theory: Concavity of utility promotes cooperation in the repeated PD Raub and Snijders (1997) used the indefinitely repeated two-person PD to derive and test the relation between utility and cooperation in prisoner’s dilemmas.4 The indefinitely repeated two-person PD is characterized by its payoffs S, P, R, and T (S < P < R < T), and by the continuation parameter w (0 < w < 1), which represents the probability that another round of 1 Chapter 2 corresponds to Fennema and van Assen (1999). 2 Chapter 4 corresponds to van Assen (1998). 3 Chapter 3 corresponds to van Assen and Snijders (2001). 4For reasons of simplicity and convenience two-person PDs were chosen. However, the results derived below

can be generalized to N-person PDs.

80

Chapter 5

the repeated PD will be played. We define a game (Γ: ∆), with Γ = (S, P, R, T, w) as a game where all payoffs are “shifted” by a value of ∆. That is, (Γ: ∆) is a game with outcomes S+∆ < P+∆ < R+∆ < T+∆, where ∆ is a real-valued constant. It is assumed that both actors have complete information and common knowledge with respect to the structure of the game, and that they have been informed about the other player’s behavior in all previous rounds of the game. Finally, it is assumed that all players behave in accordance with the same utility function u and that this utility function is common knowledge. A well-known result in game theory (e.g. Friedman, 1986, pp. 88-89; Roth and Murnighan, 1978; Taylor, 1987, Chapter 4) is that in the indefinitely repeated PD (Γ: ∆) a pair of trigger strategies is in equilibrium such that both actors choose to cooperate in all rounds if and only if w ≥ wu ( Γ : ∆ ) =

u(T + ∆ ) − u( R + ∆ ) u(T + ∆ ) − u( P + ∆ )

(1)

A trigger strategy represents conditional cooperation in the extreme. A player cooperates until a defection of the other player occurs, after which he defects in all subsequent rounds regardless of the choices of the initial defector. Equation (1) reveals that the shadow of the future, as reflected by the continuation probability w, must be large enough to make cooperation possible at equilibrium. If w is smaller than the ratio in (1) there is no equilibrium that yields only cooperative choices. Interpreting wu as a threshold for the possibility of cooperation, the relation between utility and cooperation immediately follows. If u is concave, the index wu is smaller than the ratio (T–R)/(T–P), which implies that w can be smaller for mutual cooperation to be an outcome of equilibrium behavior than it is in the case of linear utility. Conversely, if utility is convex, then wu is larger than the ratio (T–R)/(T– P) and conditions for cooperation are worse than in the case of linear and concave utility (see Raub and Snijders, 1997, for a proof). 5.2.2 Experiment: Concavity of utility promotes cooperation in the repeated PD Raub and Snijders (1997) tested the relation between agents’ risk attitudes and behavior in a repeated PD experimentally. Only a concise summary of the experiment and its results are presented here; for further details Raub and Snijders (1997), Snijders and Raub (1998), and the experiment carried out in the present study should be referred to. The outcomes P, R, and T were 4, 7, and 16 Dutch guilders respectively, and ∆ was either equal to 0 (positive PD) or –16 (negative PD). At the time, 1 Dutch guilder was worth approximately 0.55 US dollars. The value of w in the experiment was equal to (T–R)/(T–P) = 0.75, such that (1) holds for players with concave and linear utility but not for players with convex utility. A simple, but at that time convenient, measurement of utility was used. Players had to make a choice between the gamble (T, 1–w, P), yielding T with probability 1–w and P with probability w, and its expected value R. Raub and Snijders (1997) assumed that agents’ choice behavior conforms to Expected Utility theory of von Neumann and Morgenstern (1944), implying that risk aversion and risk seeking in the above gamble are equivalent to concave and convex utility respectively. Comparing agents’ risk attitudes (utilities) with their choices in the first round of both the negative and positive PD, their hypothesis that concave utility facilitates cooperation was confirmed. In the positive PD, 38% of the risk averse subjects cooperated as against 26% of the risk seeking subjects (p = 0.054); in the negative PD the observed difference was larger; with 48% of the risk averse subjects cooperating, as opposed to only

The Effect of Nonlinear Utility on Behavior in Repeated Prisoner’s Dilemmas

81

26% of the risk seeking subjects (p = 0.002). 5.2.3 Controlling for the weighting of probabilities in the assessment of utility One problem with the conclusion of Raub and Snijders (1997) is that it is not robust with respect to violations of Expected Utility theory. Empirical studies of individual decision making have demonstrated that decision makers do not usually treat probabilities linearly, as is assumed in Expected Utility theory. In general, these studies show an overweighting of small probabilities and an underweighting of intermediate to large probabilities (Kahneman and Tversky, 1979; Camerer and Ho, 1994; Camerer, 1995; Tversky and Kahneman, 1992; Wu and Gonzalez, 1996; Gonzalez and Wu, 1999). Van Assen (1998) demonstrated that the (nonlinear) weighting of probabilities has two consequences for the study of the relation between utility and cooperation in repeated prisoner’s dilemmas. Firstly, subjects may weight the continuation probability and thereby change conditions for cooperation. That is, assuming that both subjects have the same probability weight π(w) and that they underweight large probabilities, substituting π(w) < w = 0.75 in (1) implies that conditions for cooperation deteriorate in comparison to those in the case of linear probability weighting. We will return to this in the conclusion and discussion section. A second implication of probability weighting is that properties of the utility function might not be measured appropriately. Concavity and convexity are no longer equivalent to risk aversion and risk seeking respectively. Concave utility, combined with the underweighting of small probabilities, can result in risk seeking choices.5 Therefore, Raub and Snijders (1997) did not test whether concavity of utility promotes cooperation, but rather whether risk aversion promotes cooperation. To test their original hypothesis, an alternative measure of utility should be used that is robust with respect to probability weighting. All traditional utility assessment methods (see Farquhar, 1984, for a general overview) assume Expected Utility theory. Only recently Wakker and Deneffe (1996) developed a utility assessment method, the tradeoff (TO) method, that can be used to elicit utility functions under Expected Utility Theory and under other well-known theories of individual decision making that allow for nonlinear probability weighting. Examples of such theories are Rank Dependent Utility theory (e.g., Quiggin, 1982) and [Cumulative] Prospect Theory (Kahneman et al., 1979; Tversky et al., 1992). A short description of the logic of the TO method is offered here; see Wakker and Deneffe (1996) and Fennema and Van Assen (1999) for detailed explanations. The TO method requires subjects to make i = 1, ..., I comparisons between gamble (Xi, p, Rx), yielding Xi with probability p and Rx with probability 1–p, and gamble (Yi, p, Ry). The probability p and the reference outcomes Rx and Ry are constant over all I comparisons. The outcome Xi is given and the subject’s task is to specify the value of outcome Yi such that he or she is indifferent between the two gambles in comparison i. Indifference between the two gambles implies that

π x u( X i ) + π Rx u( R x ) = π y u(Y i ) + π Ry u( R y )

(2)

where πx, πy, πRx, and πRy denote the probability or decision weights for outcomes Xi, Yi, Rx, and 5 For example, assume that a subject has the concave power utility function for gains commonly found by

Tversky and Kahneman (1992), with power equal to 0.88, and let π(0.25) = 0.27 and π(0.75) = 0.73. This 0.88 0.88 0.88 subject prefers the risky gamble over its expected value since 0.27(16) + 0.73(7) > (7) .

82

Chapter 5

Ry, and u again denotes the value or utility function. The TO method demands that the four decision weights do not vary over the I comparisons, with the additional restriction that all weights are in the open interval (0, 1), and that πx = πy = π and πRx = πRy = πR. Theories of individual decision making differ in the restrictions they impose on the rank order and sign of the outcomes such that πx = πy and πRx = πRy. Expected Utility theory has no restrictions of this sort and assumes that π = p and πR = 1–p. Most restrictive is Cumulative Prospect Theory, which requires that RxRy ≥ 0 and either Xi ≥ Rx > Ry, Xi ≥ 0, or Ry > Rx ≥ Xi, Xi ≤ 0. In the present study the restrictions on the outcomes in the experiment are such that utility assessment is robust with respect to probability weighting in Cumulative Prospect Theory, Prospect Theory, and Rank Dependent Utility theory. Combining comparisons i and j leads to the following equality in utility differences: u (Y i ) − u ( X i ) = u(Y j ) − u ( X j ) = [u( R x ) − u( R y )] π R π

(3)

Note that possible probability transformations on p and 1–p are canceled out because they have identical effects in both comparisons. Properties of the utility function such as concavity can then be inferred from distances Yi – Xi. For example, presuppose that Xj+ > Xi+ ≥ 0. Then Yj+ – Xj+ > Yi+ – Xi+ is consistent with concave utility, Yj+ – Xj+ = Yi+ – Xi+ with linear utility, and Yj+ – Xj+ < Yi+ – Xi+ with convex utility. It is possible to construct a nonparametric measure for the concavity of the utility function using these comparisons of utility differences. This allows a comparison of the concavity of utility with the level of cooperation to be made, both being measured at the individual level. 5.2.4 Analyzing mixed PDs and parametric utility functions Van Assen (1998) derived predictions about the relation between utility and conditions of cooperation in “mixed dilemmas”, that is, dilemmas that have payoffs P < 0 and T > 0. A summary of his results is presented here. In order to examine the behavior of wu(Γ: ∆) in Equation (1) in mixed dilemmas, it is convenient to fix the PD outcome P = 0 and to distinguish four domains of ∆: ∆1 ≥ 0, 0 > ∆2 ≥ –R, –R > ∆3 ≥ –T, –T > ∆4. The four intervals of ∆ define positive PDs (the value of S is neglected because it is not involved in (1)), mixed PDs with one loss, mixed PDs with two losses, and negative PDs respectively. Loss aversion, denoted by λ, refers to the relative steepness of the utility function for losses in comparison to that for gains, and is modeled here by assuming that u(–x) = –λu(x) for all x ≥ 0, and λ > 1. Substituting ∆2 and ∆3 in (1) and assuming this utility function yields w ≥ wu ( Γ : ∆ 2 , λ ) = w ≥ wu ( Γ : ∆ 3 , λ ) =

u(T + ∆ 2 ) − u( R + ∆ 2 ) u(T + ∆ 2 ) + λu( − ∆ 2 ) u(T + ∆ ) + λu ( − R − ∆ 3 ) u (T + ∆ 3 ) + λu ( − ∆ 3 )

(4)

(5)

Note that conditions for cooperation in mixed dilemmas improve with increasing loss aversion, because both wu(Γ: ∆2, λ) and wu(Γ: ∆3, λ) decrease with increasing λ. In addition, one can compare the conditions for cooperation in mixed dilemmas with those for positive and negative dilemmas, if one assumes a specific utility function. Previous empirical studies have demonstrated that the power utility function (6) provides a good fit of choice data both on the individual and aggregate level (Tversky and Kahneman 1992; Gonzalez and Wu,1999;

The Effect of Nonlinear Utility on Behavior in Repeated Prisoner’s Dilemmas

Figure 5.1a

Figure 5.1b

Figure 5.1c Figure 5.1:

wu(Γ: ∆, α, β, λ) as a function of its parameters, and with T = 20, R = 10. Figures 1a, 1b, and 1c depict wu(Γ: ∆, 1, 1, λ), wu(Γ: ∆, 0.88, 0.88, λ), and wu(Γ: ∆, 0.5, 0.5, λ) respectively, for values of λ equal to 1 (highest curve), 2.25 (middle curve), and 5 (lowest curve). In each figure, the horizontal line represents wu(Γ: ∆, 1, 1, 1) = 0.5.

83

84

Chapter 5

and Fennema and van Assen, 1999; Camerer and Ho 1994): u(x) = xα u(x) = –λ(–x)β

x ≥ 0, and

(6)

x P(C| ∆ = 0) > P(C| ∆ = –20)

Note that under these assumptions no clear prediction can be made about the rank order of P(C| ∆ = –10) in H1 because this rank order depends on the relative effect of curvature and loss aversion, as can be seen in Figure 5.1. If we assume that agents’ choice behavior can be adequately described by Expected Utility theory, but we do neither assume that utility is necessarily S-shaped nor that agents are loss averse, we hypothesize that H2:

For all PDs (Γ: ∆) there is a positive relation between P(C|∆) and risk aversion in the domain of the outcomes of the PD.

Finally, when we also relax the assumption that Expected Utility theory should be valid, and use the TO method to measure utility without bias, the relation between utility and

The Effect of Nonlinear Utility on Behavior in Repeated Prisoner’s Dilemmas

85

cooperation in (1) can be tested directly. H3:

For all PDs (Γ: ∆) there is a positive relation between P(C|∆) and wu.

Assuming utility function (6), H3 can actually be separated into three hypotheses, two concerning the relation between utility curvature and cooperation and the other concerning between loss aversion and cooperation. H3a:

For (Γ: ∆2) and (Γ: ∆3) there is a positive relation between λ and P(C|∆).

H3b: For (Γ: ∆1) there is a positive relation between α and P(C| ∆). H3c: For (Γ: ∆4) there is a negative relation between β and P(C| ∆). 5.4

Experiment6

5.4.1 Subjects Subjects were recruited using flyers, advertisements in the university newspaper, and emails to students. In total 227 subjects participated, of whom 216 finished both sessions of the experiment. Most subjects were undergraduates from the University of Groningen. The age of the subjects was between 17 and 29, and 50.5% of them were male. 5.4.2 Procedure The experiment consisted of two parts. In the first part the utility of subjects was assessed, and in the second part subjects played a number of repeated PDs. A computer program was written for both parts of the experiment;7 the two parts were run on different days. We attempted to ensure that the length of time between participating in the first and second part of the experiment was at least 24 hours and at most a week. The experimental sessions took place in a large room containing 15 computers distributed in five rows of three. In the experimental sessions, both subjects participating in the first part and subjects participating in the second could be present. The subjects were explicitly told to be silent during the experiment. One experimenter was always present in the room to answer questions of the subjects. 5.4.2.1 First part of the experiment The first part consisted of five tasks. The subjects started with an extension of the utility assessment procedure used by Raub and Snijders (1997). For the second task, subjects were required to indicate their indifference point in comparisons of a gamble with a certain outcome. Subsequently, the TO method was used to elicit subjects’ utilities. The fourth task was a questionnaire. All subjects had to complete the first four tasks, and if they were completed within 45 minutes, the subjects were told to continue with a test-retest reliability task. Here the subject was given exactly the same choice situations as in the first three tasks. The subjects were instructed that in all choice situations of the tasks they should imagine that they could make a choice between gambles or a gamble and the certain outcome only once, and that the outcomes of the choice situations involved their own money. A more detailed description of each of the tasks is provided below. 6 Except for Task 2, the experiment is also described in Chapter 3. 7 The computer programs are written in Turbo Pascal 7.0 and are available from the first author upon request.

86

Chapter 5

Task 1: Extension of utility assessment procedure as used by Raub and Snijders The first task required the subjects, as in the experiment of Raub and Snijders, to make a preference comparison between the gamble (T+∆, 1–w, P+∆) and the expected value R+∆ of the gamble, for ∆ = 0, –5, –10, –20, and with w = 0.5. We refer to this way of measuring the risk preference as the “traditional method” as it represents the measurement used by Raub and Snijders (1997). To extend the measurement, two additional preference comparisons were constructed: a comparison between (T+∆, 1/3, P+∆) and R+∆, and one between (T+∆, 2/3, P+∆) and R+∆. In the analyses, we refer to this measurement as the “extended traditional method”. In the extended traditional method, the number of times a subject indicated a preference for the expected value over the gamble (0, 1, 2, or 3 times) constitutes a measure of risk aversion. A brief instruction session, including three preference comparisons between (60, p, –20) and 20 with p equal to 2/3, 0.5, and 1/3 respectively, explained the task. The order of the four sets consisting of three preference comparisons was varied among the subjects using a Latin Square design (Edwards, 1968) consisting of four orderings. Within a set, subjects always first stated their preference in the comparison with p equal to 0.5. After stating their preference in this situation, subjects had to estimate the proportion of the human population with the same preference. This assessment is taken as a crude measure for the degree to which our assumption holds that subjects believe others to have the same utility function as themselves. This question was not asked after the other two preference comparisons. After each preference comparison the subjects were asked to confirm their choice. They also had the possibility of changing answers they had given earlier. Task 2: Extra choice questions Five additional choice situations were presented to the subjects that involved a comparison between a gamble and a certain outcome. Subjects had to specify the value of zi in comparison i such that they were indifferent between the two options in the comparison. Denoting the indifference relation by ~, the comparisons were (1) 10 ~ (z1, 1/3, 0); (2) –10 ~ (0, 1/3, z2); (3) 0 ~ (z3, 1/3, –10); (4) 0 ~ (z4, 1/3, –25); (5) 10 ~ (z5, 1/2, 0). The order of the five choice situations was varied among the subjects using a Latin Square design for the first four comparisons, consisting of all possible 24 orderings. Comparison (5) was given to all subjects last. The answers given by the subjects to these comparisons, together with the responses to the comparisons in the TO method in Task 3, allowed us to estimate parametric utility functions. Appendix 5.1 explains how the power utility function was estimated from the responses in the two tasks.The indifference points zi were not determined using a direct estimation technique where a subject’s indifference point is obtained from a single response. Rather, the indifference point was established by a convergence technique that successively adjusts the value of z over a sequence of preference comparisons until the indifference point is reached (Farquhar, 1984). Task 3: The TO method The instruction of the TO method was extensive because previous research with the method has indicated that, at least initially, subjects have some problems in grasping the complexity of the choice situation (Fennema and van Assen, 1999). After the instruction a subject’s utility was measured using the convergence technique, also used in Task 2, for all gamble comparisons. The gamble comparisons are depicted in Figure 5.2.

The Effect of Nonlinear Utility on Behavior in Repeated Prisoner’s Dilemmas

XI+

87

YI+

1/3

1/3

2/3

2/3 Rx+

Ry+

Figure 5.2a Rx-

Ry-

1/3

1/3

2/3

2/3 XI-

YI-

Figure 5.2b Figure 5.2:

Gamble comparisons used to elicit subjects’ utilities in the experiment. For gains in Figure 2a, Rx+ = –5, Ry+ = –10, X1+ = 0, X2+ = 5, X3+ = 10, X4+ = 20, for losses in Figure 2b, Rx– = 20, Ry– = 50, X1– = 0, X2– = –5, X3– = – 10, X4– = –20. Outcomes Yi+ and Yi- are chosen by the subject such that (s)he is indifferent between the two gambles.

For gains, the values Rx+ = –5, Ry+ = –10, X1+ = 0, X2+ = 5, X3+ = 10, and X4+ = 20 were used, for losses Rx– = 20, Ry– = 50, X1– = 0, X2– = –5, X3– = –10, and X4– = –20. Half of the subjects started with the TO method for losses, while the other half started with the TO method for gains. The order of comparisons within both value domains was such that a subject first determined his or her indifference point for the two most extreme values X1 and X4, followed by the other two comparisons. The 24 possible orderings were varied among the subjects using a Latin Square design. Task 4: Questionnaire The responses to the questions in the questionnaire were used as controls in the analyses. Participants were asked about their age, sex, secondary school subjects, and whether they had knowledge of game theory. The questionnaire was concluded with a measure of the social

88

Chapter 5

orientation of the participant (Cain, 1998; Taylor, 1987; Messick and McClintock, 1968; McClintock and Liebrand, 1988) with six ordinal values. Social orientation acknowledges that ego’s utility can also incorporate alter’s outcomes. Because high social orientations transform a PD into a game in which defection is no longer the dominant choice, social orientation might serve as a relevant background variable in the analyses. Task 5: Test-retest reliability The choice situations in Task 1, Task 2, and Task 3 were repeated once until 45 minutes had passed since the start of the experimental session. The order of choice situations within Task 1 and Task 2 was randomized. The choice situations in Task 3 with the extreme values (X1–, X4–, X1+, X4+) were repeated first, followed by the four other choice situations. 5.4.2.2 Second part of the experiment The second part of the experiment started with an extensive instruction session. The PDs were presented to the subjects using 2x2 tables. The relation between the choices of the players and the subsequent outcomes was explained by letting participants play a repeated PD twice, with outcomes T = 60, R = 20, P = –20, and S = –40, and w equal to 0.5, as in the games of interest The subjects were told that they were not playing against another player in the lab but against a randomly selected player from a previous experiment, drawn with replacement before each repeated PD. The experiment from which players were drawn was actually that of Raub and Snijders (1997). In Appendix 5.2 we explain how we were able to model the strategies of human players accurately and realistically by making use of the responses of subjects in the 1997 experiment. A lot of effort was invested in the instruction in order to make participants in our second experiment realize that (i) they played against a human player, although the computer generated the responses, and (ii) that they were not deceived in any way (which was true). After the subjects had played the two repeated PDs during the instruction it was explained to them how their payoff would be determined after the experiment. The procedure for rewarding the subjects is described later. Finally, before playing the repeated PDs of interest, the subjects were told to imagine that the outcomes of the PDs involved their own money. After the instruction session the subject played the four repeated PDs (∆ = 0, –5, –10, –20) in a random order. The subject continued to play these four repeated PDs in a random order until 40 minutes had passed, or until (s)he had played each repeated PD twenty times. The responses in these subsequent games had no effect on the subject’s reward. Finally, the subject played a fifth repeated PD with ∆ = 5, which we included to ensure that subjects would earn a reasonable amount of money (see Reward for participation in the experiment).

The Effect of Nonlinear Utility on Behavior in Repeated Prisoner’s Dilemmas Table 5.1:

Risk averse for gains

89

Classification of risk preferences for the gains and losses domain using the extended traditional method with three choices per subject. Higher values correspond to greater risk aversion.

0 1 2 3 Total

0 11 18 7 1 37

Risk averse for losses 1 2 19 8 72 26 34 16 3 2 128 52

3 2 2 1 2 7

Total 40 118 58 8 224

5.4.2.3 Reward for participation in the experiment In the first part of the experiment subjects were given a fixed reward of 35 guilders (at that time approximately US$ 17.5) for participation. We were not concerned that using hypothetical payoffs in the choice situations would significantly bias our results because some previous studies on individual decision making in risk situations using real incentives have yielded results similar to studies using hypothetical payoffs (e.g., Tversky and Kahneman, 1992; Camerer, 1995). Some other studies (Beattie and Loomes, 1997; see Camerer and Hogarth, 1999 for a review) present evidence that incentives make subjects more risk averse. The possible effects of incentives on risk aversion are not an issue in the present study as long as these effects are similar among all participants in the experiment. To make the PDs in the second part of the experiment as realistic as possible, we chose to make the subject’s reward dependent on his or her behavior (and on the behavior of the other player). Separating the two parts of the experiment had the advantage that the “gambling with the house money effect” (Thaler and Johnson, 1990) was, in all likelihood, minimized. The subject’s reward in the second part of the experiment was determined by the random selection method. The outcome of the last round of each of the five different repeated PDs played for the first time was selected as a possible payoff. This was explained to the subjects during the instruction at the start of the second part of the experiment.. As shown by Raub and Snijders (1997), the selection of only the last round excludes wealth effects and does not affect the strategic structure of the repeated PD. At the end of the experiment the subjects drew one card from a pile of six cards. The last PD with ∆ = 5 was associated with two jesters, while the other PDs were associated with clubs, diamonds, hearts, or spades. The minimum reward for participation in both parts of the experiment was 10 guilders (=35–25), and the maximum reward was equal to 60 guilders (=35+25). 5.5

Results

We first summarize the results dealing with the measurement of utility. The results of the analyses with respect to our hypotheses of the relation between utility and cooperation are then reported.

90

Chapter 5

Table 5.2:

,

Classification of utility for the gains and the losses domain using parametric estimates of utility. Concave utility corresponds to values of α (β) smaller (larger) than 1. To make Table 2 as similar as possible to Table 1, value of α decreases in subsequent rows.

>1 1 Yi+ – Xi+), linear utility (Yi+1,+ – Xi+1,+ = Yi+ – Xi+), and convex utility (Yi+1,+ – Xi+1,+ < Yi+ – Xi+), for i = 1, 2, and 3. Similarly, but using reversed inequalities, evidence was summed in the loss domain. Response patterns were then classified by the category for which there were two or three counts of evidence. If the response patterns yielded one count of evidence in all three categories, then the pattern was considered as missing. Application of the nonparametric tradeoff measurement resulted in risk preferences that are more in accordance with S-shaped utility than the results of the extended traditional method. From the nonlinear utilities, 82.8% (130/157) were concave for gains, 53.5% (77/144) were convex for losses, and 50% (57/114) were both. Note the large difference between the extended traditional method and the tradeoff method with respect to the percentage of the participants who are classified as risk averse (29.5%) against those 8 In Chapter 3 the risk preferences of 216 subjects were considered. In this chapter the risk preferences of eight

additional subjects who did not turn up for the second part of the experiment were also considered.

The Effect of Nonlinear Utility on Behavior in Repeated Prisoner’s Dilemmas Table 5.3:

91

Correlations between measurements of risk preferences using the ET (extended traditional method), TO (nonparametric tradeoff method), and P (parametric estimates). Correlations for gains are shown in the upper right part of the table, and correlations for losses in the lower left part. One-sided plevels are shown in brackets. The diagonal shows the test-retest reliabilities of the ET and TO methods. Note that for all methods, except for the parametric estimate in the gains domain, high values correspond to greater risk aversion.

ET TO P

ET 0.50 0.042 (0.306) 0.061 (0.182)

TO 0.120 (0.049) 0.67 0.352 ( 1). The median estimate of loss aversion was equal to 1.4. The estimates of the power parameters α and β can again be used to classify participants’ utilities. In Table 5.2 the participants’ utilities are classified as convex, linear, or concave for each value domain. Note that concave utility corresponds to α < 1 and to β > 1. As in the classification using the nonparametric tradeoff measurement, the results are more similar to S-shaped utility than in the classification using the extended traditional method. Of all participants, 66.1% and 61.2% were classified as having concave utility in the gains domain and convex utility in the losses domain respectively, while 50.4% were classified as having both. The corresponding median estimates of α and β were 0.88 and 0.93 respectively. Our next step was to compute the correlations between the three previously discussed measurements of risk preferences, doing so separately for gains and losses. The correlations for gains are shown in the upper right part of the correlation matrix in Table 5.3, and the correlations for losses are shown in the lower left part. One-sided p-levels of the test of zero correlation against the alternative of a positive correlation are also shown. The table demonstrates that the correlations between the measurements of utility range from small to very small. In particular the correlations between classification using the traditional method and the other two classifications are very small. In the losses domain these correlations are not even significantly different to zero. A discussion of the low correspondence between the three methods is provided in the conclusion and discussion section.

92

Chapter 5

Table 5.4:

Random effects logistic regression of the probability of cooperation in round 1 of the repeated PD, using the extended traditional method of utility measurement. Predictor variables are risk between persons (“risk between”), risk within persons (“risk within”), and control variables “know game theory”, “social orientation”, and “no social orientation”. The improvement of the fit of the model is !2(5) = 18.57 (p = 0.0023) and the hypothesis  = 0 is rejected (!2(1) = 46.32, p < 0.0001). Pvalues of single predictors are two-sided.

Variable Risk between Risk within Know game theory Social orientation No social orientation Constant

Effect 0.578 0.011 0.787 0.107 0.878 -0.779

z-test (p-value) 2.14 (0.032) 0.074 (0.941) 2.854 (0.004) 0.979 (0.328) 2.387 (0.017) -1.380 (0.167)

5.5.2 Relation between agents' utility in gambles and their cooperation in PDs In the tests of our hypotheses we use as a measure for the likelihood of cooperation the probability of cooperation in the first round of a repeated PD. To take into account individual differences we used the items from our questionnaire (age, sex, choice of secondary school subjects, whether participants had knowledge of game theory, and the social orientation value). Only two of these variables approached significance in some of the analyses: knowledge of game theory and the social orientation value. Knowledge of game theory is a dummy variable (1=yes). The social orientation value was calculated on the basis of the preference of subjects over four ego-alter payoff configurations, and yields for each subject a measure (in categories, ranging from 1 to 7) for the degree to which (s)he values payoffs to alter. A value of 1 denotes no interest in the payoffs to alter whatsoever, while a value of 7 denotes substantial interest. A value of “0” was assigned to the 62 participants who provided preferences inconsistent with the social orientation model. These participants were assigned the value “1” for the variable “noso” (no social orientation), while the other participants were assigned the value “0”. Assuming S-shaped utility with loss aversion, Hypothesis 1 suggests that we can rank the four PDs in order according to the proportion of cooperation in the first round: P(C| ∆ = – 5) > P(C| ∆ = 0) > P(C| ∆ = –20). This hypothesis is refuted. We find percentages of cooperation of 29.2, 36.5, and 26.9 respectively. Controlling for individual characteristics (using random effects logistic regression) does not change this result. We do find that knowledge of game theory has a positive effect on the probability of cooperation (p=0.009). Subjects with knowledge of game theory have on average a probability of cooperation that is approximately 13% higher. Those for whom the measurement of social orientation was not possible have on average a probability of cooperation that is 16% higher (p=0.021). Assuming Expected Utility theory, Hypothesis 2 predicts that there is a positive relation between our non-parametric measure of risk aversion (taking values 0, 1, 2, and 3) and the probability of cooperation in the first round. Because the data are clustered within

The Effect of Nonlinear Utility on Behavior in Repeated Prisoner’s Dilemmas Table 5.5:

93

Random effects logistic regression of the probability of cooperation in round 1 of the repeated PD, using the tradeoff method to measure the utility function and compute the threshold wu. Predictor variables are wu between persons (“wu between”), wu within persons (“wu within”), and control variables “know game theory”, “social orientation”, and “no social orientation”. The improvement of the fit of the model is !2(5) = 14.86 (p = 0.0110) and the hypothesis  = 0 is rejected (!2(1) = 50.46, p < 0.0001). P-values of single predictors are two-sided.

Variable wu between wu within Know game theory Social orientation No social orientation Constant

Effect -1.750 0.286 0.799 0.135 0.938 -1.095

z-test (p-value) -0.737 (0.461) 0.478 (0.633) 2.853 (0.004) 1.202 (0.230) 2.512 (0.012) -1.051 (0.293)

persons, we use maximum-likelihood random effects logistic regression (Conaway, 1990) and introduce two variables involved with risk preferences: the between person difference in average risk preference and the within person deviation from the average personal risk preference. We expect a positive relation for both variables. People who are more risk averse across the four PDs are expected to cooperate more often and, in addition, each person is expected to cooperate more often in those repeated PDs for which he or she has a more risk averse preference. Table 5.4 displays the results of our analyses. It appears that we find only an effect of risk preferences between persons: people who are more risk averse across the four PDs, are more likely to cooperate across the PDs (p=0.016, one-sided), which is in accordance with Hypothesis 2. The non-parametric measure of risk preference varies between 0 and 3, and an increase of 1 in this variable has an average effect of 0.1 (10%) on the probability of cooperation. Therefore, the difference in the probability of probability between the subjects who are most and least risk averse is approximately 0.3, which is quite substantial. However, we do not find any effect within persons (p=0.94). One reason for this could be that there is considerable noise in the data per person, part of which is cancelled out when we average per person across games. To conclude, we believe that Hypothesis 2 is supported by the data. Once again we observe positive effects for subjects with a knowledge of game theory and for subjects whose social orientation could not be estimated. Measuring utility without bias and relaxing the assumption that Expected Utility theory should hold, Hypothesis 3 implies a positive relation between the probability of cooperation and wu. Using the estimated values of α, β, and λ, the value wu can be estimated for each person and each PD. Once again we separate the measure of risk preference in a between person and a within person part. Table 5.5 displays the results. The analysis in Table 5.5 suggests that there is no relation between the shape of the utility function and cooperation, and hence provides no support for Hypothesis 3. Both the between and the within person wu yield non-significant effects (p=0.69 and p=0.71; p=0.46

94

Chapter 5

and p=0.63). Again we observe positive effects for subjects with a knowledge of game theory and for subjects whose social orientation could not be estimated, of the same order of magnitude as in the previous analyses. Separating Hypothesis 3 into the effects of utility curvature on cooperation and the effect of loss aversion on cooperation (Hypotheses 3a, 3b, and 3c), we find some evidence that loss aversion has an effect. Using logistic regression, there seems to be no relation between α and the probability of cooperation in the PD with ∆=∆1 (p=0.66), and likewise no relation between β and the probability of cooperation in the PD with ∆=∆4 (p=0.52). Therefore, no support is provided for either Hypotheses 3b or 3c. However, there is weak evidence (using random effects logistic regression) for a relation in the expected direction between λ and the probability of cooperation in the games where ∆ equals ∆2 or ∆3. The p-value equals either 0.083 or 0.174 excluding or including control variables in the analyses. It is also interesting to note that there is even less support for the intuitively plausible assertion that loss aversion increases the probability of cooperation in all repeated PDs (as opposed to this assertion only applied to mixed PDs). Both for the gains and the losses PD, loss aversion has no statistically significant effect on cooperation (p=0.61; p=0.69). Taken together, the analyses provide no support for our third hypothesis. 5.6

Conclusion and discussion

In our research program we attempt to test game theoretical assertions about behavior in prisoner’s dilemmas by associating agent utility differences with differences in agent behavior in these dilemmas. Raub and Snijders (1997) demonstrated that, both theoretically and empirically, there is a correlation, although small, between agents’ risk preferences and their propensity to cooperate in PD games containing only losses or only gains. Their research did not take into account agents’ nonlinear weighing of probabilities. In the present study utilities were assessed by the tradeoff method which yields utility measurements unbiased by probability weighting. Van Assen (1998) derived that in some games involving both losses and gains a positive effect of loss aversion on the propensity to cooperate can be expected. The present study also tested the effect of loss aversion. With respect to the measurement of utility many more agents with S-shaped utility were found using the tradeoff method (50%) than with the traditional method (20%), and the correlation between the two measurements was smaller than expected. With respect to our hypotheses, the assumption that all subjects behave in accordance with S-shaped utility with loss aversion implies that there should be differences in the proportion of cooperation in different repeated PDs that are not apparent in the data. However, if we measure risk preferences using the extended traditional method, that is, by eliciting preferences from comparisons of gambles with their expected value, and we then correlate these risk preferences with behavior in the repeated PD, we do find evidence suggesting that risk aversion promotes cooperation. The difference is found only between persons and not within persons between different games. Surprisingly, the improvement of utility measurement by taking nonlinear probability weighting into account gives measurements of utility that barely correlate with cooperation in the repeated PD. We do find (very) weak evidence in favor of a positive effect of loss aversion on cooperation in mixed PDs. Test-retest reliabilities of responses using the traditional method and the tradeoff method, and in the extra gamble comparisons in Task 3, were only moderate (of the order of 0.6). The reliability coefficients are comparable to those found by Tversky and Kahneman (1992), who obtained an average test-retest correlation of 0.55. One can interpret these

The Effect of Nonlinear Utility on Behavior in Repeated Prisoner’s Dilemmas

95

findings in both positive and negative ways. The positive interpretation is that the moderate reliabilities suppress the true relation between utility and cooperation, which would have been stronger if utility could have been measured more reliably. The negative interpretation is that the moderate values suggest that agents do not have stable utilities or risk preferences over time, perhaps suggesting that utility does not exist as a stable agent characteristic (Shoemaker, 1982; Isaac and James, 2000). The small correlations between different measurements of utility in the present study are another source of concern. Several complementary reasons can be provided to explain these small correlations. Firstly, the moderate reliabilities of agents’ responses decrease the correlations between the methods considerably. Secondly, the large standard errors of the parameter estimates obtained using the TO method result in less reliable assessments of the agents’ utility. The large standard errors are a consequence of the fact that the parameters were estimated using a relatively small number of observations. Thirdly, the bias in the traditional method resulting from probability weighting can vary among agents, which suppresses the correlation even further. However, even when considering such suppressions, we would expect the correlations to be higher than the range obtained in the present study. On the other hand, other studies have also yielded surprisingly low correlations between different measurements of utility. A notable study is the research of Isaac and James (2000). They elicited utility for the same individuals using two different scenarios, by using the BDM procedure that is well-known in economics (Becker, DeGroot, Marschak, 1964), and by studying agents’ behavior in first price auctions with the CCRAM model (e.g., Cox, Roberson, and Smith, 1982). Isaac and James (2000) observed that agents were much more risk averse in first price auctions than in the BDM procedure, and that the power estimates of both measures were even negatively correlated, although the correlation was not significantly different to zero. Therefor, very small correlations between different measurements of utility do not seem to be abnormal. Naturally, the small correlations present a problem both for empirical measurements of utility and for theoretical research that derives predictions of agent behavior from utility considerations, where utility is usually considered to be a stable and consistent characteristic of agents. In the context of the present study, the low reliability and small correlations between methods imply that if a correlation exists between assessed utility and cooperation in the PD games, then it is very likely to be small. Several observations can be made that might be able to explain the weak relation between agents’ utility as measured by the tradeoff method and cooperation in repeated PD games. Naturally, the moderate reliability of the tradeoff method suppresses its correlation with cooperation. Moreover, biases in the selection of participants have perhaps had a confounding effect, which might also explain the differences between our measurements of utility and previous measurements in the literature. The median estimated value of the loss aversion parameter was smaller than that found in Tversky and Kahneman (1992). A difference in the designs of their experiment and ours may have caused this. In their experiment, the reward for participation was fixed and hence independent of participants’ choice behavior in the gamble comparisons. In our experiments participants could lose money in the second session of our experiment. Therefore, our experiment was less attractive to loss averse individuals. If our experiment did not attract not enough loss averse individuals, this so-called “restriction of range” effect might have influenced the relation between utility and cooperation. However, we believe that this effect should be quite small. Similarly, we do not

96

Chapter 5

believe that the lack of incentives used in the utility measurement has affected our results. There is some evidence in the literature that using incentives in eliciting utilities can make subjects slightly more risk averse (Camerer and Hogarth, 1999), but as long as this small effect does not vary among subjects, the incentive effect is of no consequence for the tests of our hypotheses. There is a related issue that could be responsible for the fact that we do find an effect of risk aversion on cooperation on the basis of the extended traditional method, but that does not apply to the tradeoff method. Under Expected Utility theory, one can derive that a relation between risk preferences (as measured by the traditional method) and cooperation should exist. We know that if we measure risk preferences without taking probability weighting into account, a bias is created. However, for the derivation of the equilibrium condition(s) in the repeated PD, we likewise assume that probability weighting does not occur,9 which implies that, in one sense, we might benefit from the fact that we “make the same mistake twice.” In this sense, it is conceivable that we have found the connection between risk aversion and cooperation to exist only if both the risk preferences and the equilibrium conditions are based on the same, perhaps mistaken, assumption that nonlinear probability weighting does not happen. Assuming that utility is a stable individual characteristic and that it can be measured reliably and without bias, the question can be asked of what size of effect of utility on cooperation can be expected? An indication of the size of this effect can be obtained from Roth and Murnighan (1978) and Murnighan and Roth (1983) who studied the effect of changes in the payoffs of indefinitely repeated PD games and in their probability of continuation w on the cooperation rate in these games. In these studies the probability of continuation w was varied between 0.105, 0.5 and 0.895 while keeping the payoffs constant. A difference in cooperation rate was observed between the condition where mutual cooperation was not at equilibrium (w = 0.105) and the two conditions where it was at equilibrium. No difference in cooperation rate was found between the w = 0.895 and w = 0.5 conditions. Roth et al. (1979) and Murnighan et al. (1983) also varied the index wu, assuming linear utility, from 0.05 to 0.93 in 9 steps, while keeping w = 0.5 constant. The Spearman rank order correlation between wu and the percentage of cooperation in the first round was equal to –0.24, suggesting that there is indeed a (weak) relation between agents’ propensity to cooperate and conditions for cooperation as measured by the wu index. From Figure 5.1 we can infer that the individual wu values in the present study are expected to vary as much as the wu value in Roth et al. (1979) and Murnighan et al. (1983). However, there the correlation was computed from average cooperation rates and not from individual cooperation as in the present study. Correlations based on averages are in general larger than correlations between individual observations because the errors of averages are smaller. This fact, together with the fact that the index wu in our experiment could not be measured very reliably, implies that a correlation of –0.24 is likely to be the maximum (absolute) value of the correlation we could have expected in our experiment. A complicating factor in our experiment might be that participants were attempting to 9 Using this simplifying assumption is necessary, because for the derivation of the equilibrium conditions one 2

has to consider that agents compare streams of discounted payoffs (such as T + w P + w P + … versus R + w R 2 + w R + …) and it is not obvious how probability weighting should affect these comparisons between different streams of payoffs.

The Effect of Nonlinear Utility on Behavior in Repeated Prisoner’s Dilemmas

97

reach an equilibrium, where behavior on the equilibrium path prescribes to alternate between cooperation and defection, yielding alternating payoffs (S, T) and (T, S). However, this is unlikely to be the case because R was larger than the average of T and S. Moreover, alternating response patterns were observed only infrequently. Finally, the fact that participants played against the computer and not against a human player could have been a cause for them to behave differently. Although the participants were explicitly and repeatedly told during the instruction that their opponent was a human player in a previous experiment selected by the computer program (see Appendix 5.2), this procedure could have led to a certain lack of reality for the participants (“there was no real person on the other side”). Participants were told that after the experiment they could obtain a description of how the computer could accurately mimic a human player from a previous experiment. Less than five participants asked for the description, hence we assume that most participants correctly believed that at least their opponent could not be distinguished from a human player in the previous experiment. To conclude, we propose three approaches for of our future research. Firstly, we are developing a learning model based on the algorithm described in Appendix 5.2 with which we can estimate and test the effects of utility and other variables on agents’ responses in the rounds following the first. This would allow for more extensive testing of the connection between utility and cooperation. Secondly, a logical extension of the theory presented in the present study would be to include incomplete information about the utility of alter, reflecting the uncertainty of ego when considering how alter is likely to behave. Finally, we could improve the theory by examining more closely the effects of agents’ nonlinear weighing of probabilities on the properties of the equilibrium conditions of mutual cooperation. In measuring the utility functions we took probability weighting into account, but there are also valid arguments that agents evaluate the probability of continuation nonlinearly, thereby changing the equilibrium conditions.

98

Chapter 5

Appendix 5.1: Estimation of utility function In the estimation procedure it was assumed that agents' individual decision making behavior conforms to Cumulative Prospect Theory (CPT) with power utility functions as in equation (6). To estimate the parameter vector (π, πR, π0.5, λ, α, β), thirteen observations were used, eight from Task 3 (y1, ..., y8) and five from Task 2 (y9, ..., y13). An iterated weighted nonlinear regression was employed to estimate the parameters for each agent separately. The parameter values were chosen such that 2

(y −yp) E=∑ i i i =1 log( y + 1) i 13

(7)

p was minimized, where y is the response predicted by CPT using the final parameter value estimates. The weights were introduced because the magnitude of the residuals is likely to depend on the size of the value of the observations (see Carrol and Ruppert, 1998, for a discussion of weighing in regression). Wu and Gonzalez (1999) estimated utilities using a similar estimation procedure with weights equal to log(yp). They report that the use of these weights did not result in a systematic pattern in the residuals but in relatively uniform errors. There are three differences between the weights used by Wu and Gonzalez (1999) and those in the present study. Firstly, absolute values of observations had to be used because our study involves losses in addition to gains. Secondly, 1 is added to the observations yi because a few individuals had yi = –1, which would otherwise yield a denominator equal to zero in (7). Finally, we chose to incorporate yi instead of its prediction yp, because using yp might result in overestimated values. This bias can result from the fact that the contribution of the residual becomes smaller for larger values of yp when using yp instead of yi.

The algorithm used to estimate the parameters was programmed in Turbo Pascal. The algorithm’s initial estimate of (π, πR, π0.5, λ, α, β) is the expected value estimate (1/3, 2/3, 1/2, 1, 1, 1). Parameter λ is then estimated while keeping the values of the other parameters constant. After updating the estimate of λ, the other five parameters are subsequently estimated and updated separately. To estimate one parameter, given the values of the other parameters, a simple bisection algorithm was used (Atkinson, 1989, Chapter 2). In the bisection it was assumed that the estimate was in the open interval (0.05, 10) for the power parameters, (0.05, 100) for the loss aversion parameter, and (0.01, 0.99) for the probability parameters, with the additional restriction that π < π0.5. The loop in which the six parameters were estimated was repeated until the estimates at the end of the loop were identical to those at the start of the loop. The fact that the estimation procedure always converged is an indication that there is one local and global minimum of E at the estimated values of the parameters. For a few subjects E was analyzed as a function of its parameters, which indeed showed that there was only one minimum for E. However, we do not have a proof that E is minimized at the estimated values of the parameters for all subjects. The estimates of λ, α, and β obtained from the procedure described above were used to generate wu values to test Hypothesis H3. The estimates of α and β were also used to test Hypotheses H3b and H3c. The interpretation of λ is not that of loss aversion if α ≠ β. In order to estimate loss aversion the estimation procedure described above was repeated with the restriction α = β. The estimate of λ resulting from this estimation procedure was used to test Hypothesis H3a.

The Effect of Nonlinear Utility on Behavior in Repeated Prisoner’s Dilemmas

99

Appendix 5.2: Simulation of other agent's strategy in repeated PDs10 The subjects were informed that they were interacting with a computer program whose behavior was indistinguishable from the behavior of subjects in the similar, previous experiment (Raub and Snijders, 1997). It was explained to the participants that they were matched to a randomly selected player from the previous experiment. A lot of effort was invested in the instruction in order to convince the subjects that they were not deceived in any way (which was true). During the instruction subjects were told that after the experiment they could obtain an explanation of the computer program from the experimenter. Only a few subjects made use of this possibility. The computer program is explained below. The computer program contains a statistical model of the probability that an agent cooperates in a given round of a PD, conditional on the history of the game, where the parameters in the model are chosen such that it provides a close fit to the agents' behavior in Experiment 1. The model is a logistic probability model Prob(C i ) =

1 1 + exp( − X i )

(8)

where Prob(Ci) denotes the probability that the agent cooperates in round i. The function Xi, denoting the propensity of actor z to cooperate in round i, is dependent on four parameters:  i −1   i −1      X i = β 0 + β D  ∑ 2( j − i + 1)(1 − y jA)  + β C  ∑ 2( j − i + 1) y jA  +  j =1   j =1    i −1   i −1  β M  β D  ∑ 2( j − i + 1)(1 − y jZ )  + β C  ∑ 2( j − i + 1) y jZ        j =1   j =1  

(9)

Variable YjZ denotes actor z’s choice in round j, YjA denotes the other actor’s choice in round j, and y equals 1 if the choice is cooperation or 0 if the choice is defection. The sums in the equation imply that the responses in round i-1-k are 2-k times as relevant for the propensity to cooperate than the responses in round i-1. The parameters have the following interpretation:

β0

Propensity to cooperate in the first round of the repeated PD

βD

Effect of defection in the previous round on the propensity to cooperate in the present round

βC

Effect of cooperation in the previous round on the propensity to cooperate in the present round

βM

Relative importance of the opponent’s choice in comparison to the subject’s choice in the previous round for the propensity to cooperate in the present round

The statistical model was fitted to the observed frequencies in round 1, 2, and 3, and to the proportions eternal mutual defection up to round 5 of the experiment of Raub and Snijders (1997). The model was able to predict the 2 ('C' and 'D') + 4 ('DD', 'DC', 'CD', 'CC') + 8 ('DDD', 'DDC', etc.) + 2 ('DDDDD' and 'CCCCC') proportions very accurately. Estimates 10 This appendix is essentially identical to Appendix 3.1.

100

Chapter 5

of the parameters β0, βD, βC, and βM were equal to –0.7369, -1.1, 1.3, and 0.85 respectively. The estimates indicate that cooperation in the previous round had a larger effect on the present response than defection, and that the subject’s own response had a larger effect on the present response than the response of the other actor. The chi-square statistic comparing expected and observed frequencies was 3.24. It is difficult to construct a statistical test because the frequencies are clearly not independent. However, even for 1 degree of freedom the obtained chi-square value is not significant, suggesting that our statistical model accurately describes the behavior of subjects in the first experiment. To use the statistical model in Experiment 2 with the fitted values of the four parameters, the computer program generates random numbers in the unit interval [0,1]. The program cooperates if the number is in the interval [0,P], where the value of P is determined by the history of the game as fas as the previous round, and defects otherwise.

6

Bargaining in Exchange Networks

Van Assen, M.A.L.M., & Girard, D. (2001). Bargaining in Exchange Networks. Submitted to Social Psychology Quarterly.

102

Chapter 6

Abstract Exchange network research focuses on the relation between outcomes of actors in exchange networks and the position of the actors in the network structure. This relation is indirect and mediated by actor bargaining on the micro-level. In the present study for the first time an account is provided of the complete bargaining process leading to the exchange outcomes, by distinguishing between two stages, an initial offer stage and a concession stage. The data of Skvoretz and Zhang (1997) are re-analyzed to test hypotheses concerning the effects on both stages of an actor's relative power in a relation, actor differences, bargaining time, and exclusion and inclusion in the previous exchange round. The analyses demonstrated that actors' initial offers can be predicted accurately by theoretical measures of the actors' relative power. The results indicate that actors base their initial offer on their relative degree in the network and learn from previous experiences. Actors' concession rates (i) depend on actors' relative power as well, but (ii) actor effects are stronger than structural network effects, and (iii) time effects are large in weak power networks. Finally, only effects of exclusion on initial offers were observed, inclusion had no effect in both stages.

Acknowledgements We would like to thank Philip Bonacich, Werner Raub, Frans Stokman, Tom Snijders, and Rolf Ziegler for their suggestions to improve the manuscript.

Bargaining in Exchange Networks 6.1

103

Introduction

Exchange network research focuses on outcome differentials that emerge when actors embedded in a network exchange with one another. Emerson (1962, 1972a, 1972b) was the first to formulate an explicit theory of exchange networks, called power-dependence theory. But only after an empirical test of the theory by Cook and Emerson (1978) did other researchers join the arena and an explosion of research on exchange networks occurred. Wellknown theories on exchange networks, next to power-dependence, are the game-theoretic core (e.g., Bienenstock and Bonacich, 1992)1, expected value (Friedkin, 1992), and network exchange-resistance (e.g. Markovsky, Willer, and Patton 1988). In these theories it is assumed that exchange is negotiated, as opposed to Molm (1997) who studies non-negotiated exchanges. Our study focuses on negotiated exchanges. Exchange network theories and the research that developed out of them mainly concentrate on the relation between the network structure and the outcome distributions over the positions in the network using different exchange rules2, and on finding models and algorithms to explain this relation. Network structure as well as outcome distributions are treated as macro-level variables without reference to the micro-processes involved in exchange networks. An important reason for neglecting the micro-processes in studies of exchange networks is that studies demonstrated that outcome distributions could be predicted reasonably well by the network structure while ignoring micro-processes and unique characteristics of the actors involved (Markovsky, 1987, p. 103). Although analyses of the data in the studies do not go beyond predicting and testing the relation on the macro-level, most theories do refer to the micro-level and micro-processes in their assumptions and in their derivation of the relationship between network structure and exchange outcomes. Power-dependence theory assumes that outcomes in an exchange shifts toward the point at which both actors depend on each other to the same degree, called the equidependency principle (Cook and Yamagishi, 1992). Bienenstock and Bonacich's game-theoretic core is implied by a set of four assumptions about actor bargaining behavior in exchange networks (Bonacich, 1998a, 1999. See also the background section). Network exchange-resistance theory predictions are partly based on an actor bargaining model proposed by Heckathorn (1980). Also, in simulations of exchange networks, processes on the micro-level play an important role. Because to simulate outcome distributions for different network structures one is obliged to construct an actor model on the micro-level. The actor models in these simulation studies contain simple behavioral rules for actors offering exchanges to other actors. The results obtained with the simulations are similar to those found in empirical 1 In later research Bienenstock and Bonacich applied other solution concepts from cooperative game theory to

exchange networks (Bienenstock and Bonacich, 1993). Therefore a better and more general name for their approach would be cooperative game theory. 2 The most common exchange rule adopted is the so-called one-exchange rule, where all positions in the

network can only make one exchange. In some studies exchange networks are included where actors on some positions in the network are allowed to make more than one exchange (e.g. Skvoretz and Zhang, 1997, and also in the study reported here). Some studies focus on, using Willer and Anderson's (1981) terminology, inclusionary connected exchange networks where actors must exchange with more than one actor to be able to profit from an exchange (e.g. Szmatka and Willer, 1995). Related to inclusionary connections, Willer and his colleagues have work in progress on another exchange rule that involves ordering (Corra, 2000). That is, an actor is only able to profit from a specified exchange after he has made another specific exchange.

104

Chapter 6

studies (Cook, Emerson, Gillmore, and Yamagishi, 1983; Markovsky, 1987; Markovsky, Skvoretz, Willer, Lovaglia, and Erger, 1993; Bonacich and Bienenstock, 1995, 1997b). These results can suggest that the simple actor models used in the simulation are an accurate description of the processes that occur on the micro-level. However, the fact that results of simulations and predictions of theories are close to observed outcome distributions does not imply that assumptions about micro-level processes are correct, because other sets of assumptions might lead to the same results and predictions. The previous discussion makes evident that network exchange research in general does not specifically test assumptions on actor bargaining (see also Bienenstock and Bonacich, 1997b). Hence the conclusion that network exchange theories and simulation models validly describe actor behavior and processes on the micro-level is premature. The main reason to study actor bargaining is that the relation between network structure and outcomes is indirect and mediated by the actors' behavior on the micro-level. If actor bargaining is not understood, we have an incomplete and in our view unsatisfactory comprehension of why and how actors on certain network positions obtain better outcomes than actors on other positions. In the present essay an empirical study of micro-level processes is reported to explicate how actor bargaining mediates the relation between network structure and outcome distribution. Other more theoretical arguments for the importance of studying micro-level processes and macro-micro-level transitions are, among others (Coleman, 1987, 1994; Markovsky, 1987; Lawler, Ridgeway, and Markovsky, 1993), provided by Coleman in the first chapter of his standard work Foundations of social theory (1990). First, because data are gathered at the actor level it is natural to begin the explanation of system behavior at the micro-level at which observations are made (Coleman, 1990, p. 3). Second, according to Coleman (1990, p. 3), "An explanation based on internal analysis of system behavior in terms of actions and orientations of lower-level units is likely to be more stable and general than an explanation which remains at the system level.". Additionally, an internal analysis can be regarded as more fundamental than an explanation that remains at the system level (Coleman 1990, p. 4). Coleman considers an explanation to be more fundamental when it provides a basis for intervention. A micro-level mediated explanation of system behavior is ordinarily more useful for intervention (Coleman, 1990, p. 3). Last but not least, because in general many more micro-level observations are made than macro-level observations, statistically more powerful tests can be constructed of the effect of an independent macro-level variable (network structure) on micro-level processes (actor bargaining and strategies) than of a relation between macro-level variables (network structure and outcome distribution). Hence the micro-level lends itself for more elaborate analyses of system dynamics and relations in the system than does the macro-level. Studies are scarce in which empirical data are analyzed on actor bargaining in exchange networks on the micro-level (Skvoretz, Willer, and Fararo, 1993; Skvoretz and Zhang, 1997; Thye and Lovaglia, 1997). Skvoretz et al. (1997) and Thye et al. (1997) concentrated on two common assumptions of actor bargaining, the effect of exclusion and inclusion of an actor at the previous exchange opportunity on that actor's exchange offers to other actors at the present exchange opportunity. Assumptions commonly made of actor bargaining in exchange networks are discussed in the Background section. Implications of the results of simulation studies and the two empirical studies for the validity of the assumptions

Bargaining in Exchange Networks

105

are outlined as well. Thye and Lovaglia (1997) and Skvoretz and Zhang (1997) focussed only on specific aspects of the bargaining process (exclusion and inclusion), ignoring other relevant variables. In the present study an attempt is made to assess the effects of a larger number of variables, by developing a model of actor bargaining in exchange networks on the micro-level where these variables are included as explanatory variables. The bargaining model is described in the Model and hypotheses section. The model distinguishes two stages in bargaining. In the first stage actors make initial exchange offers to others. Frequently, initial offers of two actors are not compatible because they together ask more than can be divided among them. In that case actors make concessions to each other to make an exchange possible. The concession stage was ignored in previous empirical studies, with the exception of Skvoretz, Willer and Fararo (1993). In the bargaining model both the initial exchange offers and the rate at which actors make concessions, the concession rate, are assumed to be dependent on several variables. In agreement with the results of previous empirical studies we allow initial exchange offers and concession rates to vary between positions and relations in the network that are in different automorphic equivalence classes. In addition we include effects of exclusion and inclusion in both bargaining stages. Furthermore, bargaining time is included to model concession rates. As opposed to effects of positions in the network, which is a structural variable, we also include a micro-level variable to allow for differences in actor characteristics. The network structure including its positions, the relations between them, and the activities attached to the positions is called the macro-structure by Lawler et al. (1993). In their terminology, the micro-structure is similar to the macro-structure but connects specific actors to the positions. By including actor characteristics as independent variables in our model we check and test whether systematic tendencies in bargaining behavior can be explained by system variables (the macro-structure) only, or also require actor characteristics. Experiments suggest that outcome distributions can be accurately predicted by the configuration of network ties without reference to characteristics of the actors involved (Markovsky, 1987). In the present paper we test the effect of actor characteristics on bargaining behavior. However, in testing effects of individual actors we adhere to Coleman's (1990) warning that a focus on individual behavior should not lead away from the primary objective to explain the functioning of the social system. Therefore, our model includes a minimal and parsimonious representation of the actor as suggested in for example Coleman (1990) and Lawler et al. (1993). In the Data section the data, taken from Skvoretz and Zhang (1997), is described that we use to test our hypotheses and model of actor bargaining behavior in exchange networks. After discussing the tests of the bargaining model and the specific hypotheses in the Results section the paper is concluded with a Conclusion and discussion section. 6.2

Background

Theories of network exchange mentioned in the introduction were mainly interested in predicting the exchange outcome distributions for given network structures. However, researchers developing these theories also conducted simulation studies parallel to their theories, in order to investigate whether the exchange outcome distributions can also be predicted by simple actor bargaining models on the micro-level. These simulation studies and

106

Figure 6.1:

Chapter 6

Experimental networks used in the analysis.

their implications are discussed in this section. Also the few empirical studies are discussed that explicitly test assumptions of these simple actor models using experimental data. To be able to understand this discussion, it is necessary to provide a description of the experimental paradigm used by researchers in the field to study exchange networks. 6.2.1 Experimental paradigm to study exchange networks Most empirical and simulation studies of exchange networks represent an exchange by a division of a prize or common resource pool.3 The value of the prize is in general 24 points for each of the exchange relations.4 This will also be the case for the research described in the present study. The prize is divided in two integer parts between the two actors in an exchange relation that carry out the exchange. Not all actors can exchange with each other. The restrictions of which actor can exchange with whom are represented by an exchange network. Exchange networks that are used in the present study are depicted in Figure 6.1. For example, the Branch31 network consists of four actors of which only the central actor has no exchange restrictions with whom he can exchange.5 The peripheral actors however can only exchange with the central actor 3 In a minority of studies on negotiated exchange, exchange is represented by an exchange of two resources that

are valued differently by the two actors involved in the exchange (e.g., Willer and Anderson, 1981). 4 In some studies other values of prices are used, or prices vary across exchange relations (e.g., Cook et

al.,1983; Bonacich and Friedin, 1998). 5 For simplicity the male pronoun is used to refer to subject and actor, which of course does not mean that

experiments do not contain female subjects.

Bargaining in Exchange Networks

107

and not with each other. Other restrictions are the number of exchanges an actor can and/or must make in an exchange round. In the exchange networks considered in the present study actors can only make a maximum of one exchange in one round, except the central actor in the Branch32 network, who can make two exchanges. Consider one exchange round. In the experiments one exchange round ends for an actor either when he made the maximum number of exchanges, or after a fixed time limit. An actor does not receive any points from an exchange relation if he was not able to make an exchange with the other actor in the relation before the end of the round. When an actor has completed an exchange his point total is increased with the number of points he made in the exchange. After the experiment an actor is usually paid according to a linear function of the number of points he gathered in the whole experiment. An actor has four alternatives in each of his exchange relations at any time between the start and end of his exchange round. First, he can send an offer to another actor in one of his relations. An offer is defined here as a proposed division of the prize, where the value of the offer denotes the number of points allocated to the actor who received the offer. Second, he can accept an offer of the other actor in one of his relations, meaning that he agrees with the proposed division. Third, the actor can confirm the acceptance of his offer by another actor, which completes the exchange. If the actor does not confirm the acceptance then the acceptance by the other actor is of no consequence and the exchange is not (yet) carried out. And fourth, the actor can choose to do nothing. Note that the bargaining situation as described by the four alternatives of the actors is less structured than alternating offer or sequential bargaining games as studied in game theory (Ochs and Roth, 1989; Kreps, 1990; Rubinstein, 1982; Osborne and Rubinstein, 1990). Experiments of exchange networks can differ in the information the actors have on the bargaining in the network.6 In the experiments of the present study actors have perfect information on the structure of the exchange network, their position in the network, and all offers and exchanges actors make in each relation in the exchange network at any time.7 The information is made public to all actors immediately after the offers or exchanges are made. Sessions, periods, and rounds organize an experiment. An experiment consists of a number of sessions. Each session is run with a different group of subjects. In each session, the number of subjects equals the number of positions in the exchange network. In sessions actors rotate among positions such that each actor occupied each position once. Thus the number of periods in a session is equal to the number of positions. Each period consists of a number of exchange rounds in which actors negotiate possible divisions of the resources. Details of the design of the experiments of the present study are explained later in the Data section.

6 Perfect information concerning network structure and bargaining between actors was provided to all actors in

most experiments carried out in the research program of network exchange theorists (Markovsky, Skvoretz, Willer), including the experiments of Skvoretz et al. (1997) analyzed in the present study. Eperiments in other research programs, like the research programs of power dependence theorists (e.g. Cook et al., 1983) and of Lawler and Yoon (e.g., 1996), provide restricted information. See Skvoretz and Burkett (1994) for a discussion and a test of the effects of information on the distribution and development of power in exchange networks. 7 In some exchange network studies an actor only knows his relations and his shares of the offers in these

relations (e.g., Cook and Emerson, 1978; Cook et al., 1983).

108

Chapter 6

6.2.2 Simulation studies A number of studies on network exchange included simulation.8 Advocates of most of the theories on network exchange carried out these studies. For example, simulations were used by Cook et al. (1983) who developed power-dependence theory, Markovsky et al. (1988) who developed network exchange-resistance theory, and Bonacich et al. (1995) who developed the game-theoretic approach. In order to simulate exchange outcome distributions for a given network structures one was obliged to develop a model of how actors bargain with each other. Although the researchers developed different theories of network exchange, the actor models they used in their simulations were remarkably similar. Assumptions of actor's behavior in exchange networks commonly made in simulation studies can be summarized by the following eight conditions (Markovsky, 1995; Markovsky et al., 1988; Skvoretz and Zhang, 1997). Note that the first four conditions deal with actor behavior (actor conditions or assumptions), while the latter four deal with the structure of the exchange situation (position conditions or definitions). (1)

all actors use identical strategies in negotiating exchanges;

(2)

actors consistently excluded from exchanges raise their offers to others;

(3)

actors consistently included in exchanges lower their offers;

(4)

actors accept the best offer they receive, and choose randomly in deciding among best tied offers.

(5)

each position is related to, and seeks exchange with, one or more other positions;

(6)

at the start of an exchange round, equal pools of positively valued resource units are available in every relation;

(7)

two positions receive resources from their common pool if and only if they exchange;

(8)

each position exchanges with at most one other position per round.

Conditions (6) to (8) determine the type of exchange relation. Condition (6) refers to the size of the prize (in most cases 24), condition (7) refers to exclusionary exchange relations, condition (8) refers to the often used one-exchange rule (see also footnote 2). These conditions will also hold for the network relations adopted in our empirical study, except that in the Branch32 network the central actor can exchange twice. Conditions (1) to (4) refer to the actor bargaining model on the micro-level in the simulations. Note, however, that in Lawler et al.'s (1993) terminology the conditions deal with the macro-structure and not the micro-structure, because all actors are assumed to behave the same. Note also that these four conditions together do not yet completely specify an operating actor model. First, no assumption or condition is included that refers to the actor's initial offers. It is common in the simulation studies mentioned to assume that an actor's initial offer is a uniformly distributed random variable. The range of the uniform distribution assumed in Cook et al. (1983) was from 1 to 23, and from 9 to 15 in Bonacich et al. (1995, 1997b). Furthermore, with the exception of Markovsky (1995), it is assumed that 8 One simulation study, of Skoretz and Lovaglia (1995) is not reviewed in this paper. The reason is that their

simulation study focuses on an explanation of exchange frequency and not on actor bargaining or outcomes in exchange networks.

Bargaining in Exchange Networks

109

they obtain what they wanted plus half of the remaining prize. Second, often no assumptions are made of concessions that actors make when their initial offers are irreconcilable for exchanging. In Markovsky (1987) an actor obtains the average of what he offers himself and what is offered to him if the two are irreconcilable. In other studies actors simply accept their most profitable offer (condition (4)). If the sum of the actors' offers is larger than 24, it is assumed that they obtain what they are offered plus half the difference of 24 and the sum of the offers. The simulation studies with the conditions and assumptions stated above produced exchange outcome distributions similar to those found in empirical studies (Cook et al., 1983; Markovsky, 1987; Markovsky et al., 1993; Bonacich et al., 1995, 1997b). The observed similarity between the simulated data and the empirical data suggests that the set of assumptions on actors' bargaining is a reasonably good approximation of the processes that occur at the micro-level. In order to verify the latter suggestion one is obliged to test whether other sets of assumptions lead to the same simulated outcomes. In his early and important simulation study Markovsky (1987) carried out a test of this kind. Markovksy's (1987) study was motivated by the large difference in outcome distributions in empirical studies between strong power and weak power network structures. A strong power network is a network where a group of actors cannot be excluded and some other actors are inevitably excluded, like in the Branch31 and Branch32 in Figure 6.1. In these network structures the central actor A is not excluded and exchanges once (Branch31) or twice (Branch32), necessarily leaving out two or one peripheral actors. A weak power network is a network where actors might but need not be excluded. For example, in the Line4 structure in Figure 6.1 actors A and D are excluded when B and C exchange, but not when B exchanges with A and C with D. In strong power networks the central actor receives almost all the resources when he makes a deal with either one of the peripheral actors, while the actors that are never excluded in weak power networks only obtain a small advantage over the peripheral actors. Markovsky (1987) therefore suggested that network structure alone is a necessary and sufficient condition to generate these extreme outcome distributions. To test this assertion, rather than holding the actors constant and varying the network structure as in previous research, Markovsky held the network constant and varied the actor strategies. For his simulations Markovsky (1987) took a Branch21 (or three actor line (Line3)) structure, where the central actor can exchange with only one of the two peripherals. In the first of 19 simulations Markovsky (1987) made assumptions in agreement with the assumptions stated above. He assumed that all three actors in the Line3 made offers to all actors they could exchange with. The increase in an offer following inclusion and decrease in an offer following exclusion was equal to one. He additionally assumed that initial offers in the first round were random. Instead of making concessions actors were assumed to agree upon an exchange equal to the average of their offers. Given these assumptions Markovksy (1987) replicated the results of empirical research and other simulation studies: a network developed in which the central actor receives almost all resource points when he made an exchange.9 9 However, it must be noted that there is an inconsistency in Markovsky (1987) between the algorithm in the

Appendix and the text of the article. Markovsky (1987, p. 105) states that in the actor model of Experiment 1 an actor offers one recourse unit more to the central actor after exclusion, and offers one unit less after inclusion in the previous round. However, the algorithm depicted in the Appendix does not contain the decrease after

110

Chapter 6

In the other 18 simulations Markovsky systematically varied the strategies of the actors, but using identical strategies for all actors in the network structure, therefore not violating assumption (1). Variations on the actor strategies used in Experiment 1 where among others (i) make random offers and delete assumptions (2) and (3), (ii) restrict the offer ranges of the central actor, and delete assumptions (2) and (3), (iii) the central actor prefers to exchange with one of the peripherals, (iv) stubbornness in the periphery, meaning that sometimes the periphery rejects offers from the central actor. The results of his simulation analyses demonstrated that the variations of the actor strategies had a large effect on the resource distribution, therefore rejecting the hypothesis that a strong power structure alone is enough to develop a resource distribution extremely in favor of the central actor. Markovsky also suggested that the assumptions concerning the reactions to exclusion and inclusion in the previous round were important in generating the extreme resource distributions. The merit of Markovsky's work is to show that network structure at the macro-level and actor strategies at the micro-level work together and are needed to realize the exchange outcome distribution. Markovsky (1987, p. 113) concludes "That we may choose to 'see' processes at only one level of analysis does not reduce the explanatory potential of what others see at different levels. Each perspective offers unique and potentially valuable insights.". Although we agree with Markovsky's conclusion, we argue that the results of his study and also of the simulation studies based on the actor model outlined above with the eight conditions are unsatisfactory for several reasons. First, the assumptions in these simulation studies of actor's behavior in exchange networks are unrealistic. The variations of the actor strategies studied by Markovsky in his simulations 2-19 are not likely to be used by human actors in network exchange experiments. Moreover, neglecting the concession stage with condition (4) or simplifying it by averaging irreconcilable offers is not realistic. Inspecting offers between actors in network exchange experiments shows that a lot of offers and concessions are made before the actors agree upon an exchange (or not). Finally, assuming a uniform distribution for initial offers is not realistic. On the contrary, we will demonstrate in this study that there is a regularity in both initial offers and concessions that can be associated with the network structure in which the actors bargain. Another unrealistic assumption of the actor model is condition (1), which states that all actors use identical bargaining strategies, independent of network structure and the actor's position in this structure. Bonacich et al. (1997b) suggested that actors' strategies might vary depending on network structure and their position in the network. In their simulations they distinguish between accommodative versus exploitative strategies. Accommodative strategies maximize the probability of inclusion by, for example, simply accepting the best offer the actor receives. An exploitative strategy, for example, lowers offers to other actors after it has inclusion for the peripheral actors. Hence the actor model in the Appendix does not embody condition (3). Omission of condition (3) has extensive implications for the results of the simulations. It can be easily demonstrated that the actor model with both an increase and decrease of one unit after respectively exclusion and inclusion never yields strong power development in a Branch21 network. For example, assume that both peripheral actors offer 12. In the following rounds an actor's offer will never be higher than 13 because then he will be included and lower his offer in the next round. In this example the central actor will always be included and lower his offers, resulting in a equilibrium exchange outcome distribution with at most 18 (23/2 +13/2) units for the central actor and 6 units for one of the peripheral actor. Note that this distribution differs substantially from strong power outcome distributions, which are much more in favor of the central actor. Hence the strong power outcome distributions in Markovsky's (1987) first experiment are obtained by omitting the inclusion effect of the peripheral actors, and not with including an inclusion effect as suggested by Markovsky in the text of his 1987 article.

Bargaining in Exchange Networks

111

been included. Bonacich et al.'s (1997b) simulations demonstrate that it is more rational for actors that cannot be excluded, like the central actor in the strong power networks, to have an exploitative strategy. On the other hand accommodative strategies benefit more actors that can be excluded, such as the peripheral actors in strong power branches. This study will provide an empirical test of Bonacich et al.'s (1997b) hypothesis that central actors are more exploitative and peripheral actors are more accommodative. The previous discussion reveals that although the simulations lead to reasonably accurate predictions of observed exchange outcome distributions, the actor model is not an accurate description of how actors bargain at the micro-level. Another problem with the actor models proposed in the simulation studies is that they are not derived from the theories of network exchange. One exception is the game-theoretic core. Bonacich (1998a, 1999) demonstrated that if actors behave in agreement with four behavioral conditions, then the exchanges agreed upon must be in the core solution of cooperative game theory. The first two conditions are identical to conditions (1) and (2) above, but conditions (3) and (4) are replaced by (Bonacich, 1999) with: (3')

actors who are included in an exchange in the previous round make the same offers as in the previous round;

(4')

an actor leaves a current partner if and only if he receives a strictly superior offer from another.

The simulations of Bonacich et al. (1995, p. 318, endnote 3) and Markovsky (1987, see our footnote 5) confirm that an actor model without effects of inclusion like in assumption (3') yields accurate simulated outcome distributions. The fact that actor models with different assumptions on actor behavior all have considerable predictive success indicate that these simulation studies are not sufficient to draw conclusions upon the strategy actors use in network exchange. To do that, one must explicitly test the assumptions themselves. Skvoretz and Lovaglia (1997) and Thye et al. (1997) did precisely that. These empirical studies are discussed in the next section. 6.2.3 Empirical studies The previous section demonstrates that condition (2), raise offers after exclusion, and condition (3), lower offers after inclusion, have an important function in actor models of network exchange. Markovsky (1987) suggested that both are important in establishing extreme outcome distributions in strong power networks. But Bonacich's (1998a, 1999) game-theoretical core and simulations (Bonacich et al., 1995), and an analysis of Markovsky's (1987) algorithm (see our footnote 5) suggest that inclusion effects may not be a necessary condition. Given the importance of conditions (2) and (3) in both theory and simulation of actor models of network exchange, it is surprising that they have been studied only recently. Of the two studies we first describe Thye et al. (1997). Instead of being interested in increasing and decreasing offers after respectively exclusion and inclusion, Thye et al. (1997) were interested in the likelihood of an increase or decrease of the offers as a result of an inclusion or exclusion. Thye et al. (1997) hypothesized that following an inclusion (exclusion) actors are more (less) likely to increase than to decrease their request. They also derived that the likelihood of decreasing their request after exclusion is greater than the likelihood of increasing their request after inclusion in an

112

Chapter 6

exchange. Moreover, they expected that actors were more likely to increase their request to actors with whom they exchanged than to actors with whom they did not exchange. Thye et al. (1997) tested the effects of inclusion and exclusion for several network structures, including the Line4 and Stem used also in our study (see Figure 6.1). To test their hypotheses, they modified the exchange relation that is traditionally used in experiments as described in the Experimental paradigm to study exchange networks section (Thye et al., 1997). Actors cannot modify their offers freely, but can change the offer in each relation they made in the previous round by only -1, 0, or 1. After each actor has made his offer the negotiation outcome was either (i) a split, if the sum of both offers was larger than 30 points, actors received their requested amount and split the profit surplus, (ii) an agreement if the sum of both offers equals 30, (iii) a stalemate and no exchange if the sum is smaller than 30. Note that because of this experimental procedure where negotiation is necessarily restricted to only one offer, bargaining and concession behavior, present in traditional empirical network exchange research, is eliminated. The results of Thy et al. (1997) demonstrated that there were relatively low but significant likelihoods to decrease an offer when included in an exchange. As expected, the likelihood of an increase of an offer after exclusion was much greater. Actors were also more likely to exploit the exchange partner than the non-exchanging alternative partner. Finally, these results were independent of the position the actors occupied in the network. Thye et al.'s (1997) results are important in that they show that the effects of inclusion and exclusion on the likelihood of changing are different. However, the difference between the setup of traditional network exchange experiments and the setup of their study is considerable, which could mean that their results are to some extent a peculiarity of their setup. Skvoretz and Zhang (1997), the study we describe now, had an experimental setup, which was identical to those, used in traditional network exchange research. Skvoretz and Zhang (1997) also studied the effects of inclusion and exclusion. However, they were not interested in the likelihood of changes in offers. Their dependent variables were the average increase in offers after experiencing exclusion in the previous round, and the average decrease in offers after being included in an exchange. Note that Skvoretz et al. (1997) also largely eliminates the concession and bargaining behavior of actors. Instead of modeling all successive offers between actors before they agree upon an exchange or not, they compute the average of these offers and use that as the dependent variable, leaving out the dynamics of the bargaining process. They modeled changes in average offers by two functions. Offers after being included in an exchange in a previous round were modeled by10 Yt+1 = Yt - α(Yt-Min)

(1)

where Yt is the offer at round t, Min is the minimum (1) the actor can get in the exchange, and α is the parameter expressing the size of the effect of inclusion in the previous round. Offers after being excluded were modeled by Yt+1 = Yt + β(Max-Yt)

(2)

10 Because Skvoretz et al. (1997) defined actors' offers as proposals to themselves and not as proposals to others

(see section Experimental paradigm to study exchange networks) like in the present study, equations (1) and (2) in Skvoretz et al. (1997) are changed accordingly.

Bargaining in Exchange Networks

113

where Max is the maximum (23) the actor can get in the exchange, and β is the parameter expressing the size of the effect of exclusion in the previous round. In line with Thye et al. (1997), Skvoretz et al. (1997) hypothesized that both parameters were different from zero and that the effect of exclusion was larger than the effect of inclusion, that is, β > α. Skvoretz et al. (1997) tested the effects of inclusion and exclusion for several network structures. All these network structures, that is, Branch31, Branch32, Kite, Line4, and Stem are depicted in Figure 6.1. The data of Skvoretz et al. (1997) will also be used to test the hypotheses in our study. Skvoretz et al. (1997) results corroborated their hypotheses, which were also in line with Thye et al. (1997): Effects of inclusion were on average small, exclusion effects were on average large(r).11 Their analyses revealed interesting other results as well. First, the estimated effects of inclusion and exclusion are significantly different for experienced and inexperienced actors. Second, and more important, the effects of inclusion and exclusion interact with network position and network structure. In strong power networks peripheral actors were less inclined to decrease their offers after being included in a previous exchange, than in weak power structures, or than central actors in a strong power structure. Also the effect of inclusion for the central actor is larger than the effect of exclusion for peripheral actors. All these additional results imply that, as opposed to the commonly made assumption in condition (1) of simulations of actor models (1), actor strategies are not identical, but vary over actors in different positions in different structures. Moreover, the strategies vary according to the predictions of Bonacich et al. (1997); strategies of actors in strong power positions are primarily exploitative (inclusion effect but no exclusion), strategies of actors in weak power positions are primarily accommodative (no inclusion effect but substantial exclusion effect). To summarize the Background section, the simulation studies of actor models of network exchange suggested that inclusion and exclusion effects steer the development of the outcome distributions. The empirical studies demonstrated that the predicted effects of inclusion and exclusion take place, that exclusion effects are larger, and that these effects interact with the actor's position in the network structure. Unfortunately, however, Thye et al. (1997) and Skvoretz et al. (1997) only focussed on effects of inclusion and exclusion, neglecting other main variables of actor bargaining in exchange networks. That is, they do not provide a model or explanation of the whole bargaining process. More specific, they modeled neither actors' initial offers nor concessions that actors make to each other and the dynamics therein. Skvoretz, Willer and Fararo (1993) alone studied concession-making behavior in exchange networks. But they only considered frequencies and not magnitudes of concessions. One of their findings was that the frequency of concessions of the central actor in the strong power Branch31 was smaller than of the peripheral actors, which was not true in the weak power Line4. In the next section a model at the micro-level is developed and hypotheses are derived dealing with effects of variables on both initial offers and (the magnitude of) concessions.

11 Note that these results are not in agreement with the set of assumptions of Bonacich (1998a, 1999), who

assumed that there is no effect of inclusion. Bonacich (1999) realized that both studies discussed here found evidence for effects of inclusion. He however neglects them and justifies it by stating that these effects are small.

114 6.3

Chapter 6 Models and hypotheses

In the models we assume that bargaining can be divided in two stages. A first stage in which actors make their initial offer and a longer second stage which embodies reactions to these initial offers until an agreement is reached or time runs out. These stages will be called respectively initial stage and concession stage. It is common to distinguish between initial offers and concession making behavior in bargaining research (Siegel and Fouraker, 1960; Bacharach and Lawler, 1981; Lawler and Ford, 1995; Morley and Stephenson, 1977) and bargaining models (Cross, 1965, 1969; Coddington, 1968; Heckathorn, 1980).12 The resemblance between these studies is that they focus on bilateral exchange or monopoly (see Lawler et al., 1995 for a brief review of both theory and empirical findings). The present study distinguishes itself from bilateral exchange research in that actor bargaining can be and is related to the positions of the actor in the network. More generally, in this section the hypotheses are derived on the relation between our dependent variables initial offer and concession rate and the independent variables actor position, bargaining time, exclusion and inclusion in the previous exchange round, and actor differences. Initial offers need to be defined before elaborating how to model them. Consider an exchange relation between actors i and j who can exchange with each other in multiple rounds. In each round the actors i and j can make only one exchange. An offer from i to j in an exchange round is defined as an initial offer when (i) it is the first offer from i to j in this exchange round, and (ii) there is not yet an offer from j to i in this exchange round. The second condition is very important here. Only one offer in a relation can be initial, because it is assumed that the first offer of j to i is already affected by the initial offer of i to j. That is, j's offer to i can already be a concession of j to i from his 'true latent' initial offer to the offer of i to j.13 Hence the number of initial offers in the initial bargaining stage in a network structure is never larger than the number of relations in the network structure. Concession making behavior is simply defined as the behavior following initial offers leading to an exchange or to the end of the bargaining period with no exchange being made. The corresponding dependent variable concession rate Cij(t) of actor i to actor j at time t is defined if, at this time point, there exists a most recent offer Nij(t) and immediately preceding offers Oij and Oji with offer Oji made after Oij but before Nij(t). The concession rate is defined by C ij (t ) =

N ij (t ) − Oij 24 − O ij − O ji

(3)

The assumption in (3) is that concessions are not dependent on the absolute but on the relative differences in the offers of the two actors. Skvoretz and Zhang (1997) make a similar assumption in their tests of the effects of exclusion and inclusion on the change in average offer with (1) and (2). However, they do not control for the offers made by the other actor because (1) and (2) contain the absolute maximum and minimum offer. 12 Initial offer is called (initial) level of aspiration in the bargaining literature (e.g., Lawler et al., 1995; Morley et al., 1977; Siegel et al., 1960). 13 Note that the actor who makes the initial offer in an exchange relation does not necessarily have a first

mover's advantage, like in alternating offer or sequential bargaining games (see e.g., Ochs and Roth, 1989; Kreps, 1990; Roth, 1983; Rubinstein, 1982), because the initial offer does not restrict the time or value of the other actor's offers in any way.

Bargaining in Exchange Networks Table 6.1:

115

AER and AEAB classes in the network structures used in our study.

Network structure Branch31/ Branch32

AER classes (AB,AC,AD)

AEAB classes (AB,AC,AD), (BA,CA,DA)

Kite

(AB,AC,AD,AE), (BC,DE)

(AB,AC,AD,AE), (BC,CB,DE,ED), (BA,CA,DA,EA)

Line4

(AB,CD), (BC)

(BA,CD), (BC,CB), (AB,DC)

StemEx/ StemIn

(AB), (AC,AD), (CD)

(AB), (AC,AD), (CD,DC), (CA,DA), (BA)

The most important explanatory variable in our model of initial offers and concession rates is the automorphically equivalent actor bargaining (AEAB) class. Two positions in a network are called automorphically equivalent if the positions have identical structural locations in the network (Wasserman and Faust, 1994, pp. 469-473).14 We define relations m and n to be automorphically equivalent if actor im in relation m is automorphically equivalent to actor in in relation n, and actor jm in relation m is automorphically equivalent to actor jn in relation n. The second column of Table 6.1 lists the automorphically equivalent relations (AERs) for all the network structures used in our study. For example, in the Line4 the relations AB and CD are automorphically equivalent because actors A and D, and B and C, are automorphically equivalent. If it is assumed that there are no differences between actors in their characteristics then bargaining behavior can be expected to be identical in all relations in an AER class. That is, in the Line4 structures actors A and D can be expected to exhibit identical bargaining behavior, and actors B and C as well. Because the two actors in each of the relations AB and CD are not expected to behave similarly, this AER class designates two AEAB classes. These classes consist of AERs where the actors listed first in the relation are expected to display identical bargaining behavior. For example, the SER class (AB,CD) in the Line4 structure designates two AEAB classes (AB,DC) and (BA,CD). If two actors in an AER are on structurally equivalent positions, then the relation designates one AEAB class containing both actors' behavior.15 For example, B and C in AER class (BC) in the Line4 are structurally equivalent, which designates one AEAB class containing both BC and CB. The AEAB classes for each network structure are listed in the last column of Table 6.1. To test for effects of AEAB class on actor bargaining we allow both initial offers and concession rates to vary between AEAB classes. As opposed to the structural variable in the last paragraph our bargaining models also contain an explanatory variable on the actor or individual level. Differences in the bargaining processes and exchange outcomes of actors might be based on differences in structural 14 Instead of automorphically equivalent, Borgatti and Everett (1992: p. 291) use the terminology structurally

isomorphic. 15 Two actors in a network are called structurally equivalent if they have identical ties to and from all other

actors in the network (e.g. Wasserman and Faust, 1994, pp. 468-469).

116

Chapter 6

advantages of the positions they occupy as well as on the strategic success of the position occupants (Lawler et al., 1993). Therefore the models also include parameters corresponding to effects of individual actors. Consequently, for the first time actor effects and structural effects on bargaining can be compared and an answer is provided to the question on whether actor differences (micro-structure) or position differences (macro-structure) are more important determinants of concession making behavior dynamics. Two other explanatory variables are already familiar: inclusion and exclusion in the previous exchange round. Both variables can have an effect on initial offers as well as on concession rates. Another variable included in the model of concession making behavior but not in the model of initial offers is the time when the offer is made. 6.4

Hypotheses

Our main hypothesis relates power in AEAB classes to average initial offers and average concession rates. A simple measurement of power in the AEAB classes distinguishes three different ordinal values. Power in an AEAB is high if the first actor in the relation can never be excluded but the second actor can. Power is equal when both actors in the relation are structurally equivalent. Power is low if the second actor can never be excluded but the first actor can. Unfortunately not all AEAB classes fall in these three categories. Not contained are AEAB classes consisting of relations with positions that are not structurally equivalent and where none of these two positions are safe from exclusion. In our study the Kite contains two such AEAB classes, (AB,AC,AD,AE) and (BA,CA,DA,EA). For simplicity we assume that the ordinal value of power in these AEAB classes is equal because none of the actors has a large structural advantage. The hypothesis is that there is a negative relationship between the power in a AEAB class and the average initial offer of the actors in the first position in the relations in that class, and a negative relation between the power in the class and the average concession rate of the actors in that position. We assume that actors accurately perceive the structural advantages of their position relative to the other positions in their exchange relations. If they perceive themselves to be in a more advantageous position than the other actor in the exchange relation, they are more inclined to offer less, and they are more inclined to make only small concessions to the other actor. Hence we expect a spurious positive relationship between average concession rates and average initial offers induced by the actors' perceptions of the network structure. Actors' perception of structural advantages of their position is not the only mechanism that directs their bargaining behavior. Previous experience is important as well. Experiences might correct inaccurate perceptions of structural advantages of positions. Evidence for the importance of previous experience is reported by Skvoretz and Zhang (1997). They found different effects of inclusion and exclusion for experienced and inexperienced actors in the Stem. Because experience of actors can make a difference we will distinguish between the data obtained from experienced actors (StemEx) and inexperienced actors (StemIn) as well. Listed below are the hypotheses of effects of AEAB class on the average initial offer of the first actor to the second actor in the relation in the class. No ordinal predictions are stated for the classes that have equal ordinal power values, which is denoted by the ‘~’ sign in Hypothesis 1a below.

Bargaining in Exchange Networks Hypothesis 1a:

117

Effects of AEAB class on the average initial offer and the average concession rate.

Branch31/ Branch32

(AB,AC,AD) < (BA,CA,DA)

Kite

(AB,AC,AD,AE) ~ (BA,CA,DA,EA) ~ (BC,CB,DE,ED)

Line4

(BA,CD) < (BC,CB) < (AB,DC)

Stem

(AB) ~ (AC,AD) < (CD,DC) < (CA,DA) ~ (BA)

Both the Stem and the Kite contain tied AEAB classes. Can we derive predictions with another method that distinguishes power in these tied classes? And can predictions be derived for differences in average initial offers and average concession rates between networks, instead of within networks as in Hypothesis 1a? To be able to answer both questions another more refined measure of power should be constructed. We construct a measurement based on the Exchange-Seek Likelihood procedure to find breaks in networks (Simpson and Willer, 1999).16 With this measurement we are able to derive a stronger hypothesis than Hypothesis 1a. The Exchange-Seek Likelihood procedure assumes that actors seek an exchange randomly among those to whom they are connected. The exchange-seek likelihood (ESL) of an actor is the likelihood that the actor is included conditional on this random search. It is calculated as the sum of the relative proportions of mutual exchange seeks involving that actor (see Simpson and Willer, 1999, for a more detailed explanation). The likelihoods, obtained from Simpson and Willer (1999), are also shown for the networks in Figure 6.1. Note that the simple power measurement can be derived from the ESLs. Strong power classes are formed by relations where the ESL of the first and second actor equal respectively 1 and less than 1, and reversed for low power classes, while actors in relations of equal power classes have equal ESL's or both have ESL values smaller than 1. To derive stronger hypotheses on the relation between power in AEAB classes and bargaining behavior, the difference in ESL values of the positions in a AEAB class is used as measure of the power in the AEAB class. The ESL values suggest that power values tied to the simple measurement within and between networks are different with respect to average initial offers and average concession rates. Using differences in ESL as a measure of the power in AEAB classes larger average initial offers would be expected, for example, in the Stem for class (AB) than for class (AC,AD), and similarly larger values would be expected for (BA) than for (CA,DA). Applying the measure to differences between networks, we expect larger average initial offers for class (BA,CA,DA) in the Branch31 (ESLs are 1/3 and 1) than for class (BA,CA,DA) in the Branch32, where A can make two exchanges in one round. A similar reasoning holds for the relations between power and average concession rates. The reasoning is summarized in Hypothesis 1b. Hypothesis 1b:

Effects of AEAB class on the average initial offer and the average concession rate within and between networks.

16 Only after analyzing the data we came up with the Exchange-Seek Likelihood procedure as another possible

explanatory mechanism to explain differences in initial offers and concession rates. For editorial reasons we incorporated hypothesis 1b in the Model and Hypotheses section instead of the Discussion section.

118

Chapter 6

A

There is a negative relation between power in a AEAB class as calculated by the ESL procedure and average initial offers of the first actor in the relations in that class, both within and between networks.

B

There is a negative relation between power in a AEAB class as calculated by the ESL procedure and average concession rates of the first actor in the relations in that class, both within and between networks.

Previous empirical and simulation studies reported in the Background section suggest that exclusion and inclusion have effects on bargaining. In correspondence with the results of these studies we hypothesize that exclusion (inclusion) in a previous exchange opportunity leads to both a higher (lower) initial offer and a higher (lower) concession rate in the next exchange opportunity. Furthermore, also in line with previous studies, we expect an interaction with network position. Exclusion effects are expected to be largest in AEAB classes low in power, and inclusion effects are expected to be largest in AEAB classes high in power. Hypothesis 2: Effects of inclusion and exclusion on the average initial offer and the average concession rate A1

The average initial offer decreases after inclusion in a previous exchange opportunity, and there is a positive relation between the size of the decrease and power in AEAB classes.

A2

The average initial offer increases after exclusion in a previous exchange opportunity, and there is a negative relation between the size of the increase and power in AEAB classes.

B1

The average concession rate decreases after inclusion in a previous exchange opportunity, and there is a positive relation between the size of the decrease and power over AEAB classes.

B2

The average concession rate increases after exclusion in a previous exchange opportunity, and there is a negative relation between the size of the increase and power over AEAB classes.

Two remaining hypotheses deal with the relationship between time and concession rates and with actor differences in concession rates and initial offers. (Real) Time is not a relevant explanatory variable for initial offers because by definition these offers are all made early in the bargaining process. Because an exchange is better than no exchange, we expect that the average concession rate increases as time approaches the end of the bargaining period. This is in accordance with the deadline effect observed by Roth, Murnighan, and Shoumaker (1988) in bargaining experiments. In the experiments summarized by Roth et al. (1988) the frequency of agreements or exchanges grows exponentially over time, meaning that the second derivative of the time effect is positive, with a peak at the deadline of the bargaining round. Hypothesis 3:

Effect of time on the average concession rate.

The average concession rate increases as time approaches the end of the bargaining period.

Bargaining in Exchange Networks

119

The previous studies discussed in the Background section assumed that actors are similar in their behavior (condition (1)), or that behavior is similar when controlling for position in the network (Bonacich et al.,1997b; Skvoretz et al.,1997; Thye et al.,1997). The latter assumption is also implicit in the data analyses of the experiments, in which all exchanges of one actor on a position together with all exchanges of other actors on that same position are treated as replications of each other (e.g., Skvoretz and Willer, 1993). However, none of these studies in fact tested for actor differences or compared them to position differences, that is differences in behavior related to network structure. Corresponding to the studies mentioned, our null hypothesis is that there are no actor differences in their average initial offers and average concession rates after controlling for the effects of the other explanatory variables. Hypothesis 4: Effect of actor on average initial offer and on average concession rate. A

There are no actor differences in average initial offer.

B

There are no actor differences in average concession rate.

6.5

Data

Data to infer processes of actor bargaining on the micro-level come from experiments using the five different networks depicted in Figure 6.1. The same data is also used by Skvoretz et al.(1997) to test their hypotheses on the effects of exclusion and inclusion.17 The Branch networks are strong power networks, and the Line4 and Stem (StemIn and StemEx) networks are weak power networks. The Kite is a special network structure because there is always one actor excluded and each actor can be the unlucky one. The organization of the experiment is outlined in the subsection Experimental paradigm to study exchange networks. An exchange round lasted 5 minutes. The experiment was run on computers with the ExNet program. Subjects were in separate rooms and could not communicate with each other except by making offers via the computer. The network, the subject's position in the network, and all offers in all relations in the network were shown on each subject's screen (perfect information). The data for each experiment is obtained from eight sessions of the Branch31, Branch32, Line4, and Stem (32 subjects per experiment) and six sessions of the Kite (30 subjects). Four sessions of the Stem involved actors who had experience with other exchange experiments, while the other four did not. Because Skvoretz et al.(1997) found differences in the behavior of the experienced and inexperienced actors the two sets, StemIn and StemEx, are analyzed separately here as well. See Skvoretz et al. (1997) for a more detailed description of the experiments. Because actors interacted through a computer network all offers were saved together with the time when they were made. The number of offers obtained from each experiment is shown in the second column of Table 6.2, with the total number of offers equal to N’ = 21,446. To prepare the data to test our hypotheses one simplifying assumption was made. Consider a relation between two actors where the first actor sends more than one offer to the second before it is answered by an offer of the second actor. To calculate the concession rate of the first actor one can either interpret his offers separately, or one can interpret them as one 17 We thank John Skvoretz and the research group at the University of South Carolina for their permission to use their data.

120

Chapter 6

Table 6.2:

Summary of the number of cases for the analyses of the data of the experiments. The second column shows the total number of offers in the experiments (N'). The third column shows the number of offers after deleting offers not used in the analyses (N). The number of cases that remain to test our hypotheses on concession rates (N[R]), initial offers (N[I]), and effects of inclusion (N[Iin]) and exclusion (N[Iex]) on initial offers are presented in respectively columns four to seven.

Branch31 Branch32 Kite Line4 StemEx StemIn Total

N' 1,361 4,130 4,407 6,149 3,393 2,006 21,446

N 1,005 3,178 3,511 5,433 2,902 1,482 17,511

N[R] 231 1,985 1,801 4,299 2,173 789 11,278

N[I] 93 96 171 94 62 63 579

N[Iin] 95 167 289 170 117 119 957

N[Iex] 128 66 64 26 11 23 318

single concession. Presuming that actors attempt to correct mistakes in their offers by immediately sending another offer, it is reasonable to interpret succeeding offers as one concession. Furthermore, by interpreting them as one concession we automatically control for the number of concessions made by the actors in the relation. Thus, both actors in the relation make the same number of concessions minus or plus one. Both arguments led us to interpret succeeding offers as one single concession. Conforming to this interpretation our data sets were modified by deleting in each round for each relation an offer from actor i to actor j when it was followed by another offer from i to j. The third column of Table 6.2 shows the number of remaining offers for each of the network structures, with the total number equal to N = 17,511. Note that in total 81.65% of the offers remain, suggesting that only a small percentage of the offers were followed by offers from the same actor to the same exchange partner. Explanations of the number of cases in the other columns of Table 6.2 are provided in the Results section. 6.6

Results

Results of the analyses of the data are discussed separately for the dependent variables initial offer and concession rate. 6.6.1 Initial offers Choices needed to be made to select the initial offers from the data. We were very strict in our selection. To begin with, a first offer from actor i to actor j in a particular round is not considered as an initial offer if it followed a first offer of j to i. Consequently, in each round there is at most one initial offer for each relation. Only initial offers in the first round, of which there are 597 (see column 5 of Table 6.2), are used to test for effects of AEAB class. Initial offers in later rounds are likely to be affected by exclusion or inclusion in the previous round(s) and are therefore only used to estimate the effects of inclusion and exclusion. Only initial offers made by the same actor to the same partner in two subsequent rounds were used in the analyses of the effect of inclusion and exclusion. In this way 957 and 318 observations remained to test for effects of respectively inclusion and exclusion (see column 6 and 7 of

Bargaining in Exchange Networks Table 6.3:

121

Effects of (power in) AEAB class on and actor differences in average initial offers. Classes are ordered with respect to power from top to bottom for each network structure. The LR test of AEAB class has number of degrees of freedom equal to the number of AEAB classes in the network minus one. The LR test of the random actor effect always has one degree of freedom.

Network

AEAB class

Effect of AEAB class (SE)

Random actor effect (SE)

4.3 (1.15) 17.6 (1.25)

LR test AEAB class (p-value) 74.1 (< 0.001)

1.44 (0.99)

LR test random actor effect (p-value) 0.62 ( 0.430)

Branch31

(AB,AC,AD) (BA,CA,DA)

Branch32

(AB,AC,AD) (BA,CA,DA)

6.3 (1.44) 11.8 (1.47)

12.91 (< 0.001)

2.24 (0.81)

2.56 (0.110)

Kite

(AB,AC,AD,AE) (BC,CB,DE,ED) (BA,CA,DA,EA)

7.0 (0.94) 9.2 (1.01) 10.8 (0.95)

16.37 (< 0.001)

2.35 (0.48)

16.78 (< 0.001)

Line4

(BA,CD) (BC,CB) (AB,DC)

7.4 (1.01) 8.5 (1.22) 9.7 (1.14)

4.62 (0.100)

0.76 (1.24)

0.10 (0.751)

StemEx

(AB) (AC,AD) (CD,DC) (CA,DA) (BA)

5.2 (1.13) 8.6 (2.02) 9.9 (1.47) 10.4 (1.36) 13.6 (2.02)

19.12 (0.001)

0 (2.40)

0.00 (1.000)

StemIn

(AB) (AC,AD) (CD,DC) (CA,DA) (BA)

7.9 (1.05) 8.0 (1.53) 8.0 (1.28) 10.4 (1.21) 10.3 (1.49)

9.84 (0.043)

0.54 (1.09)

0.07 (0.800)

Table 6.2). To test for the effects of AEAB class on and for actor differences in average initial offer the following model was fitted to the data of each network structure: Iijk = βk + θi + εijk

(4)

Initial offer Iijk from i to j in AEAB class k is equal to the effect of AEAB class k (βk), plus the random effect of actor i (θi), plus the residual or measurement error εijk. The analysis of the

122

Chapter 6

random effects linear model (4) resulted in (maximum likelihood) estimates of the variance of the actor effects and the average initial offers for each of the AEAB classes, controlling for the dependence between observations of the same actor. Table 6.3 shows separately for each network structure (a) the estimated average initial offers with their standard errors (SE) for all AEAB classes (column 3), the results of the Likelihood Ratio (LR) test that there is no effect of AEAB class (column 4), (b) the standard deviation of actor effects with its standard error (column 5) and LR test (column 6). The LR statistic is asymptotically χ2 distributed with the number of degrees of freedom either equal to the number of AEAB classes minus one for the AEAB class effect or equal to one in the case of the random actor effect. Hypothesis 1a stated that there should be differences in the average initial offers between the AEAB classes for all network structures except the Kite. The hypothesis is confirmed for the Branch31, Branch32, StemEx and StemIn structures. Note also that the order of means is as predicted by Hypothesis 1a: classes high in power have smaller averages than equal power classes, which have smaller averages than classes low in power. The order of averages for the Line4 is also as predicted by Hypothesis 1a, but the averages are not statistically significant at the 0.05 level. Differences in the averages for the Kite are larger than expected, they are even significant at the 0.001 level. It is as if the actors in the Kite perceive the central actor to have considerable power because it has more relations or opportunities to exchange than the peripheral actors, although these opportunities do not guarantee the central actor to be included. Hypothesis 1bA, a stronger version of Hypothesis 1a, states that there is a negative relation between power in a class and the average initial offer in that class. The data provide strong support to Hypothesis 1bA. Table 6.3 shows that all average initial offers within a network structure are in the predicted order, except for a small negligible deviation for the StemIn. Also across network structures there is a strong negative relation between power of a class and average initial offers. The rank order correlation between average initial offer and classes’ power over all 20 classes is -0.906 (p < 0.001), where power in AEAB classes is calculated as the difference between the ESLs of the positions in the relations in the class. As an extension of Hypothesis 1a and Hypothesis 1bA three additional tests were carried out. First, Hypothesis 1bA and the averages in Table 6.3 suggest that differences in average initial offers for the Branch31 are larger than for the Branch32. This hypothesis is 2 confirmed (χ (2) = 38.98, p < 0.001). In a second test differences between the StemEx and StemIn are compared. Combining the initial offers for the experienced and inexperienced actors for each class does not decrease the fit of the model significantly (χ2(5) = 8.00, p = 0.156). Finally, the data of the Kite, Line4, StemEx, and StemIn structures were grouped in the three power values suggested by Hypothesis 1a. That is, AEAB classes were assigned to high, equal and low power classes such that the sixteen classes were reduced to only three. We tested whether this strong simplification of the model led to a decrease in fit. This was the case (χ2(13) = 31.72, p < 0.003). The model misfit is primarily produced by the unexpected differences in the average initial offers for the Kite. The AEAB classes (AB,AC,AD,AE) and (BA,CA,DA,EA) can also be assigned to respectively high and low power classes. The alternative assignment is motivated by the fact that both classes are not of equal power because positions in the class differ in ESL value, and because it seems that the central actor in the Kite perceives himself to have considerable power. Incorporating the new assignment of the AEAB classes of the Kite the model with three power classes described the data of the

Bargaining in Exchange Networks

123

Kite, Line4, StemEx, and StemIn as well as the elaborated model with the sixteen AEAB classes (χ2(13) = 12.00, p = 0.528). The results of this test and of the other tests make us conclude that the power in AEAB classes is indeed an important and relevant predictor of the average initial offer of actors in relations in these classes. Moreover, the results strongly suggest that in exchange networks that are not of strong power, patterns of average initial offers are similar across networks within the same AEAB class. In the discussion and conclusion section the results are related to actors’ perceptions and experiences of their power. Two interesting observations were made in this regard. There is convincing evidence from our study that actors in the Branch31 initially underestimated the power of the central actor A and updated their estimate from experience. A significant increase in initial offers of peripheral actors to the central actor was observed over periods from 13.8 in the first round in the first period to 17.1, 18.5, 19.9, in the first round of respectively periods two, three, and four (χ2(3) = 17.99, p < 0.001). A similar effect was observed in the Kite. The initial offers of the central actor A to the other actors seemed to increase (5.3, 3.6, 6.0, 13.0, 7.9) over periods (χ2(4) = 14.54, p = 0.006). Note that differences in averages in the Kite were larger than can be expected from the small power differences, which is in agreement with the hypothesis that here also actors realize the central actor has not as much power as initially thought. Hypothesis 4A states that there are no actor differences in average initial offer. Standard deviations of random actor effects and their standard errors are shown in column 5 of Table 6.3. Column six shows that actor differences in initial offers were not significant, except for the Kite. Again the data for the Kite are not as expected. Perhaps the actor differences for the Kite can be attributed to actor differences in perceptions of the power of the central actor in the Kite. Nevertheless, in general Hypothesis 4A, which states that there are no actor differences in average initial offer can be accepted. The random effects model to estimate effects of inclusion and exclusion is given by Iijk,r - Iijk,r-1 = βk + θi + εijk

(5)

The model is the same as (4), but the dependent variable is now the difference between two initial offers Iijkr and Iijkr-1 in subsequent rounds (2 ≤ r ≤ 4). The model is fitted separately to subsequent initial offers enclosing respectively inclusion and exclusion (957 and 318, see two last columns of Table 6.2). To begin with, the hypothesis was tested that inclusion (the first part of Hypothesis 2A1) and exclusion (first part of Hypothesis 2B1) have a main effect on average initial offer, averaged over all AEAB classes, that is, model (5) was fitted with only one average β. Table 6.4 presents for all network structures the main effects β of inclusion and exclusion, its standard error, and its significance. There is no evidence in favor of Hypothesis 2A1 that there is an effect of inclusion on initial offers. For all networks there is a very small and insignificant change in the average initial offer as a result of inclusion in a previous exchange round. However, there is compelling evidence in favor of Hypothesis 2A2 that exclusion affects initial offers. Focussing on the main effects of exclusion, in five of the network structures exclusion increased average initial offers by more than one point. Four of these effects were significant at the 0.05 level. Only in the Kite structure was no main effect of exclusion was observed. Not only was evidence for general inclusion effects absent, the second part of Hypothesis 2A1 that inclusion effects are positively related to power in a AEAB class within

124

Chapter 6

Table 6.4:

Network Branch31 Branch32 Kite Line4 StemEx StemIn

Main effects of inclusion and exclusion on average initial offer. Test statistics are approximately normally distributed. Tests are one-sided. Effect of inclusion (SE) -0.16 (0.30) 0.10 (0.40) 0.04 (0.19) -0.18 (0.30) -0.02 (0.36) 0.09 (0.31)

z-test inclusion (p-value) -0.52 (0.302) 0.24 (0.594) 0.23 (0.590) -0.61 (0.272) -0.05 (0.481) 0.30 (0.616)

Effect of exclusion 1.13 (0.47) 1.79 (0.75) -0.07 (0.65) 1.54 (1.02) 4.73 (1.64) 1.98 (0.77)

z-test exclusion (p-value) 2.43 (0.008) 2.39 (0.009) -0.11 (0.555) 1.50 (0.067) 2.88 (0.002) 3.49 (0.005)

a network structure received no support as well. None of the six significance tests had pvalues smaller than 0.25. With respect to exclusion effects for different AEAB classes, exclusion effects could not be tested for the central actor in the Branch31 and Branch32 because the central actor was never excluded, and not for the Line4 because actors B and C were excluded only three times. For the StemIn and StemEx no differences in exclusion effects were found for the different AEAB classes. Surprisingly, marginally significant differences were found for the Kite (p = 0.056). Although the main effect of exclusion was negligible, the exclusion effect was large for AEAB class (AB,AC,AD,AE) (average increase of 2.81, z = 2.04, p = 0.042). This result is in agreement with our suspicion stated above that central actors in the Kite perceive themselves to be high in power resulting in a small initial offer, and after exclusion correct their own power’s estimate and their initial offers. In short, summarizing the results of the analysis with respect to Hypothesis 2A, there was evidence for only a main exclusion effect, which in general did not differ between AEAB classes, and no evidence for inclusion effects on the average initial offer. 6.6.2 Concession rates The concession rate was defined in (3) as the ratio Cij(t) between i’s change in offer at time t divided by the difference between j’s offer to himself and i’s previous offer. The fourth column of Table 6.3 shows the number of cases N[R] used for the analyses for each network. The numbers in this column are considerably smaller than the numbers in the third column. That is, many offers could not or are not used in our analysis of concession rates. First, of course, initial offers N[I] but also other first offers cannot be used because there was not a previous offer Oij. Second, offers are excluded for which Cij(t) is not defined, that is, the offers of which the denominator of Cij(t) is zero. These offers are either (i) offers where one actor reconfirmed the offer he previously made to conclude the exchange, or (ii) offers where one actor did not reconfirm and conclude the exchange but changed his/her mind and proposed another offer. Of the remaining cases a large majority (87.5%) had a concession rate 0 ≤ Cij ≤ 1, where the value 0 and 1 correspond to no concession and giving in completely respectively. Offers with extreme values of Cij(t) were excluded from the analyses. One extreme represents all the offers where an actor gave in more than the other actor asked for, thus where Cij(t) > 1. We interpreted these cases (1.1%) as mistakes. The other extreme represents offers where instead of a concession, an offer was made that increased the disagreement between the two actors (Cij < -1; 1.4%). Offers where -1 ≤ Cij < 0 (11%) were

Bargaining in Exchange Networks Table 6.5:

125

Bivariate tests of effects of explanatory variables round, session, time (linear and quadratic), and AEAB class on average concession rate, controlling for (random) actor effects. LR tests and their significance (between brackets) are shown.

Network Branch31

Round χ (3) = 0.70 (0.874)

Period χ (3) = 6.76 (0.080)

Time: linear, quadratic χ2(1) = 5.02 (0.025) χ2(1) = 0.71 (0.399)

AEAB class χ2(1) = 9.20 (0.002)

Branch32

χ2(3) = 3.65 (0.302)

χ2(3) = 3.22 (0.360)

χ2(1) = 0.98 (0.323) χ2(1) = 0.26 (0.610)

χ2(1) = 13.15 (< 0.001)

Kite

χ2(3) = 1.05 (0.789)

χ2(4) = 12.07 (0.017)

χ2(1) = 1.75 (0.186) χ2(1) = 0.86 (0.354)

χ2(2) = 5.14 (0.077)

Line4

χ2(3) = 6.38 (0.094)

χ2(3) = 5.94 (0.115)

χ2(1) = 0.74 (0.390) χ (1) = 43.33 (< 0.001)

χ2(2) = 10.92 (0.004)

StemEx

χ2(3) = 2.52 (0.471)

χ2(3) = 4.11 (0.250)

χ2(1) = 2.54 (0.111) χ2(1) = 32.5 (< 0.001)

χ2(4) = 3.60 (0.463)

StemIn

χ2(3) = 4.40 (0.222)

χ2(3) = 6.37 (0.095)

χ2(1) = 9.63 (0.002) χ2(1) = 12.76 (< 0.001)

χ2(1) = 2.72 (0.606)

2

2

2

included to reduce a (positive) bias in the estimates of the average concession rate. To obtain an overview of the effects of the variables, a number of bivariate tests were carried out with concession rate as the dependent variable and one other variable as explanatory variable. In these bivariate analyses effects of dependencies on concessions by the same actor are controlled for, again by using a random actor effects model such as (4) and (5). Then a random actor effects model is estimated which contains the most important explanatory variables, the AEAB class and time. The results of the latter random actor model are presented after discussing the results of the bivariate analyses and its implications for our hypotheses. The results of bivariate tests on effects of single explanatory variables are summarized in Table 6.5. Two variables derived from the experimental design are round and period. The effect of round can be regarded as the accumulation of the effects of inclusion and exclusion over all actors occupying all positions. The second column demonstrates that for all structures there was no combined effect of inclusion and exclusion on concession rates averaged over actors and over positions. Effects of exclusion and inclusion are tested separately below. The other design variable period, presented in the third column, in general did not have an effect on average concession rate as well. Average concession rates only varied across periods in the Kite (p = 0.017), but there was no systematic trend in the effect of period. The effect of time on concession rates was measured as a linear and quadratic effect. The fourth column of Table 6.5 depicts the LR test of a linear effect, and the LR test of a model with both the linear and the quadratic effect versus a model with only the linear effect. Time had a strong,

126 Table 6.6:

Chapter 6 Effects of (power in) AEAB class on the average concession rate. Classes are ordered with respect to power from top to bottom for each network structure. Average concession rates and their standard error (between brackets) are shown. LR tests (significance between brackets) of a difference in these averages are shown in Table 6.5. Network

Class

Branch31

(AB,AC,AD) (BA,CA,DA)

Effect of AEAB class (SE) 0.013 (0.085) 0.312 (0.059)

Branch32

(AB,AC,AD) (BA,CA,DA)

0.183 (0.025) 0.262 (0.024)

Kite

(AB,AC,AD,AE) (BC,CB,DE,ED) (BA,CA,DA,EA)

0.151 (0.033) 0.161 (0.024) 0.217 (0.030)

Line4

(BA,CD) (BC,CB) (AB,DC)

0.127 (0.023) 0.187 (0.027) 0.143 (0.023)

StemEx

(AB) (AC,AD) (CD,DC) (CA,DA) (BA)

0.145 (0.035) 0.136 (0.042) 0.140 (0.035) 0.189 (0.041) 0.132 (0.035)

StemIn

(AB) (AC,AD) (CD,DC) (CA,DA) (BA)

0.284 (0.060) 0.207 (0.071) 0.267 (0.054) 0.306 (0.064) 0.240 (0.060)

especially quadratic, effect on average concession rates in the weak power structures Line4, StemEx, and StemIn. The variable AEAB class (sixth column) had only an effect in the Line4 structure and in the strong power networks Branch31 and Branch32. Hypothesis 1 states that AEAB class has an effect on the average concession rate. More specifically, it states that there is a negative relation between power in an AEAB class and the average concession rate of the first actor in the relations in that class. In Table 6.6 the average concession rates are presented for all AEAB classes, like in Table 6.3 for the average initial offers. The results of the strong power structures Branch31 and Branch32 are as expected from Hypotheses 1a and Hypothesis 1bB. The powerful central actor on average made smaller concessions than the peripheral actors, with the averages of the Branch32 in between the

Bargaining in Exchange Networks

127

averages of the Branch31. The order of averages for the only other structure for which AEAB class had an effect, the Line4, was not in agreement with Hypothesis 1a and Hypothesis 1bB. Actors B and C on average made larger concessions to each other than actors in other relations. A partial explanation for this finding can be that actors B and C made less offers to each other than actors in other relations (3.06 times as little) where concessions were split in more smaller parts. The order of averages for the Kite is as expected, the more power in the AEAB class the smaller the average concession rate. However, differences in average concession rate were small and not significant, which is also as expected because power differences between AEAB classes are small. Power differences in the Stem are considerable according to the ESL measure, but the average concessions were neither significantly different nor in the predicted order, for both the StemIn and StemEx. In particular a difference in average concession rates would be predicted between classes (BA) and (AB) of the Stem, but this difference was small (and with the wrong sign). A possible explanation for the absence of the expected difference is that in the large majority of cases (93.1%) C and D exchanged with each other and did not pay much attention to A. In that case A had to make an exchange with B, without the power to exclude B, making them equal power actors. Hence A had to decrease his offers and to increase his concessions to B in agreement with his loss in power. Although the patterns of differences in averages in the StemEx and StemIn were similar, averages in the StemIn were considerably larger. Inexperienced actors were much more impatient, thus making fewer offers in an agreed upon exchange. These differences between the two Stem networks and the deviations from Hypothesis 1a and Hypotheses 1bB in the other weak power networks led to a moderately negative relation between AEAB classes’ power and average concession rate over network structures. The Spearman rank order correlation was -0.420 (t = 1.96, p = 0.033). Note that the relation between power in AEAB class and average initial offer was much stronger. In contrast, the relationship between power and average concession rate was confounded by actor experience, frequencies in which actors made offers, and frequencies of exchanges with each other. The random effects model to estimate effects of inclusion and exclusion is given by Cijkr(t) = βk + θi + δinc,k + δexc,k + εijk

(6)

In (6) the dependent variable is the concession rate at t, conditional on AEAB class (βk), random actor effect (θi), random measurement error (εijk), and two times k dummies δinc,k and δexc,k. Dummies δinc,k and δexc,k represent respectively the effect of inclusion and exclusion in AEAB class k in a previous round. Both dummy variables are zero for concessions in the first round. Hypothesis 2 states that exclusion of an actor in an exchange round leads to a higher average concession rate in the next round for that actor, while inclusion has the reverse effect. The bivariate analyses of the effect of variable round showed that there is no aggregated effect of inclusion and exclusion over positions and over actors. In a first series of analyses the main effects of inclusion (δinc) and exclusion (δexc) were estimated and tested. The results of these analyses are summarized in Table 6.7 for each network structure. The second column shows the results of the combined test δinc = 0 and δexc = 0, while the other columns present the effects of inclusion and exclusion separately. The second column shows that in general there were no differences in average concession rate in the first round, after exclusion in a previous

128

Chapter 6

Table 6.7:

Effects of inclusion and exclusion on average concession rates. The second column tests differences in average concession rates between categories ‘no previous round’, ‘exclusion in previous round’, and ‘inclusion in previous round’. The other columns show one-sided z-tests of the differences between the first category and respectively the inclusion and exclusion categories

. Network

LR test (p-value)

Branch31

χ (2) = 0.42 (0.812) 2 χ (2) = 6.04 (0.049) 2 χ (2) = 0.74 (0.691) 2 χ (2) = 4.56 (0.102) 2 χ (2) = 3.03 (0.220) 2 χ (2) = 3.89 (0.143)

Branch32 Kite Line4 StemEx StemIn

2

Effect of inclusion (SE) 0.041 (0.106) 0.029 (0.025) -0.020 (0.028) -0.028 (0.013) -0.013 (0.018) 0.080 (0.041)

z-test inclusion (p-value) 0.39 (0.652) 1.18 (0.883) -0.72 (0.238) -2.13 (0.017) -0.73 (0.233) 1.97 (0.975)

Effect of exclusion (SE) 0.072 (0.112) 0.084 (0.034) 0.001 (0.038) -0.025 (0.020) 0.040 (0.037) 0.053 (0.069)

z-test exclusion (p-value) 0.64 (0.262) 2.46 (0.007) 0.02 (0.494) -1.26 (0.900) 1.10 (0.136) 0.77 (0.222)

round, and after inclusion in a previous round. Hence Hypothesis 2B1 and Hypothesis 2B2 with respect to the main effects of exclusion and inclusion are not supported. The third and fifth columns mirror the latter conclusion. Only for two structures was the main effect significant; a main effect of exclusion in the Branch32, and a main effect of inclusion in the Line4. Fitting and testing (6) with interactions between AEAB class and inclusion/exclusion showed that in general effects of exclusion and exclusion in a class were also not related to power in that class. Hence also Hypothesis 2B1 and Hypothesis 2B2 were not supported. Of all two times twenty effects only a few were as expected. Significant inclusion effects (p < 0.05) were found in class (BC,CB) in the Line4, and (AC,AD) in the StemEx. Significant exclusion effects were found in class (BC,CB) in the Line4, (AB) in the StemIn, and (CD,DC) in the StemEx. Concession rates were also modeled simultaneously with the explanatory variables which are crucial in our hypotheses and which turned out to be important in prediction. For these reasons variables period, round, and effects of exclusion/inclusion are not included in the model. Variables included are time, AEAB class, and actor effects, leading to the following random actor model of concession rate:18 Cijk(t) = βk + γ1t + γ2t2 + θi + εijk

(7)

18 Another variable which could be important to model concession rate i to j is the previous concession rate of j

to i. Analyses were carried out where (6) was expanded with this previous concession rate, but this variable only had a marginal effect in the Kite (p = 0.018). Hence there is no effect of the size of the previous concession of j to i on i's concession to j after controling for AEAB, time, and actor effects.

Bargaining in Exchange Networks Table 6.8:

129

LR tests (significance between brackets) of effects on average rd th concession rates of AEAB class (3 column), actor differences (4 th column), and time (5 column). The second column shows the improvement of the general model’s fit in comparison to the model with only random actor effects and one intercept.

Network

General model

AEAB class

Branch31

χ (3) = 15.66 (0.001) 2 χ (3) = 14.29 (0.003) 2 χ (4) = 8.06 (0.089) 2 χ (4) = 53.88 ( 0. The arbitrary unit of measurement implies that interpersonal comparisons of utility are impossible (see Luce and Raiffa, 1957, 2.7, for a discussion). That is, utility gains or losses cannot be meaningfully compared among actors. Researchers in the field of social exchange research vary in how they deal with the theoretical impossibility of interpersonal comparisons of utility. For example, powerdependence theory solves the indeterminacy of the exchange rate by using the equidependence principle, which assumes that actors choose the exchange rate such that the actors' utility gains are equal to each other. That is, if A gives ∆y to B and receives ∆x from B, then the values of x and y are such that uA(x)∆x - uA(y)∆y = uB(y)∆y - uB(x)∆x (Cook and Emerson, 1978, p. 723; Emerson, 1981; Emerson, Cook, Gillmore, and Yamagishi 1983, p.

Two Representations of Negotiated Exchange: A Review and Comparison

153

1235). Other researchers question the equidependence solution because it presupposes interpersonal utility comparisons (Heckathorn, 1983a, p. 1211, 1983b, 1985, p. 151). The equiresistance or RKS solution of exchange-resistance theory is based on the impossibility of these comparisons and is invariant with respect to linear transformations of utility. Other researchers argue against the reduced exchange representation because it leads to confusion about whether invalid interpersonal comparisons of utility are being made (Skvoretz, Willer, and Fararo 1993, p. 113). However, as explained later in this section, the reduced exchange representation does not imply interpersonal comparisons of utility. Although theorists may be correct that valid interpersonal comparisons of utility are impossible (however, see also e.g. Friedman, 1986; Luce and Raiffa, 1957; Shubik, 1982, p. 112), it is an empirical fact that participants in experiments do attempt to make these comparisons. In bilateral monopoly experiments where actors had full information about each other's possible payoffs, there is a strong tendency for agreements to be close to an equal division of payoffs (Felsenthal and Diskin, 1982; Nydegger and Owen, 1975; Roth and Malouf, 1979; Roth, 1983; Schellenberg, 1988; Siegel and Fouraker, 1960). Bargainers in these experiments made interpersonal comparisons of the payoffs, thereby contradicting the property of independence of equivalent utility representations (Heckathorn, 1978; Roth and Malouf, 1979): equal payoff does not imply equal utility, because the units of utility of different actors' payoffs cannot be compared to each other. In the present study there are three reasons to assume that actors cannot and do not make interpersonal comparisons of utility. Firstly, in situations of partial information about other actors’ payoffs, actors are not able to make interpersonal comparisons of payoffs and therefore have to agree upon an exchange rate by different means (Heckathorn, 1978; Roth and Malouf, 1979). Secondly, in Section 7.4 the kernel and RKS solutions are discussed, which assume that the solution is invariant with respect to linear transformations of utility. Finally, the discussion of the relations between the reduced and non-reduced exchange representation becomes more general when interpersonal comparisons of utility are considered to be invalid. The relations involved in the special case where interpersonal comparisons of payoffs are assumed to be possible are contained in this general discussion. Employing the assumptions of linear utility and the impossibility of interpersonal comparisons of utility, exchange possibilities in bilateral monopoly are derived in Section 7.3.2 and the inequivalence of the non-reduced and reduced representations is demonstrated. In Section 7.3.3 an additional assumption is required when the exchange possibilities of a larger group of actors in an exchange network are examined. An exchange network describes the exchange relations between the actors. A relation between two actors signifies that they are able to make a profitable exchange with each other. The additional assumption employed in 7.3.3 is the so-called 1-exchange rule. This rule implies that an actor in the network can exchange only once, irrespective of the number of relations he has in the network. 7.3.2 Exchange possibilities in bilateral monopoly The actors' utilities and endowments must be specified in order to determine the exchange possibilities in a bilateral monopoly. Firstly, take a particular amount of commodity y and define this amount as the unit of commodity y. Repeat the procedure for commodity x. Then determine the actors' relative utilities of a unit of the two goods, i.e., uA(x)/uA(y) and uB(x)/uB(y). Since, by the assumption of invalid interpersonal comparison of utilities, the unit

154

Chapter 7

of utility is not determined, without loss of generality the utility of a unit of commodity y is chosen to be equal to 1 for both actors. Let the flows of goods in the exchange be denoted by ∆x and ∆y, both being positive, and the exchange rate by z = ∆y/∆x. A first result is that actor A and B will not profit from an exchange if both have the same relative utilities of the two goods, i.e., when uA(x)= uB(x). The reason is that both inequalities (A gives ∆y and receives ∆x), uA = uA(x)∆x - ∆y > 0 and uB = ∆y - uB(x)∆x > 0, cannot be satisfied at the same time, and neither can the inequalities (A gives ∆x and receives ∆y) uB = uB(x)∆x - ∆y > 0 and uA = ∆y - uA(x)∆x > 0. Therefore, the actors' relative utilities must be different in order for the exchange to be profitable for both A and B. A profitable exchange can then occur under two circumstances: Firstly, when uB(x) > uA(x) and EA(x) > 0, EB(y) > 0, and secondly, when uA(x) > uB(x) and the endowments EB(x) > 0, EA(y) > 0. Without loss of generality it is assumed that A is relatively more interested in x than B, and gives ∆y to B while receiving ∆x in the exchange. It can be derived that, under the assumptions specified above, an exchange rate z between A and B is profitable, i.e., uA = uA(x)∆x - ∆y > 0 and uB = ∆y - uB(x)∆x ≥ 0, or uA ≥ 0 and uB > 0, if uB(x) ≤ z ≤ uA(x)

(1)

The actors' utilities resulting from the exchange and the relations between the reduced and non-reduced exchange representations vary across the possible values of the ratio EA(y)/EB(x). Three possible values of the ratio need to be distinguished in a bilateral monopoly: (I) uB(x) ≥ EA(y)/EB(x); (II) uA(x) > EA(y)/EB(x) > uB(x); (III) EA(y)/EB(x) ≥ uA(x). For each situation an example is constructed. In each situation the endowments are EA(x) = EB(x)= 4, and EA(y)= EB(y)= 10. The utility coefficients of the commodities for the actors in each situation are shown in Table 7.1. 7.3.2.1 Bilateral monopoly, (I) uB(x) ≥ EA(y)/EB(x) In situation (I) the exchange rate z cannot be smaller than EA(y)/EB(x), because z ≥ uB(x). For each profitable value of z both actors' utilities cannot be increased when A gives all of EA(y) to B and receives EA(y)/z. That is, all exchange rates uB(x) ≤ z ≤ uA(x) with ∆y = EA(y) are in the core. For z ≥ EA(y)/EB(x), the utility of A, uA, is equal to EA(y)[uA(x)/z - 1], while uB = EA(y)[1uB(x)/z]. These utilities are summarized in the first row of Table 7.2. The maximum utility of A, denoted by uAM, is obtained at z = uB(x) and is equal to EA(y)[uA(x)/uB(x)-1]. The maximum utility of B, uBM, is obtained at z = uA(x) and is equal to EA(y)[1-uB(x)/uA(x)]. Both maximum utilities are summarized in the second column of Table 7.2. The question is now asked whether the direct exchange situation (I), represented as a non-reduced exchange, can equivalently be represented by reduced exchange. If so, it should be possible to choose the units of the actors' utility such that uA + uB is a constant. At the most extreme exchange rates, uA + uB = uAM + 0 = 0 + uBM, hence the sum of utilities can only be a constant when uAM = uBM. The maximum utilities can be equalized by multiplying uB by uA(x)/uB(x). If it can then be demonstrated that the sum of uA + uB is constant for all exchange rate values z, then we have proven that direct exchange situation (I) can be represented by reduced exchange. Expanding uA + uB yields u A + u B = E A ( y )(

u A ( x) u ( x) u A ( x) u ( x) ) − 1) + E A ( y )(1 − B = E A ( y )( A − 1) z z uB ( x) uB ( x)

(2)

Two Representations of Negotiated Exchange: A Review and Comparison Table 7.2:

155

Utility as a function of exchange rate z and maximum utility for A and B in bilateral monopoly situations (I), (II), and (III).

uAM uBM UA if z > EA(y)/EB(x) UB if z > EA(y)/EB(x)

(I) u A ( x) − 1) E A ( y )( uB ( x) uB ( x) ) E A ( y )(1 − uA ( x) uA ( x) − 1) E A ( y )( z uB ( x) ) E A ( y )(1 − z

UA if z ≤ EA(y)/EB(x) UB if z ≤ EA(y)/EB(x)

(II) x ( )( u A ( x ) − u B ( x )) EB uB ( x) ) uA ( x) uA ( x) − 1) E A ( y )( z uB ( x) ) E A ( y )(1 − z E B ( x )(u A ( x ) − z ) E B ( x )( z − u B ( x )) E A ( y )(1 −

(III) E B ( x )(u A ( x ) − u B ( x )) E B ( x )(u A ( x ) − u B ( x ))

E B ( x )(u A ( x ) − z ) E B ( x )( z − u B ( x ))

which is indeed independent of the value of z. An example of situation (I) is shown in Table 7.1, with uA(x) = 4, uB(x) = 3, EA(y) = EB(y) = 10, and EA(x) = EB(x) = 4. Figure 7.1a depicts the Edgeworth box for this situation. The core or contract curve of the situation contains all points on the line from (6.5, 0) to (7.333, 0), where the first coordinate corresponds to z = 4 and the second to z = 3. If uB is multiplied by uA(x)/uB(x) = 4/3, then all points on the contract curve are characterized by uA + uB = 10(4/3 - 1) = 3.333. The utility gains of both actors for different values of uB(x) ≤ z ≤ uA(x) in exchange situation (I) can be depicted in a so-called utility space (Shubik, 1982: pp. 110-112). Figure 7.2a depicts the utility space of (I) after multiplying the utility of b by 4/3. All profitable exchanges are bounded above by the contract curve. If it is assumed that the outcome of the actors' exchange is on the line (in the core), then exchange situation (I) can be represented by a reduced exchange or a common resource pool split with the value of the pool equal to 3.333. 7.3.2.2 Bilateral monopoly, (II) uA(x) > EA(y)/EB(x) > uB(x) Suppose that uA > z ≥ EA(y)/EB(x), as in situation (I). Exchanges where A gives all EA(y) to B are then in the core, with uA = EA(y)[uA(x)/z-1] and uB = EA(y)[1-uB(x)/z]. The maximum utility of B is obtained at z = uA(x) and is equal to uBM = EA(y)[1-uB(x)/uA(x)], as in situation (I). However, situation (II) differs from situation (I) because in (II) the exchange rate z can also be smaller than EA(y)/EB(x). If z < EA(y)/EB(x) then the exchanges most profitable to A and B, that is, the exchanges in the core, are those where B gives all EB(x) to A. When this is the case, uA = EB(x)[uA(x)-z] and uB = EB(x)[z-uB(x)]. The maximum utility of A is obtained at z = uB(x) and is equal to uAM = EB(x)[uA(x)-uB(x)]. All utilities and maximum utilities are shown in the third column of Table 7.2. To prove that the direct exchange situation (II) cannot be represented as a reduced exchange, the unit of utility of B is firstly chosen such that uAM = uBM. This is accomplished if uB is multiplied by uA(x)EB(x)/EA(y). The second part of the proof consists of demonstrating that uA + uB is not constant but depends on the value of the exchange rate z. For example, if

156

Chapter 7

Figure 7.2a

Figure 7.2b

Figure 7.2c Figure 7.2:

Utility spaces of the examples in Table 7.1 for bilateral exchange situations (I) (Figure 7.2a), (II) (Figure 7.2b), and (III) (Figure 7.2c). The actors’ utilities in the RKS and equal gain solutions are obtained by crossing the “RKS & Equal Gain” line with the contract curve. The actors’ utilities in the Nash solution are obtained by determining the intersection of the contract curve and the equal utility product or “Nash” curve.

Two Representations of Negotiated Exchange: A Review and Comparison

157

z < EA(y)/EB(x) then20 u A + u B = E B ( x )(u A ( x ) − z ) + E B ( x )( z − u B ( x )) u A ( x )

E B ( x) = E A ( y) (3)

E B ( x )[u A ( x )(1 − u B ( x )

E B ( x) z E ( x) ) + (u A ( x ) B − 1)] E A ( y) E A ( y)

In situation (II) uA(x) > EA(y)/EB(x), hence uA + uB is increasing with z when uB(x) < z ≤ EA(y)/EB(x) and uAM = uBM. Similarly, it can be proven that uA + uB decreases with z if EA(y)/ EB(x) < z < uA(x) and uAM = uBM. An example of situation (II) can be constructed simply by changing the actors' utilities in the example of situation (I). For example, take uA(x) = 3 and uB(x) = 2. This situation has already been discussed in the previous subsection "An example of bilateral negotiated exchange", and is depicted in Figure 1b. It should be remembered that the core consists of two connected straight lines, one from (7.333, 0) to (8, 0), and the other from (8, 0) to (8, 2). The utility of A is increasing from left to right, and from bottom to top. The utility space of the example, after multiplying uB by uA(x)EB(x)/EA(y) is depicted in Figure 2b. Note that uA + uB > uAM = uBM = 4 for all uB(x) < z < uA(x). The two straight lines in both Figure 1b and 2b intersect where z = EA(y)/EB(x). 7.3.2.3 Bilateral monopoly, (III) EA(y)/EB(x) ≥ uA(x) In situation (III) the core only contains exchanges where B gives all EB(x) to A. The utility gains corresponding to these exchanges are uA = EB(x)(uA(x)-z) and uB = EB(x)(z-uB(x)). The maximum utilities are equal, that is, uAM = uBM = EB(x)(uA(x)-uB(x)). The proof that the sum of utilities is constant is straightforward: u A + u B = E B ( x )(u A ( x ) − z ) + E B ( x )( z − u B ( x )) = E B ( x )(u A ( x ) − u B ( x ))

(4)

Therefore, direct exchange situation (III) can be represented as a reduced exchange. An example of situation (III) (see Table 7.2) is created if the actors' utilities are changed to, for example, uA(x) = 4 and uB(x) = 3. The Edgeworth box and utility space of the example are depicted in Figure 7.1c and Figure 7.2c respectively. An actor's maximum utility gain in this example is equal to 4. 7.3.3 Exchange networks The previous sections demonstrate that direct and non-reduced exchange are equivalent to reduced exchange if the condition uA(x) > EA(y)/ EB(x) > uB(x) does not hold, that is, they are equivalent in situations (I) and (III), but not in (II). In the bilateral exchange situations (I) and (III), the units of utility can always be chosen such that the sum of utilities is equal to a particular constant, for example 24. However, the matter can become more complicated if more exchange relations are considered simultaneously. If the exchange relations involve the same actor, then the size of the common resource pool can no longer be varied independently across these relations. The interdependence of the sizes of different resource pools shared by an actor with other actors is caused by the fact that the actor's relative interests in the goods are fixed. An implication of the resource pool size dependence is that, although resource 20 The proof is similar in the case of z ≥ E (y)/E (x), and is left to the reader to work through. A B

158

Figure 7.3:

Chapter 7

Exchange networks used to illustrate predictions of network exchange theories in reduced and non-reduced exchange. The reduced exchange representation that corresponds best to the examples of the non-reduced representations in Table 7.3 has relations with the values c > d > a > b.

pools are constant within one relation, it is very likely that they have different values across relations, as is illustrated by the following example. Consider an exchange 'triangle' ABC, depicted in Figure 7.3, where all three actors are endowed with goods x and y and have different relative interests in these goods. Moreover, assume that each actor can only exchange once (the 1-exchange rule), and that A’s and B's endowments and utilities are represented by the example of situation (III). Finally, assume that C is 1.5 times more interested in x than in y. The question is whether all exchanges can be represented by reduced exchange with common resource pools equal to 24. The exchange relations between A and C, and A and B, can both be represented by reduced exchange of size 24 when either EA(y)/EC(x) ≥ uA(x) = 2, or EA(y)/EC(x) ≤ uB(x) = 1. However, it is impossible to choose the actors' unit of utility such that both exchange relations can be simultaneously

Two Representations of Negotiated Exchange: A Review and Comparison

159

transformed into reduced exchange of size 24.21 The sizes of reduced exchange relations can be equalized by introducing other goods in the exchange relations. For example, B and C exchange goods v and w, and A and C exchange goods t and s. Introducing other goods makes the model less parsimonious. Moreover, if the actors exchange the same goods x and y, then the restriction that the actors can make only one exchange is a natural one. However, if the exchanges involve different goods and are hence logically independent, in principle all three exchanges can be carried out simultaneously. With a few exceptions (Bonacich and Friedkin, 1998; Cook and Emerson, 1978; Stolte and Emerson, 1977), almost all network exchange research has dealt with sets of reduced exchange relations of equal size, mostly equal to 24. One reason why researchers have chosen equally valued relations is to disentangle the effects of the exchange structure on outcome distributions and the effects of the values of exchange relations (Bonacich and Friedkin, 1998). The analyses above demonstrate that sets of reduced exchange relations of equal size are very unlikely or can even be impossible. Therefore, I agree with Bonacich and Friedkin (1998, p. 170): "that it is time to break free of the research paradigm ... and to examine a wider spectrum of exchange structures, including those in which the values of relations are unequal.". Unequally valued exchange relations are a natural consequence of direct exchange, and because exchange relations of the kind in situation (II) cannot be represented as reduced exchange, direct non-reduced exchange is best suited to the examination of the effect of differently valued relations on actor behavior and outcomes in exchange. 7.4

Consequences of representation inequivalence

7.4.1 Introduction As shown in Section 7.3, direct exchange cannot always be represented by reduced exchange. Even if direct exchange can be represented by reduced exchange, then in general the values of exchange relations in the network are unequal. However, most small group exchange research in sociology and bilateral monopoly studies in economics use the reduced exchange representation. The question that follows is, to what extent can the results of both theoretical and empirical studies on exchange be generalized or extended to direct exchange? The third objective of the present study is to answer this question by examining the consequences of 21 The maximum utilities of A and B in the example of situation (III) were equal to 4. Therefore, the utilities of

both actors must be multiplied by 6 to transform the exchange relation into a reduced exchange of size 24. Consider now the exchange relation between A and C. One case in which the sum of utilities is constant is when EA(y)/EC(x) ≥ 2. The maximum utility gain of actor A is then 6EC(x)(uA(x)-uC(x)) = 3EC(x). However, because EA(y) = 10, EC(x) can never be larger than 5 under the constraint EA(y)/EC(x) ≥ 2. Therefore, uAM and the sum of utilities can never be larger then 15. Consider now the other case, EA(y)/EC(x) ≤ 1. The maximum utility of A is then equal to 6EA(y)[uA(x)/uC(x)-1] = 20. Therefore, the exchange relation between A and C cannot be represented as a reduced exchange with the same size as the relation between A and B. Similarly, it can be shown that the same condition holds for the relations that B has with A and C, and the relations that C has with A and B. However, introducing other goods in the exchange relations can equalize the sizes of reduced exchange relations. For example, B and C exchange goods v and w, and A and C exchange goods t and s. Introducing other goods makes the model less parsimonious. Moreover, if the actors exchange the same goods x and y, then the restriction that the actors can make only one exchange is a natural one. However, if the exchanges involve different goods and are hence independent, in principle all three exchanges can be carried out simultaneously.

160

Chapter 7

this inequivalence by comparing predictions of exchange rates in reduced exchange networks with predictions in non-reduced exchange networks. The predictions using three well-known theories of network exchange, core theory (7.4.2), power-dependence theory (7.4.3), and exchange-resistance theory (7.4.4), are calculated. The conclusions with respect to the comparisons of the predictions for both representations across the three theories are provided in Section 7.4.5. In Section 7.4.6 the implications of previous experimental research using the reduced representation for research on non-reduced exchange are discussed. The exchange networks that are analyzed using the three theories are described below in Section 7.4.1.1. In order to derive predictions for these networks some additional assumptions, described in Section 7.4.1.2, must be made. Finally, in Section 7.4.1.3 some general remarks on the selection of the three theories and on their derivations are given. 7.4.1.1 The networks In order to identify the possible consequences of another, non-reduced, exchange representation for theories of network exchange, predictions resulting from three theories are calculated for examples of some simple and well-known network structures. The network structures investigated are shown in Figure 7.3. They include the simple bilateral exchange or Line2 structure (Figure 3a), the Triangle (Figure 3b), the Line3 structure (Figure 3c), and the Line4 structure (Figure 3d). All structures, except for the Triangle, have been investigated intensively using the reduced exchange representation, using equally valued relations, and under the so-called 1-exchange rule. The main goal in network exchange research is to identify how an actor’s exchange outcomes and his power depend on his position in the exchange network. Exchange networks can be classified into three distinct types: equal power networks, weak power networks, and strong power networks. The network structures in Figure 3 represent, given the 1-exchange rule and equally valued relations, the simplest examples of each of the three types of networks. In equal power networks (Line2 and Triangle) all actors obtain equal outcomes on average, for example, all actors obtain 12 from the exchange. In weak power networks, some actors (B and C in the Line4) obtain on average more than other actors (A and D in the Line4), for example. B obtains 14 and A obtains 10. In strong power networks on average some actors gain almost everything (B in the Line3), while others gain almost nothing (A and C in the Line3), for example, B obtains 23 and A obtains only 1. Theories of network exchange are applied to these structures below, but using the non-reduced exchange representation with unequally valued relations. The structures are defined by the actors' initial endowments and their utilities, as presented in Table 7.1 (the Line2, or bilateral exchange) and Table 7.3 (other structures). For example, A in the Line4 cannot exchange profitably with C (C does not have commodity x, that in which A is relatively more interested than C) or D (D and A have exactly the same relative interests in each other's goods). Checking all pairs of actors and their exchange possibilities in this way leads to the structures in Figure 3. Note also that each individual actor has the same characteristics in all structures. The only exception is that A has 4 units of commodity x in the Triangle, and 0 units in the other structures. The exchange relations AB, AC, BC, and CD in all structures are cases of exchange situations (I), (III), (II), and (I) respectively.

Two Representations of Negotiated Exchange: A Review and Comparison Table 7.3:

Triangle

Example of a non-reduced exchange representation of the Triangle, Line3, and the Line4. Initial endowments and utilities are denoted by E and U respectively.

E U

Line3

E U

Line4

161

E U

X Y X Y X Y X Y X Y X Y

A 4 10 3 1 0 10 3 1 0 10 3 1

B 8 0 2 1 8 0 2 1 8 0 2 1

C 0 20 4 1 0 20 4 1 0 20 4 1

D

12 0 3 1

The network structures cannot be represented by reduced exchange because in each structure there is one relation (BC) of situation (II). However, in order to compare the predictions made by the theories of network exchange for both representations, a reduced representation must be constructed that corresponds as closely as possible to the examples in Table 7.3. It is assumed in all structures that A and B (AB), A and C (AC), B and C (BC), and C and D (CD), divide a, b, c, and d units respectively. Analyzing each relation separately and calculating the maximum gains of the actors in the relations AB, AC, BC, and CD, yields (uAM = 5, uBM = 10/3), (uAM = 4, uCM = 4), (uBM = 10, uCM = 16), and (uCM = 20/3, uDM = 5) respectively. If the exchange relations for each actor are ordered with respect to the maximum gain he can obtain from these relations, and if it is assumed that the actors prefer the relation with the higher gain, then A prefers AB over AC, B prefers BC over AB, C prefers BC over CD, while D has only one option. The reduced exchange representation corresponding as closely as possible to this non-reduced representation provides the values c > d, and c > a > b for the exchange relations. To provide ordered values for AB and CD in the Line4, it should be noted that CD is a better alternative to C than AB is to B, both in absolute values (10/3 for B and 20/3 for C) and in relative values (the maximum of BC is three times that of AB for B, and the maximum of BC is 2.4 times that of CD). Therefore, d > a is assumed in the reduced representation of the Line4. 7.4.1.2 Additional assumptions The analyses of the exchange relations in Section 7.3 were based on three assumptions: linear utility in endowments, impossibility of interpersonal comparison of utilities, and the 1exchange rule. One additional assumption is required to derive predictions using exchangeresistance and power-dependence theory. This is the assumption that all actors in the exchange network have complete information about all actors’ endowments and relative utilities, and about the network structure, that is, who can exchange with whom. Empirical research suggests that the amount of information is relevant to the speed of power

162

Chapter 7

development in repeated exchange but not to the allocation of outcomes in the long run term (Willer, 1992; Skvoretz and Burkett, 1994). This empirical finding is also a demonstration of a well-established fact that solutions of a repeated game can be completely different to solutions of so-called one-shot games (e.g., Kreps, 1990; Friedman, 1986; Taylor, 1987). Indeed, the number of exchange transactions in a relation might affect the predictions (and, naturally, actor behavior) in exchange situations. Most studies on network exchange do not distinguish between one-shot and repeated exchange. However, to simplify the analyses, it is assumed in the present study that actors in an exchange relation exchange only once. The predictions made by the theories of network exchange have been tested in experiments using a common setting and bargaining protocol of how and when actors in a network can make offers and exchanges with other actors. This common setting and bargaining protocol is described in Section 6.2.1. Although the theories of network exchange used do not precisely demand this setting and protocol, they present the context within which the theories are tested. Therefore, it is recommended that Section 6.2.1 be read before continuing to read the remainder of the text. 7.4.1.3 General remarks on the theories of network exchange and their derivations Predictions of exchange rates are calculated using adaptations of three well-known theories of network exchange: core theory, power-dependence, and exchange-resistance. The fact that these theories are better known than other theories of network exchange is not the only reason why these theories have been selected for our analyses. A more important reason is that there is one-to-one correspondence between predictions of these theories with solution concepts of cooperative game theory, the core, the kernel, and the RKS solution. Because the RKS solution is often related to the better known Nash solution, the Nash solution for bilateral exchange is also derived. The predictions made by the theories for the reduced and the non-reduced representations of the structures in Figure 3 are depicted in Table 7.4 and Table 7.5. It is relatively easy to derive general predictions for the reduced representations of the examples in Table 7.3, that is, for unspecified values of c > d > a > b, because the reduced representation implies that an exchange relation has a constant value. Therefore the general predictions for the reduced representation are presented in Table 7.4 and Table 7.5. It is much more laborious to derive and present general predictions in non-reduced exchange networks for all possible different endowments and utilities because a relation does not necessarily have a single constant value. Therefore only the predictions for the non-reduced representation of the examples in Table 7.3 with these particular values of utility and endowment are presented in Table 7.5. However, since it was not too laborious to derive the general predictions for the non-reduced representation of bilateral exchange, these predictions are presented in Table 7.4. It should be emphasized that the objective of the present study is not to derive general predictions for a large number of different networks, simple and complicated, based on different possible endowments and utilities. Predictions are only derived in order to identify the consequences of the representation of exchange on the predictions of the theories, by comparing them to predictions of the same theories applied to reduced exchange. However, predictions using power-dependence and exchange-resistance theory for the non-reduced exchange networks were calculated using algorithms based on that of Cook and Yamagishi

Two Representations of Negotiated Exchange: A Review and Comparison Table 7.4:

163

General predictions of profit points for A (reduced exchange representation) and the exchange rate (non-reduced exchange representation) in the Line2. Line2 (I): uB(x) ≥ EA(y)/EB(x)

Core: Reduced

[0, a]

Line2 (II): uA(x)> EA(y)/EB(x) > uB(x) Impossible

Core: Non-reduced

[uB(x), uA(x)]

[uB(x), uA(x)]

[uB(x), uA(x)]

Power-dependence (kernel): Reduced

a/2

Impossible

a/2

Power-dependence (kernel): Non-reduced

u A ( x) + uB ( x ) 2

u A ( x) + uB ( x ) 2

u A ( x) + uB ( x ) 2

a/2

Impossible

a/2

Exchangeresistance (RKS/Nash): Reduced Exchangeresistance (RKS): Non-reduced

2 u A ( x) uB ( x) u A ( x) + uB ( x )

Nash: Non-reduced

2 u A ( x) uB ( x) u A ( x) + uB ( x )

E A ( y) ] E B ( x) E A ( y) u A ( x) + E B ( x)

u A ( x )[u B ( x ) +

See Appendix 7.2

Line2 (III): EA(y)/EB(x) ≥ uA(x) [0, a]

u A ( x) + uB ( x ) 2 u A ( x) + uB ( x ) 2

(1992), which in principle can be applied to any exchange network of any complexity. These algorithms are discussed only briefly because they are not within the scope of the present study. 7.4.2 Core theory 7.4.2.1 Background and principles of the core From a microeconomic and game theoretical point of view, the core is the first candidate when looking for solutions of exchange rates in exchange markets in general and network exchange in particular. The core, previously defined as that set of possible final endowments that cannot be improved upon by any coalition of actors, is crucial to the analysis of cooperative game theory and is intrinsic to the Edgeworth solution of the contract curve

164

Chapter 7

Table 7.5:

Predictions of the exchanges and exchange rates (z) in the Triangle, Line3, and Line4, derived for the reduced representation (general predictions) and non-reduced representation (specific predictions) of the examples in Table 7.3. Triangle c < a+b: empty c ≥ a+b: C gets b, B gets a, and B and C divide c-a-b

Line3 B gets a, B and C divide c-a

Core: Nonreduced

zBC in [29/12, 10/3] ~ [2.417, 3.333]

zBC in [29/12, 4] ~ [2.417, 4]

Powerdependence: Reduced

AB, AC, BC resp. [a+b-c]/2, [a-b+c]/2; [b+a-c]/2, [b-a+c]/2; [c+a-b]/2, [c-a+b]/2;

B gets a + 2(c-a)/3, C gets (c-a)/3

AB, BC, CD resp. [a-c+d]/4, [3a+c-d]/4; [c+a-d]/2, [c-a+d]/2; [3d-a+c]/4, [d+a-c]/4;

zBC = 20/7 ~ 2.857

zBC = 52/15 ~ 3.467

zBC = 2.871

c ≤ a+b: solution equal to the kernel c > a+b: B gets c(c-b)/(2c-a-b), C gets c(c-a)/(2c-a-b)

B gets c2/(2c-a), C gets (c2-ca)/(2c-a)

c ≤ a+d: solution can be found iteratively c > a+d: B gets c(c-d)/(2c-a-d), C gets c(c-a)/(2c-a-d)

zBC = 2.815

zBC = 2.933

zBC = 2.781

Core: Reduced

Powerdependence: Nonreduced Exchangeresistance: Reduced

Exchangeresistance: Nonreduced

Line4 B and C together get at least c; if c ≥ a+d, B gets a, C gets d, and B and C divide surplus c-d-a; else exchanges AB and CD are also in the core zBC in [29/12, 3] ~ [2.417, 3]

problem. It is therefore no surprise that Bonacich and Bienenstock in their work first turned to the core concept for the solution of the exchange rates in exchange networks. In order to find the core solution, the characteristic value function is constructed. This function is the cornerstone of cooperative game theory. It is a set function that provides each coalition with what it can guarantee for itself irrespective of the behavior of actors outside the coalition. If there is transferable utility, i.e., when utility can be transferred from one actor to other actors, then the function assigns one value to each coalition. When utilities are not transferable the function is a collection of vectors in the utility space of actors (Shubik, 1982, pp. 134-135). Transferable utility does not imply interpersonal comparison of utilities but

Two Representations of Negotiated Exchange: A Review and Comparison

165

constant-sum games. Transferable utility means that a set of utility scales for the actors exists and that there are infinitely divisible goods such that the changes in individual utilities which result when these goods are transferred conserve the total utility sum (Luce and Raiffa, 1957, p. 181). Therefore, reduced exchange can be characterized by a characteristic value function with transferable utility. However, the previous section shows that direct or non-reduced exchange cannot be characterized as such, because in general it is not a constant-sum game. Bonacich and Bienenstock presuppose the reduced exchange representation and hence that exchanges are constant-sum games. Therefore, their derivations of the outcomes in exchange networks are based on the characteristic value function with transferable utility. In order to derive the core solution of direct exchange the characteristic value function without transferable utility must be employed (see for an overview and references, Shubik, 1982, pp. 134-135, 145-157). Hildenbrand and Kirman (1988) analyze non-reduced direct exchange using the core concept. They show that if actors’ utility functions are concave, then the exchange economy is convex and the core of the economy is always non-empty (Hildenbrand and Kirman, 1988, Chapter 4). That is, there is always a (set of) distribution(s) of endowments that cannot be improved upon by any coalition of actors. However, Hildenbrand and Kirman’s result does not take into account exchange restrictions, such as the 1-exchange rule and the network structure, as in the exchange situations considered here and in network exchange research. 7.4.1.2 Core theory applied In the Line2 structure the core simply includes all exchanges that yield utilities to the two actors that cannot be improved upon by either of them in other exchanges. Therefore, in the case of reduced exchange, all divisions of a units are in the core, and in the case of nonreduced exchange all exchange rates in the interval [uB(x), uA(x)], including the borders, are in the core. However, there is one important difference between the two representations that does not only hold for core theory but also for the other theories. The reduced exchange representation implies that the exchange is efficient, that is, all 24 units are divided between the two actors. In non-reduced exchange, however, the actors might agree upon an exchange with a rate in the core (e.g., z = 2.5 in situation (II)), but which does not transfer the maximum number of units to the other actor (e.g., A gives y = 5 to B, and B gives x = 2 to A, instead of the efficient exchange where A gives y = 10 to B, and B gives x = 4 to A). The structure of the core in the other networks does not differ between the two exchange representations. In the Triangle the core is empty, except for when one exchange relation is more profitable to the two actors than the other two exchange relations. The excluded actor cannot then offer an exchange to one of the other actors which is more profitable to them than the exchange already made between these two actors. This condition is stated in the first row of Table 7.5. If one of the (reduced exchange) relations, for example BC, yields c ≥ b+a, then the core contains exchange BC where B receives at least b, C receives at least c and all possible divisions of the additional c-b-a units. In the case of the non-reduced exchange representation, the core can also contain exchange BC with all exchange rates for which both B receives more than the maximum he can get in AB, and C receives more than the maximum he can get in AC. An illustration of this case is provided by the example in Table 7.3. If B and C exchange at a rate zBC in the closed interval [2.417, 3.333] both B and C obtain more than the maximum of what one of them can get by

166

Chapter 7

exchanging with A.22 In the Line3, a strong power network, B has an advantage over the other actors. B obtains a utility gain which is at least the maximum he can obtain from his least profitable exchange relation. That is, in the case of the reduced representation, if a < c, then core theory predicts an exchange BC where B receives at least a, and B and C divide the remaining c - a units. The structure of the core is similar in the non-reduced exchange representation. In the example of Table 7.3, C offers an exchange to B which is more profitable to B than all exchanges B can make with A. Therefore, all exchanges BC with 2.417 ≤ zBC ≤ 4 are in the core. The difference to the core in the Triangle is that in the Line3 3.333 < zBC ≤ 4 is also included in the core because B has no alternative exchange partner as in the Triangle. The structure of the core is also similar for both representations of the Line4. In both cases the solution is described by those exchanges for which B and C both obtain at least as much as they can obtain by exchanging with each other. In some possible situations the exchange BC yields more to B and C than their exchanges with A and D respectively. All exchanges BC are then in the core for which B obtains at least a (in the case of non-reduced exchange, uBM in AB) and C obtains at least d (uCM in CD). In the example of Table 7.3, the core consists of exchanges between B and C with zBC in the closed interval [2.417, 3]. Note that the core in this example is smaller than that in the example of the Triangle. The core excludes the exchange rates that provide B with the largest gain. Because C’s exchange alternative in the Line4 (D) is better than his exchange alternative in the Triangle (A), B must offer more to C in the Line4 than in the Triangle to outbid C’s alternative exchange partner. 7.4.3 Power-dependence theory or the kernel 7.4.3.1 Background and principles of power-dependence theory or kernel Emerson (1962, 1972a, 1972b, 1976) developed the first theory of network exchange, known as power-dependence theory. This theory has been extended and tested in a number of studies (e.g., Bonacich and Friedkin, 1998; Cook and Emerson, 1978; Cook and Gillmore, 1984; Cook and Yamagishi, 1992; Emerson et al., 1983; Skvoretz and Willer, 1993; Stolte and Emerson, 1977). Bonacich and Friedkin (1998) proved that power-dependence predictions are identical to the kernel of an exchange situation. In contrast to the core, power-dependence theory provides point predictions of who exchanges with whom and at what exchange rate.23 Another difference is that this theory can predict exchange outcomes outside the core, or when there is no core. Another very important difference to the core and exchange-resistance theory is that it assumes interpersonal comparisons of utility. That is, if utilities of actors are multiplied by a positive constant, then the power-dependence or kernel predictions change whereas the predictions of the two other theories do not change. The idea of the kernel is that in each exchange relation the surplus of one actor is equal to the surplus of the other actor. The surplus of an actor is defined as the gain of that actor relative to what he would obtain from his best alternative exchange relation. This balance of equal surplus in a relation, applied to all exchange relations, characterizes the 22 At z = 2.417 B receives 3.333, which is equal to the maximum he can gain in his relation with A. At higher BC

exchange rates B receives more than 2.417. At zBC = 3.333 C receives 4, the maximum he can obtain by exchanging with A, and at lower exchange rates he gains more. 23 In principle the kernel can contain more than one prediction (Shubik, 1982, pp. 342-347). I suspect that in an exchange economy with linear utilities the kernel contains only one point, but I did not prove this proposition.

Two Representations of Negotiated Exchange: A Review and Comparison

167

kernel solution. For example, consider the Line3 with unequally valued relations c > a. Applying the kernel to this structure yields two equations in two unknows. If uAB and uBC denote A's utility in his exchange with B, and B's utility in his exchange with C respectively, then the two equations are: uAB

=

uBC - (a - uAB) =

(a - uAB) - uBC

(5a)

c - uBC

(5b)

In (5a), A's surplus is equal to that which he obtains from AB (uAB), because he has no alternative. B's surplus in (5a) is (a - uAB) minus that which he obtains from his best alternative (uBC). Equation (5b) is formed similarly. Solving these two simultaneous equations yields uAB = (a-c)/3 and uBC = a + 2(c-a)/3, hence if c > a then uAB < 0 and exchange BC occurs with B obtaining a + 2(c-a)/3 and C obtaining (c-a)/3. 7.4.3.2 Power-dependence theory or kernel applied In general, the kernel solution is obtained by solving a set of simultaneous equations such as (5) in the case of the reduced exchange representation of the Line3. Cook and Yamagishi (1992) constructed an iterative algorithm to solve the equations in the case of equally valued reduced exchange relations. However, after some small changes the algorithm can also be applied to unequally valued exchange relations and to non-reduced exchange. To explain the algorithm briefly, (see Cook and Yamagishi, 1992, for a detailed explanation), in the first iteration for each exchange relation the kernel solution (equal additional gain on top of the best alternative gains) is calculated assuming that the best alternative for both actors provides no gain. Therefore, the first iteration yields the kernel solution for each relation as if it is not embedded in a network structure. In the second iteration, the calculation of the kernel solution for each relation is then based upon the gain of the best alternative of both actors in the previous iteration. The possible third and subsequent iterations are analogous to the second iteration. The process stops when it has converged, i.e., when the output of the iteration is identical to its input.24 The only difference in applying the algorithm to both representations is that in the case of reduced exchange for each relation the kernel solution is a split of the resource pool in that relation, while in the case of non-reduced exchange the kernel solution refers to an exchange rate of two goods. The algorithm is applied to the four structures of Figure 3, assuming both the reduced and non-reduced representations. In simple structures, such as the ones considered here, the general algebraic solution can be derived analytically without many problems when the structures are represented by reduced exchange. The general solution is presented in Table 7.4 for bilateral exchange and in Table 7.5 for the simple exchange networks. It is assumed here that all relations in the network can be represented by a constant-sum game, or reduced exchange, which is not true for the examples in Table 7.3. The kernel solutions of the examples in Table 7.3 are presented in Table 7.5. Power-dependence theory or the kernel solution, predicts that both actors in the Line2 obtain equal utility gains. In the case of reduced exchange both actors gain a/2 units, and in the case of non-reduced exchange both obtain equal gains if the exchange rate is equal to the 24 The algorithm converged in all its applications to find either the kernel or the equiresistance solution

(discussed below) of a particular network structure given the 1-exchange rule. Although my intuition tells me that the algorithm always converges, I do not yet have a proof of its convergence.

168

Chapter 7

average of uA(x) and uB(x) for each of the three situations (I), (II), and (III) (see Table 7.4).25 The proof of the latter statement is directly obtained by equating the utilities of uA and uB as stated in Table 7.3. The kernel solution of the examples of bilateral exchange in Table 7.1 are z = 3.5 (situation (I)), z = 2.5 (situation (II)), and z = 3.5 (situation (III)). In Figure 2 the utility spaces of the examples in Table 7.1 are shown after multiplying the utility of B by a factor such that uAM = uBM. Substituting the modified utilities of B in the formula of footnote 25 yields kernel solutions equal to 24/7 ~ 3.429, 27/11 ~ 2.455, and 3.5 for situations (I). (II), and (III) respectively. These solutions are also shown in Figure 7.2. The kernel solution of the Triangle in Table 7.3 is (zAB = 5, zAC = 2, zBC = 2.857). Both exchanges AB and AC are not executed because they yield negative utilities for A. Both B and C have more to offer to each other than they can obtain from A, as is explained above in the section on the core solution. The exchange between B and C yields 6 units of utility to B, which corresponds to 60% of his maximum gain, and 8 units of utility to C, which is 50 % of his maximum gain. The slight advantage of B over C in the exchange is the result of the fact that A prefers AB to AC. The kernel solution of reduced exchange also displays this effect. According to Table 7.5, B gains [c+a-b]/2, which is more than c/2 when a > b. Therefore, the example suggests that the structure of the kernel solution is identical for both representations. The only difference between the two representations is that the proportional utility gains in the non-reduced representation can add up to more than 1 (100%). The kernel solution for the Line3 yields an extra advantage for B over C because C looses his alternative exchange partner A. At the exchange rate zBC = 52/15 ~ 3.467, B obtains 8.462 utility units (85%) and C obtains 3.077 (19%). In the Line4, C again has an extra exchange alternative, D, which improves C’s position in comparison to the Line3. The kernel solution of the Line4 is (zAB = 3.588, zBC = 2.871, zBC = 4.357). Only the exchange between B and C provides positive utility for both actors, with 6.066 to B (61%) and 7.869 (49%) to C. Note that C obtains less in the Line4 than in the Triangle, although he has a higher maximum utility gain from an exchange with D in the Line4 than with A in the Triangle. Assuming that d > a, the kernel solution in the reduced representation of the Line4 would predict that C obtains more than 50% of the value of the relation c ([c-a+d]/2 > c/2). Therefore, the kernel solutions in the two representations of the Line4 differ structurally with respect to who is more powerful in the relation BC, although the difference in predictions does not appear to be large. 7.4.4 Exchange-resistance theory and the RKS and Nash solution 7.4.4.1 Background and principles of exchange-resistance theory and the RKS and Nash solution Exchange-resistance theory was developed in the early eighties by Willer and Anderson (1981), which was followed by a large number of extensions and tests of the theory (see for references footnote 14). Willer and Anderson (1981, Chapter 1, footnote 3) acknowledged that the theory makes use of the resistance equation as developed by Heckathorn (1980). However, Willer and Anderson (1981, Chapter 1, footnote 4), together with other network 25 The general formula for the exchange rate z in isolated bilateral exchange, when u (y) and u (y) are not A B

necessarily equal to 1, is

( x) + uB ( x) z = uA . u A ( y ) + uB ( y )

Two Representations of Negotiated Exchange: A Review and Comparison

169

exchange theorists (see footnote 13), did not realize that Heckathorn's equal-resistance bargaining solution is identical to the well-known RKS solution in fixed-threat two-person cooperative game theory. In this subsection the resistance concept and its relations to both the RKS and Nash solution are explained and illustrated by applying the solutions to the examples of bilateral exchange in Table 7.3. In the next subsection, ways in which exchangeresistance theory could be applied to unequally valued exchange relations in both the reduced and non-reduced exchange representations are suggested. As explained in subsection 7.2.3, Exchange research in economics (2), several approaches evolved in order to identify a solution from the indeterminate range of possible exchange rates. A first approach, cooperative game theory, specifies a set of axioms which serves to identify one outcome of a game as the solution. The two best known solutions are the Nash solution and the RKS solution in two-person fixed-threat cooperative game theory. The sets of axioms that characterize each of the two solutions consist of four axioms, one of them being different in each set.26 Some theorists have criticized the axiom of independence of irrelevant alternatives on which the Nash solution is based (Heckathorn, 1978, 1980; Kalai and Smorodinsky, 1975; Luce and Raiffa, 1957), while others have criticized the monotonicity axiom underlying the RKS solution (Felsenthal and Diskin, 1982). Experiments have been conducted to test both solutions (Felsenthal and Diskin, 1982; Heckathorn, 1978, 1980; Nydegger and Owen, 1975; Roth and Malouf, 1979; Schellenberg, 1988), and modifications of the two solutions have been proposed on the basis of the experimental results (Felsenthal and Diskin, 1982; Schellenberg, 1988). Summarizing the experimental findings, the results are mixed with respect to which solution is superior to the other, and both solutions do not seem to do very well in predicting the final bargaining outcome. However, because both cooperative solutions are supported by concession models from the second approach, non-cooperative game theory, both solutions are in general considered to be credible and plausible predictions. The Nash solution from two-person fixed-threat cooperative game theory is that outcome which maximizes the product of utility gains of both actors. Denoting the status quo or conflict payoff by c, then the exchange rate z that is identified by Nash as the solution in a bilateral monopoly is that for which the product (uA(z)-uAC)(uB(z)-uBC) is maximized. Harsanyi (1956; also see, for example, Cross, 1969; Luce and Raiffa, 1957) showed that the concession model of Zeuthen (1930) results in the same prediction as the Nash solution. Similarly, Heckathorn (1980) showed that a concession model slightly different to that of Zeuthen yields the RKS solution from two-person fixed threat cooperative game theory. An exchange rate z is an RKS solution of a bilateral monopoly when the proportional utility gain is equal 26 Both sets include three assumptions: (1) Pareto optimality, which means that there are no outcomes that

provide higher utility to both players than the solution. (2) Symmetry, meaning that the solution remains unchanged if players' labels are changed. (3) Invariance under linear transformations of utility, which means that the solution does not require interpersonal comparison of utilities. The fourth axiom that characterizes the Nash solution is (4') Independence of irrelevant alternatives. This axiom states that if (a) two different bargaining games have the same status quo points and the trading possibilities of one game contain all possibilities of the other, and (b) the game with the smaller set of possibilities contains the solution of the other game, then the solutions of the two games must be identical. The fourth axiom of the RKS solution is the axiom of (4') Monotonicity. This axiom states that if, for every utility that player 1 may demand, the feasible utility level that player 2 can simultaneously reach is increased, then the utility level assigned to player 2 according to the solution should also be increased (Kalai and Smorodinsky, 1975). See for example Felsenthal and Diskin (1982), Heckathorn (1978, 1980), and Luce and Raiffa (1957) for a more elaborate discussion of the axioms.

170

Chapter 7

for both actors. More formally, that value of z for which both 1 − RA =

u A ( z ) − u AC u B ( z ) − u BC = = 1− R B u AM − u AC u BM − u BC

(6)

and for which 1-RA is a maximum, where uiM and uiC denote the maximum and conflict utility of actor i respectively. Heckathorn (1980) refers to RA and RB as A's and B's resistances to an exchange with exchange rate z. In his concession model, the actor who is least resistant to the trade with exchange rate z makes concessions until the other actor becomes the least resistant. In this model the process continues until both actors' resistances are equal and minimal. Heckathorn shows that the RKS solution is identical to his equiresistance solution. The study of Willer and Anderson (1981), and a large number of other studies, were based on the equiresistance solution; however, most of them formulated resistance, and hence equiresistance, differently. In Appendix 7.1 the equivalence of the three formulations is proved. The original RKS formulation of equal proportional utility gains is used below in applying exchange-resistance theory to the network structures. 7.4.4.2 Exchange-resistance theory and the RKS and Nash solution applied to bilateral exchange relations Both the Nash and RKS solution can easily be calculated in the case of bilateral monopoly situations (I) and (III). In both cases the utility space (see Figures 2a and 2c) is a straight line and exchange can be represented as reduced exchange, that is, a division of a constant sum of utility. In these cases both the Nash and RKS solutions are equal to that exchange rate z where both A and B obtain half of the maximum of their utilities.27 The problem is to find the exchange rates that correspond to these utility gains. In the case of reduced exchange, where there is no exchange rate, the solution is simply equal to a/2. In the case of non-reduced exchange, the exchange rates can be found by equating uA/uAM to uB/uBM, where the functions uA, uB, uAM, uBM are presented in Table 7.2. This results in solutions (see also Table 7.4) equal to the arithmetic mean of uA(x) and uB(x) in situation (III), which is equal to 3.5, and solutions equal to the harmonic mean of uA(x) and uB(x) in situation (I), which is equal to 24/7. Note that in the reduced exchange representation the Nash, RKS, and equidependence solutions are identical, but in general are not identical in the reduced exchange representations. The Nash and RKS solutions are equal in situations (I) and (III), and also equal to the equidependence or kernel solution in situation (III) when uA(y) = uB(y) = 1. The equidependence solution is equal to the RKS and Nash solution in situation (I) when uAM = uBM, as in the examples depicted in Figure 2. The differences between the three solutions are greater in situation (II). The equivalence of the Nash and RKS solution no longer holds in situation (II). The RKS solution, which again can be found by equating uA/uAM to uB/uBM, is presented in the middle column of Table 7.2.28 The Nash solution is (together with its proof) presented in Appendix 27 The proof is as follows. Because both solutions are invariant under linear transformations of utility, the utility coefficients can be chosen such that the sum of utility gains equals 1 for all exchanges on the contract curve. The Nash solution maximizes the product of uB(1-uB), with uB ≤ 1, which occurs at uB = 0.5, or half of the maximum utility gain. The RKS solution is found by intersecting the contract curve uB = 1-uA with the equiresistance line (uB - uBC)/(uBM - uBC) = uB = uA. The two curves intersect at the middle of the line uB = 1-uA, at uB = uA = 0.5. 28 Equating u /u to u /u for both z ≥ E (y)/E (x) and z < E (y)/E (x) yields, after simplifications, the solution A AM B BM A B A B

presented in Table 4.

Two Representations of Negotiated Exchange: A Review and Comparison

171

7.2, because it is too spacious to fit in Table 7.2. Depending on the values of uA(x), uB(x), and EA(y)/EB(x), the Nash solution of situation (III) can be equal to the Nash and RKS solution of situation (I), or to the Nash and RKS solution of situation (III), or to EA(y)/EB(x). To summarize this section, analyses of the most simple network structure, the bilateral exchange relation, already demonstrate that there are major differences between the theories' predictions for reduced exchange and for non-reduced exchange. In the case of reduced exchange the predictions of power-dependence theory, the RKS solution, and the Nash solution are all equal to a/2, half of the fixed price. However, in the case of non-reduced exchange, the predictions can be different, depending on the exchange situation. In situation (II), which cannot be represented by reduced exchange, all predictions are different.29 Therefore, the conclusion seems justified that the reduced exchange representation is not only an invalid representation for some exchange situations, but that it also eliminates aspects of the exchange situation that cause the theories' predictions to be different to each other. 7.4.4.3 Exchange-resistance theory and the RKS solution applied to more complex networks The Nash solution can easily be generalized to include an exchange economy with more than two actors when there are no restrictions on who can exchange with whom. If there are n actors in the economy then the Nash solution is that allocation of endowments that maximizes the product of utility gains of all actors (e.g., Osborne and Rubinstein, 1990, p. 23; Roth, 1979a). There is some controversy over whether the RKS solution can also be generalized to exchange economies without restrictions and more than two actors. Roth (1979a) proves that if there are more than two players no solution exists which satisfies the assumptions on which the two-person RKS solution is based. However, Heckathorn and Carlson (1981) demonstrate that the RKS solution can be generalized to more than two actors by modifying the monotonicity axiom (see footnote 26) of the RKS solution. The question can be asked whether the Nash and RKS solutions can be used in exchange economies with exchange restrictions, that is, exchange networks. Some studies (Lovaglia, Skvoretz, Willer, and Markovsky, 1995, p. 131; Patton and Willer, 1990, p. 37; Szmatka and Willer, 1995, p. 126) remark that the scope of the resistance model (RKS solution) is broader than that of the Nash solution. However, both require a theory of the way in which the maximum utilities and status quo or conflict utilities in each exchange relation depend on the network structure. When such a theory has been developed, the answer to the question is yes. Exchange-resistance theory and its derivatives (see footnote 14 for references) are theories of how conflict and maximum utilities depend on network structure. Until now exchange-resistance theory has primarily been applied to reduced exchange with equally valued exchange relations. Bonacich and Friedkin (1998, p. 161) note that exchangeresistance theory does not deal with unequally valued relations. However, as also remarked upon by Willer (1999, pp. 295-297), the generalization of exchange-resistance theory to include unequally valued relations in either reduced or non-reduced exchange is 29 The differences between the solutions are small for the examples in Table 1. One might argue that the differences are so small that they are not relevant in practice. However, in other examples the differences can be very large. Consider for example an exchange relation AB with uA(x) = 10, uB(x) = 0.5, and uA(y) = uB(y) = EA(y) = EB(x) = 1. The kernel, equiresistance and core solutions of the exchange rate are 5.25, 15/11, and 1 respectively, yielding utilities (uA, uB) equal to respectively (0.905, 0.905), (6.333, 0.633), and (9, 0.5).

172

Chapter 7

straightforward. A basis for the generalization of exchange-resistance theory based on the RKS solution is constructed below. One way to define the equiresistance solution in an exchange network is to construct the utility gains in all relations in such a way that in all relations the proportional utility gain of both actors is equal, relative to their maximum conflict utility and to the maximum utility in their relation. Therefore, the primary difference between the kernel and equiresistance solutions is that in the kernel solution both actors’ surplus is equal in all relations, while in the equiresistance solution both actors’ proportional utility gain is equal in all relations. In order to apply the algorithm developed by Cook and Yamagishi (1992) to find the equiresistance solution, some additional assumptions must be made. Consider an exchange relation AB and denote A’s and B’s maximum conflict utility by uAC and uBC. respectively An actor’s (maximum) conflict utility is equal to the utility he would have obtained from an exchange with his best alternative exchange partner. If both uAC < uAM and uBC < uBM, then the equiresistance exchange rate or split of resource pool can be calculated in a straightforward fashion. However, some additional assumptions are required for a possible exchange relation where the conflict utility of at least one actor is equal to or higher than his maximum utility in this exchange relation. It is assumed here that if uAC ≥ uAM and uBC < uBM, then A obtains uAM in AB and both proportional utility gains are defined as zero. Therefore, AB yields A a conflict utility equal to uAM in his other relations. If both uAC ≥ uAM and uBC ≥ uBM, then it is assumed that exchange relation AB is not used and has no effect on the other exchange possibilities in the network. Therefore, AB yields a conflict payoff of zero to both A and B in all their relations. This assumption corresponds to the so-called network breaks in the literature (e.g., Lovaglia, Skvoretz, Markovsky, and Willer, 1995; Simpson and Willer, 1999).30 The structure of the algorithm used to find the equiresistance solution with the additional assumptions stated above is similar to that used to find the kernel solution. In the first iteration of the algorithm, the equiresistance solution is calculated for all exchange relations under the restriction that all conflict utilities are equal to zero. In the second iteration, the equiresistance solution is calculated for all exchange relations, with each actor’s conflict utility in a relation equal to the maximum of the utilities for his alternative exchange relations in the previous iteration. The possible third and subsequent iterations are similar to the second iteration. The process stops when it has converged, that is, when all utility gains and conflict utilities do not change with further iteration. Analogous to the algorithm used to find the kernel solution, this algorithm can be used to find the equiresistance solution in exchange networks based on either the reduced or non-reduced exchange representation. The algorithm is applied to the non-reduced exchange representation of the examples in Table 7.3. The general equiresistance solutions of the reduced representation of the examples in Table 7.3 are derived analytically. The solutions for both representations are shown in Table 7.5. Beginnig with the solutions for the reduced exchange representation, consider a 30 In the literature a network break refers to an exchange relation that is never, or hardly ever, used. It is then also assumed that this relation does not affect the other exchange possibilities in the network. However, both actors in the network break as a rule gain less in their alternative exchange relation than the maximum they can obtain in their network break. In our “network break” definition, both actors gain more in their alternative exchanges than they could possibly obtain in their network break. Therefore, our “network break” definition is not as strong or restrictive as the usual network break definition.

Two Representations of Negotiated Exchange: A Review and Comparison

173

reduced exchange relation where B and C can divide m units. Assume that B’s and C’s conflicts uBC and uCC are both smaller than m. Equating B’s proportional gain to C’s proportional gain then yields an equiresistance solution which provides the following utilities to both actors: uB =

m( m − u CC ) 2m − u BC − u CC

and

uB = m - uA

(7)

Equation (7) can be directly applied to all network structures since B and C exchange with each other in all of them. The solutions of the networks can be obtained by substituting the correct values of the conflict utilities and m. In the Triangle uCC = b and uBC = a, in the Line3 uCC = b and uBC = 0, and in the Line4 uCC = d and uBC = a, while m = c in all networks. Comparing the general kernel solutions to the general equiresistance solutions, it is observed that in all three networks the kernel assigns greater utility to the more advantaged actor than the equiresistance solution. Utility uB for the solution of the kernel is larger than uB for the equiresistance solution in the Line3 when c > a, in the Triangle when a > b, and in the Line4 when a > d. Finally, note that the exchange-resistance solution is equal to the kernel solution in the Triangle if c ≤ a+b. If c ≤ a+b then the equiresistance solution provides the same utility to an actor in his two exchange relations, hence the proportional utility gains of all actors in all relations are also equal to zero. The exchange rates and utilities in the solutions for the non-reduced exchange representation of the examples in Table 7.3 are calculated using the algorithm. In the Triangle zBC = 2.815, uB = 5.789 (58%) and uC = 8.421 (53%). Deleting AC from the network improves B’s outcome [zBC = 2.933, uB = 6.364 (64%) and uC = 7.273 (45%)], while adding CD to the network again improves C’s outcome [zBC = 2.781, uB = 5.616 (56%) and uC = 8.767 (55%)]. There are two remarks with respect to these solutions. Firstly, note that for the non-reduced exchange representation the kernel also assigns larger utilities to the more advantaged B than the equiresistance solution. The difference is especially large in the Line3 (85% versus 64%). Secondly, uB is smaller in the Line4 than in the Triangle, which is in agreement with the equiresistance solution for reduced exchange and the observation that C’s alternative exchange partner in the Line4 (D) is better than in the Triangle (A). Note that this is not true in the kernel solution, but it is always true for the core and equiresistance solutions. Improving the conflict utility of one actor in a relation results in a deterioration of the situation for the other actor in the relation, that is, a deletion of his most profitable solutions from the core, and a lower utility in the equiresistance solution, which can be derived from (7) in the case of a non-reduced exchange representation. 7.4.5 Conclusions with respect to the comparisons of the predictions for both representations across the three theories Summarizing the results of sections 7.4.2 to 7.4.4, the predictions of three theories have been derived for three exchange networks. The exchange networks are based upon the non-reduced exchange representation and cannot be represented by reduced exchange. However, reduced exchange representations have been derived that are as similar as possible to the original nonreduced exchange representations of the exchange networks. Some conclusions can be derived from the comparisons of predictions of the theories for each representation and comparisons of each theory for both representations. The conclusions are provisional because only a few networks, and only one non-reduced representation of them, have been studied.

174

Chapter 7

One conclusion is that the interaction between the theories’ predictions and the type of exchange representation is small. That is, the structure of the predictions of the three theories is similar for both representations, and the differences between the theories are similar for each representation. The only interaction is observed in the kernel solution. Improving one actor’s conflict utility results in a higher utility for him in the reduced representation, but this is not necessarily so in the non-reduced representation. Therefore, it seems that the largest difference between applying the theories to each exchange representation lies in the difficulty of deriving analytically the solutions in the non-reduced exchange representation. 7.4.6 Consequences of representation inequivalence for experimental research One practical reason for using the reduced exchange representation in experiments instead of the non-reduced representation is that it is probably easier to understand for the subjects. In a reduced exchange relation an actor always gains a proportion of the fixed prize, and the other actor obtains 1 minus this proportion. In a non-reduced exchange relation actors both have to give away and receive resources. Two difficulties for subjects arise that are not present in reduced exchange. Firstly, in non-reduced exchange the actors must be careful that they do not lose utility by giving too much, which is not possible in the reduced exchange representation. Secondly, in non-reduced exchange subjects might carry out an inefficient exchange. That is, it is possible that their exchange will not be on the contract curve and that it could have been improved by another exchange that would have been better for both actors. Another practical reason in favor of using the reduced exchange is related to the previous one. An exchange network in a reduced exchange representation is easily presented in one figure on a computer screen or on one piece of paper, but this is not the case for a nonreduced exchange representation. Every reduced exchange relation can be represented by a line with a number attached to it that signifies the number of units that the actors can divide in this relation (see Figure 3). The non-reduced representation requires at least eight numbers to represent an exchange relation.31 It is clear that a non-reduced representation of an exchange network requires more space on either a computer screen or paper than a reduced exchange representation of the same network. Although there are practical reasons in favor of using reduced exchange, the analyses in 7.3 Relations between representations of negotiated exchanges demonstrate that reduced exchange is not a theoretically valid representation of direct negotiated exchange. Therefore the non-reduced representation is required in experiments to most accurately test actor behavior in direct negotiated exchange. Furthermore, the analyses in 7.4 Consequences of representation inequivalence also demonstrate that the reduced representation eliminates aspects of the exchange situation that cause the predictions of different theories of network exchange to vary with respect to each other. Therefore, experiments using the non-reduced representation would also be better in distinguishing the different theories from each other. Some studies have already used the non-reduced exchange representation in their experiments. One example is the study of Brennan (Chapter 9 in Willer and Anderson, 1981). The non-reduced exchange representation in Brennan’s study is also discussed by Willer (1984), Szmatka and Willer (1995) and Willer (1999, pp. 37-46). In this representation two types of actors are distinguished. An actor of one type, referred to as f, has 33 units of 31 Two actors have two characteristics, utilities and endowments, with respect to two goods.

Two Representations of Negotiated Exchange: A Review and Comparison

175

commodity blue, and an actor of the other type g has 10 units of commodity white. Actor f values both goods equally, while white has no value to g. In Brennan’s experiments there is always one actor of type f, while the number of actors of type g differs between 3 and 5. It is of interest here whether the experiments are a case of non-reduced exchange situation (I), (II), or (III). It should be remembered that situation (II) is most interesting because this situation in particular distinguishes reduced from non-reduced exchange. An analysis of Brennan’s exchange representation (see Willer and Anderson, 1981; Willer, 1999, pp. 37-46) shows that it is not of situation (II). On the contract curve an actor of type g exchanges all of his 10 units of commodity white for 1 to 9 units of commodity blue. The contract curve in utility space can be represented by a straight line uf = 10 – ug, connecting the 11 discrete points representing all possible profitable exchange rates. Therefore, Brennan’s experiments are formally equivalent to reduced exchange. Other examples of experiments using the non-reduced exchange representation are the “widget experiments” of Willer (1999, pp. 291-294), constructed to demonstrate power at a distance. In the exchange networks used in these experiments, some actors can have a widget, which is a commodity of one unit that is not valued by these actors but highly valued by some other actors who, in turn, have a number of money units that are valued by all actors. These two types of actors cannot exchange directly, and must exchange indirectly via exchanges with a third type of actors who also do not value the widget. The non-reduced representation in these experiments is also not of situation (II), because in all exchanges the widget must be transferred completely. However, the introduction of the non-reduced representation was necessary to model a flow of resources through the network and, consequently, power at a distance. To summarize this section, although there has been some experimental research into exchange using the non-reduced exchange representation, this research has only focused on exchange situations which in principle can also be represented by reduced exchange. Therefore, experimental research on the most interesting case of direct exchange, exchange situation (II), has not yet been carried out. 7.5

Advantages of the non-reduced exchange representation

The previous section showed that there are two fundamental advantages of using the nonreduced over the reduced representation of direct exchange. Firstly, the reduced representation cannot always represent direct exchange, and secondly, the reduced representation eliminates aspects of the exchange situation that cause the predictions of different theories of network exchange to vary with respect to each other. The fourth objective of the present study is to demonstrate that there are a number of other advantages of the non-reduced representation in the sense that it can be applied to a much broader range of issues both inside and outside the field of exchange research. Some of the issues that demand investigation are also mentioned in Willer’s recent book (Willer, 1999, pp. 289-299) on network exchange theory. A suggestion is made here about how these issues can be investigated by making use of the non-reduced representation. The versatility of the nonreduced representation is directly related to a previously hot debate in exchange theory about involving individual and structural determinants of actor behavior in exchange relations. Homans (1958, 1971, p. 376) asserted that no general laws are specifically social and that only psychological general propositions are needed to explain human activity. The social

176

Chapter 7

exchange theory developed by Homans proposes that social structures contain nothing that is not a quality of individuals or a consequence of individual qualities (Willer, 1999, p. 5). Willer (1999, p. 6) asserts that, according to social exchange theory, social structures are determined by individual behavior but have no effect on this behavior. However, as opponents of social exchange theory argue, network exchange experiments using the reduced exchange representation convincingly demonstrate that network structure has a strong impact on the divisions of fixed prizes that are not related to actor differences. Note that individuals and their characteristics play no role in the reduced exchange representation, because an exchange relation is not defined in terms of individual characteristics. A relevant reply from a social exchange theorist to the above argument would be the question of where the exchange relation comes from when it is not related to individuals and their characteristics. Direct exchange relations must presuppose individuals who have a common interest in the exchange, naturally because they have different characteristics such as endowments and utilities. Individuals with different characteristics can also be the cause of exchange networks as studied by theories of network exchange. For example, the networks in Figure 3 are defined by the actors’ endowments and utilities without imposing artificial restrictions that one actor cannot exchange with another; some actors will not exchange simply because it would not be profitable to them. Therefore, I would agree with social exchange theorists that individuals and their characteristics are an important cause of exchange networks. Naturally, as network exchange experiments show, these networks in turn have an effect on the behavior of the actors in their exchange relations. Providing an interpretation of exchange networks in terms of individuals and their characteristics, and hence using the non-reduced exchange representation, makes it possible to study the evolution of exchange networks from these individuals and their characteristics. Until now network exchange research has dealt with the network structure as an exogenous variable which was fixed in the experiments, but has not dealt with the evolution of exchange relations over time. This hiatus can be explained by the observation that research on network evolution is either impossible or unrealistic using the reduced exchange representation (see also Willer 1999, pp. 290-291).32 It should be remembered that in the reduced representation an exchange relation is an exogenously determined (by the experimenter) characteristic of a pair of actors, and not the endogenous dynamic result of actor characteristics such as their endowments and utilities. One way to study network evolution using the non-reduced representation would be to provide a number of actors with endowments and utilities and to allow them to interact repeatedly with each other without explicit restrictions. The network structure is provided only implicitly in the distribution of endowments and utilities. For example, four actors could be provided with the endowments and utilities of the example of the Line4 structure in Table 7.3 without presenting them with the characteristics of the other actors. By repeatedly interacting with the other actors, they would have to find out who their possible and best exchange partners are, if any. Note that this experimental setting resembles to a considerable degree the reality of the evolution of social ties in general, and in particular that of exchange relations. In a new environment, for example after having moved to a new city, one must also interact with a number of individuals to find out their characteristics and consequently whether they can become one’s friends. The stability of exchange networks and 32 Leik (1992) and Willer and Willer (2000) have indicated how changes in the network structure based on the reduced exchange representation could be studied experimentally.

Two Representations of Negotiated Exchange: A Review and Comparison

177

commitment (Kollock, 1994) can also be investigated in a straightforward fashion using the non-reduced exchange representation. These can be assessed by changing the endowments or utilities of one or more of the actors, and then by observing if and how exchange patterns change in the network. Another application of exchange where the non-reduced representation is required is when resource flows in exchange networks must be modeled. Resource flows have to be modeled when studying power-at-a-distance (see Willer, 1999, pp. 291-294). In the reduced exchange representation an actor divides a resource pool with an adjacent actor in the network structure, and the obtained resources cannot be used in further exchanges with other adjacent actors. Therefore, reduced exchange networks restrict the exercise of power to adjacent positions. However, exchange networks in reality need not to be so restrictive. When material goods are exchanged, an actor can decide to exchange the goods he has obtained in order to achieve an even higher utility gain. Using the non-reduced exchange representation power-at-a-distance can be studied straightforwardly. Willer (1999) devised the widget experiments discussed in the previous section to study power-at-a-distance. It may be recalled that the exchange representation in the widget experiments is a special case of the nonreduced representation. Resource flows are also intrinsic to generalized exchange. In the introduction it was already noted that by definition generalized exchange cannot be represented using reduced exchange. However, generalized exchange can be modeled and tested experimentally naturally using the non-reduced exchange representation. For example, consider a network structure with four actors where both A and C can interact with either B or D, but A cannot interact directly with C, and neither can B interact with D (the structure can be referred to as a Square). Additionally, assume that each actor has his own unique resource and that although pairs A and C and B and D cannot interact with each other, within a pair they value their own resources much more than those of the other pair. The only way for the actors to obtain considerable utility is then to transfer all of their resources to an adjacent actor who does not value his resource, in the hope that the other actors in the Square will do the same. An interesting feature of this setup is that the free-riding of the intermediary, B, who does not transfer the resources of A from A to C, can be made more attractive to B by increasing B’s relative utilities of the resources of A and C. The non-reduced exchange representation can also be used for other social interactions, like coercion and conflict. Together with exchange, coercion and conflict form three different types of social relation or transactions (Willer, 1999, p. 26; Szmatka and Mazur, 1998). While exchange is characterized by bilateral resource flows that provide gains to both sides, in coercive relations one actor has the potential to transfer unilaterally a resource that harms the other. In conflict relations both actors have this potential. Conflict and coercive relations can be studied using the non-reduced representation when actors are not only endowed with positively evaluated goods, but also with goods that have negative value to other actors when transferred to them. Willer (1984, 1997; Willer and Szmatka, 1993) made use of the non-reduced representation to study and experimentally test coercion in networks. Szmatka and Mazur (1998) and Szmatka, Skoretz, Sozanski, and Mazur (1998) study conflict in networks but do not use the non-reduced exchange representation. All the above studies of coercion and conflict assumed negotiated transactions. However, both conflict and coercion can also be, and are most naturally be studied as non-negotiated

178

Chapter 7

transactions Molm (see her 1997 book for a review), in her research on non-negotiated or reciprocal exchange and coercion, also made use of a special case of the non-reduced representation. In the paradigm that she and her colleagues use in experiments, actors can unilaterally transfer a resource to an adjacent actor in the network, which is either evaluated positively or negatively by the other actor. Therefore, in contrast to the reduced representation, the non-reduced representation can also be used to study reciprocal exchange relations in networks. Most network exchange research uses the so-called 1-exchange rule. This rule specifies that an actor at each position can make a maximum of only one exchange, a rule that is artificial in most exchange situations. With the non-reduced exchange representation the 1exchange rule is superfluous: actors in an exchange network can be allowed to exchange with anyone they want until they believe that their optimal endowment has been reached. A final application of the non-reduced exchange representation is the exchange of goods with externalities. Externalities occur when an exchange between two actors also affects the utility of actors that are not involved in the exchange. Life is abundant with situations involving exchanges with externalities, for example, situations where an actor exchanges on behalf of a group to which he belongs, such as his household, his company, or any other group that is affected by the actor’s exchanges. Perhaps the most frequent and recognizable example is when the wife (husband) buys a commodity at a price (exchange rate) at which the husband (wife) would not have considered buying it, but also has to ‘suffer’ the consequences of the purchase. The non-reduced representation can be used to study the effect of externalities in exchange networks by incorporating in an actor’s utility not only his endowments but also the endowments of other actors. This approach has already been used in research on collective decision making by Stokman and van Oosten (1994). In this research actors exchange voting positions which indirectly affects the utility of all actors when the final decisions, a weighted average of the voting positions, are made. To provide an example of how externalities can be studied in exchange networks, consider the example of the Line3 structure in Table 7.3. Assume that H is the husband, W is the wife, and S the seller of a commodity of 8 units that is valued by both H and W. Furthermore, assume that H and W together have 30 units of y (e.g., money) but that at a moment in time H has 10 and W 20. Who will buy the good of S and at what exchange rate? Analyzing the situation learns that S has no power or advantage over H and W, as opposed to the situation where there were no externalities, and that it is better for both H and W when the least interested of them gets all of y and exchanges with S to obtain as many units of x as possible. This example demonstrates that network effects on exchange outcomes can be expected to be completely different for exchanges with and without externalities.

Two Representations of Negotiated Exchange: A Review and Comparison

179

Appendix 7.1: Equivalence of equiresistance formulations The equiresistance or RKS solution in non-reduced bilateral exchange is defined as the value of the exchange rate z for which

1 − RA =

u A ( z ) − u AC u B ( z ) − u BC = = 1− R B u AM − u AC u BM − u BC

(I)

and 1-RA and 1-RB are a maximum. Heckathorn (1980) proves that the equiresistance solution can also be defined as that value of z for which both

RA =

u AM − u A ( z ) u BM − u B ( z ) = = RB u AM − u AC u BM − u BC

(II)

and RA is a minimum. Equation (I) can be obtained from (II) by subtracting both sides of Equation (II) from zero and then adding 1 to both sides. Both operations preserve the identity of both sides of Equation (II), hence both equations identify the same solution. Other studies (e.g., Willer and Anderson, 1981; Willer and Skvoretz, 1997b) identify the equiresistance solution as the value of z for which both

RA = '

u AM − u A ( z ) u BM − u B ( z ) = = R 'B u A ( z ) − u AC u B ( z ) − u BC

(III)

and R'A is a minimum. Equation (I) is obtained by adding 1 ([ui(z)-uiC]/ [ui(z)-uiC]) to both sides of Equation (III) and taking the reciprocal of both sides. Both operations preserve the identity of both sides of Equation (III), hence all three equations can be used to find the RKS solution.

180

Chapter 7

Appendix 7.2: Nash solution in bilateral monopoly situation (II) The Nash solution is found by maximizing the product of utility gains. If z < EA(y)/EB(x) = q then the product is (a) [uA(x)/z-1][1-uB(x)/z], as in situation (I), else the product is (b) [uA(x)z][(z-uB(x)], like in situation (III), with (a) = (b) for z = q. Let z1 and z3 denote the exchange rates where (a) and (b) respectively obtain their maxima. In Table 7.3 z1 (situation (I)) and z3 (situation (III)) are presented as a function of uA(x) and uB(x). Both (a) and (b) are parabolic functions. Therefore, the maximum of (a) is obtained at z = z1 if z1 < q, or at z = q if z1 ≥ q. Similarly, the maximum of (b) is obtained at z = z3 if z3 ≥ q, or at z = q if z3 < q. Consequently, four situations can be considered depending on the values of z1, z3, and q: (i)

z3 > q ≥ z1

(ii)

z1 > q and z3 > q

(iii)

q ≥ z1 and q ≥ z3

(iv)

z1 > q ≥ z3

Consider first situation (i). The maximum of (a) and (b) is obtained at z = q, hence the Nash solution in (i) is equal to z = q = EA(y)/EB(x). In situation (ii) the maximum of (b) is obtained at z3 = q and the maximum of (a) is obtained at z = z1. Because (a) is larger for z = z1 than for z = q, the Nash solution in situation (ii) is equal to z1, or equal to the Nash and RKS solution of bilateral monopoly situation (I). In situation (iii) the maximum of (a) is obtained at z = q, while the maximum of (b) is obtained at z = z3. Therefore, the Nash solution in situation (ii) is equal to z3, or equal to the Nash and RKS solution of bilateral monopoly situation (III). The fourth situation is impossible. Substituting in (iv) the solutions z1 and z3 in terms of utilities as presented in Table 7.3 yields 2uA(x)uB(x)/[uA(x)+uB(x)] > [uA(x)+uB(x)]/2, which would imply that 0 > [uA(x)-uB(x)]2. Situations (i) to (iii) can also be formulated in terms of uA(x) and uB(x) by substituting the values of z1 and z3. Making the substitutions and rearranging the equations yields the following Nash solutions in situations (i) to (iii) as a function of uA(x) and uB(x): (i)

If 2q – uB(x) < uA(x) ≤ [uB(x)q]/[2uB(x)-q] and q/2 < uB(x) ≤ q, or if uA(x) > 2q – uB(x) and uB(x) ≤ q/2 then z = q = EA(y)/EB(x)

(ii)

If uA(x) > [uB(x)q]/[2uB(x)-q] and q/2 < uB(x) ≤ q then z = z1 =

(iii)

If uA(x) ≤ 2q – uB(x) then

2 u A ( x) uB ( x) u A ( x) + uB ( x )

Two Representations of Negotiated Exchange: A Review and Comparison z = z3 =

181

u A ( x) + uB ( x ) 2

It can be seen that (i) to (iii) are mutually exclusive and exhaustive, by noting that 2q – uB(x) < [uB(x)q]/[2uB(x)-q] for uB(x) < q, which is always the case in situation (III).

8

Epilogue

184

Chapter 8

Epilogue 8.1

185

Introduction

A central problem in economics and sociology is that of explaining or predicting phenomena at the macro-level. One possible mode of explanation or prediction of macro-level phenomena entails examining processes internal to the system that involve actors at the micro-level. Rational choice theorists often argue that it is important that the micro-level component remains simple. If the internal analysis based on a standard rational choice model yields an inaccurate prediction of a macro-level outcome, they often avoid adding complexity to the model of the actor at the micro-level. In addition, rational choice theorists are satisfied if their model yields an accurate prediction of a macro-level outcome without having tested the descriptive accuracy of the rational actor model at the micro-level. However, I argued in Chapter 1 that the micro-level should receive more attention and I formulated two guidelines for when adding complexity at the micro-level should be considered. Briefly, the first guideline states that if the prediction of a macro-level outcome is insufficiently accurate, adding complexity should be considered when it has substantial effects on predictions of the macro-level outcome. The second guideline states that even when the prediction of the macro-level outcome is accurate, adding complexity should be considered because a necessary condition for understanding a macro-level outcome is to have at least an approximately accurate actor model at the micro-level. In this dissertation complexities were added to the model of the actor on the micro-level in order to better predict and understand two macro-level phenomena that receive a lot of attention in social science research in general, and sociological research in particular: outcomes of social dilemmas (chapters 3 to 5) and outcomes in exchange networks (chapters 6 and 7). Adding complexity to the rational choice model can have substantial effects on the predictions of outcomes of social dilemmas. Chapters 3 to 5 examine cooperation in a special social dilemma, the indefinitely repeated two-person prisoner’s dilemma (PD). It can be derived from game theory, an important branch of rational choice theory, that conditions for cooperation in the PD become more favorable when the utility of both actors becomes more concave in the range of the outcomes of the PD. If a standard game theoretic analysis is applied to a social dilemma, it is natural to start with the simple assumption that an actor’s utilities are linear in his own outcomes in the social dilemma. However, empirical research on individual decision making under risk demonstrates that in general utility is not linear and that individuals differ with respect to their utility. Therefore, following the first guideline, in chapters 3 to 5 we allowed actors to have different nonlinear utilities and we derived from game theory and tested experimentally the relation between the concavity of an actor’s utility function and his propensity to cooperate in a PD. In order to test this relation an assessment of the concavity of actors’ utilities was required. One of the utility assessment methods used in these chapters, the tradeoff method, is tested and related to other methods in Chapter 2. Outcomes in exchange networks have been predicted with reasonable accuracy by a large number of theories that have fundamentally different assumptions with respect to micro-level processes like actor bargaining. Only a handful of experimental studies have focused on only a few aspects of actor bargaining in exchange networks. Therefore, there is an incomplete understanding of the effect of the exchange network on actor bargaining, and of how actor bargaining leads to exchange outcomes. Following the second guideline, in Chapter 6 an experimental study was carried out to examine which variables determine actor bargaining behavior, and to estimate and test the effects of these variables.

186

Chapter 8

The actor model used in exchange research in general and in network exchange research in particular was examined in a more fundamental way in Chapter 7. In exchange research an exchange relation between two actors is commonly represented as an opportunity to divide a common resource pool, called the reduced exchange representation. In the reduced representation the exchange relation is an exogenously determined characteristic of a pair of actors. In Chapter 7 the non-reduced exchange representation was re-introduced from pure exchange research in standard microeconomics. In the non-reduced representation, exchange relations are the endogenous dynamic result of two actor characteristics; endowments and utilities. In Chapter 7 the two exchange representations were compared and the consequences of the re-introduction of the non-reduced representation were examined. In the following text the main findings are summarized and critically examined. Moreover, some possible directions for future research are suggested, with an emphasis on those directions I want to pursue myself. This exercise is performed separately for each of the research topics in this dissertation: measuring the utility by means of the tradeoff method (Chapter 2), the relation between actors’ utility and their behavior in PDs (Chapter 3 to Chapter 5), bargaining in exchange networks (Chapter 6), and two representations of negotiated exchange: a review and comparison (Chapter 7). 8.2

Measuring utility by means of the tradeoff method

8.2.1 Theory The standard rational choice model of individual decision making under risk, Expected Utility, assumes that actors choose the alternative with the highest expected utility. An alternative’s utility is a linear combination of the utilities of the alternative’s possible outcomes with weights equal to the objective probabilities of these outcomes. However, empirical research on individual decision making under risk has demonstrated that actors often distort objective probabilities, and hence that Expected Utility is not an adequate descriptive theory of how actors make decisions. A consequence of probability distortions is that the equivalence under Expected Utility between concave utility (convex utility) and risk aversion (risk seeking), i.e., the preference of the expected value of a gamble above the gamble itself, no longer holds. Another consequence is that traditional utility assessment methods, which assume Expected Utility, yield biased assessments of utilities. Wakker and Deneffe (1996) developed the tradeoff (TO) method, which allows the assessment of utility independently of probability distortions. They employed the TO method to assess utility of gains, but not that of losses. 8.2.2 Results The study described in Chapter 2 employed three methods to assess the utility of both gains and losses: the well-known certainty-equivalence (CE) method which is sensitive to probability distortions, and two different versions of the TO method. Evidence in favor of Sshaped utility was obtained using all three methods. For all methods, the utility of gains was classified as concave for the vast majority of subjects, and the utility of losses was significantly more classified as convex than concave. However, the utilities assessed using the two versions of the TO method were significantly different, displaying a new violation of procedure invariance. That is, the latter finding demonstrates (again) that utility depends on the assessment method that is used. It was argued that both S-shaped utility and the violation of procedure invariance show that utility measurement is primarily responsive to a

Epilogue

187

diminishing sensitivity to increasing absolute differences from the reference point. 8.3

The relation between actors’ utility and their behavior in PDs

8.3.1 Theory All hypotheses tested in chapters 3 to 5 were derived from a theory that can be summarized in one sentence: conditions for cooperation in the indefinitely repeated two-person PD become more favorable when the utility of both actors becomes more concave in the range of the outcomes of the PD. The theory was interpreted in two different ways; within persons and between persons. Firstly, if an actor’s utility is more concave in the range of the outcomes of another PD of which all outcomes are shifted by a constant, then his propensity to cooperate in the other PD should be higher (within persons). Secondly, between persons, if an actor’s utility in the range of the outcome of a PD is more concave than the utility of a second actor, then the propensity of the first actor to cooperate should be higher (between persons). Three hypotheses were formulated with respect to the relation between actors’ cooperation in the first round of a PD and their utility. The hypotheses differ with respect to the number of utility assumptions on which they are based. The first hypothesis is based on the strongest, and the third hypothesis is based on the weakest utility assumptions. The first hypothesis (H1) presupposed Expected Utility and that actors have S-shaped utility with loss aversion. H1 stated that the proportion of cooperation is greatest in one mixed PD with both positive and negative outcomes, less in a positive PD, and least in a negative PD. Because of the assumptions on utility that were made, no utility assessment was required to test the nullhypothesis of H1. The second hypothesis (H2) presupposed Expected Utility and stated that for all PDs there is a positive relation between the proportion of cooperation and risk aversion in the range of the outcomes in the PD. In order to test the null-hypothesis of H2, risk aversion was assessed using a traditional method, which yields biased utilities if Expected Utility is violated. In the third hypothesis (H3), the assumption of Expected Utility was relaxed, allowing for probability distortions. H3 stated that for all PDs there is a positive relation between the proportion of cooperation and concavity of utility in the range of the outcomes of the PD. The TO method was employed in order to test the null-hypothesis of H3. The hypotheses were tested in two experiments. Experiment 1, which tested H1 and H2 for either negative or positive PDs, was carried out by Raub and Snijders (1997). Experiment 2 was carried out by van Assen and Snijders. 8.3.2 Results In both experiments the traditional utility assessment method classified the majority of subjects as risk seeking for losses, and it also classified the majority of subjects as risk seeking for gains. These findings do not seem to be in agreement with S-shaped utility. However, the unbiased TO method generated response patterns that were more in line with Sshaped utility and with the results of the TO methods reported in Chapter 2. Unfortunately, test-retest reliabilities of the utility assessment methods were only moderate, and correlations between the different assessment methods were small. The small correlations again demonstrate that utility depends on the assessment method used, and hence again present evidence for a violation of procedure invariance. In general, only weak evidence was found in favor of the three hypotheses with respect to the relation between actors’ cooperation in the first round of a PD and their utility.

188

Chapter 8

H1 was only partly confirmed; the proportion of cooperation was larger when the outcomes of the PD were positive than when they were negative, although, the proportion of cooperation in the mixed PD was not the largest. H2 was partly confirmed. In Experiment 1 a positive effect of risk aversion on cooperation was found, whereas in Experiment 2 an effect of risk aversion was found between persons, but not within persons. Finally, in general no support for H3 was found, although weak evidence in favor of a positive effect of loss aversion was obtained. At first we were surprised by these generally weak relationships between actors’ cooperation and their utility. However, two good reasons were put forward as to why only weak relationships could have been expected. Firstly, the fact that the reliabilities of utility assessment methods are only moderate and the fact that there are only small correlations between them weaken the possible relation between utility and cooperation to a considerable extent. Secondly, other studies have improved conditions for cooperation by manipulating the shadow of the future of a PD, that is, the probability w of continuation of the PD in the following round. Although in these studies w was varied from very small to very large values, again only a weak relationship between actors’ propensity to cooperate and conditions for cooperation was obtained as well. 8.3.3 Evaluation and suggestions for future research Because it is well-known that actors’ utilities differ and because according to the standard rational actor model utility has substantial effects on predictions of outcomes of social dilemmas, we added complexity (shape of utility) to the rational actor model, thereby following the first guideline. Although the effects on predictions were large, no large effects were observed in the experiments. The absence of large effects could be at least partly explained by the problems in assessing utility. However, the absence of large effects also raises the question of whether the standard rational actor should be abandoned in favor of another actor model. The standard rational model may still be considered as a good normative model of how actors should behave. However, following the second guideline, if one pursues to understand how actors behave in PDs and if one wants to obtain an accurate descriptive model of the actor behavior, then the standard rational model presumably must be abandoned. I have planned future studies in both the normative and descriptive direction. Although we allowed for actor differences in utility, we assumed in our analyses that an actor considers the utility of the other actor to be the same as his own. A logical extension of the rational actor model is to add complexity by including incomplete information about the utility of the other actor, reflecting uncertainty about what the other actor is likely to do. For example, one can assume that an actor does not know the other’s utility but knows the probability distribution of utility of all actors in the population from which the other actor is drawn randomly. Preliminary analyses based on this assumption suggested several interesting results. Uncertainty about the other’s utility worsens conditions for cooperation; only the most risk averse actors cooperate, and the sucker’s payoff also affects conditions of cooperation. In particular the last result is interesting. The theoretical analyses in chapters 3 to 5 demonstrated that the sucker’s payoff is irrelevant for conditions of cooperation; however, empirical research shows that the value of the sucker’s payoff does have an effect on actors’ cooperation (e.g., Rapoport and Chammah, 1965). Although the more complex model might possibly improve the prediction and understanding of actors’ behavior in social dilemmas, I am more inclined to consider it as a normative model of how actors should

Epilogue

189

behave in the case of uncertainty about the other’s utility. Another possibility for expanding the standard rational choice model is to examine more closely the effects of actors’ probability weighting on the properties of the equilibrium condition. In measuring the utility functions we took probability distortions into account, but there are also good arguments that actors evaluate the probability of continuation nonlinearly, thereby changing the equilibrium conditions. For example, in Section 4.3 it was already demonstrated that weighing of the continuation probability of the PD has effects on the conditions for cooperation. One other approach is to abandon the standard rational model completely and to attempt to construct a model that accurately describes actor behavior in repeated social dilemmas. A descriptive alternative to the standard rational model is the statistical model presented in appendices 3.1 and 5.2. This model was used to mimic human subjects from Experiment 1 in order to increase the efficiency of the experimental procedure. It is a learning model that yields the probability of an actor cooperating in a particular round of a PD, and is conditional on the history of the game. I prefer to interpret this probability as the proportion of actors that would cooperate in a round of a PD, conditional on the history of the game. Initially, the model was only constructed for practical reasons. However, to our surprise the model could produce very accurate predictions of the proportions of cooperation for the first five rounds, conditional on the history of the game. Furthermore, its parameters can be interpreted in a substantive way. The parameter estimates based on the data in Experiment 1 revealed that the odds of cooperation became 3.67 times as large after an actor’s cooperative response, and 0.33 times as large after an actor’s defection. Furthermore, the estimates showed that the effect of the previous responses of the other actor was 0.85 times as large as the effect of the actor’s own previous response. In the near future we plan to apply this model to the data of Experiment 2. Other parameters can and will be added to the statistical model to allow for differences in the behavior of actors in different games or in the behavior of actors belonging to different groups. It should be noted that the model allows us to estimate and test the effects of utility on actors’ responses in all rounds following the first, which we were not able to do satisfactorily in chapters 3 to 5. It would also be interesting to apply the model to other repeated games and to examine how the characteristics of these games determine actor behavior in them. 8.4

Bargaining in exchange networks

8.4.1 Theory The research reported in Chapter 6 describes the results of the first attempt ever to account for the complete bargaining process in exchange networks. The bargaining process was divided into two stages, an initial offer stage and a concession stage. On the basis of previous research, hypotheses were derived concerning the effects on the dependent variables initial offers and concession rates of the following four independent variables: actor’s relative power in the exchange relation, unspecified actor characteristics, bargaining time, and exclusion and inclusion in the previous exchange round. Concerning initial offers, it was hypothesized that on average inclusion decreases the initial offer to the actor in the relation, exclusion increases it, and an actor’s relative power decreases it. Concerning concession rates, it was hypothesized that on average inclusion decreases the concession rate, exclusion increases it, an actor’s relative power decreases it, and bargaining time increases it. Finally,

190

Chapter 8

because of actor differences with respect to certain unknown but relevant characteristics, it was hypothesized that there are actor differences in the average initial offer and the average concession rate. 8.4.2 Results Concerning initial offers, effects of exclusion and of the actor’s relative power were observed, but not of inclusion or actor differences. Actors’ initial offers could be predicted surprisingly accurately by the Exchange-Seek Likelihood (ESL) measure of an actor’s relative power. The results indicate that actors base their initial offer on their relative degree in the network and that they learn from previous experiences. Concession rates also depended on the actor’s relative power, but the effects of unspecified actor characteristics were in general stronger than structural network effects as measured by actors’ relative power. No effects of exclusion or inclusion were found. Surprisingly, parabolic effects of time were observed in weak power networks. The decrease of concession rates in the first half of the bargaining period could be explained by the fact that some actors made large concessions early in the bargaining process, probably because they were afraid of exclusion. 8.4.3 Evaluation and suggestions for further research A large number of theories with fundamentally different assumptions concerning micro-level processes such as actor bargaining were able to predict outcomes in exchange networks with reasonable accuracy. Therefore, following the second guideline, actor bargaining in exchange networks was examined in order to improve our understanding of how it leads to the exchange outcomes. In my opinion, the study in Chapter 6 succeeded in improving our understanding by yielding some unexpected results. For example, the results of this study invalidate the myth that exclusion and inclusion are important variables and have large effects on actor bargaining. However, the understanding of how bargaining produces exchange outcomes is far from complete. Too few different exchange networks were examined for unequivocal conclusions concerning the role of learning and of actors’ perceptions of their power in bargaining to be drawn. Moreover, the understanding of actor bargaining might be improved by comparing it in exchange networks with different exchange rules (e.g., exclusionary versus inclusionary, or ordering). To provide a deeper understanding of actor bargaining, psychological research could be carried out to investigate which actor characteristics are mainly responsible for actor differences in actor bargaining. Perhaps, after having carried out some of the studies suggested here, a model of boundedly rational actors that can predict both exchange outcomes and actor bargaining at the micro-level could be developed. Another approach is not to focus on how actors bargain in exchange networks, but on how actors should bargain to maximize their own profits, that is, the normative approach. We already use the normative approach in intervention studies of collective decision making (Stokman, van Assen, van der Knoop, and van Oosten, 2000). In these studies it is assumed that actors (among other bargaining strategies) exchange voting positions on issues in order to improve their outcomes. In the context of exchange networks, the normative approach requires the optimal strategy of an actor to be found, conditional on his position and on the strategies of the other actors in the exchange network. Using this approach, actor strategies can be modeled by a combination of rules concerning how the actor at a particular position makes his initial offer, how he reacts to inclusion and exclusion, and how he makes

Epilogue

191

concessions depending on bargaining time. These normative analyses can provide a benchmark for actor bargaining, which can be compared to how actors actually bargain in exchange networks. 8.5

Two representations of negotiated exchange: a review and comparison

8.5.1 Theory and results In the comparison of two micro-models of exchange, the reduced and non-reduced exchange representations, four results were derived. Firstly, it was demonstrated that, in contrast to what most researchers on exchange in sociology believe, there is a close or even a one-to-one correspondence between how game theory and theories of network exchange in sociology solve the indeterminacy of exchange outcomes. More specifically, there are one-to-one relationships of power-dependence and exchange-resistance theory with the kernel (see also Bonacich and Friedkin, 1998) and the RKS solution from game theory (see also Heckathorn, 1980) respectively. The other results, in contrast to the first result, are directly concerned with the differences between the two exchange representations. The second and most important result is that the two representations are not equivalent in contrast to what is commonly believed, and that the most often employed reduced representation is an invalid representation of exchange in some well-defined situations. Thirdly, predictions of exchange outcomes for both representations of exchange were derived using three theories of network exchange: core, power-dependence, and exchange-resistance. These derivations demonstrated that the predictions are most different to each other when exchange cannot be represented in the reduced form. However, the implications of the two different representations for the predictions of exchange outcomes in general were considered to be secondary. Fourthly and finally, it was argued that the non-reduced exchange representation should be reintroduced in exchange research. This should be done not only because it is a valid representation of exchange, but also because it is more general than the reduced representation in the sense that it can be applied to a much broader range of issues in exchange research. Examples include the evolution of exchange networks, commitment in exchange relations, generalized exchange, power-at-a-distance, and exchange of externalities. It was also argued that the nonreduced representation can also be used for social interactions other than exchange, such as coercion and conflict. 8.5.2 Suggestions for further research I propose a research project based on the non-reduced exchange representation with the aim of explaining exchange outcomes in recurrent exchange relations involving actors in small groups. In the proposed project, exchange situations are represented by actors who are distinguishable with respect to their endowments and utilities for commodity bundles, in similar fashion to non-reduced pure exchange in classical microeconomics. The pure exchange representation will be the basis of a general experimental paradigm. The paradigm makes it possible and will be used to investigate several issues in recurrent exchange relations both separately and simultaneously. These issues refer to the effects of network structure, actor strategies and actor characteristics, externalities, and network evolution. More specifically, the project will attempt to answer four questions: (1)

What is the effect of network structure on outcomes in exchange relations?

Predictions derived from core, power-dependence, and exchange-resistance theory will be

192

Chapter 8

tested in experiments based on the non-reduced exchange representation. (2)

To what extent do outcomes in exchange relations differ between exchanges with and without externalities?

The same network structures that were used in order to answer question (1) will be used in experiments to answer question (2) after adding externalities to exchanges. The same theories will also be used to derive predictions. (3)

What is the effect of actor strategies and actor characteristics on the outcomes the actors obtain in exchange relations?

This question is an extension of the study in Chapter 6, and examines how actor bargaining at the micro-level produces the relationship between the structure of the exchange network and the outcomes of pure exchange rather than of reduced exchange. (4)

How do network structures of exchange relations develop over time (network evolution)?

The three theories will be employed to predict which exchange relations will develop, and to predict the effect of changes in actors' endowments and incentive structure on the stability of the exchange network and commitment in exchange relations. All experiments will be conducted in a common general experimental paradigm. Subjects obtain endowments at the start of a round in an experiment. Their payoff or utility is determined by a linear combination of the commodities they have at the end of that round. In the case of exchanges with externalities utility also depends (linearly) on changes in the commodities of others, and thus on other exchanges. By varying endowments and utility for commodities among individuals, any kind of profitable exchange network can be constructed. In the experiments the network structure of exchange relations will either be fixed and conveyed to the subjects or, when network evolution is studied, subjects themselves will have to detect and optimize possible profitable exchange relations. The project is innovative in five respects. (1) The inappropriate reduced representation of exchange currently employed in social exchange research will be replaced by the valid non-reduced representation that reflects the true nature of exchange, on which (2) one general experimental paradigm will be based. This paradigm makes it possible (3) to study simultaneously the effects on the outcomes in exchange relations of network structure, externalities, actor strategies and characteristics, and network evolution. Making use of one single common experimental paradigm allows a theoretically meaningful comparison of the effects of these variables on the outcomes to be made. Most importantly, the general paradigm enables us (4) to study (a combination of) important and relevant determinants of exchange outcomes that have rarely been studied before (such as network evolution and the effect of network structure on exchanges with externalities). Finally, (5) the theory of exchange will make use of insights from sociological theories of network exchange, game theory, and psychology. 8.6

Conclusions

The present study defends the viewpoint that the actor model at the micro-level should receive more attention than it generally does in attempts to account for macro-level phenomena. Two guidelines for research were formulated for when adding complexity to the

Epilogue

193

actor model at the micro-level should be considered; firstly, when predictions of macro-level outcomes are inaccurate, and secondly, when one wants to understand macro-level outcomes. In line with the first guideline, cooperation in repeated PDs was studied and related to actors’ utility of the outcomes of the PD. In line with the second guideline, bargaining behavior as a determinant of outcomes in exchange networks was studied. After having discussed the research on these two topics, the time has come to discuss the implications of this research for the claim that the actor model at the micro-level should receive more attention, and on the two consequently derived two guidelines. Under the assumptions of rational choice theory it was derived in chapters 3 to 5 that utility has substantial effects on the predictions of cooperation in social dilemmas. The evidence that actors differ with respect to their utility is abundant. Therefore, from a normative point of view (how an actor should behave), the first guideline in general and the effects of differences in the utility of actors at the micro-level on predictions of macro-level outcomes in particular, are highly relevant to the social sciences. However, from a descriptive point of view (how an actor actually behaves), the first guideline in general and differences in utility in particular do not seem to be highly relevant to the social sciences. Two causes for the discrepancy between the normative and descriptive points of view can be identified. Firstly, we find it to be very difficult to reliably assess the characteristics of actors that are relevant according to normative theory. Secondly, we find that the behavior of actors in many situations is not rational (see also Section 1.2). Therefore, one should not believe that macrolevel outcomes have been understood when a normative theory yields accurate predictions of macro-level outcomes. The present research shows that for understanding macro-level outcomes the second guideline is relevant to the social sciences indeed. The learning model of social dilemmas described in appendices 3.1 and 5.2 shows how the history of the game and the defection and cooperation of both ego and alter influence the actors’ behavior and consequently the outcome of the social dilemma. The model of bargaining behavior in exchange networks suggests that the actors’ relative degree is a relevant variable for explaining actors’ bargaining behavior. One would not have obtained both insights if one would have persisted with the normative or rational model of actor behavior at the microlevel. The review and comparison of the reduced and non-reduced representations of exchange present further evidence with respect to the beneficial value of adding complexity to the actor model at the micro-level. It is shown that the simple actor model as assumed by the reduced representation is suited to explaining the effect of network structure on outcomes in exchange networks. However, it is also shown in Section 7.5 that a more complex actor model, as assumed by the non-reduced exchange representation, is needed to model and account for a much broader range of issues both inside and outside the field of exchange research. The fact that the more complex representation is much more generally applicable than the simple reduced representation is strongly related to the principle of sufficient complexity, a recent insight of Lindenberg (2001). One interpretation of this principle is that in order in order to prevent that a simple actor model assumes away interesting macro-level phenomena that require an explanation, complexity should be added to the actor model to allow the simultaneous investigation and explanation of multiple macro-level phenomena using the same actor model. For example, the non-reduced representation allows attempts to develop a unitary theory of social relations (exchange, coercion, and conflict) to be made and

194

Chapter 8

allows comparisons of the effects of network structure, externalities, network evolution, and actor strategies and actor characteristics on these different social relations to be made.

References Abdellaoui, M. (2000). Parameter-free elicitation of utilities and probability weighting functions. Management Science, 46, 1497-1512. Akerlof, G.A., & Yellen, J.L. (1985). Can small deviations from rationality make significant differences to economic equilibria? American Economic Review, 75, 708-720. Allais, M. (1953). Le comportement de l'homme rationnel devant le risque: Critique des postulats et axiomes de l'école américaine. Econometrica, 21, 503–546. Atkinson, K.E. (1989). An introduction to numerical analysis. New York: Wiley. Bacharach, S.B., & Lawler, E.J. (1981). Bargaining: power, tactics, and outomes. San Francisco: Jossey-Bass. Baron, J. (1997). Confusion of relative and absolute risk in valuation. Journal of Risk and Uncertainty, 14, 301–309. Beattie, J., & Loomes, G. (1997). The impact of incentives upon risky choice experiments. Journal of Risk and Uncertainty, 14, 155-168. Becker, G.S. (1976). The economic approach to human behavior. Chicago and London: The University of Chicago Press. Becker, G.M., DeGroot, M.H., & Marschak, J. (1964). Measuring utility by a single-response sequential method. Behavioral Science, 9, 226-232. Becker, O., & Huschens, S. (1988). Bounded rational strategies in sequential bargaining: an experiment and a learning by evolution strategy. In R. Tietz, W. Albers, & R. Selten (Eds.), Bounded rational behavior in experimental games and markets. Proceedings of the Fourth Conference on Experimental Economics (pp. 129-141). Berlin: Springer-Verlag. Bentham, J. (1789). Principles of morals and legislation. Oxford: Clarendom Press. Berejikian, J. (1992). Revolutionary collective action and the agent-structure problem. American Political Science Review, 86, 647-657. Berg, C., & Panther, S. (1998). Network exchange as a non-cooperative bargaining game. Unpublished manuscript. Bernoulli, D. (1738). Specimen theoria novae de mensura sortis. Commentarii Academiae Scientiarum Imperialis Petropolitanae, 5, 175–192. Translated into English by L. Sommer (1954), Exposition of a new theory on the measurement of risk. Econometrica, 22, 23–36. Bienenstock, E.J., & Bonachich, P. (1992). The core as a solution to exclusionary networks. Social Networks, 14, 231-243. Bienenstock, E.J., & Bonachich, P. (1997). Network exchange as a cooperative game. Rationality and Society, 9, 37-65. Bienenstock, E.J., & Bonacich, P. (1993). Game-theory models for exchange networks: experimental results. Sociological Perspectives, 36, 117-135. Bishop, R.L. (1963). Game-theoretic analyses of bargaining. Quarterly Journal of Economics, 77, 559-602. Blau, P.M. (1964). Exchange and power in social life. New York: Wiley. Bonacich, P. (1987). Power and centrality: A family of measures. American Journal of Sociology, 92, 1170-1182.

196 Bonacich, P. (1992). Power in positively connected networks: a critical review. Advances in group processes, 9, 21-40. Bonacich, P (1995). Four kinds of social dilemmas within exchange networks. Current Research in Social Psychology, 1, 1-7. Bonacich, P. (1998a). A behavioral foundation for a structural theory of power in exchange networks. Social Psychology Quarterly, 61, 185-198. Bonacich, P. (1998b). The evolutionary stability of strategies in exchange networks. Current Research in Social Psychology, 3, 12-34. Bonacich, P. (1999). An algebraic theory of strong power in negatively connected exchange networks. Journal of Mathematical Sociology, 23, 203-224. Bonacich, P., & Bienenstock, E.J. (1993). Assignment games, chromatic number, and exchange theory. Journal of Mathematical Psychology, 17, 243-259. Bonacich, P., & Bienenstock, E.J. (1995). When rationality fails: unstable exchange networks with empty cores. Rationality and Society, 7, 293-320. Bonacich, P., & Bienenstock, E.J. (1997a). Latent classes in exchange networks: Sets of positions with common interests. Journal of Mathematical Psychology, 22, 1-28. Bonacich, P., & Bienenstock, E.J. (1997b). Strategy in exchange networks: exploitation versus accommodation. In J. Szmatka, J. Skvoretz, & J. Berger (Eds.), Status, Network, and Structure: Theory Construction and Theory Development. Stanford: Stanford University Press. Bonacich, P., & Friedkin, N.E. (1998). Unequally valued exchange relations. Social Psychology Quarterly, 61, 160-171. Borgatti, S.P., & Everett, M.G. (1992). Graph colorings and power in experimental exchange networks. Social Networks, 14, 287-308. Braun, N. (1994). Restricted access in exchange systems. Journal of Mathematical Sociology, 19, 129-148. Braun, N., & Gautschi, T. (2000). Who gets how much in which relation? A non-cooperative bargaining approach to exchange networks. Unpublished manuscript. Burgess, R.L., & Nielsen, J.M. (1974). An experimental analysis of some structural determinants of equitable and inequitable exchange relations. American Sociological Review, 39, 427-443. Burke, P.J. (1997). An identity model for network exchange. American Sociological Review, 62, 134-150. Cain, M.J.G. (1998). An experimental investigation of motives and information in the prisoner’s dilemma game. Advances in Group Processes, 15, 133-160. Camerer, C. (1995). Individual decision making. In J.H. Kagel, & A.E. Roth (Eds.), The handbook of experimental economics (pp. 587-703). Princeton, N.J.: Princeton University Press. Camerer, C.F., & Ho T.H. (1994). Violations of the betweenness axiom and nonlinearity in probability. Journal of Risk and Uncertainty, 8, 167-196. Camerer, C.F., & Hogarth, R.M. (1999). The effects of financial incentives in experiments: a review and capital-labor-production. Journal of Risk and Uncertainty, 19, 7-42. Carrol, R.J., & Ruppert, D. (1988). Transformations and weighting in regression. New York:

References

197

Wiley. Coddington, A. (1968). Theories of the bargaining process. London: Allen & Unwin. Cohn, R.A., Lewellen, W.G., Lease, R.C., & Schlarbaum, G.G. (1975). Individual investor risk aversion and investment portfolio composition. Journal of Finance, 30, 605–620. Coleman, J.S. (1972). Systems of social exchange. Journal of Mathematical Sociology, 2, 145-163. Coleman, J.S. (1987). Psychological structure and social structure in economic models. In R.M. Hogarth, & M.W. Reder (Eds.), Rational choice. The contrast between economics and psychology (pp. 181-186). Chicago: University of Chicago Press. Coleman, J.S. (1990). Foundations of social theory. Cambridge, MA: Harvard University Press. Coleman, J.S. (1994). A rational choice perspective on economic sociology. In N.J. Smelser, & R. Swedberg (Eds.), The Handbook of Economic Sociology (pp. 166-180). Princeton, NJ: Princeton University Press. Conaway, M.R. (1990). A random effects model for binary data. Biometrics, 46, 317-328. Cook, K.S. (1995). Social relationships and group processes. In K.S. Cook, G.A. Fine, & J.S. House (Eds), Sociological perspectives on social psychology (pp. 203-235). Boston, MA: Allyn and Bacon. Cook, K.S., & Emerson, R.M. (1978). Power, equity and commitment in exchange networks. American Sociological Review, 43, 721-739. Cook, K.S., Emerson, R.M., Gillmore, M.R., & Yamagishi, T. (1983). The distribution of power in exchange networks: theory and experimental results. American Journal of Sociology, 89, 275-305. Cook, K.S., & Gillmore, M.R. (1984). Power, dependence, and coalitions. Advances in group processes, 1, 27-58. Cook, K.S., Gillmore, M.R., & Yamagishi, T. (1980). Does micro-economic theory need a defense? American Sociological Review, 45, 351-353. Cook, K.S., Gillmore, M.R., & Yamagishi, T. (1986). Point and line vulnerability as bases for predicting the distribution of power in exchange networks: Reply to Willer. American Journal of Sociology, 92, 445-448. Cook, K.S., & Yamagishi, T. (1992). Power in exchange networks: a power-dependence formulation. Social Networks, 14, 245-265. Corra, M. (2000). Applying resistance to ordering in exchange networks: a theoretical extension. Current Research in Social Psychology, 5, 84-96. Cox, J.C., Roberson, B., & Smith, V.L. (1982). In V. L. Smith (Ed.). Research in Experimental Economics. Greenwich, Conn.: JAI Press. Cox, J.C., Smith, V.L., & Walker, J.M. (1988). Theory and individual behavior of first-price auctions. Journal of Risk and Uncertainty, 1, 61-69. Cramer, G. (1728). Letter from Cramer to Nicholas Bernoulli. Translated into English by L. Sommer in D. Bernoulli, (1954), Exposition of a new theory on the measurement of risk. Econometrica, 22, 23–36. Cross, J.G. (1965). A theory of the bargaining process. The American Economic Review, 55, 67-94.

198 Cross, J.G. (1969). The economics of bargaining. New York: Basic Books. Cross, J.G. (1977). Negotiation as a learning process. Journal of Conflict Resolution, 21, 581606. Currim, I.S., & Sarin, R.K. (1984). A comparative evaluation of multiattribute consumer preference models. Management Science, 30, 543–561. Davidson, D., Suppes, P., & Siegel, S. (1957). Decision making: An experimental approach. Stanford: Stanford University Press. Chapter 2 has been reprinted in W. Edwards, & A. Tversky (Eds.), (1967), Decision making (pp. 170-207). Harmondsworth: Penguin. Debreu, G. (1959). Theory of value: an axiomatic analysis of economic equilibrium. New York: Wiley. Delquié, P. (1993). Inconsistent tradeoffs between attributes: New evidence in preference assessment biases. Management Science, 39, 1382–1395. Durkheim, E. (1951, [1897]). Suicide. Glencoe, Illinois: Free Press. Durkheim, E. (1964, [1895]). The rules of sociological method. New York: The Free Press of Glencoe. Dyer, J.S., & Sarin, R.K. (1982). Relative risk aversion. Management Science, 28, 875–886. Edgeworth, F.Y. (1881). Mathematical physics. Fairfield: Kelley. Edwards, A.L. (1968). Experimental design in psychological research (3rd ed.). London: Holt, Rinehart, & Winston. Edwards, W. (1954). The theory of decision making. Psychological Bulletin, 51, 380–417. Edwards, W. (1962). Subjective probabilities inferred from decisions. Psychological Review, 69, 109–135. Ekeh, P. (1974). Social exchange theory: the two traditions. Cambridge, MA: Harvard University Press. Emerson, R.M. (1962). Power-dependence relations. American Sociological Review, 27, 3140. Emerson, R.M. (1972a). Exchange theory, part I: A psychological basis for social exchange. In J. Berger, M. Zelditch Jr., & B. Anderson (Eds.), Sociological Theories in Progress, 2,.(pp. 38-57). Boston: Houghton-Mifflin. Emerson, R.M. (1972b). Exchange theory, part II: Exchange relations and networks. In J. Berger, M. Zelditch Jr., & B. Anderson (Eds.). Sociological Theories in Progress, 2,.(pp. 58-87). Boston: Houghton-Mifflin. Emerson, R.M. (1976). Social exchange theory. Annual Review of Sociology, 2, 335-361. Emerson, R.M. (1981). Social exchange theory. In M. Rosenberg, & R.H. Turner (Eds.), Social psychology: sociological perspectives (pp. 30-65). New York: Basic Books. Emerson, R.M., Cook, K.S., Gillmore, M.R., & Yamagishi, T. (1983). Valid predictions from invalid comparisons: Responses to Heckathorn. Social Forces, 61, 1232-1247. Erev, I., & Wallsten, T.S. (1993). The effect of explicit probabilities. Journal of Behavioral Decision Making, 6, 221–241. Eysenck, M.W., & Keane, M.T. (1990). Cognitive psychology. A student’s handbook. London: Lawrence Erlbaum.

References

199

Farquhar, P.H. (1984). Utility assessment methods. Management Science, 30, 1283-1300. Felsenthal, D.S., & Diskin, A. (1982). The bargaining problem revisited: Minimum utility point, restricted monotonicity axiom, and the mean as an estimate of expected utility. Journal of Conflict Resolution, 26, 664-691. Fennema, H, & van Assen, M.A.L.M. (1999). Measuring the utility of losses by means of the tradeoff method. Journal of Risk and Uncertainty, 17, 277-295. Fischhoff, B., Slovic, P., & Lichtenstein, S. (1980). Knowing what you want: Measuring labile values. In T. Wallsten (Ed.), Cognitive Processes in Choice and Decision Behavior (pp. 119–141). Hillsdale, NJ: Erlbaum. Fishburn, P.C., & Kochenberger, G.A. (1979). Two-piece von Neumann-Morgenstern utility functions. Decision Sciences, 10, 503–518. Flache, A. (1996). The double edge of networks: an analysis of the effect of informal networks on cooperation in social dilemmas. Amsterdam: Thesis Publishers. Friedkin, N.E. (1992). An expected value model of social power: Predictions for selected exchange networks. Social Networks, 14, 213-229. Friedkin, N.E. (1993). An expected value model of social exchange outcomes. Advances in group processes, 10, 163-193. Friedkin, N.E. (1995). The incidence of exchange networks. Social Psychology Quarterly, 58, 213-221. Friedman, J.W. (1986). Game theory with applications to economics. New York: Oxford University Press. Friedman, M. (1953). Methodology of positive economics. Chicago: University of Chicago Press. Friend, I., & Blume, M.E. (1975). The demand for risky assets. American Economic Review, 65, 900–922. Galanter, E. (1980). Utility functions for nonmonetary events. American Journal of Psychology, 103, 449–470. Gillmore, M.R. (1987). Implications of generalized versus restricted exchange. In K.S. Cook (Ed.), Social Exchange Theory (pp. 170-189). Newbury Park, CA: Sage. Gode, D.K., & Sunder, S. (1993). Allocative efficiency of markets with zero-intelligence traders: market as a partial substitute for individual rationality. Journal of Political Economy, 101, 119-137. Gonzalez, R., & Wu, G. (1999). On the shape of the probability weighting function. Cognitive Psychology, 38, 129-166. Green, P.E. (1963). Risk attitudes and chemical investment decisions. Chemical Engineering Progress, 59, 35–40. Haltiwanger, J., & Waldman, M. (1985). Rational expectations and the limits of rationality: an analysis of heterogeneity. American Economic Review, 75, 326-341. Hardin, R. (1982) Collective action. Baltimore, MD: Johns Hopkins University Press for resources for the future. Harsanyi, J.C. (1956). Approaches to the bargaining problem before and after the theory of games. Econometrica, 24, 144-157.

200 Harsanyi, J.C. (1980). Analysis of a family of two-person bargaining games with incomplete information. International Journal of Game Theory, 9, 65-89. Hechter, M. (1987). Principles of group solidarity. Berkeley: University of California Press. Heckathorn, D.D. (1978). A paradigm for bargaining and a test of two bargaining models. Behavioral Science, 23, 73-85. Heckathorn, D.D. (1980). A unified model for bargaining and conflict. Behavioral Science, 25, 261-284. Heckathorn, D.D. (1983a). Extensions of power-dependence theory: The concept of resistance. Social Forces, 61, 1206-1231. Heckathorn, D.D. (1983b). Valid and invalid interpersonal comparisons: response to Emerson, Cook, Gillmore and Yamagishi. Social Forces, 61, 1248-1259. Heckathorn, D.D. (1984). A formal theory of social exchange: process and outcome. Current Perspectives in Social Theory, 5, 145-180. Heckathorn, D.D. (1985). Power and trust in social exchange. Advances in Group Processes, 2, 143-166. Heckathorn, D.D., & Carlson, R. (1981). A possibility result concerning n-person bargaining games (Tech. Rep. No. 6). Kansas City: University of Missouri, Department of Mathematics. Hershey, J.C., Kunreuther, H.C., & Schoemaker, P.J.H. (1982). Sources of bias in assessment procedures for utility functions. Management Science, 28, 936–953. Hershey, J.C., & Schoemaker, P.J.H. (1985). Probability versus certainty equivalence methods in utility measurement: Are they equivalent? Management Science, 31, 1213– 1231. Hildenbrand, W., & Kirman, A.P. (1988). Equilibrium analysis. Amsterdam, Netherlands: North-Holland. Holt, C.A. (1995). Industrial organization: a survey of laboratory research. In J.H. Kagel, & A.E. Roth (Eds.), The handbook of experimental economics (pp. 349-443). Princeton, N.J.: Princeton University Press. Homans, G.C. (1958). Social behavior as exchange. Journal of American Sociology, 62, 597606. Homans, G.C. (1974). Social behavior: Its elementary forms. New York: Harcourt Brace Jovanovich. Huber, P.J. (1967). The behavior of maximum-likelihood estimates under non-standard conditions. Proceedings of the Fifth Berkely Symposium on Mathematical Statistics and Probability, 1, 221-233. Isaac, R.M., & James, D. (2000). Just who are you calling risk averse? Journal of Risk and Uncertainty, 20, 177-187. Johnson, E. & Schkade, D. (1989). Bias in utility assessments: Further evidence and explanations. Management Science, 35, 406–424. Kagel, J.H. (1995). Auctions: a survey of experimental research. In J.H. Kagel, & A.E. Roth (Eds.), The handbook of experimental economics (pp. 501-585). Princeton, N.J.: Princeton University Press. Kahneman, D. (1988). Experimental economics: a psychological perspective. In R. Tietz, W.

References

201

Albers, & R. Selten (Eds.), Bounded rational behavior in experimental games and markets. Proceedings of the Fourth Conference on Experimental Economics (pp. 11-18). Berlin: Springer-Verlag. Kahneman, D., Knetsch, J.L., & Thaler, R.T. (1987). Fairness and the assumptions of economics. In R.M. Hogarth, & M.W. Reder (Eds.), Rational choice. The contrast between economics and psychology (pp. 101-116). Chicago: University of Chicago Press. Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47, 263-291. Kahneman, D., & Tversky, A. (1984). Choices, values and frames. American Psychologist, 39, 341-350. Kalai, E. (1977). Proportional solutions to bargaining situations: interpersonal utility comparisons. Econometrica, 45, 1623-1630. Kalai, E., & Smorodinsky, M. (1975). Other solutions to Nash’s bagaining problem. Econometrica, 43, 513-518. Kappelhoff, P. (1990). Power in exchange systems: A new look at the Coleman-model of collective action. Unpublished manuscript. Kihlstrom, R.E., Roth, A.E, & Schmeidler, D. (1981). Risk aversion and solutions to Nash’s bargaining problem. In O. Moeschlin, & D. Pallaschke (Eds.), Game theory and mathematical economics (pp. PAGINAS). Amsterdam: North Holland. Kollock, P. (1994). The emergence of exchange structures: an experimental study of uncertainty, commitment, and trust. American Journal of Sociology, 100, 313-345. Kreps, D.M. (1990). A course in microeconomic theory. New York: Harvester Wheatsheaf. Krzystofowicz, R., & Duckstein, L. (1980). Assessment errors in multiattribute utility functions. Organizational Behavior and Human Performance, 26, 326-348. Laughhunn, D.J., Payne, J.W., & Crum, R. (1980). Managerial risk preferences for belowtarget returns. Management Science, 26, 1238–1249. Lawler, E.J., & Ford, R. (1995). Bargaining and influence in conflict situations. In K.S. Cook, G.A. Fine, & J.S. House (Eds.), Sociological perspectives on social psychology (pp. 236-256). Boston, MA: Allyn and Bacon. Lawler, E.J., Ridgeway, C., & Markovsky, B. (1993). Structural social psychology and the micro-macro problem. Sociological Theory, 11, 268-290. Lawler, E.J., & Thye, S.R. (1999). Bringing emotions into social exchange theory. Annual Review of Sociology, 25, 217-244. Lawler, E.J., & Yoon, J. (1993). Power and the emergence of commitment behavior in negotiated exchange. American Sociological Review, 58, 465-481. Lawler, E.J., & Yoon, J. (1996). Commitment in exchange relations: test of a theory of relational cohesion. American Sociological Review, 61, 89-108. Lawler, E.J., & Yoon, J. (1998). Network structure and emotion in exchange relations. American Sociological Review, 63, 871-894. Lawler, E.J., Yoon, J., Baker, M.R., & Large, M.D. (1995). Mutual dependence and gift giving in exchange relations. Advances in group processes, 12, 271-298. Ledyard, J.-O. (1995). Public goods: A survey of experimental research. In Kagel, J.H., & Roth, A.E. (Eds.), The handbook of experimental economics (pp. 111-194). Princeton, NJ:

202 Princeton University Press. Leik, R.K. (1992). New directions for network exchange theory: strategic manipulation of network linkages. Social Networks, 14, 309-323. Lévi-Strauss, C. (1949). Les structures élémentaires de la parenté. Paris: Presses Universitaires de France. Libby, R., & Fishburn, P.C. (1977). Behavioral models of risk taking in business decisions: A survey and evaluation. Journal of Accounting Research, 15, 272–292. Lichtenstein, S., & Slovic, P. (1971). Reversals of preference between bids and choices in gambling decisions. Journal of Experimental Psychology, 89, 46–55. Liebrand, W.B.G., and McClintock, C.G. (1988). The ring measure of social values: a computerized procedure for assessing individual differences in information processing and social value orientation. European Journal of Personality, 2, 217-230. Lindenberg, S. (1985). An assessment of the new political economy: it’s potential for the social sciences and sociology in particular. Sociological Theory, 3, 99-114. Lindenberg, S. (1992). The method of decreasing abstraction. In J.S. Coleman, & T.J. Fararo (Eds.), Rational choice theory. Advocacy and critique (pp. 3-20). Newbury Park: Sage. Lindenberg, S. (1998). The influence of simplification on explananda: phenomenon-centered theories in the social sciences. In H-P. Blossfeld, & G. Prein (Eds.), Rational choice theory and large-scale analysis. Boulder, CO: Westview Press. Lindenberg, S. (2001). Social rationality versus rational egoism. To appear in J. Turner (Ed.) Handbook of sociological theory. Plenum. Lovaglia, M.J., Skvoretz, J., Markovsky, B., & Willer, D. (1995). Assessing fundamental power differences in exchange networks: Iterative GPI. Current Research in Social Psychology, 1, 8-15. Lovaglia, M.J., Skvoretz, J., Willer, D., & Markovsky, B. (1995). Negotiated exchanges in social networks. Social Forces, 74, 123-155. Luce, R. D., & Fishburn, P.C. (1991). Rank- and sign-dependent linear utility models for finite first-order gambles. Journal of Risk and Uncertainty, 4, 29–59. Luce, R.D., & Raiffa, H. (1957). Games and decisions. New York: Wiley. Malinowski, B. (1922). Argonauts of the western Pacific. New York: E.P. Dutton. Markovsky, B. (1987). Toward multilevel sociological theories: Simulations of actor and network effects. Sociological Theory, 5, 100-115. Markovsky, B. (1992). Network exchange outcomes: limits of predictability. Social Networks, 14, 267-286. Markovsky, B. (1995). Developing an exchange network simulator. Sociological Perspectives, 38, 519-545. Markovsky, B. (1997). Network games. Rationality and Society, 9, 67-90. Markovsky, B., Skvoretz, J., Willer, D., Lovaglia, M.J., & Erger, J. (1993). The seeds of weak power: An extension of network exchange theory. American Sociological Review, 58, 197-209. Markovksy, B., Willer, D., & Patton, T. (1988). Power relations in exchange networks. American Sociological Review, 53, 220-236.

References

203

Markowitz, H.M. (1952). The Utility of Wealth. Journal of Political Economy, 60, 151–158. Marsden, P.V. (1981). Introducing influence processes into a system of collective decisions. American Journal of Sociology, 86, 1203-1235. Marsden, P.V. (1983). Restricted access in networks and models of power. American Journal of Sociology, 4, 686-717. Marsden, P.V. (1987). Elements of interactor independence. In K.S. Cook (Ed.), Social Exchange Theory (pp. 130-148). Newbury Park, CA: Sage. Marshall, A. (1920). Principles of Economics: an introductory volume (8th ed.). London: MacMillan. McClintock, C.G. & Liebrand, W.B.G. (1988). Role of interdependence structure, individual value orientation, and another’s strategy in social decision making: A transformational analysis. Journal of personality and social psychology, 55, 396-409. McCord, M.& De Neufville, R. (1986). Lottery equivalents: Reduction of the certainty effect problem in utility assessment. Management Science, 32, 56–60. Messick, D.M., & McClintock, C.G., (1968). Motivational basis of choice in experimental games. Journal of experimental social psychology, 4, 1-25. Michaels, J.W., & Wiggins, J.A. (1976). Effects of mutual dependency and dependency assymetry on social exchange. Sociometry, 39, 368-376. Michener, H.A., Cohen, E.D., & Sørensen, A.B. (1975). Social exchange: predicting interpersonal outcomes in four-event, three person systems. Journal of Personality and Social Psychology, 32, 283-293. Michener, H.A., Cohen, E.D., & Sørensen, A.B. (1977). Social exchange: predicting transactional outcomes in five-event, four person systems. American Sociological Review, 42, 522-535. Mizruchi, M.S., & Potts, B.B. (1998). Centrality and power revisited: actor success in group decision making. Social Networks, 20, 353-387. Molm, L.D. (1994). Dependence and risk: transforming the structure of social exchange. Social Psychology Quarterly, 57, 163-176. Molm, L.D. (1997). Coercive power in social exchange. Cambridge: Cambridge University Press. Molm, L.D., Peterson, G., Takahashi, N. (1999). Power in negotiated and reciprocal exchange. American Sociological Review, 64, 876-890. Molm, L.D., Quist, T.M., & Wisely, P.A. (1993). Reciprocal justice and strategies of exchange. Social Forces, 72, 19-43. Molm, L.D, Quist, T.M., & Wisely, P.A. (1994). Imbalanced structures, unfair strategies: power and justice in social exchange. American Sociological Review, 59, 98-121. Molm, L.D., Takahashi, N., & Peterson, G. (2000). Risk and trust in social exchange: an experimental test of a classical proposition. American Journal of Sociology, 105, 13961427. Molm, L.D., & Wiggins, J.A. (1979). A behavioral analysis of the dynamics of social exchange in the dyad. Social Forces, 57, 1157-1179. Morley, I., & Stephenson, G. (1977). The social psychology of bargaining. London: Allen & Unwin.

204 Murnighan, J.K., & Roth, A.E. (1983). Expecting continued play in prisoner's dilemma games. Journal of Conflict Resolution, 27, 279-300. Nash, J.F. (1950). The bargaining problem. Econometrica, 18, 155-162. Nash, J.F. (1953). Two-person cooperative games. Econometrica, 21, 128-140. Nydegger, R.V., & Owen, G. (1975). Two-person bargaining: An experimental test of the Nash axioms. International Journal of Game Theory, 3, 239-249. Ochs, J., & Roth, A.E. (1989). An experimental study of sequential bargaining. The American Review, 79, 355-384. Officer, R.R., & Halter, A.N. (1968). Utility analysis in a practical setting. American Journal of Agricultural Economics, 50, 257–277. Osborne, M.J., & Rubinstein, A. (1990). Bargaining and markets. San Diego: Academic Press. Parker, S., & Schneider, B. (1988). Conjoint scaling of the utility of money using paired comparisons. Social Science Research, 17, 277–286. Patton, T, & Willer, D. (1990). Connection and power in centralized exchange networks. Journal of Mathematical Psychology, 16, 31-49. Pen. J. (1952). A general theory of bargaining. Journal of American Economic Review, 42, 24-42. Popper, K.R. (1959). The logic of scientific discovery. London: Hutchinson. Pratt, J.W. (1964). Risk aversion in the small and in the large. Econometrica, 32, 122–136. Quattrone, G.A., & Tversky, A. (1988). Contrasting rational and psychological analyses of political choice. American Political Science Review, 82, 719–736. Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behaviour and Organization, 3, 323-343. Raiffa, H. (1953). Arbitration schemes for generalized two-person games. In H.W. Kuhn, & A. Tucker (Eds.), Contribution to the theory of games. II: Annals of mathematics studies. Princeton, N.J.: Princeton University Press. Rapoport, A., & Chammah, A.M. (1965). Prisoner’s dilemma. A study in conflict and cooperation. Ann Arbor: The University of Michigan Press. Raub, W. (1984). Rationale Akteure, institutionelle Regelungen und Interdependenzen. Untersuchungen zu einer erklärenden Soziologie auf strukturell-individualistischer Grundlage. Frankfurt am Main: Lang. Raub, W., & Snijders, C. (1997). Gains, losses, and cooperation in social dilemmas and collective action: the effects of risk preferences. Journal of Mathematical Sociology, 22, 263-291. Raub, W., & Voss, T. (1981). Individuelles Handeln und gesellschaftliche Folgen. Das individualistische Programm in den Sozialwissenschaften. Darmstadt: Luchterhand. Roth, A.E. (1979a). An impossibility result concerning n-person bargaining games. International Journal of Game Theory, 8, 133-164. Roth, A.E. (1979b). Proportional solutions to the bargaining problem. Econometrica, 47, 775-778. Roth, A.E. (1983). Toward a theory of bargaining: an experimental study in economics.

References

205

Science, 220, 687-691. Roth, A.E. (1995). Introduction to experimental economics. In J.H. Kagel, & A.E. Roth (Eds.), The handbook of experimental economics (pp. 3-109). Princeton, N.J.: Princeton University Press. Roth, A.E., & Malouf, M.W.K. (1979). Game theoretic models and the role of information in bargaining: an experimental study. Psychological Review, 86, 574-594. Roth, A.E., & Murnighan, J.K. (1978). Equilibrium behavior and repeated play of the prisoner's dilemma game. Journal of Mathematical Psychology, 17, 189-198. Roth, A.E., & Murnighan, J.K. (1982). The role of information in bargaining: An experimental study. Econometrica, 50, 1123-1142. Roth, A.E., Murnighan, J.K., & Shoumaker, F. (1988). The deadline effect in bargaining: Some experimental evidence. American Economic Review, 78, 806-823. Roth, A.E., & Shoumaker, F. (1983). Expectations and reputations in bargaining: An experimental study. The American Economic Review, 73, 362-372. Rubin, J.Z., & Brown, B.R. (1975). The social psychology of bargaining and negotiation. New York: Academic Press. Rubinstein, A. (1982). Perfect equilibrium in a bargaining model. Econometrica, 50, 97-109. Russel, T., & Thaler, R.H. (1985). The relevance of quasi rationality in competitive markets. The American Economic Review, 75, 1071-1082. Samuelson, P.A. (1937). A note on measurement of utility. Review of Economic Studies, 4, 155–161. Savage, L.J. (1954). The Foundations of Statistics. New York: Wiley. Schellenberg, J.A. (1988). A comparitive test of three models for solving “the bargaining problem”. Behavioral Science, 33, 81-96. Schelling, T.C. (1978). Micromotives and Macrobehavior. New York: Norton. Shoemaker, P.J.H. (1982). The expected utility model: Its variants, purposes, evidence, and limitations. Journal of Economic Literature, 20, 529-563. Shubik, M. (1982). Game theory in the social sciences: concepts and solutions. Cambridge, MA: MIT Press. Shubik, M. (1984). Game theory in the social sciences: a game-theoretical approach to political economy. Cambridge, MA: MIT Press. Siegel, S., & Fouraker, L. (1960). Bargaining and group decision making: experiments in bilateral monopoly. New York: McGraw-Hill. Simon, H.A. (1987). Rationality in psychology and economics. In R.M. Hogarth, & M.W. Reder (Eds.), Rational choice. The contrast between economics and psychology (pp. 2540). Chicago: University of Chicago Press. Simpson, B., & Willer, D. (1999). A new method for finding power structures. In Willer (Ed.), Network Exchange Theory (pp. 270-284). Westport, CT: Praeger. Skvoretz, J., & Burkett, T. (1994). Information and the distribution of power in exchange networks. Journal of Mathematical Sociology, 19, 263-278. Skvoretz, J., & Fararo, T.J. (1992). Power and network exchange: an essay toward theoretical unification. Social Networks, 14, 325-344.

206 Skvoretz, J., & Lovaglia, M.J. (1995). Who exchanges with whom: Structural determinants of exchange frequency in negotiated exchange networks. Social Psychology Quarterly, 58, 163-177. Skvoretz, J., & Willer, D. (1991). Power in exchange networks: Setting and structural variations. Social Psychology Quarterly, 54, 224-238. Skvoretz, J., & Willer, D. (1993). Exclusion and power: a test of four theories of power in exchange networks. American Sociological Review, 58, 801-818. Skvoretz, J., Willer, D., & Fararo, T.J. (1993). Toward models of power development in exchange networks. Sociological Perspectives, 36, 95-115. Skvoretz, J., & Zhang, P. (1997). Actors’ responses to outcomes in exchange networks. Sociological Perspectives, 40, 183-197. Slovic, P. & Lichtenstein, S. (1983). Preference reversal: A broader perspective. American Economic Review, 73, 596–605. Smidts, A. (1997). The relationship between risk attitude and strength of preference: A test of intrinsic risk attitude. Management Science, 43, 357–370. Smith, V.L. (1962). An experimental study of competitive market behavior. The Journal of Political Economy, 70, 111-137. Smith, V.L. (1982). Microeconomic systems as an experimental science. American Economic Review, 72, 923-955. Smith, V.L., Williams, A.W., Bratton, W.K., & Vannoni, M.G. (1982). Competitive market institutions: double auctions vs. sealed bid-offer auctions. American Economic Review, 72, 58-77. Snijders, C. (1996) Trust and commitments. Amsterdam: Thesis Publishers. Snijders, C., & Raub, W. (1998). Revolution and risk. Pardoxical consequences of risk aversion in interdependent situations. Rationality and Society, 10, 405-425. Starmer, C. & Sugden, R. (1989). Violations of the independence axiom in common ratio problems: An experimental test of some competing hypotheses. Annals of Operations Research, 19, 79–101. Stigler, G.J. (1950). The development of utility theory: I; II. Journal of Political Economy, 58, 307–327; 373–396. Reprinted in Alfred N. Page (1968), Utility Theory: A Book of Readings (pp. 55-119). New York: Wiley. Stokman, F.N., & van Assen, M.A.L.M., van der Knoop, J., & van Oosten, R.C.H. (2000). Strategic decision making. Advances in Group Processes, 17, 131-153. Stokman, F.N., & Van Oosten, R. (1994). The exchange of voting positions: An objectoriented model of policy networks. In B. Bueno de Mesquita, & F.N. Stokman (Eds.), European community decision making. Models, applications, and comparisons (pp. 105127). New Haven: Yale University Press. Stolte, J.F. (1990). Power processes in structures of dependence and exchange. Advances in group processes, 7, 129-150. Stolte, J.F., & Emerson, R.M. (1977). Structural inequality: position and power in exchange structures. In R.L. Hamblin, & J.H. Kunkel (Eds.), Behavioral Theory in Sociology (pp. 117-138). New Brunswick: Transaction Books. Swalm, R.O. (1966). Utility Theory. Insights into risk taking. Harvard Business Review, 47,

References

207

123–136. Szmatka, J., & Mazur, J. (1998). Power distribution in conflict networks: an extension of elementary theory to conflict networks. Advances in group processes, 15, 187-211. Szmatka, J., Skvoretz, J., Sozanski, T., & Mazur, J. (1998). Conflict in networks. Sociological Perspectives, 41, 49-66. Szmatka, J., & Willer., D. (1995). Exclusion, inclusion, and compound connection in Exchange networks. Social Psychology Quarterly, 58, 123-132. Szpiro, G.G. (1986). Measuring risk aversion: An alternative approach. Review of Economics and Statistics, 68, 156−159. Taylor, M. (1987). The possibility of cooperation. Cambridge: Cambridge University Press. Thaler, R.H. (1987). The psychology and economics conference handbook: Comments on Simon, on Einhorn and Hogarth, and on Tversky and Kahneman. In R.M. Hogarth, & M.W. Reder (Eds.), Rational choice. The contrast between economics and psychology (pp. 95-100). Chicago: University of Chicago Press. Thaler, R.H., & Johnson, E.J. (1990). Gambling with the house money and trying to break even: The effects of prior outcomes on risky choices. Management Science, 36, 643-660. Thibaut, J., & Kelley, H.H. (1959). The social psychology of groups. New York: John Wiley and Sons. Thye, S.R. (2000). A status-value theory of power in exchange relations. American Sociological Review, 65, 407-432. Thye, S.R., & Lovaglia, M.J. (1997). Responses to social exchange and social exclusion in networks. Social Forces, 75, 1031-1047. Tietz, R. (1988). Experimental economics: Ways to model bounded rational bargaining behavior – Introductory remarks to the fourth conference on experimental economics. In R. Tietz, W. Albers, & R. Selten (Eds.), Bounded rational behavior in experimental games and markets. Proceedings of the Fourth Conference on Experimental Economics (pp. 310). Berlin: Springer-Verlag. Tversky, A. (1975). A critique of expected utility theory. Descriptive and normative considerations. Erkenntnis, 9, 163-173. Tversky, A. & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185, 1124–1131. Tversky, A., & Kahneman, D. (1981). The framing of decisions and the psychology of choices. Science, 211, 453-458. Tversky, A., & Kahneman, D. (1987). Rational choice and the framing of decisions. In R.M. Hogarth, & M.W. Reder (Eds.), Rational choice. The contrast between economics and psychology (pp. 67-94). Chicgo: University of Chicago Press. Tversky, A., & Kahneman, D. (1991). Loss aversion in riskless choice: a reference dependent model. Quarterly Journal of Economics, 106, 1039-1061. Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5, 297-323. Tversky, A., Slovic, P., & Kahneman, D. (1990). The causes of preference reversal. American Economic Review, 80, 204–217. Van Assen, M.A.L.M. (1996). Eliciting utilities when probabilities are distorted and eliciting

208 decision weights independently from outcome evaluations. Unpublished master's thesis. Nijmegen, Netherlands: Universtity of Nijmegen. Van Assen, M.A.L.M. (1998). Effects of individual decision theory assumptions on predictions of cooperation in social dilemmas. Journal of Mathematical Sociology, 23, 143-153. Van Assen, M.A.L.M., & Snijders, C. (2001, accepted for publication). Effects of risk preferences in social dilemmas: a game-theoretical analysis and evidence from two experiments. In Ramzi, Suleiman, Budescu, Fischer & Messick (Eds.), Contemporary Psychological Research on Social Dilemmas. Van Lange, P.A.M. (1999). The pursuit of joint outcomes and equality in outcomes: An integrative model of social value orientation. Journal of Personality and Social Psychology, 2, 337-349. Van Lange, P.A.M., Liebrand, W.B.G., Messick, D.M., & Wilke, H.A.M. (1992). Social dilemmas: the state of the art. Introduction and literature review. In W.B.G. Liebrand, D.M. Messick, & H.A.M. Wilke (Eds.), Social dilemmas. Theoretical issues and research findings (pp. 3-28). Oxford, Perganon Press. Viscusi, W. K. (1989). Prospective reference theory: Toward an explanation of the paradoxes. Journal of Risk and Uncertainty, 2, 235–264. Von Hecker, U. (1998). Resource-based power differentials and value-increasing behavior options. Unpublished manuscript. Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton: Princeton University Press. Wakker, P.P. (1994). Separating marginal utility and probabilistic risk aversion. Theory and Decision, 36, 1–44. Wakker, P.P., & Deneffe, D. (1996). Eliciting von Neumann-Morgenstern utilities when probabilities are distorted or unknown. Management Science, 42, 1131-1150. Walder, A.G. (1994). Implications of loss avoidance for theories of social movements. Mimeo, Harvard University. Walker, J.M., Smith, V.L., & Cox, J.C. (1990). Inducing risk neutral preferences: An examination in a controlled market environment. Journal of Risk and Uncertainty, 3, 5-24. Wasserman, S., & Faust, K. (1994). Social network analysis: methods and applications. Cambridge: Cambridge University Press. Weber, E.U., & Milliman, R.A. (1997). Perceived risk attitudes: relating risk perception to risky choice. Management Science, 43, 123-144. Weesie, J. (1994). Social orientations in symmetric 2x2 games. ISCORE paper, no. 17. Utrecht University. Whitmeyer, J.M. (1994). Social structure and the actor: the case of power in exchange networks. Social Psychology Quarterly, 3, 177-189. Whitmeyer, J.M. (1997a). The power of the middleman – a theoretical analysis. Journal of Mathematical Sociology, 22, 59-90. Whitmeyer, J.M. (1997b). Applying general equilibrium analysis & game theory to exchange networks. Current Research in Social Psychology, 2, 13-23. Whitmeyer, J.M. (1999a). Interest-network structures in exchange networks. Sociological

References

209

Perspectives, 42, 23-47. Whitmeyer, J.M. (1999b). Convex preferences and power inequality in exchange networks: an experimental study. Rationality and Society, 11, 419-442. Whitmeyer, J.M. (2000). Measuring power in exchange networks. Unpublished manuscript. Willer, D. (1984). Analysis and composition as theoretic procedures. Journal of Mathematical Psychology, 10, 241-269. Willer, D. (1986). Vulnerability and the location of power positions: Comment on Cook, Emerson, Gillmore, and Yamagishi. American Journal of Sociology, 92, 441-444. Willer, D. (1987). Theory and the empirical investigation of social structures. New York: Gordon and Breach Science Publishers. Willer, D. (1992). Predicting power in exchange networks: a brief history and introduction to the issues. Social Networks, 14, 187-211. Willer, D. (Ed.) (1999). Network exchange theory. Westport, CT: Praeger Press. Willer, D., & Anderson, B. (1981). Networks, exchange and coercion: The elementary theory and its applications. Elsevier, North Holland. Willer, D., Lovaglia, M.J., & Markovsky, B. (1997). Power and influence: A theoretical bridge. Social Forces, 76, 571-603. Willer, D. & Markovsky, B. (1993). Elementary theory: Its development and research program. In J.Berger, & M. Zelditch Jr. (Eds.), Theoretical research programs: Studies in the growth of theory (pp. 323-363).Stanford, CA: Stanford University Press. Willer, D., Markovsky, B., & Patton, T. (1989). Power structures: derivations and applications of elementary theory. In J. Berger, M. Zelditch Jr., & B. Anderson (Eds.). Sociological Theories in Progress, 3 (pp. 313-353). Boston: Houghton-Mifflin. Willer, D., & Patton, T. (1987). The development of network exchange theory. Advances in Group Processes, 4, 199-242. Willer, D., & Skvoretz, J. (1997a). Games and Structures. Rationality and Society, 9, 5-35. Willer, D., & Skvoretz, J. (1997b). Network connection and exchange ratios: Theory, predictions, and experimental tests. Advances in Group Processes, 14, 199-234. Willer,D., & Szmatka, J. (1993). Cross-national experimental investigations of elementary theory: Implications for the generality of the theory and the autonomy of social structure. Advances in Group Processes, 10, 37-81. Willer, R., & Willer, D. (2000) Exploring dynamic networks: hypotheses and conjectures. Social Networks, 22, 251-272. Wippler, R., & Lindenberg, S. (1987). Collective phenomena and rational choice. In J.C. Alexander, B. Giesen, R. Muench, & N.J. Smelser (Eds.), The micro-macro link (pp. 135151). Berkely: University of California Press. Wolf, C., & Pohlman, L. (1983). The recovery of risk preferences from actual choices. Econometrica, 51, 843–850. Wu, G., & Gonzalez, R. (1996). Curvature of the probability weighing function. Management Science, 42, 1676-1690. Yamagishi, T., & Cook, K.S. (1990). Power relations in exchange networks: A comment on network exchange theory. American Sociological Review, 55, 297-300.

210 Yamagishi, T., & Cook, K.S. (1993). Generalized exchange and social dilemmas. Social Psychology Quarterly, 56, 235-248. Yamagishi, T., Gillmore, M.R., & Cook, K.S. (1988). Network connections and the distribution of power in exchange networks. American Journal of Sociology, 93, 833-851. Yamaguchi, K. (1996). Power in networks of substitutable and complementary exchange relations: a rational-choice model and an analysis of power centralization. American Sociological Review, 61, 308-332. Yamaguchi, K. (2000). Power in mixed exchange networks: A rational choice model. Social Networks, 22, 93-121. Young, O.R. (1975). Bargaining: formal theories of negotiation. Urbana: University of Illinois Press. Zeuthen, F. (1930). Problems of monopoly and economic welfare. London: Routhledge & Sons.

Samenvatting (Summary in Dutch) Verhandelingen over modellen van actoren in ruilnetwerken en sociale dilemma’s 1.

Inleiding

Een belangrijke taak van de economie en de sociologie is om fenomenen op het macroniveau te verklaren of te voorspellen. Fenomenen op het macroniveau zoals bestudeerd in de sociologie en de economie zijn bijvoorbeeld revoluties en prijzen van goederen. Deze en andere macroniveau fenomenen kan men proberen te verklaren en te voorspellen door middel van een analyse van onderliggende processen die betrekking hebben op de actoren (individuen of organisaties) op het microniveau. Deze analyse wordt veelvuldig uitgevoerd in de sociale wetenschappen zoals economie, politicologie en sociologie, met behulp van de rationele keuzetheorie. De rationele keuze theorie is ook het uitgangspunt van het merendeel van het onderzoek dat gerapporteerd is in dit proefschrift. De basisaanname van de theorie is dat een actor dat gedrag vertoont dat zijn utiliteit maximaliseert (zijn belangen het best behartigt). Onderzoekers die de rationele keuzetheorie aanhangen beargumenteren vaak dat het belangrijk is om de microniveau component in de verklaring van het macroniveau fenomeen zo eenvoudig mogelijk te houden. Dit komt er op neer dat deze onderzoekers (i) vermijden het model van de actor op het microniveau uit te breiden als een analyse gebaseerd op dit model leidt tot onnauwkeurige voorspellingen van uitkomsten op macroniveau, en (ii) tevreden zijn als hun analyse leidt tot nauwkeurige voorspellingen van uitkomsten op macroniveau, zonder te weten of het model een goede beschrijving geeft van de onderliggende microprocessen. Echter, in Hoofdstuk 1 wordt beargumenteerd dat het model van de actor op het microniveau meer aandacht verdient. Op basis van de twee hierboven weergegeven observaties zijn twee richtlijnen geformuleerd die aangeven wanneer het model van de actor dient te worden uitgebreid: (I)

Als de voorspelling van een uitkomst op macroniveau onvoldoende nauwkeurig is en wanneer de uitbreiding van het model van de actor substantiële effecten heeft op voorspellingen van die uitkomst.

(II)

Voor het begrijpen van uitkomsten op macroniveau, want een model van de actor waarmee onderliggende microniveau processen goed worden beschreven is hiervoor een noodzakelijke voorwaarde.

In dit proefschrift worden deze richtlijnen toegepast om twee fenomenen op macroniveau, die veel aandacht hebben gekregen in de sociale wetenschappen in het algemeen en in de sociologie in het bijzonder, beter te voorspellen en te begrijpen. De eerste richtlijn wordt toegepast om uitkomsten van sociale dilemma’s (Hoofdstuk 3 tot en met Hoofdstuk 5) beter te voorspellen. De tweede richtlijn wordt toegepast om uitkomsten in ruilnetwerken (Hoofdstuk 6 en Hoofdstuk 7) beter te begrijpen.

212 Een sociaal dilemma is een situatie waar gemeenschappelijke belangen en individuele belangen van actoren in de situatie met elkaar conflicteren. Het meest bekende voorbeeld van een sociaal dilemma is het zogenaamde ‘prisoner’s dilemma’ (dilemma van gevangenen) of ‘PD’. Een illustratie van een PD is de situatie waar twee actoren een keuze hebben tussen de volgende twee opties. De eerste optie, ook wel coöperatie genoemd, is een investering die f10,- kost voor degene die de investering maakt en f20,- oplevert voor de ander. De tweede optie is dat geen investering wordt gemaakt. Gaan we er van uit dat de keuzes eenmalig zijn dan is het in het individuele belang van beide actoren geen investering te maken. Immers, door te investeren, in het algemeen coöpereren genoemd, wordt een verlies geleden van f10,-. Echter, het is in het gemeenschappelijke belang van beiden om allebei te coöpereren. Dan wint men namelijk allebei f10,- (20 – 10), wat meer is dan wanneer men allebei niet investeert en quitte speelt. In Hoofdstuk 3 tot en met Hoofdstuk 5 wordt onderzocht in welke condities actoren coöpereren in onbepaald herhaalde PDs die steeds met een voor beide actoren bekende constante kans wordt herhaald. Veel eerdere onderzoeken naar sociale dilemma’s nemen aan dat de waardering of utiliteit van uitkomsten equivalent is aan de eigen uitkomsten in het sociaal dilemma, zoals bijvoorbeeld geld. Echter, empirisch onderzoek naar individuele besluitvorming heeft laten zien dat deze equivalentie niet opgaat en dat individuen verschillen met betrekking tot de concaviteit van hun utiliteiten. Concave utiliteit correspondeert met een afname in extra utiliteit als gevolg van een constante toename in de uitkomst, naarmate de uitkomst hoger wordt (in wiskundige termen correspondeert concaviteit met een negatieve tweede afgeleide van utiliteit). Bij convexe utiliteit neemt de extra utiliteit toe naarmate de uitkomst hoger wordt (positieve tweede afgeleide van utiliteit). Uit de speltheorie, een belangrijke stroming binnen de rationele keuzetheorie, kan worden afgeleid dat condities voor coöperatie in een PD gunstiger worden naarmate de utiliteit in het domein van de uitkomsten van het PD meer concaaf wordt. Daarom, daarmee richtlijn (I) opvolgend, hebben we in Hoofdstuk 3 tot en met Hoofdstuk 5 de precieze relaties afgeleid en getoetst tussen de utiliteit van actoren en hun coöperatie in verschillende onbepaald herhaalde PDs. Om deze relatie te kunnen toetsen is het noodzakelijk de concaviteit van de utiliteit van actoren te meten. Eén van de methoden om utiliteit te meten, de zogenaamde ‘tradeoff’ methode, hebben we getoetst en gerelateerd aan andere methoden in Hoofdstuk 2. Een ruilnetwerk is een verzameling van ruilrelaties tussen verschillende paren van actoren. Een ruilrelatie wordt voorgesteld als een mogelijkheid van twee actoren om na samenwerken allebei een winst te behalen. Een voorbeeld van een ruilnetwerk met een eenvoudige structuur is de situatie waarin een handelaar in tweedehands auto’s kan kiezen zijn auto te verkopen aan twee potentiële kopers. Uiteindelijk zal de handelaar de auto met winst verkopen aan de hoogste bieder die in de regel echter ook winst maakt door uiteindelijk minder te betalen dan de prijs die hij over heeft voor de auto. Uitkomsten, wie ruilt met wie tegen welke ruilvoet, kunnen goed worden voorspeld door een groot aantal theorieën. Echter, deze theorieën zijn gebaseerd op fundamenteel verschillende aannames met betrekking tot microniveau processen, zoals het onderhandelingsproces tussen actoren. Slechts een handvol onderzoeken is verricht naar een klein aantal aspecten van onderhandelen tussen actoren in ruilnetwerken. Er is daarom een onvolledig begrip van het effect van de structuur van het ruilnetwerk op het onderhandelen

Samenvatting

213

tussen actoren en hoe dit onderhandelen de uiteindelijke ruiluitkomsten bepaalt. Daarom, daarmee richtlijn (II) opvolgend, hebben we een empirisch onderzoek uitgevoerd in Hoofdstuk 6 om te onderzoeken welke variabelen het onderhandelen tussen actoren bepalen en om de effecten van deze variabelen te schatten en te toetsen. Het model van de actor zoals gebruikt in onderzoek naar ruilen in het algemeen en naar ruilnetwerken in het bijzonder, hebben we op een meer fundamentele manier geanalyseerd in Hoofdstuk 7. In onderzoek naar ruilen wordt een ruilrelatie meestal gerepresenteerd als een mogelijkheid om een gezamenlijke winst te verdelen. In termen van ons voorbeeld, als de waarde van de auto voor de handelaar gelijk is aan f5000,- en voor de hoogste bieder f9000,-, dan wordt de ruilrelatie gereduceerd tot een verdeling van een gezamenlijke buit van f4000,-. In deze representatie, gereduceerde representatie genoemd, is een ruilrelatie een exogeen bepaalde eigenschap van een paar actoren. In Hoofdstuk 7 wordt de niet-gereduceerde representatie uit micro-economisch onderzoek naar ruilhandel opnieuw geïntroduceerd. In de niet-gereduceerde representatie zijn ruilrelaties het endogene en dynamische resultaat van twee eigenschappen van actoren, namelijk hun bezit en hun waardering of utiliteit van hun bezit. In termen van ons voorbeeld, de handelaar heeft een auto met een utiliteit gelijk aan die van f5000,-. De hoogste bieder heeft geen auto maar wel geld en zijn utiliteit van de auto is voor hem gelijk aan die van f9000,-. In Hoofdstuk 7 worden de twee representaties met elkaar vergeleken en de verschillen op een rijtje gezet. Na deze algemene uiteenzetting over de verschillende onderwerpen die in dit proefschrift aan de orde komen, volgt nu een uiteenzetting van de belangrijkste bevindingen in dit proefschrift. De bevindingen worden afzonderlijk beschreven voor elk van de onderwerpen: utiliteitsmeting door middel van de tradeoff methode (Hoofdstuk 2), de relatie tussen utiliteit van actoren en hun gedrag in PDs (Hoofdstuk 3 tot en met Hoofdstuk 5), onderhandelen in ruilnetwerken (Hoofdstuk 6) en twee representaties van ruilen: een overzicht en een vergelijking (Hoofdstuk 7). 2.

Utiliteitsmeting door middel van de tradeoff methode

Het standaardmodel van rationele keuzetheorie van individuele besluitvorming, ‘Expected Utility’ (EU) of verwachte utiliteit, veronderstelt dat actoren dat alternatief kiezen met de hoogste verwachte utiliteit. De utiliteit van een alternatief is een gewogen gemiddelde van de utiliteiten van de uitkomsten van het alternatief met gewichten gelijk aan de kansen op deze uitkomsten. Echter, empirisch onderzoek laat zien dat actoren kansen vertekenen. Dus EU is geen goede beschrijvende theorie van individuele besluitvorming. Een gevolg van kansvertekeningen is dat concave utiliteit (convexe utiliteit) en risico-aversie (risicozoeken) niet langer hetzelfde zijn zoals in EU. Risico-aversie (risicozoeken) wil zeggen dat het gewogen gemiddelde of de verwachte waarde van een alternatief geprefereerd (niet geprefereerd) wordt boven het alternatief zelf. Een ander gevolg is dat de traditionele methoden van utiliteitsmeting die allemaal EU veronderstellen geen zuivere utiliteitsmetingen opleveren. Wakker en Deneffe (1996) ontwikkelden daarom de tradeoff (TO) methode, een methode die het mogelijk maakt om de utiliteiten te meten ondanks eventuele kansvertekeningen. Zij pasten de TO methode toe om utiliteit te meten van winsten, maar niet van verliezen. Het onderzoek beschreven in Hoofdstuk 2 paste drie methoden toe om utiliteit te meten van zowel winsten als verliezen: twee verschillende versies van de TO methode en de

214 bekende ‘certainty-equivalence’ (CE) methode die gevoelig is voor kansvertekening. We vonden bewijs voor S-vormige utiliteit met alle drie de methoden. Dat wil zeggen, de ruime meerderheid van personen had concave utiliteit van winsten en een kleine meerderheid had convexe utiliteit van verliezen. Echter, utiliteiten zoals gemeten door de twee versies van de TO methode waren significant verschillend van elkaar. Men noemt dit een schending van procedure-invariantie. Er werd beargumenteerd dat zowel deze schending als S-vormige utiliteit bewijs zijn dat utiliteitsmeting vooral ontvankelijk is voor de verminderde gevoeligheid van individuen voor toenemende absolute verschillen met het referentiepunt. 3.

De relatie tussen utiliteit van actoren en hun gedrag in PDs

3.1

Theorie

Alle hypothesen in Hoofdstuk 3 tot en met Hoofdstuk 5 worden afgeleid van een theorie die kan worden samengevat in één zin: voorwaarden om te coöpereren in een onbepaald herhaald PD zijn gunstiger naarmate de utiliteit van de twee actoren meer concaaf is in het domein van de uitkomsten van het PD. De theorie werd op twee verschillende manieren geïnterpreteerd: binnen personen en tussen personen. Ten eerste, als utiliteit meer concaaf is in het domein van de uitkomsten van een andere PD dat wordt verkregen door een constante op te tellen bij alle uitkomsten, dan zou de neiging tot coöperatie in het andere PD groter moeten zijn (binnen personen). Ten tweede, als de utiliteit van een persoon meer concaaf is dan dat van een andere persoon, dan zou de neiging tot coöperatie van de andere persoon minder moeten zijn (tussen personen). Drie hypothesen werden geformuleerd over de relatie tussen coöperatie van actoren in de eerste ronde van een PD en hun utiliteit. De hypothesen verschilden met betrekking tot het aantal aannames dat gemaakt werd over utiliteit, waarbij het aantal aannames afneemt met het nummer van de hypothese. De eerste hypothese (H1) ging uit van EU en S-vormige utiliteit met afkeer van verliezen. H1 stelde dat de proportie coöperatie het grootste is in een gemengde PD met zowel positieve als negatieve uitkomsten, kleiner in een positieve PD met alleen positieve uitkomsten en het kleinst in een negatieve PD met alleen negatieve uitkomsten. Voor het toetsen van H1 was geen utiliteitsmeting noodzakelijk. De tweede hypothese (H2) ging ook uit van EU en stelde dat er voor elk PD een positieve relatie bestaat tussen de proportie coöperatie en risico-aversie in het domein van de uitkomsten van het PD. Om de nulhypothese van H2 te toetsen werd risico-aversie gemeten met een traditionele methode van utiliteitsmeting die onzuivere metingen van utiliteit oplevert als EU onjuist is. In de derde hypothese (H3) werd rekening gehouden met eventuele kansvervormingen. H3 stelde dat er voor elk PD een positieve relatie bestaat tussen de proportie coöperatie en de concaviteit van utiliteit in het domein van de uitkomsten van het PD. De TO methode werd toegepast om de nulhypothese van H3 te toetsen. Alle drie de hypothesen werden getoetst in twee experimenten. In Experiment 1, uitgevoerd door Raub en Snijders (1997), werden H1 en H2 getoetst voor negatieve en positieve PDs. Alle drie de hypothesen werden getoetst voor negatieve, gemengde en positieve PDs in Experiment 2, uitgevoerd door Van Assen en Snijders. 3.2

Resultaten

In beide experimenten was volgens de traditionele methode van utiliteitsmeting een meerderheid van de personen risicozoekend voor verliezen, maar ook was een meerderheid

Samenvatting

215

van de personen risicozoekend voor winsten. Deze bevindingen lijken niet in overeenstemming met S-vormige utiliteit. Echter, de zuivere TO methode leidde tot bevindingen die meer in lijn zijn met S-vormige utiliteit. Helaas waren de test-hertest betrouwbaarheden van de utiliteitsmetingen niet hoog en de correlaties tussen de metingen van de verschillende methoden waren zelfs laag. De lage correlaties laten opnieuw zien dat utiliteit afhangt van de toegepaste utiliteitsmeting. In het algemeen werd slechts zwak bewijs gevonden met betrekking tot de hypothesen over de relatie tussen coöperatie van actoren in de eerste ronde van een PD en de vorm van hun utiliteit. H1 werd slechts gedeeltelijk bevestigd; de proportie coöperatie was hoger voor het positieve dan voor het negatieve PD, maar was niet het hoogst voor het gemengde PD. Ook H2 werd gedeeltelijk bevestigd. In Experiment 1 werd een positief effect gevonden van risico-aversie op coöperatie, in Experiment 2 werd een effect van risico-aversie gevonden tussen personen maar niet binnen personen. Tenslotte werd er in het algemeen geen ondersteuning gevonden voor H3, al werd er wel zwak bewijs gevonden voor een positief effect van afkeer tegen verliezen. Hoewel we in eerste instantie verrast waren door het geringe bewijs met betrekking tot de drie hypothesen, waren we toch in staat twee plausibele redenen aan te geven voor dit geringe bewijs. Ten eerste verzwakten de geringe betrouwbaarheden en lage correlaties tussen de utiliteitsmetingen de mogelijke relatie tussen utiliteit en de neiging tot coöperatie aanzienlijk. Ten tweede vonden andere onderzoeken die rechtstreeks de condities voor coöperatie gunstiger maken ook slechts een zwakke relatie tussen de neiging tot coöperatie en utiliteit. In contrast met de relatief slechte voorspellingen met het standaardmodel van rationele keuzetheorie staan de zeer nauwkeurige voorspellingen van de proporties coöperatie in alle rondes van het PD met een statistisch model, gepresenteerd in appendix 3.1. Er werd beargumenteerd dat dit statistisch model zeer geschikt is om te onderzoeken hoe bepaalde eigenschappen van sociale dilemma’s gedrag van actoren beïnvloeden, niet alleen in PDs maar ook in andere sociale dilemma’s. 4.

Onderhandelen in ruilnetwerken

4.1

Theorie

In Hoofdstuk 6 werd voor de eerste keer in het betreffende onderzoeksgebied een poging ondernomen het gehele onderhandelingsproces in ruilnetwerken te verklaren door twee fasen in het proces te onderscheiden; de fase van de eerste aanbieding en de daaropvolgende concessiefase. De concessiefase is noodzakelijk wanneer twee actoren niet meteen tot een overeenstemming komen over de ruilvoet. In de concessiefase kunnen de twee actoren tegemoetkomingen (concessies) aan elkaar doen om alsnog een overeenstemming over de ruilvoet te bereiken. Op basis van voorgaand onderzoek werden hypothesen afgeleid met betrekking tot de effecten op eerste aanbiedingen en concessies van de volgende vier onafhankelijke variabelen: de relatieve structurele macht van een actor in de ruilrelatie als gevolg van de posities van de actoren in het netwerk, ongespecificieerde actorkarakteristieken, onderhandelingstijd, en exclusie en inclusie in de voorgaande mogelijkheid om te ruilen. We veronderstelden dat de hoogte van de eerste aanbieding aan de andere actor in de relatie gemiddeld genomen afnam na inclusie, toenam na exclusie en lager is naarmate de relatieve structurele macht van de aanbieder hoger is. En we veronderstelden

216 dat concessies in een relatie gemiddeld genomen afnemen na inclusie, toenemen na exclusie, lager zijn naarmate de relatieve structurele macht van de aanbieder hoger is en de tijd dat men reeds onderhandelde toeneemt. Tenslotte, omdat actoren verschillen met betrekking tot ongespecificeerde maar relevante actorkarakteristieken, veronderstelden we dat actoren gemiddeld verschillen in hun eerste aanbiedingen en concessies. 4.2

Resultaten

Aangaande eerste aanbiedingen werden er effecten gevonden van exclusie en van de relatieve structurele macht van de actor in de relatie, maar niet van inclusie en actorverschillen. De eerste aanbiedingen van een actor konden verrassend goed worden voorspeld door een bepaalde maat (ESL) van de relatieve macht van de actor. De resultaten suggereerden dat actoren hun eerste aanbiedingen baseren op het aantal ruilrelaties dat ze hebben en dat ze leren van ervaringen bij voorgaande ruilen. Concessies hingen ook af van de relatieve structurele macht van de actor, maar actorverschillen in concessies waren in het algemeen sterker dan verschillen in concessies als gevolg van machtsverschillen tussen actoren. Effecten van inclusie en exclusie op concessies werden niet gevonden. Verrassend was dat er een parabolisch effect van tijd op concessies gevonden werd in netwerken waar actoren weinig verschillen in relatieve structurele macht. De afname van concessies in het begin van de onderhandelingstijd werd verklaard door het feit dat enkele actoren in het begin grote concessies maken en snel ruilen, waarschijnlijk omdat ze bang waren niet betrokken te worden in een ruil. We zijn van mening dat ons begrip is toegenomen over de over de effecten van het onderhandelingsproces in ruilnetwerken op uitkomsten. Maar ook denken we dat het onderhandelingsproces moet worden onderzocht in andere ruilnetwerken om ondubbelzinnige conclusies te kunnen trekken over de rol van leren en de perceptie van actoren van hun structurele macht in het netwerk. 5.

Twee representaties van ruilen: een overzicht en een vergelijking

In de Hoofdstuk 7 werden twee micromodellen van ruilen met elkaar vergeleken, de gereduceerde en de niet-gereduceerde representatie. Vier resultaten werden afgeleid. Ten eerste is er, in tegenstelling tot wat de meeste onderzoekers van ruilen in de sociologie denken, een sterke of zelfs één-op-één relatie tussen oplossingen vanuit de speltheorie en vanuit ruilnetwerktheorieën inzake de onbepaaldheid van ruiluitkomsten. Om precies te zijn, er is een één-op-één relatie tussen ‘power-dependence’ (machtsafhankelijkheids-) theorie en ‘exchange-resistance’ (ruil-resistentie-) theorie en respectievelijk de kernel (zie ook Bonacich en Friedkin, 1998) en de RKS oplossing uit de speltheorie (zie ook Heckathorn, 1980). De andere resultaten, in tegenstelling tot het eerste resultaat, hangen direct samen met verschillen tussen de twee representaties. Het tweede en meest belangrijke resultaat is dat beide representaties niet, zoals men denkt, equivalent zijn. De twee representaties in ons voorbeeld van de autohandelaar zijn theoretisch gezien wel equivalent, maar in Hoofdstuk 7 is aangetoond dat de meest gebruikte gereduceerde representatie in enkele welomschreven situaties geen geldige representatie is van ruilen. Ten derde zijn voor beide representaties voorspellingen van ruiluitkomsten afgeleid uit drie ruiltheorieën, namelijk de core, powerdependence en exchange-resistance theorie. De afleidingen lieten zien dat de voorspellingen het meest van elkaar verschillen als de twee representaties niet equivalent zijn. Tenslotte beargumenteerde ik dat de niet-gereduceerde representatie ook opnieuw moet worden

Samenvatting

217

gebruikt in ander ruilonderzoek. Niet alleen omdat het een geldige representatie geeft van een ruil, maar ook omdat het een meer algemene representatie is in de zin dat het kan worden toegepast in een groot scala van belangrijke kwesties in onderzoek naar ruilen. Als voorbeelden kunnen worden genoemd de evolutie van ruilnetwerken, betrokkenheid in veelgebruikte ruilrelaties, gegeneraliseerde ruilen tussen meer dan twee actoren, en ruilen met externaliteiten of effecten op de utiliteit van actoren die niet betrokken zijn bij de ruil. Ook gaf ik aan dat de niet-gereduceerde representatie kan worden gebruikt voor andere sociale interacties dan ruilen, zoals dwang en conflict. In de toekomst ben ik van plan met de nietgereduceerde representatie onderzoek te verrichten naar de simultane effecten op ruiluitkomsten in netwerken van actoreigenschappen, netwerkstructuur, externaliteiten en netwerkevolutie.