Modeling, Rationality, Morality, and Evolution (Vancouver Studies in Cognitive Science, vol. 7)

MODELING RATIONALITY, MORALITY, AND EVOLUTION Edited by Peter A. Danielson

Vancouver Studies in Cognitive Science is a series of volumes in cognitive science. The volumes will appear annually and cover topics relevant to the nature of the higher cognitive faculties as they appear in cognitive systems, either human or machine. These will include such topics as natural language processing, modularity, the language faculty, perception, logical reasoning, scientific reasoning, and social interaction. The topics and authors are to be drawn from philosophy, linguistics, artificial intelligence, and psychology. Each volume will contain original articles by scholars from two or more of these disciplines. The core of the volumes will be articles and comments on these articles to be delivered at a conference held in Vancouver. The volumes will be supplemented by articles especially solicited for each volume, and will undergo peer review. The volumes should be of interest to those in philosophy working in philosophy of mind and philosophy of language; to those in linguistics in psycholinguistics, syntax, language acquisition and semantics; to those in psychology in psycholinguistics, cognition, perception, and learning; and to those in computer science in artificial intelligence, computer vision, robotics, natural language processing, and scientific reasoning.

VANCOUVER STUDIES IN COGNITIVE SCIENCE Forthcoming volumes Volume 8

Visual Attention Editor, Richard Wright Psychology Simon Eraser University

Volume 9

Colour Perception: Philosophical, Psychological, Artistic, and Computational Perspectives Editor, Brian Funt School of Computing Science Simon Eraser University

SERIES EDITORS General Editor Steven Davis, Philosophy, Simon Fraser University Associate General Editors Kathleen Akins, Philosophy Department, Simon Fraser University Nancy Hedberg, Linguistics, Simon Fraser University Fred Popowich, Computing Science, Simon Fraser University Richard Wright, Psychology, Simon Fraser University

EDITORIAL ADVISORY BOARD Susan Carey, Psychology, Massachusetts Institute of Technology Elan Dresher, Linguistics, University of Toronto Janet Fodor, Linguistics, Graduate Center, City University of New York F. Jeffry Pelletier, Philosophy, Computing Science, University of Alberta John Perry, Philosophy, Stanford University Zenon Pylyshyn, Psychology, Rutgers University Len Schubert, Computing Science, University of Rochester Brian Smith, System Sciences Lab, Xerox Palo Alto Research Center, Center for the Study of Language and Information, Stanford University

BOARD OF READERS William Demopoulos, Philosophy, University of Western Ontario Allison Gopnik, Psychology, University of California at Berkeley Myrna Gopnik, Linguistics, McGill University David Kirsh, Cognitive Science, University of California at San Diego Frangois Lepage, Philosophy, Universite de Montreal Robert Levine, Linguistics, Ohio State University John Macnamara, Psychology, McGill University Georges Rey, Philosophy, University of Maryland Richard Rosenberg, Computing Science, University of British Columbia Edward P. Stabler, Jr., Linguistics, University of California at Los Angeles Susan Stucky, Center for the Study of Language and Information, Stanford University Paul Thagard, Philosophy Department, University of Waterloo

Modeling Rationality, Morality, and Evolution

edited by Peter A. Danielson

New York Oxford OXFORD UNIVERSITY PRESS 1998

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Library of Congress Cataloging-in-Publication Data Modeling rationality, morality, and evolution / edited by Peter Danielson. p. cm. — (Vancouver studies in cognitive science: v. 7) Includes bibliographical references. ISBN 0-19-512549-5 (alk. paper). - ISBN 0-19-512550-9 (pbk.: alk. paper) 1. Ethics. 2. Rational choice theory. 3. Ethics, Evolutionary. 4. Prisoner's dilemma game. I. Danielson, Peter, 1946- . II. Series. BJ1031.M63 1998 110-dc21 98-27140 CIP

Printing 9 8 7 6 5 4 3 2 1 Printed in Canada on acid-free paper

Acknowledgments I would like to thank the Social Sciences and Humanities Research Council of Canada, the Office of the Dean of Arts and the Publications Committee at Simon Fraser University, and the Centre for Applied Ethics at the University of British Columbia for their generous support of the volume and of the Seventh Annual Cognitive Science Conference, Modeling Rational and Moral Agents (Vancouver, Canada, February 1994), at which many of the papers were delivered. Many people helped to organize the conference and prepare the volume. My greatest thanks go to Steven Davis, who first suggested this project and invited me to serve as local chairman of the conference. Steven provided knowledge, support, and superb organizing skills at every point. The VSCS Committee provided help in organizing, advertising and running the conference. Fred Popowich deserves special thanks for helping me to keep in touch with the Internet community. Thanks to Tom Perry, head of the Cognitive Science program, Tanya Beaulieu, who did a remarkable job with local organizing, Lindsey Thomas Martin, who prepared the camera-ready copy from the authors' disks and texts, and Edward Wagstaff, who did the copy-editing and proofreading. I would also like to thank my colleague Leslie Burkholder who helped to plan the conference and my research assistant Chris MacDonald for help with proofreading. Thanks to all who attended the conference and contributed to its lively discussions, and to the authors of the volume for their cheerful co-operation and willingness to get material to me on time. The following publishers were kind enough to allow me to reprint material: Andrew Irvine, "How Braess' Paradox solves Newcomb's Problem" is reprinted from International Studies in the Philosophy of Science, 7:2, with the permission of Carfax Publishing. Paul Churchland, "The Neural Representation of the Social World" is exerpted from chapters 6 and 10 of P. M. Churchland, The Engine of Reason, the Seat of the Soul: A Philosophical Essay on the Brain (Cambridge, 1994: Bradford Books/MIT Press). Reprinted with the permission of MIT Press. David Schmidtz, "Moral Dualism" contains material from Rational Choice and Moral Agency (Princeton, NJ: Princeton University Press, 1995) with permission of Princeton University Press.

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Contents Acknowledgments Contributors 1

Introduction 3 Peter A. Danielson

RATIONALITY

2

Rationality and Rules Edward F. McClennen

13

3

Intention and Deliberation David Gauthier

41

4 Following Through with One's Plans: Reply to David Gauthier 55 Michael E. Bratman 5

How Braess' Paradox Solves Newcomb's Problem A. D. Irvine

67

6

Economics of the Prisoner's Dilemma: A Background Bryan R. Routledge

7

Modeling Rationality: Normative or Descriptive? 119 Ronald de Sousa

92

MODELING SOCIAL INTERACTION

8

Theorem 1 137 Leslie Burkholder

9

The Failure of Success: Intrafamilial Exploitation in the Prisoner's Dilemma 161 Louis Marinoff

10 Transforming Social Dilemmas: Group Identity and Co-operation 185 Peter Kollock

11 Beliefs and Co-operation 210 Bernardo A. Huberman and Natalie S. Glance 12 The Neural Representation of the Social World 236 Paul M. Churchland MORALITY

13 Moral Dualism 257 David Schmidtz 14 Categorically Rational Preferences and the Structure of Morality 282 Duncan Macintosh 15 Why We Need a Moral Equilibrium Theory William J. Talbott

302

16 Morality's Last Chance 340 Chantale LaCasse and Don Ross EVOLUTION

17 Mutual Aid: Darwin Meets The Logic of Decision Brian Skyrms

379

18 Three Differences between Deliberation and Evolution Elliott Sober 19 Evolutionary Models of Co-operative Mechanisms: Artificial Morality and Genetic Programming 423 Peter A. Danielson 20 Norms as Emergent Properties of Adaptive Learning: The Case of Economic Routines 442 Giovanni Dosi, Luigi Marengo, Andrea Bassanini and Marco Valente

408

Contributors Andrea Bassanini, Faculty of Statistics, University "La Sapienza," Rome Michael E. Bratman, Philosophy Department, Stanford University Leslie Burkholder, Department of Philosophy, University of British Columbia Paul M. Churchland, Department of Philosophy, University of California, San Diego Peter A. Danielson, Centre for Applied Ethics and Department of Philosophy, University of British Columbia Ronald de Sousa, Department of Philosophy, University of Toronto Giovanni Dosi, Department of Economics, University "La Sapienza/' Rome and IIASA, Laxenburg David Gauthier, Department of Philosophy, University of Pittsburgh Natalie S, Glance, Rank Xerox Research Centre, Meylan, France Bernardo A. Huberman, Dynamics of Computation Group, Xerox Palo Alto Research Center A. D. Irvine, Department of Philosophy, University of British Columbia Peter Kollock, Department of Sociology, University of California, Los Angeles Chantale LaCasse, Department of Economics, University of Ottawa

Duncan Macintosh, Department of Philosophy, Dalhousie University Luigi Marengo, Department of Economics, University of Trento, and IIASA, Laxenburg Louis Marinoff, Department of Philosophy, The City College of CUNY Edward F. McClennen, Department of Philosophy, Bowling Green State University Don Ross, Department of Philosophy, University of Ottawa Bryan R. Routledge, Graduate School of Industrial Administration, Carnegie-Mellon University David Schmidtz, Departments of Philosophy and Economics, University of Arizona Brian Skyrms, Department of Philosophy, University of California, Irvine Elliott Sober, Department of Philosophy, University of Wisconsin, Madison William J. Talbott, Department of Philosophy, University of Washington Marco Valente, Faculty of Statistics, University "La Sapienza," Rome

Modeling Rationality, Morality, and Evolution

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1

Introduction Peter Danielson This collection began as a conference on Modeling Rational and Moral Agents that combined two themes. First is the problematic place of morality within the received theory of rational choice. Decision theory, game theory, and economics are unfriendly to crucial features of morality, such as commitment to promises. But since morally constrained agents seem to do better than rational agents - say by co-operating in social situations like the Prisoner's Dilemma - it is difficult to dismiss them as simply irrational. The second theme is the use of modeling techniques. We model rational and moral agents because problems of decision and interaction are so complex that there is much to be learned even from idealized models. The two themes come together in the most obvious feature of the papers: the common use of games, like the Prisoner's Dilemma (PD), to model social interactions that are problematic for morality and rationality. The presentations and discussion at the conference enlarged the topic. First, many of the resulting papers are as much concerned with the modeling of situations, especially as games, as with the details of the agents modeled. Second, evolution, as a parallel and contrast to rationality, plays a large role in several of the papers. Therefore this volume has a broader title and a wider range of papers than the conference. Both of the original themes are broadened. On the first theme, contrasts between rationality and morality are complemented by the contrast of rationality and evolution and the effect of evolution on norms. On the second theme, the papers appeal to a wide range of models, from Irvine's abstraction that spans decision theory, game theory, queuing theory, and physics, to the particular and specialized models of minds in Churchland, and working models of the evolution of strategies in Danielson and Dosi et al. The Papers The papers are organized into four sections: Rationality, Modeling Social Interaction, Morality, and Evolution to capture some common elements. Here I will sketch the rationale for these groupings emphasizing the connections between the papers. 3

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Peter Danielson

Rationality The volume opens with a central theme in cognitive science: planning. Agents' plans do not fit well within the received theory of rational choice, according to which one should choose the best option at each occasion for choice. The moralized interpretation of the pull of a plan - as a commitment or promise - worsens this problem. In the opening paper Edward McClennen develops his well-known work of revising the received theory to account for dynamic choice. He concludes "that a viable version of consequentialism will be a version of rule consequentialism, in which the notion of a rational commitment to extant rules has a central place" (McClennen, p. 32).1 Many of the papers focus on David Gauthier's account of how rational choice should be modified to allow for commitments to plans. In the next two papers, Michael Bratman and Gauthier debate the extent of the revision needed: "Bratman wants to replace the desire-belief theory of intention in action with what he calls the planning theory. But he does not go far enough; he accepts too much of the desire-belief model to exploit fully the resources that planning offers" (Gauthier, p. 40). Arguably, one of the biggest advances in our understanding of the Prisoner's Dilemma was the development of an analogous problem in decision theory: Newcomb's problem.2 A great advantage of abstract modeling techniques is the ability to draw strong analogies across widely different subject areas. Andrew Irvine argues that Braess' paradox (in physics) is structurally identical to Newcomb's problem. He sets out a fascinating matrix of analogies to educate our intuitions and show how four so-called paradoxes can be similarly resolved. Since many of the papers argue that the received theory of rational choice needs revision, it is important to set out what the received theory is. Bryan Routledge provides a review of the relevant economics literature. This section ends with Ronald de Sousa's paper, which discusses the interplay of normative and descriptive interpretations of rationality. De Sousa proposes a role for emotions in framing rational strategies and accounting for normativity. There are links between de Sousa's discussion and Macintosh's interest in fundamental values as well as many connections between de Sousa's discussion of evolution and the work in the fourth section. Modeling Social Interaction Robert Axelrod's The Evolution of Cooperation has stimulated much interest in the area of modeling interaction in mixed motive games like the Prisoner's Dilemma. The first two papers in this section build on Axelrod's work. Louis Marinoff works most closely within Axelrod's framework, elaborating Axelrod's tournament-testing device using

Introduction

5

Axelrod's own game, the Iterated Prisoner's Dilemma. He illuminates one of the reasons why correlated strategies of conditional co-operation are problematic. Leslie Burkholder explains one source of complexity in his generalization of Axelrod's theorem 1: given the variety of possible opponents in a tournament environment, no one strategy is best even for the one-shot PD. The third paper in this section tests a game model against actual interaction of human subjects. Peter Kollock's experimental setting provides evidence of a gap between the payoffs offered and the utilities of the subjects. He finds "that people transform interdependent situations into essentially different games . . . [T]he motivational basis for many social dilemma situations is often best modeled by an Assurance Game rather than a Prisoner's Dilemma" (Kollock, p. 208). While most of the models in this collection focus on interaction between pairs of agents, Bernardo Huberman and Natalie Glance use a more complex, many player game. They introduce modeling techniques for allowing agents to form expectations for dealing with their complex social situation. They argue that co-operation requires (1) limits on group size and (2) "access to timely information about the overall productivity of the system" to avoid "complicated patterns of unpredictable and unavoidable opportunistic defections" (Huberman and Glance, p. 235). Paul Churchland's paper bridges this section to the next on Morality. He sketches how neural nets might represent social situations and thereby show "how social and moral knowledge .. . might actually be embodied in the brains of living biological creatures" (Churchland, p. 238). The resulting "portrait of the moral person as a person who has acquired a certain family of perceptual and behavioural skills contrasts sharply with the more traditional accounts that picture the moral person as one who has agreed to follow a certain set of rules ... or alternatively, as one who has a certain set of overriding desires ... Both of these more traditional accounts are badly out of focus" (Churchland, p. 253).

Morality Where the opening papers of the first section reconstructed rationality with a view to morality, this section begins with three papers that reconstruct moral theory with a view to rationality. For example, we have characterized the tension in the Prisoner's Dilemma as one between rationality and morality, but David Schmidtz's theory of moral dualism gives both the personal and interpersonal elements a place within morality. His sketch of moral dualism has the attractive feature of remaining "structurally open-ended" (Schmidtz, p. 258). Moving in a different direction, Duncan Macintosh questions the instrumental

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Peter Danielson

account of values assumed by most attempts to rationalize morality. He argues that, in addition to instrumental reasons to change one's values, there are metaphysical grounds for criticizing some "first values" as well. (Cf. de Sousa for another discussion of "foundational values.") William Talbott connects rational choice and morality in a different way. He criticizes conditionally co-operative strategies - such as Gauthier's constrained maximization - as deeply indeterminate. (This connects to the discussion of correlation in Skyrms and Danielson.) He sees the solution for this problem in the equilibrium techniques central to game theory. In contrast with the first three papers in this section, which are constructive, LaCasse and Ross claim to be decisively critical. "The leading advantage of this attempt to find (as we shall say) 'economic' foundations for morality is that the concepts to which it appeals have been very precisely refined. As usual, however, this sword is doubleedged: it permits us to show, in a way that was not previously possible, that the concept of morality is ill-suited to the task of understanding social co-ordination" (LaCasse and Ross, p. 342). Hence their title: "Morality's Last Chance." Evolution Evolution provides another perspective on the rationality of co-operation. If co-operation is irrational, why is it found in nature? How can co-operation evolve given the analogy between rationality and evolution? "Selection and [rational] deliberation, understood in terms of the usual idealizations, are optimizing processes. Just as the (objectively) fittest trait evolves, so the (subjectively) best actions get performed. This isomorphism plays an important heuristic role in the way biologists think about the evolutionary system" (Sober, p. 407). Skyrms (p. 381) puts the problem neatly, having noted the fact of co-operative behaviour in a wide range of species: "Are ground squirrels and vampire rats using voodoo decision theory?" Skyrms' solution turns on the role of correlated as contrasted with random pairing of strategies. If co-operators have a greater chance of playing with each other rather than playing with other members of the population, then procedures - like Jeffries' The Logic of Decision - that recommend co-operation are not defective. As Skyrms notes, there is a wide-ranging literature on detection of similarity in biology and game theory. Skyrms's model requires that correlation determines which strategies are paired, while some of the literature, including Gauthier and Danielson in this volume, use detection to determine strategy given random pairing. But when recognition of similar agents is used to generate moves in a game, complications arise (cf. Talbott and Marinoff).

Introduction

7

Sober also discusses how correlation distinguishes evolution and deliberation, and adds two additional reasons for differences. First, he argues, the conclusion of the puzzling case of the PD with known finite length is different for evolutionary and rational game theory. Second, rational deliberation has a role for counterfactual reasoning that is missing from the evolutionary case. The final two papers descend to a lower level of modeling, where an evolutionary mechanism is used actually to construct various agents and strategies. Dosi et al. opens with a good survey of the evolutionary computing approach to modeling. Both papers use the genetic programming method to construct functions to test in interaction. Danielson's test case is the Extended Prisoner's Dilemma (see the story in Gauthier and critical discussion in LaCasse and Ross), and the successful players deploy correlated, conditional, co-operative strategies. Dosi et al. argue for theoretical limits on rational choice and illustrate the robustness of norms with a model of learning pricing procedures in an oligopolistic market. Some Methodological Reflections Prisoner's Dilemma The game upon which most of the papers in this volume focus is the two-person Prisoner's Dilemma. Irvine recounts the classic story to which the game owes its name; Skyrms notes the variation played by two clones. In the round-table discussion that ended the conference, there was sharp criticism of the large role played by this one game. One must agree that both from the point of view of rationality and morality, it is a mistake focus on a single game as if it represented the only problem in the theory of rational interaction. From the moral side, the PD is very simple; its single, equal, optimal outcome obviates problems of distributive justice. Therefore, on the rational side, the PD does not demand any bargaining. None the less, there is something to be said for the emphasis on this one game. First, while the Prisoner's Dilemma is not the only problem in the relation of rationality to morality, it is both crucial and controversial. Crucial, because were there no situation in which morally motivated agents did better, there would be little hope of a pragmatic justification of morality. Controversial, because, as the papers collected here attest, there is strong disagreement about the correct account of the "simple" PD situation. While McClennen, Gauthier, and Talbot defend the rationality of a commitment to co-operate, Bratman, Irvine, LaCasse and Ross, and Skyrms all criticize this conclusion.

8

Peter Danielson

Second, focusing on a single puzzle case has the advantage of unifying discussion. We are fortunate that each author has not invented an idiosyncratic model. Much beneficial critical interaction is gained by use of a common model. Third, the appearance of focus on a single game is a bit deceptive; the papers employ several variants on the Prisoner's Dilemma. While Irvine, Kollock, Macintosh, Talbott, and Skyrms all focus on the classic, one-play game, McClennen, Gauthier, Bratman, Danielson, and LaCasse and Ross employ an extended version. Marinoff focuses on the Iterated game, and Glance and Huberman and Dosi et al. employ an n-player variant. These variations form a family of models, but differ in crucial respects. For example, the gap between rationality and cooperation is narrowed in the Iterated and Extended versions of the game, and there are bargaining problems in the n-player variants. Finally, it must be admitted that emphasizing the Prisoner's Dilemma has the risk of oversimplifying morality. The moral virtue needed to overcome the compliance problem in the PD is very rudimentary, ignoring much of moral interest. Indeed, as Burkholder reminds us, the connection between game models like the PD and morality is not simple. It is not always moral to co-operate and immoral to defect in a situation modeled as a PD. The oversimplification of morality is a risk of an approach to ethics that begins with rationality. From this perspective, even simple social norms look moral in their contrast to act-by-act rational calculations (cf. Dosi, p. 442). The subtler discussions of morality in section three counterbalance this tendency to oversimplification.

Modeling "This is one of the reasons why the recent artificial network models have made possible so much progress. We can learn things from the models that we might never have learned from the brain directly. And we can then return to the biological brain with some new and betterinformed experimental questions to pose .. ."(Churchland, p. 245). Similarly, artificial models of social evolution allow us to study the evolution of norms and dispositions in ways we might never have studied the historical evolution of real norms and dispositions. Abstract models have two faces. From one side, they simplify their subjects, allowing arguments and analogies to flow between apparently distant domains. The Irvine paper is the best example of the power of abstraction, but notice as well how McClennen and Bratman are able to apply the act/rule distinction from utilitarianism more generally. This allows Bratman to apply ]. J. C. Smart's criticism of rule worship to "plan worship" in the case of rationality (Bratman p. 10). The other side

Introduction

9

of models - especially computerized models - is their generative power, which throws up myriad concrete examples and counterexamples. The discussions in Churchland, Huberman and Glance, Danielson, and Dosi et al. are all informed by the unexpected results that crop up when thought experiments get somewhat more realized. Of course, we must not confuse these artifacts with the real subject matter, as Kollock, Churchland, and Dosi et al. remind us: "stylized modeling exercises ... complement more inductive inquiries from, e.g., social psychology and organizational sciences" (Dosi et al. p. 458).

Application Finally, the essays that follow are theoretical - extremely so - not practical. None the less there is reason to think that a unified theory of rationality and morality would have a wide and beneficial application. At the end of his conference paper, McClennen held out some tantalizing possibilities for an evolutionary social process leading to the success of a new and morally informed conception of rational agency: Beyond theory, it is worth pondering on what might be the effect of a course of study in which the issue of what rationality requires in such choice situations was not begged in favour of [an] extremely limiting sort of model... Suppose, in particular, that a more concerted effort were made to make individuals aware of the complex nature of decision-making over time, and in interactive situations with other agents, and at least to mark out concretely the advantages to be realized by those who could resolutely serve as their own agents, and choose within the context of co-operative schemes in a principled fashion. One might then reasonably expect to see this more efficient mode of dynamic decision-making drive out more costly precommitment and enforcement methods, and this through nothing more than what economists like to describe as the ordinary competitive process.3

Notes 1 Page references in the text refer to articles in this volume. 2 See Campbell and Sowden (1985), which is organized around this analogy. Richmond Campbell's introduction, "Background for the uninitiated" may also be useful to readers of this volume. 3 Edward McClennen, "Rationality and Rules," as delivered at the Seventh Annual Cognitive Science Conference, Modeling Rational and Moral Agents, Vancouver, 11-12 February 1994, note 55.

References Campbell, Richmond, and L. Sowden (eds.) (1985). Paradoxes of Rationality and Co-operation. Vancouver: University of British Columbia Press.

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Rationality

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2

Rationality and Rules Edward F. McClennen

1. Introduction and Statement of the Problem I propose to explore to what extent one can provide a grounding within a theory of individual, rational, instrumental choice for a commitment to being guided by the rules that define a practice. The approach I shall take - already signalled by the use of the term "instrumental" - is Humean, rather than Kantian, in spirit. That is, my concern is to determine to what extent a commitment to such rules can be defended by appeal to what is perhaps the least problematic sense of rationality that which is effectively instrumental with respect to the objectives that one has deliberatively adopted. In speaking of the "rules of a practice," I mean to distinguish such rules from those that are merely maxims.1 The latter are generally understood to summarize past findings concerning the application of some general choice-supporting consideration to a particular case. Thus, taking exception to such a rule can always be justified, in principle at least, by direct appeal to the underlying choice-supporting considerations. Correspondingly, a person is typically understood to be entitled to reconsider the correctness of a maxim, and to question whether or not it is proper to follow it in a particular case. The rules of a practice have a very different status. While the rule itself may be defended by appeal to various considerations, those who participate in a practice cannot justify taking exception to this rule on a particular occasion by direct appeal to those considerations. Correspondingly, those who participate in the practice are not at liberty to decide for themselves on the propriety of following the rule in particular cases. The question that I want to address, then, is how a commitment to abide by the rules of practices can be defended. More specifically, I shall be concerned with how such a commitment could be defended by reference to what would effectively promote the objectives of the participants, that is, within the framework of a model of rational, instrumental choice. Now one natural starting point for such a defence is the consideration that in a wide variety of political, economic, and social settings, it will be mutually advantageous for participants not only if 13

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Edward F. McClennen

various practices are adopted, but if each participant voluntarily accepts the constraints of the rules defining these practices. If mutual gains are to be secured, and if individuals are unable voluntarily to accept the constraints of the rules, more or less coercive sanctions will be needed to secure their compliance. But sanctions are an imperfect, or, as the economist would say, a "second-best" solution. They typically ensure only partial conformity, consume scarce resources, destroy personal privacy, and deprive participants of freedom. In a wide variety of settings, then, efficiently organized interaction requires that persons voluntarily respect constraints on the manner in which they pursue their own objectives - that is, mutually accept an "ethics of rules." There is, however, significant tension between this conclusion and the basic presupposition of contemporary rational choice theory, namely, that rational choice is choice that maximizes preferences for outcomes. The tension arises because the objectives that one seeks to promote by participating in a practice are typically realized by the participatory actions of others. Given that others will in fact participate, one can often do better yet by acting in some specific, non-participatory manner. In short, one confronts here a version of the familiar public goods problem. In such cases, any given participant can easily have a preference-based reason for violating the rule. This being so, it would seem that from the perspective of a theory of instrumental rationality, a "practice" could never be conceived as setting more than a guideline or "maxim" for choice. But this implies that the standard model of instrumental reasoning does not provide a secure footing for a rational commitment to practice rules.2 What is needed, then, is a solid way of rebutting this argument from the public goods nature of co-ordination schemes based on practices. I propose to begin somewhat obliquely, however, by exploring a quite distinct, but related, type of problem. In many cases what one faces is not interpersonal conflict - as suggested by the public goods story - but intrapersonal conflict. That is, in many situations the self appears to be divided against itself, and this, moreover, as a result of reasoning, both naturally and plausibly, by reference to the consequences of its own choices. The story to be told here can be clarified by studying some simple, abstract models of rational choice. What emerges is that the problem of an isolated individual making a rational commitment to rules turns out to be rooted in a way of thinking about individual rational choice in general, a way that is so deeply ingrained in our thinking as to virtually escape attention altogether, yet one that can and should be questioned. I shall suggest, then, that it is here that we find a model for the problem that arises at the interpersonal level, and one that offers an important insight into how the interpersonal problem can be resolved.

Rationality and Rules

15

The analysis commences in Section 2, with an exploration of some standard models of intrapersonal choice - models which suggest a serious limitation to the standard way of interpreting consequential reasoning. Section 3 develops the thesis that it is not consequentialism as such, but only an incremental version of consequentialism, that generates the problem. This paves the way for a presentation of an alternative, and more holistic or global way of thinking about consequences. Section 4 argues for the instrumental superiority of this alternative conception. Section 5 extends these results to problems of interdependent choice. In Section 6, these results are then explicitly brought to bear on the rationality of accepting an "ethics of rules" - of accepting the constraints of practices. 2. Intrapersonal Choice Consider everyday situations in which you find yourself conflicted between more long-range goals and present desires. You want to reduce your weight, but right now what you want to do is have another helping of dessert. You want to save for the future, but right now you find yourself wanting to spend some of what could go into savings on a new stereo. The logic of this type of situation can be captured by appeal to a very simple abstract model, in which you must make a pair of choices in sequence (Figure 1). This neatly particularizes into a situation in which you face a problem of a change in your preferences, if we suppose an intelligible story can be told to the effect that at time t0 you will prefer outcome o3 to o2 and o2 to o4/ but that at time tl you will prefer o4 to o3/ and would prefer o3 to o2, if the latter were (contrary to fact) to be available at that time.3

Figure 1: A simple sequential choice problem

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Consider now the plan that calls for you to move to the second choice node and then choose path a3 over a4. Call this plan a1-a3. Since at t0 you prefer the outcome of this plan, o3, to the outcome of choosing path «2 outright, namely o2, you might be inclined to pursue the former rather than the latter. Upon arriving at node 2, however, you will (so the argument goes) choose path a4 over a3, since, by hypothesis, you will then prefer o4 to o3. That is, you will end up abandoning the plan you adopted. To do this is, on the standard account, to choose in a dynamically inconsistent manner. Being dynamically inconsistent involves more than just changing plans in midstream. Having decided upon a plan, you may acquire new information that calls upon you, as a rational person, to alter your plans. In such a case there is no inconsistency.4 The charge of dynamic inconsistency arises when the change is not predicated on receipt of new information, and where, indeed, it is one that you should have anticipated. That is, more careful reflection would have led you to realize that the plan you are inclined to adopt is one that you will subsequently abandon. To be dynamically inconsistent, then, is to be a myopic chooser: you shortsightedly fail to trace out the implications of the situation for what you will do in the future. Being dynamically inconsistent in this sense means that your future self ends up confounding the preferences of your earlier self. As it turns out, however, myopia involves something worse. A myopic approach makes you liable to what are clearly, from a consequential perspective, unacceptable outcomes. The extensive literature on Dutch-books and money-pumps shows that myopic choosers can be "tricked" into accepting bets and making other choices that result in a sure net loss of scarce resources.5 Typically, this involves your being willing to pay to give up one option in exchange for another, and then, after certain events have occurred, being willing to trade once again, for another fee, in such a way that the net effect is that you end up being exposed all along to the original option, and thus have paid fees twice to no purpose. When matters are viewed from this perspective, it is not just one or the other of the selves whose preferences are confounded. Rather, both selves stand to loose as a result of myopia. Moreover, since the myopic chooser's loss is the exploiter's sure gain, myopic choosers must expect, at least in an entrepreneurial world, that they will be exploited: others will be eager to do business with them. All of this makes for a powerful pragmatic argument against myopic choice. Such unfortunate consequences can be avoided, however, by choosing in a sophisticated manner. To be sophisticated is to first project what you will prefer, and thus how you will choose, in the future, and then reject any plan that can be shown, by such a projection, to be one that

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you would subsequently end up abandoning. Such plans, so the argument goes, are simply not feasible. For the problem given in Figure 1, plan a1-a3 is not a feasible plan. Despite your preference at t0 for outcome o3, the corresponding plan is one that you will not end up executing, and thus o3 cannot be secured. Given this, and given your preferences at the first-choice node, you should choose a2 outright, and thereby realize o2. To be sophisticated, then, is to tailor your ex ante choice of a plan to your projection of what you will prefer, and hence choose, ex post. Just what this implies will vary from case to case. Thus, for example, plan a2 might involve giving irrevocable instructions to a hired agent to choose o3 rather than o4 at the second-choice node. That is, you may be able to achieve dynamic consistency by precommitting.

3. Assessing Consequences Myopic and sophisticated choice share certain things in common. Both make an explicit appeal to the principle of choosing plans so as to maximize with respect to your preferences for outcomes. That is, your assessment of the alternatives available is presumed to turn on your preferences for the outcomes realizable by your choices. Call this consequentialism.6 More importantly, however, in both myopic and sophisticated deliberation the assessment of consequences is perceived to take place in an incremental manner. What is relevant for deliberation and choice at a given node in a decision tree is not what preferences you had at the outset, when you first contemplated the whole decision tree, but simply those preferences that you just happen to have, at the given node, with respect to outcomes still realizable by your action.7 Now, certainly part of what is involved in this perspective is altogether plausible. On the standard preference (or desire) and belief model, it is your preferences together with your beliefs that causally determine your choice of an action. Intuitively, there can be no causal action at a distance. If preferences are to play a causal role, it must be the preferences you have (together with your beliefs) now, that determine your choice now. Notice, moreover, that such deliberation is consistent with having "backward-looking" concerns or commitments of various types.8 It may be, for example, that in certain cases what you now prefer to do is take your marching orders from some previous self. Alternatively, and less subserviently, you might now prefer to participate in a co-ordination scheme with earlier selves. The point, however, is that what counts in all such cases are the backward-regarding preferences that are entertained by you now.9 There is, however, a distinct and much more problematic assumption that appears to be implicit in this way of construing rational deliberation. This is that such a present concern for choosing to co-ordinate

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your choice now with a choice made previously - to choose in a manner consistent with a prior commitment - cannot itself be grounded in a process of rational deliberation. That is, it would appear that, on the standard account, the logic of rational deliberation precludes that a preference on the part of your present self for co-ordinating its choice with the choices of earlier selves might issue from, as distinct from setting the stage for, rational deliberation itself. By way of seeing what is at issue here, it will prove helpful, first of all, to consider the case in which you do not have a present preference - deliberative or otherwise - for co-ordinating choice now with choice made earlier. In such a case, your present self will take the choices it previously made as simply setting constraints upon what outcomes are still possible - in exactly the same way that it will also take past natural events as setting such constraints. Suppose now that this situation also holds for your future self - that is, when it becomes the present self, it has no preference for co-ordinating with its earlier selves. This, in turn, implies that there could be no point to your present self trying to coordinate with its future self. On the assumption that this state of affairs exists for each present self, what is left open to you at each moment is not co-ordination with, but, at best, strategic adjustment to, the (hopefully) predictable behaviour of your future self.10 Each of your timedefined selves, then, will think of itself as an autonomous chooser, who has only to take cognizance of the choices that your other time-defined selves have made or will make, in the very same way that it must take cognizance of other "natural" events that can affect the outcome of its choices. That is, each of your time-defined selves will deliberate in an autarkic manner.11 Consider now the situation in which you do have a past regarding preference. Suppose you prefer, now, that you act in conformity with a plan that you initiated at some earlier point in time. How are we to understand your coming to have such a preference? A variety of explanations here are possible, of course. Two possible explanations were briefly mentioned in Section 1 above (note 2), where appeal was made to models of genetic and social transmission. That is, your backwardlooking concern for consistency may be the result of certain experiences triggering inborn dispositions; alternatively, such a concern may come about as a result of a process of socialization. Neither of these roads, however, lead us back to a model in which your several selves are deliberatively linked in any fashion. These linkages are due to non-deliberative causal, as distinct from essentially deliberative, processes.12 A pressing matter for consideration here, then, is under what conditions this sort of preference could arise as the result of a deliberative process. Now clearly there are going to be cases in which you deliberatively

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decide to act in conformity with a previously adopted plan. For example, you have made a bet with someone that you could carry through with your plan, and now want to collect on that bet. Here the explanation is straightforwardly pragmatic. Your deliberative preference for acting in conformity with a previously adopted plan is based on wanting to secure a certain (future) reward. Correspondingly, at the earlier point in time when the plan was to be adopted, you had a sufficient sense of your continuing pragmatic interests to know that you could count on your subsequent self carrying through. That is, your choice at the earlier point in time is shaped by the sense that you will be able to respond in the appropriate manner at the subsequent point in time, and your subsequent choice is shaped by your sense of what you committed yourself to do at the earlier point. The earlier choice is, then, conditioned by an expectation; and the subsequent choice is conditioned by knowledge of what your earlier self has chosen. So here is a model in which a backward-looking preference is deliberatively based. The pattern here is, moreover, paradigmatic of such deliberatively based preferences. Specifically, on the standard account, looking backward is justified by looking forward. Notice that the relevant backward-looking preference in the case just discussed is the one that is held by the subsequent self. What, correspondingly, is the relevant forward-looking preference? The logic of the perspective under consideration is that it must be the preference held by the earlier, not the subsequent, self. It is the earlier self's interest in securing the reward that motivates it to attempt the co-ordination, not any interest that the subsequent self has. At the same time, however, that in which the earlier self has an interest is not its receiving the reward: the reward, by definition, comes later. So it must be supposed that the earlier self has an interest in what happens to its own subsequent self (or selves), that it has, in this sense, an other-regarding concern regarding its own future self (or selves). And given this, it is also clear that the only thing that could motivate the subsequent self to engage in a backward-looking exercise, is its interest in the reward (or its interest in what accrues to even more subsequent selves). The problem is that nothing ensures that the earlier self's concern for the subsequent self (or selves) coincides with the future self's own concerns. This can easily be overlooked, if one casually supposes that both the earlier and the subsequent self are concerned to promote the wellbeing or interests of the subsequent selves, and that "well-being" or "interest" can be given some sort of objective interpretation. On that reading, it would then follow that any lack of coincidence between their respective concerns would be due merely to misinformation (on the part of one or the other of the involved selves). But nothing in the

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logic of preferences of the relevant sort assures that the only problem is one of misinformation. The prior self may simply have a different future-regarding agenda than the agenda of the subsequent self. The prior self may be concerned with long-range considerations, while the agenda of the subsequent self may turn out to be, at the moment that counts, markedly myopic - the focus being on satisfying a momentary concern (say, an urgent desire).13 Nothing is essentially changed if we postulate some ongoing prudential concern. Your present self must reckon with the consideration that your future selves, while themselves prudentially oriented, are so from their own perspective. Whether any future self can have a commitment to abide by plans previously settled upon is a matter of what can be prudentially justified from the perspective of that future self. That is, what prudence dictates at any given point in time is a matter of what can be shown to serve longer-range interests from that specific time forward. This means that any given present self has no assurance that it is party to an arrangement to which future selves are committed and in which its own. concerns will be addressed. It all depends upon what, from some future vantage point, turns out to be prudential. This is built into the very logic of the concept of prudence: at any given point in time, what is or is not the prudential thing to do concerns what still lies in the future, not what lies in the past. Consider now what happens if there is a divergence between present and future motivating concerns, between present and future preferences. The model that emerges is not one in which selves are to be understood as compromising for the sake of mutual advantage. Rather, one of two alternative remedies is assumed to be appropriate. The structure of the problem may be such that the prior self is in a position to impose its will upon the subsequent self. This yields the model of precommitment that was discussed in the previous section, and which figures centrally in Elster's Ulysses and the Sirens. Alternatively, the structure of the problem may be such that no precommitment is possible: the prior self may have to recognize that, since the other self chooses subsequently, and with a view to maximizing from its own perspective, it has no choice but to anticipate this independent behaviour on the part of the subsequent self, and adjust its own choice behaviour accordingly, that is, to the reality of what it must expect the other self to do. In either case, then, the implicit model of choice is the one already encountered in the discussion of the case in which each present self has no backward-looking concern: that is, the model of the autarkic self. Suppose, finally, that there is as a matter of fact coincidence between present and future concerns. Clearly in this case there can be an adjust-

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ment of choices, and a convergence of expectations. But note that no less in this case, than in the case just discussed, each self is presumed to maximize from its own perspective, given what it expects the other self (or selves) to do. That is, each, from its own perspective, comes to choose and comes to expect the others to choose, in a manner that yields the requisite sequence of behaviours. But the model that is applicable here, then, is simply the mtrapersonal analogue to the model of "co-ordination" that is central to virtually all work in game theory, on interpersonal interaction, namely, the model in which the choices of the respective selves (distinct individuals) are in equilibrium. Within the framework of this model, the respective selves do not negotiate or bargain and thus compromise their respective interests or concerns with a view to reaching an arrangement that is mutually beneficial; rather, it just happens that the choice made by each maximizes with respect to that self's concern, given its expectation concerning how each other self will choose.14 Here, once again, then, the appropriate model is the one already encountered, in which each self proceeds to choose in a thoroughly autarkic manner. One can capture these conclusions in a somewhat more formal manner in the following way. It is natural, of course, to think of the initial node of a decision problem as such that whatever set of events led you to that node, those events simply fix the given against which, in accordance with consequential considerations, you now seek to maximize with respect to your present preferences for outcomes realizable within the tree. What the incremental, autarkic perspective presupposes is that each and every choice node in a decision tree presents you with just this sort of problem. Consider any choice node nt within a decision tree, T, and the truncated tree T(«;) that you would confront, were you to reach choice node n(, that is, the set of subplans that you could execute from HJ on, together with their associated outcomes. Now construct a decision problem that is isomorphic to the original tree from the node n{ onward, but which contains no choice nodes prior to njr although it can contain reports as to how as a matter of fact you resolved earlier choice options. That is, in this modified tree, n, is now the node at which you are first called upon to make a choice. Call this modified version of the original tree T(n0 -»«;). The controlling assumption, then, is this: Separability. The subplan you would prefer at a given node n, within a given tree T (on the assumption that you reach that node) must correspond to the plan that you would prefer at the initial node na = n( in the modified tree T (nQ —> n,) 15

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Separability, then, requires coincidence between choice at ni in T and choice at n0 = n{ in T(n0 —> nt• ). On the standard account, it is consequentialism, as characterized above, that determines choice at n0 = n{ in T (MO —> M; ) itself. But it is separability itself that drives the autarkic approach to deliberation and decision. It disposes you to reconsider, at each new point in time, whatever plan you originally adopted, and to settle upon a new plan on the basis of whatever, then and there, you judge will maximize your present preferences with respect to outcomes still available.

4. Dynamic Consistency Re-examined Satisfaction of both consequentialism and separability does not ensure that you choose in a dynamically consistent fashion. This is, as we saw above, the lesson to be learned from being a myopic chooser. As a myopic chooser you satisfy both consequentialism and separability, but you adopt plans only to abandon them later. If you are committed to deliberating in a separable manner, you can achieve dynamic consistency by deliberating in a sophisticated manner. That is, you can confine choice at any given node to (sub)plans that are feasible. Correspondingly, the appropriate criterion of feasibility here is simply this: a (sub)plan at node ni is to be judged feasible if and only if it is consequentially acceptable at each and every successor node to n{ at which it directs choice. Feasibility, then, can be determined by working recursively backward, by starting at each potentially last choice point in the decision tree, and then moving backward through the tree until one reaches the initial choice point. At each such point, you are to ask what you would choose there from a separable perspective, and then fold in that information in the form of a judgment of what is feasible at the immediately preceding choice point.16 To illustrate, consider once again the decision problem in Figure 1, where it is assumed that o3 is preferred to o2 and o2 is preferred to o4 at time £„, but o4 is preferred to o3 at time tlf and that these are the only relevant preferences. Here, consequentialism plus separability generates the standard results. Given this ordering of outcomes at f,, consequentialism implies that at tl in T(n0 -> MJ), you will prefer the truncated plan «4 to the truncated plan a3. And this, in turn, by appeal to separability, implies that, at tl in T, you will prefer a4 to a3. That is, a3 is consequentially unacceptable. This, in turn, implies that at n0 the plan al—a3 is not feasible, even though, at that time you consequentially prefer it to the other plan, flj-fl4. Feasibility considerations, then, determine that only o4 and o2 are realizable by (rational) choice on your part, and since at M0 you prefer o2 to o4, consequentialism determines that the rational choice is plan a2.

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This is certainly one way to achieve dynamic consistency. In effect, what you do is tailor your plans to what you would expect to do in the future, given various contingencies (including your present choice of an action). In principle, however, dynamic consistency can be achieved in a radically different way. What dynamic consistency requires is an alignment between earlier choice of a plan and subsequent choice of remaining subplans. Rather than regimenting present choice of a plan to projected future choice, the required alignment can be secured, in principle, in just the reverse manner, by regimenting future choice to the originally adopted plan. Let us call an agent who manages to achieve consistency in this way resolute. Conceptually, being resolute involves being committed to carrying out the plan that you initially selected. For example, with regard to Figure 1, to be resolute is to choose, and then proceed to execute, plan «1-fl3. Being resolute does not mean being unconditionally committed to execute a chosen plan. It allows for changing plans in the light of new information. All that it requires is that if, on the basis of your preference for outcomes, you adopt a given plan, and if unfolding events are as you had expected them to be, you then proceed to execute that plan.17 It is consistent with being resolute that you simply tyrannize, at some point in time, over your own later selves. Were this to be the only way in which resolute choice was to be understood, this would surely cast doubt on your ability to be deliberatively resolute. Even if your present self desired to tyrannize over your future self, in the absence of some special hypothesis concerning the content of the preferences of your future self, what possible rational ground could your future self have for accepting such a regimen? In such a case, it would seem that you must, if you are realistic, expect that your deliberative resolve will unravel.18 But resoluteness need not be understood to be achievable only in this way. It can function, not as a regimen that your earlier self attempts to impose upon some later self, but as the means whereby the earlier and the later self co-ordinate their choices in a manner that each judges acceptable. The suggestion, to the exploration of which I shall shortly turn, is that such co-ordination can, in certain cases, prove mutually advantageous to one's several time-defined selves, and that this paves the way for a reappraisal of the rationality of being deliberatively resolute. It must be acknowledged, however, that deliberative resoluteness, no matter how it is based, cannot be squared with the separability principle. In Figure 1, the agent who adopts and then resolutely executes the plan a1-a3, despite being disposed to rank a4 over a3, in the context of a modified decision problem in which the initial choice node is at node 2, violates the separability principle, which requires the choice of «4.19 Since many are convinced that separability is a necessary condition of

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rational choice, they conclude that the model of resolute choice must be rejected. It recommends plans that are simply not feasible, at least for a rational agent. Conceptually, however, what we have here is a conflict between a method of achieving dynamic consistency and an alleged principle of rational choice. How is that conflict to be adjudicated? One could, of course, make an appeal to intuition at this point. Unfortunately, the continuing debates within the field of decision theory over the last fifty years suggest that intuitions of this sort tend not to be interpersonally transferable.20 Under such circumstances, an appeal to intuitions is essentially a rhetorical move.21 A more promising approach, I suggest, is to consider once again the kind of pragmatic perspective to which appeal has already been made. Consider once again the case that can be made for being sophisticated rather than myopic. The argument is that the myopic self is liable to being exploited in a manner that works to its own great disadvantage, and since what it stands to lose are resources that any of its time-defined selves could put to use, here is a thoroughly pragmatic argument, and one that can be addressed to each successive self, for being sophisticated rather than myopic. What, then, can be said from this sort of perspective regarding the present issue, namely, the comparative rationality of sophisticated and resolute choice? It is sophisticated rather than resolute choice that can be criticized from this perspective. To illustrate, consider once again the version of the problem in Figure 1 in which plan a2 constitutes a precommitment strategy of paying someone else to execute a choice of o3 over o4 at the second-choice node. This is the best option consistent with the separability principle. But if you are resolute you can realize the very same outcome, without having to pay an agent. On the assumption that each of your time-defined selves prefers more to less money, to reason in a separable manner is to create a real mtrapersonal dilemma for yourself, in which "rational" interaction with your own future selves leads to an outcome that is intrapersonally suboptimal, or "second-best." That is, each time-defined self does less well than it would have done, if the selves had simply co-ordinated effectively with each other. Other values must be sacrificed as well. Precommitment devices limit your freedom, since they involve placing yourself in situations in which you do not chose, but have choices made for yourself. Moreover, they expose you to the risks associated with any procedure that is inflexible. In contrast, the resolute approach is not subject to any of these difficulties. Scarce resources do not have to be expended on precommitment devices or to pay agents; you are the one doing the choosing, and you retain the option of reconsideration insofar as events turn out to be different from what you had anticipated.

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Here, then, is a thoroughly pragmatic or consequentialist consideration in favour of being resolute, and against being sophisticated. There is a class of sequential problems in which acceptance of separability generates choice behaviour that is intrapersonally suboptimal, and where this unfortunate consequence can be avoided by choosing in a resolute manner. In at least some cases, I suggest, a more holistic but still consequentialist perspective can be marshalled in support of being resolute as opposed to being sophisticated, and hence, in support of a relaxation of the standard separability assumption.22 The pragmatic argument just rehearsed does not establish that it is always rational to be resolute. It only makes the case for being resolute in certain cases in which there are comparative advantages to being resolute rather than sophisticated, as measured in terms of standard "economic" values of the conservation of scarce resources, freedom, and flexibility. Moreover, nothing has been said about what, within a non-separable framework, a full theory of rational, intrapersonal, sequential choice would look like. At the very least what is needed, in addition, is a theory of what constitutes a fair bargain between one's different, time-defined selves. All that has been argued so far is that there are contexts within which being resolute is a necessary condition of rational, sequential choice. But even this limited conclusion has two connected, and quite powerful, implications. First, insofar as weakness of will is a manifestation of the agent's having succumbed to the "Siren's Song" of incremental reasoning, it may really be a sign of imperfect rationality; and, thus, second, talk of precomrnitment and the like in such cases is really best understood as addressed to those who are not fully rational.23 To understand the kind of case in which being resolute is pragmatically defensible is also to understand the relation between resoluteness and consequentialism. Being resolute involves, by definition, adopting a two-level, deliberative approach to consequentially oriented choice. At the first level, in settling upon a plan of action, you will compare the consequences of the various available plans, and reject all plans that fail the test of intrapersonal optimality. That is, consequentially oriented considerations will guide you to adopt plans as a means of effectively co-ordinating between your time-defined selves. But, at the second level, with respect to specific choices to be made as you move through a decision tree, the plan actually adopted will then set constraints on subsequent choice. That is, you will take the plan adopted as regulative of choice. Finally, these last remarks suggest that the model of resolute, as distinct from separable, choice provides an account of rationality in terms of which one can make sense of, and defend, a rational commitment to practice rules. But the story that has

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been told so far concerns only the intrapersonal problem that arises for the isolated individual. Before turning specifically to the issue of practices, it will prove useful to explore the implications of resolute choice for cases of interpersonal interaction.

5. Interpersonal Choice under Ideal Conditions What light does the foregoing analysis shed on the problems of interpersonal choice with which I began? I shall focus here on the logically special, but very important case of interactive games that are played under the following "ideal" conditions: (1) all of the players are fully rational; and (2) there is common knowledge of (a) the rationality of the players, (b) the strategy structure of the game for each player, and (c) the preferences that each has with respect to outcomes.24 Just what is implied by (1) remains to be spelled out, of course. (2) commits us to the assumption that there is no asymmetry in the information available to the different players. In particular, any conclusion reached by a player, regarding what choice to make, can be anticipated by the others: there are no hidden reasons. Here is a simple game of this type, one that involves the players choosing in sequence, rather than simultaneously (where the pair of numbers in parentheses to the right of each listed outcome o;, gives A's and B's preference ranking, respectively, for that outcome - with a higher number indicating that the outcome is more preferred). See Figure 2.

Figure 2: An assurance game

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Each of the outcomes o3 through o6 can be achieved by A and B coordinating on this or that plan. By contrast, outcome o2 can be reached by a unilateral move on A's part. Once again, for the sake of the argument to be explored, «2 can be interpreted as a "precommitment" plan whereby B can be assured that if she chooses bl in response to alr A will then respond with a3. That is, realizing o2 amounts to realizing o3, although at the cost of an agency fee to be paid to a third party from funds, say, contributed by A. Note also that this interactive situation has embedded in it, from node 2 on, a sequential version of the classic Prisoner's Dilemma game. Given the specified preference rankings for outcomes, and the standard consequentialist assumption that plans are to be ranked according to the ranking of their associated outcomes, plan a-^-b^a^ might seem to be the most likely candidate for a co-ordination scheme. To be sure, A prefers the outcome associated with the plan a1-b1-a4, but it is unrealistic to suppose that B would agree to co-ordinate on that plan. On the standard view, however, under ideal conditions (of mutual rationality and common knowledge), talk of voluntarily co-ordinating their choices is pointless. Suppose A were to set out to implement the first stage of such a co-ordination scheme, by selecting av and suppose, for some reason or other, B were to reciprocate with b1. In such a case, so the argument goes, A would surely select «4. In effect, plan a^-b^-a^ is simply not feasible: it calls upon A to make a choice that A knows he would not make, and, under ideal conditions, B knows this as well. Moreover, B would end up with her least preferred outcome, as the result of a failed attempt at co-ordination. Suppose, alternatively, that A were to select alf and B were to respond - for the reasons just outlined - by protectively selecting &2: under these conditions, A's best response at node 4 would be «6, and each would then end up with a second least preferred outcome. Once again, all of this is common knowledge. Against the background of these subjunctively characterized conclusions, then, A's best opening choice is not alf but a2, yielding for each a third least preferred outcome. That is, the equilibrium outcome - and projected solution - for rational preference maximizers is o2. Notice, however, that the problem in Figure 2 neatly mirrors the essential features of the intrapersonal problem given in Figure 1. The outcome associated with al-bl-a3 is preferred by each to the outcome associated with a2. But, according to the story just told, the former outcome is not accessible. Why? Under conditions of common knowledge, and on the standard analysis, A cannot expect B to co-operate. Why? A cannot plead that B is basically disposed to be non-co-operative. B's maximizing response to an expectation that A will co-operate is to cooperate herself. A's expectation that B will play defensively derives

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solely from the consideration that B must expect that A will, if and when node 3 is reached, choose a4, not a3. Thus, A's quarrel is with himself; or perhaps we should say (given the analysis of the last section), with his own future self I

What this suggests, of course, is that the analysis of intrapersonal conflict applies to this situation as well. As already indicated, consequentialism can be invoked to argue that preferences for outcomes are controlling. And once again, it can be noted that this does not, in itself, settle the question of what would qualify as a rational choice for A at node 3. What is requisite, in addition, is an assumption to the effect that A will conceptualize his situation at node 3 as one in which his own past choice behaviour, and the choice behaviour of B constitute givens against which he must maximize with respect to his preferences for outcomes still realizable at node 3. That is, A functions as an autarkic chooser at node 3. Thus, the argument turns once again on the separability assumption formulated above in Section 2. It is consequentialism together with separability that yields the standard result regarding how A must choose. The conclusion of the last section is that separability in the context of intrapersonal choice must be rejected. That conclusion carries over to the context of the type of sequential problem just discussed, where interaction is sequential and takes place under conditions of common knowledge. Here as before, A's commitment to separability precludes both A and B from realizing gains that could otherwise be realized. Within this framework, then, separability cannot be taken as an unqualified condition of rational choice. Given common knowledge, the rational choice for A is to be resolute, and since B will anticipate that A will choose in this manner, her best move is to respond with b1 for an outcome which ranks second-best (4,4) on each participant's preference ordering. Now this line of reasoning is subject to an important extension. Consider first a very minor modification of the interactive situation presented in Figure 2, according to which A and B must jointly decide at node 1 either to employ some sort of enforcement mechanism (a2), or to reach an agreement on a co-operative strategy, say, Cj-bj-^, which will govern the remainder of the interactive situation and which is to be voluntarily executed by each. Once again, of course, this interactive situation has embedded in it, from choice node 2 on, a sequential version of the classic Prisoner's Dilemma game. And once again the standard argument is that the rational outcome of this interactive situation must be o2 - for all the reasons just rehearsed in connection with the interactive situation in Figure 2. Were A and B to agree at node 1 to co-ordinate their choices at nodes 2 and 3, that would provide no deliberative

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reason for A to choose «3 rather than «4 at node 3, unless, of course, A just happened to prefer to follow through on such an agreement. Now, modify the situation once more to that given in Figure 3. This is a sequential game, the second stage of which involves a simultaneous choice problem. At node 1, A and B must not only agree upon a plan (say, to jointly choose "co-operation" at the second stage) but also decide whether to institute an enforcement mechanism that will ensure that this agreement is binding (plan c2) or whether to proceed forward and attempt to voluntarily implement their agreement (q). Once again, both players would be better off choosing cl and then mutually honouring their agreement to co-ordinate, than they would be by agreeing to an enforcement mechanism. But, if they were to agree upon cv what they then face is a classic, symmetrical version of the Prisoner's Dilemma, in which both the dominance principle and the equilibrium requirement mandate that each defect. Under these conditions, O2is the rational outcome of this interactive situation, even when it is played under the ideal conditions of common knowledge. What does the work here is not just the consequentialist assumption that each player is disposed to choose so as to maximize preferences for outcomes, but, rather, once again, an assumption about how such consequential reasoning is anchored. In effect, the problem of simultaneous interpersonal choice is conceptualized in the same manner that both the problem of intrapersonal choice and the problem of interpersonal sequential choice is conceptualized, notwithstanding that choices are now to be made simultaneously, rather than sequentially. That is, once again, the supposition is that you will chose in an autarkic manner. To see this, factor out the consideration that choices are to be made simultaneously, and focus on a sequential game, where the other person plays first and you play second. In this case, as we have already seen,

Figure 3: A joint assurance game

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the supposition is that you will reactively maximize your present preferences for outcomes against the given behaviour of the other person. That is, the choice behaviour of the other person is taken to be a given, just like the given choice behaviour of your past self, and not a choice that calls for a co-ordinating move on your part. Now consider the same situation except that choices are to be made simultaneously. Here you cannot take the choice behaviour of the other person as a given. But suppose that you are in a position to make an estimate of how that other person will choose. Then the supposition is, once again, that you will reactively maximize your present preferences for outcomes, this time against your best estimate of how the other player will choose.25 In short, just as the distinction between your own past choice behaviour and the past choice behaviour of another person is strategically insignificant, so also is the distinction between choosing after another has chosen, and choosing simultaneously. In the latter case, you are simply thrown back on having to maximize against your best estimate of how the other person will choose.26 What is implicit in this way of thinking can be captured, then, in a separability principle that parallels the one for intrapersonal choice problems: Separability (for two-person, interpersonal, synchronous choice): Let G be any two-person game, and let D be a problem that is isomorphic to G with respect to the strategy and payoff structure of the game for both you and the other player, except that the choice behaviour of the other player has been fixed at a certain value that you either know or can reliably estimate - so that what you face, in effect, is a situation in which all that remains to be resolved is your own choice of a strategy. In this case, what you choose in G must coincide with what you choose in D.27 However intuitively acceptable this principle is, within the context of ideal games it is subject to precisely the objection raised against the intrapersonal separability principle. As the classic Prisoner's Dilemma game illustrates, persons who are disposed to choose in this fashion simply do less well, in a significantly large class of ideal games, than those who are disposed to reason from a non-separable perspective, and with a view to realizing the gains that can be secured from effective co-operation. Here, then, is another context within which there is a pragmatic argument against separable and in favour of resolute choice.28 There are a number of plausible extensions of resolute reasoning, to ideal games involving more than two players, and to iterated games

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played under ideal conditions. That is, the logic of the argument is not confined to the two-person, "one-shot" case. All of the interactive situations just considered, however, are overly simple in one important respect: there is only one outcome that is Pareto-efficient relative to the standard, non-co-operative, equilibrium solution. What has been offered, then, is at best only a necessary condition of an adequate solution concept for ideal games. What is needed is a theory of co-operative (as distinct from non-co-operative) games, that is, a well-developed theory of (explicit and/or tacit) bargaining, for selecting among outcomes that are Pareto-efficient relative to the equilibrium outcomes of the game (or some other appropriate baseline), and which are themselves Pareto-optimal.29 Moreover, for any theory of bargaining that can serve as a normative guide to help people avoid suboptimal outcomes, perhaps the key issue is what constitutes a fair bargain.30 There is also the very important question concerning to what extent the results for ideal games can be extended to games played under more realistic conditions, where players may be uncertain as to the rationality of the other players and where various informational asymmetries obtain. Here, questions of assurance are bound to loom large, even for agents who are otherwise predisposed, as a result of rational deliberation, to co-operate. However, all of this pertains more to the question of the scope of the argument just rehearsed, and the more pressing concern now is to see what implication this sort of argument has for the problem originally posed - the problem of whether an agent can make a rational commitment to act subject to the constraint of practice rules. 6. Rules, Resoluteness, and Rationality I argued in Section 1 that the standard model of rational choice does not provide a secure footing for the rationality of choosing subject to the constraints of practice rules. What I now want to argue is that the alternative model presented in the intervening sections opens the door to understanding and defending a rational commitment to practice rules. Consider first the concept of a practice. One can mark in the abstract concept of being resolute a model for just the sort of hierarchical structure that characterizes practice-constrained choice, both for the kinds of practices that the isolated self may adopt, but also for many of those practices that structure our interactions with others. One has only to observe that a practice can be understood to be a type of plan, and to recall that it is precisely the resolute self that is capable of taking a plan that has been adopted as regulative of future choice, even in the face of what would otherwise count as good reasons to choose differently. But why is a practice to be taken as regulative? Because this is what is needed if individuals are to co-ordinate their actions, or if the

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isolated individual is to co-ordinate actions over time. For co-ordination to take place, it is not enough that each does what each judges to be "best"; nor is it even enough that each conforms to some rule that each judges would best serve the ends in question, if all were to conform to it. To the contrary, co-ordination requires a mutual structuring of activity in terms of a prior, established rule having both normative and positive import: that is, a rule to which all are expected to adhere, and to which it is expected that all (or most) will in fact adhere.31 The rules defining a practice, then, are to be understood as prior to the choices of action that arise under it in both a temporal and a normative sense. There is temporal priority, because what is regulative is a rule that is already established. There is normative priority because the rule takes precedence, at least in principle, over any countervailing choice supporting consideration that can arise at the level of choice of action within the context of a situation to which the rule is applicable. The logic of practice rules, so conceived, then, involves the notion that one cannot decide to overrule such a constraint in a given situation to which the practice rule applies by directly appealing to whatever considerations could be adduced in support of the practice itself. Those who participate in such a practice abdicate, in effect, their "right" to make decisions case by case by direct appeal to such underlying considerations. The sense in which a practice rule is prior to, and establishes nondiscretionary constraints on choice, is already provided for in the model of resolute choice - in the notion that choice in certain sequential decision problems is constrained by a prior decision to pursue a plan, or a prior (tacit or explicit) understanding as to how choices by quite different individuals are to be co-ordinated. That is, the account of nonseparable deliberation and choice explored in previous sections provides a model of the kind of intentional co-ordination that is essential to adopting and choosing subject to the constraints of a practice. As I argued at the close of Section 4, the intrapersonal co-ordination problem is resolved by adopting a two-level approach to deliberation and choice. At the first level, consequentially oriented considerations will lead one to adopt a specific plan; and at the second level, the plan that is in fact adopted will set constraints on subsequent choice. In this setting, what is relevant to subsequent intrapersonal choice is not what plan one might have adopted, or what plan it would have been best for one to adopt (by reference to some underlying consideration), but what plan one did in fact adopt. Correspondingly, what is relevant in certain interpersonal decision problems is not what plan the participating individuals might have adopted, or what plan it might have been

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best to adopt (once again, by reference to some underlying consideration), but what plans are already in place. In each case, then, there is both a positive or "fact of the matter" and a normative dimension to the reference point that emerges for deliberation and decision: what functions as directive for choice is the plan that as a matter of fact has been chosen. This conceptual account holds, it should be noted, even if resoluteness is conceived as merely the imposition, by the earlier self, of a regimen that the later self accepts, or, in the case of interpersonal choice, of a pure convention, among a group of people, regarding how each is to constrain choice in certain situations. Whether a given practice is fully justified turns, of course, on what arguments can be constructed for the rules themselves. What I have argued is that the logic of interactive (intrapersonal or interpersonal) situations is typically such that practice rules are required for the effective promotion of the objectives of the participants. The notion is that there are cases in which the concerns of each cannot be served unless the future is tied down and plans co-ordinated in advance. In such cases each person's deciding what to do by reference to her own concerns, case by case, will lead to confusion, and the attempt to co-ordinate behaviour simply by each trying to predict the behaviour of the others will fail.32 When this is the case, one can appeal to the model of non-separable deliberation and choice, to show that a commitment to practice rules can be defended pragmatically, by reference to consequences that are assessed from a non-separable, global perspective. Nothing need be presupposed here regarding what each takes to be the objectives that such a co-ordination scheme is to serve. In particular, there is no reason to suppose that there is some one or more objectives that all participants share. Divergence with respect to ends can be offset by convergence with respect to means - by a shared sense that the objectives of each can be more effectively promoted by the adoption of a co-ordination scheme. Correspondingly, there is no need to introduce some ad hoc assumption about persons just happening to attach value to choosing in accordance with such rules. Nor are such persons "rulebound" in a way that can be criticized from the perspective of a theory of consequential choice.33 The story to be told here can pivot fully and uncompromisingly on consequential concerns. It can be a story of individuals who come to regulate their interactions with themselves over time, and with one another, in accordance with constraints to which time-indexed selves, or distinct individuals, can mutually assent, and who do this from nothing more than a sense of the enhanced power that such a new form of activity gives them with respect to furthering their own projects and interests.34

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7. Conclusion I have sought to argue here a number of things. First, the standard way of thinking about rationality in both intrapersonal and interpersonal contexts unacceptably fails to yield a theory that can render intelligible the notion of having a commitment to practice rules, much less provide for the rationality of being so committed. Second, this feature of the standard theory can be traced back to a basic presupposition of virtually all contemporary accounts of rationality, namely, that consequential reasoning inevitably takes place within the framework of a separability principle. Third, there is a distinct account that renders the notion of a commitment to rules both intelligible and rational: the resolute model. Finally and more ambitiously, I have sought to show that the resolute model can be defended by appeal to consequentialism itself. The notion is that a consequential argument can be constructed for adopting a more holistic or global approach to deliberation and choice, and this entails, in turn, that in certain cases one should deliberatively suspend the separability principle. In terms of the more familiar notion of practices, the conclusion is that a commitment to practice rules can be defended endogenously, from within a consequentially oriented framework. Alternatively put, the logical structure of intrapersonal and interpersonal co-ordination problems is such that a viable version of consequentialism will be a version of rule consequentialism, in which the notion of a rational commitment to extant rules has a central place.

Acknowledgments I am especially indebted to the following for helpful comments during the preparation of this paper: Bruno Verbeek, David Schmidtz, Christopher Morris, Mike Robins, and the graduate students in two separate seminars that were conducted at Bowling Green State University. Versions of this paper were read at a CREA Conference held in Normandy, France, in June of 1993, and at a Conference on Modeling Rational and Moral Agents, held at Simon Fraser University, Vancouver, Canada, in February of 1994. Notes 1 The manner in which I shall characterize the distinction between maxims and the rules of practices overlaps, but does not quite coincide with the way in which that distinction is drawn in J. Rawls, "Two concepts of rules," Philosophical Review, 64 (1955): 3-32. Following B. J. Diggs, "Rules and utilitarianism," American Philosophical Quarterly, \ (1964): 32-44,1 want to focus on a somewhat broader class of practices than Rawls does. 2 The qualifier, "rational," is crucial here. 1 am not denying the obvious fact that one can offer other accounts of an individual's commitment to practice

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rules. At least all of the following are possible: (1) some persons simply have a preference for acting subject to the constraints of such rules; (2) the commitment is the result of an essentially "unconscious" process of socialization; (3) the disposition to make such commitments is the upshot of an evolutionary process. The problem with (1) is that it is ad hoc. Explanation (2) is the one most favoured, it would seem, by economists. See, for example, K. Arrow, "Political and Economic Evaluation of Social Effects and Externalities," in M. Intriligator (ed.), Frontiers of Quantitative Economics (Amsterdam: North-Holland, 1971), pp. 3-31; and M.W. Reder, "The place of ethics in the theory of production," in M. J. Boskin (ed.), Economics and Human Welfare: Essays in Honor of Tibor Scitovsky (New York: Academic Press, 1979), pp. 133-46. On this sort of account, of course, one does have a rational motive for ensuring that others are socialized, since a steady commitment on the part of others to an ethics of rules will typically work to one's own advantage. More recently, explanation (3) has been promoted by both economists and philosophers. For a powerfully argued version of this sort of explanation, see R. H. Frank, Passions within Reason: The Strategic Role of the Emotions (New York: W. W. Norton, 1988). I do not deny the relevance of either (2) or (3). My concern is rather with the nearly ubiquitous presupposition that such a commitment could not arise as the result of rational deliberation. That presupposition, I shall argue, is implausible. For a sense of the extraordinary range of stories that can be told here, see, for example, R. H. Strotz, "Myopia and inconsistency in dynamic utility maximization," Review of Economic Studies, 23 (1956): 149-58; P. Hammond, "Changing tastes and coherent dynamic choice," Review of Economic Studies, 43 (1976): 159-73, and "Dynamic restrictions onmetastatic choice," Economics, 44 (1977): 337-50; M. E. Yaari, "Endogenous changes in tastes: A philosophical discussion," Erkenntnis, 11 (1977): 157-96; J. Elster, Ulysses and the Sirens: Studies in Rationality and Irrationality (Cambridge: Cambridge University Press, 1979); E. F. McClennen, Rationality and Dynamic Choice: Foundational Explorations (Cambridge: Cambridge University Press, 1990); and G. Ainslie, Picoeconomics (Cambridge: Cambridge University Press, 1993). For an illuminating discussion of planning in the context of changing information, see M. Bratman, Intention, Plans and Practical Reason (Cambridge: Harvard University Press, 1987). See F. P. Ramsey, "Truth and probability," in R. B. Braithwaite (ed.), Foundations of Mathematics and Other Logical Essays (London: Routledge & Kegan Paul, 1931), pp. 156-98; D. Davidson, J. McKinsey, and P. Suppes, "Outlines of a formal theory of value, I," Philosophy of Science, 22 (1955): 60-80; F. Schick, "Dutch Bookies and Money Pumps," Journal of Philosophy, 83 (1986): 112-19; and E. F. McClennen and P. Found, "Dutch Books and Money Pumps," Theory and Decision (forthcoming). Pertinent discussions of consequentialism are to be found in P. Hammond, "Consequential Foundations for Expected Utility," Theory and Decision, 25

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(1988): 25-78; E. E McClennen, Rationality and Dynamic Choke: Foundational Explorations (Cambridge: Cambridge University Press, 1990), pp. 144-46, in particular; I. Levi, "Consequentialism and sequential choice," in M. Bacharach and S. Hurley, Foundations of Decision Theory (Oxford: Basil Blackwell, 1991), pp. 92-122; and J. Broome, Weighing Goods (Oxford: Basil Blackwell, 1991), pp. 1-16,123-26. 7 The qualifier "still" is important here, since as you move through the tree, certain opportunities are foregone; paths that were in fact not taken lead to outcomes that are, then, no longer possible. 8 Formally speaking, it would seem that concerns of this sort can. be captured within a consequentialist framework by working with a more permissive notion of what counts as an outcome. Someone who now prefers to make choices that are consistent with choices made earlier can be said to view the path by which they reach a given outcome (in the more ordinary sense of that term) as part of the outcome. See the references in note 6. Once again, however, what is relevant are the preferences entertained by you now. 9 By the same token, of course, nothing on this account mandates that you have such preferences. That is, it is also possible that you have virtually no, or at best only a very imperfect, commitment to past decisions. 10 The point is that each of your future selves stands to its past selves in just the same relation that your present self stands to your past selves. But as just observed, in the case in question, the present self does not conceptualize its deliberative problem in such a way that deliberation could issue in a decision to co-ordinate with its past self. 11 Autarky ordinarily implies not just independence but also self-sufficiency. What will emerge shortly is that the time-defined self does less well by exercising independent choice than it would by entering into a co-operative scheme of choice. In this respect, it can realize only an imperfect form of self-sufficiency. 12 If I have not misunderstood D. Davidson's argument in "Deception and Division," (in J. Elster, ed., The Multiple Self [Cambridge: Cambridge University Press, 1986], pp. 79-92), this appears to be the position that he adopts when he argues that there is no reasoning that extends across the boundaries of the divided self, only causal or power relations. Since I propose to challenge this assumption, it seems clear to me that our views on both rationality and, for example, weakness of the will, significantly diverge. I must leave to another paper, however, the task of sorting out and comparing our respective viewpoints. 13 This is the case upon which Ainslie focuses in Picoeconomics. Once again, space considerations preclude my exploring the relation between my own account of dynamic intrapersonal choice and that which is to be found Ainslie's most interesting and insightful work. 14 I shall return to this point in Section 5, below. For the present it perhaps will suffice to remark that the concept of choices that are in equilibrium is cen-

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tral to the work of T. Schelling, The Strategy of Conflict (Cambridge: Harvard University Press, 1960), David Lewis, Convention: A Philosophical Study (Cambridge: Harvard University Press, 1969), and virtually all those who have subsequently explored the nature of co-ordination games. The condition formulated here constitutes a generalization of the one formulated in my Rationality and Dynamic Choice. For a much fuller treatment of the more technical details of this, see my Rationality and Dynamic Choice, chaps. 6-8. Supposing that preference tout court determines choice, could it be argued that if you are resolute, you will face no preference shift at all: what you end up doing, ex post, is what you prefer to do, both ex ante and ex post? To leave the matter there, I think, would be to provide an account suited only to what Sen characterizes as "rational fools." See Amartya Sen, "Rational fools: A critique of the behavioral foundations of economic theory," Philosophy and Public Affairs, 6 (1977): 317-44. A more plausible approach would involve an appeal to the notion of counter-preferential choice, or of secondorder preferences. See, in particular, J. Raz, Practical Reason and 'Norms (Princeton, NJ: Princeton University Press, 1990), ch. 1, on exclusionary reasons; H. G. Frankfurt, "Freedom of the Will and the Concept of a Person" Journal of Philosophy, 67 (1971): 5-20; and Amartya Sen, "Rational fools." It is interesting to note, in contrast, that tyranny is exactly what the sophisticated self achieves, by the device of precommitment. Ulysses before he hears the Sirens does not respect the preferences of Ulysses after he hears the Sirens, and once he precommits, his later self has no choice but to accept the constraints imposed by his earlier self. His resolve, then, does not unravel; but this is simply because he has tied his hands in advance. Once again, of course, as discussed earlier, you might just happen to be the sort of person who values choosing in a manner that is consistent with earlier choices made. Given preferences of the type in question, however, you have no need to be resolute in the sense with which I am concerned: ordinary motivations carry you through. In such cases, it surely makes more sense to invoke a principle of tolerance, and let each theorist nurse his or her own intuitions. On this reading, however, separability has only limited, inter-subjective standing, that is, standing only within the circle of the committed. Some will insist that the separability principle, which validates sophisticated rather than resolute choice, speaks to a fundamental requirement of consistency, and thus that -which appeals to pragmatic considerations can have no force here. The root notion is presumably that there must be a match between what you are prepared to choose at some particular node in a decision tree, and what you would choose in the modified version of the problem, in which that choice node becomes the initial choice node. But why is such a match required? If one appeals to intuition here, then one simply arrives back at the point already identified in the text, where argument leaves off

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and rhetoric begins. Moreover, in this instance my sense is that the "intuition" which underpins the separability principle is essentially the product of a confusion arising from the manner in which two distinct conditions consequentialism and separability - are intertwined, and that if any condition is intuitively secure, it is consequentialism rather than separability. 22 Some, of course, will be inclined to resist this conclusion. But those who resist must embrace the rather paradoxical position that a fully rational person, faced with making decisions over time, will do less well in terms of the promotion of these standard values than one who is capable of a special sort of "irrationality." The literature of the last two decades bears testimony to the great number who have, with varying degrees of reluctance, had to embrace this odd conclusion. For a sampling, see M. E. Yaari, "Endogenous changes in tastes"; J. Elster, Ulysses and the Sirens; D. Parfit, Reasons and Persons (Oxford: Clarendon Press, 1984); and R. Nozick, The Nature of Rationality (Princeton: Princeton University Press, 1993). 23 It is instructive to consider how Elster approaches this issue in Ulysses and the Sirens. A central claim of that book is that perfect rationality involves a capacity to relate to the future, not simply in the sense of being able to look farther ahead, but also being able to wait and employ indirect strategies. That is, it involves being able to say no to an attractive, short-run advantage, a local maximum, in order to achieve something even better, a global maximum. Elster also argues, however, that human beings manifest this capacity imperfectly and, thus, have to settle for the second-best strategy of precommitment. Now, precommitment is a form of sophisticated choice. How, then, are we to understand the first-best form of reasoning, the one that lies beyond the reach of the human deliberator? The global maximum, it would seem, is precisely what can be achieved by the resolute chooser. Why does Elster conclude that the global maximum is beyond our reach? Because we are subject to weakness of will. What accounts for the latter? Elster offers a variety of suggestions, but he also - following a suggestion of D. Davidson - concludes that weakness of will is a form of surdity in which the causal processes of the mind operate behind the back of the deliberating self. This is framed, moreover, by an insistence that to provide an explanation of weakness of will is different from offering a strategy for overcoming it. In contrast, the account I have offered interprets at least one form of weakness of will not as a surdity, but as a matter of an error in deliberation, arising from a conceptual conflation of consequentialism with separability - of confusing a particular manner of reasoning from consequences, with reasoning in general with respect to consequences. This diagnosis, moreover, contrary to both Davidson and Elster, does appear to smooth the way to a cure. To grasp that there is a confusion here is to realize that there is an alternative, and more consequentially defensible approach to dynamic choice, which is captured in the notion of being resolute when this works to the mutual advantage of one's several time-defined selves.

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For a more detailed discussion of the relation between the argument pursued here and Sister's work, see E. F. McClennen, Rationality and Dynamic Choice, Section 13.7. I focus just on these games so as to keep the exposition within bounds. This class of games, however, is pivotal for the whole theory of games. This, then, is a fixed point of the standard theory of games: if you are rational your choice must be a preference-maximizing response to what, at the moment of choice, you expect the other player to do. There is a huge literature on refinements in, and modifications of, this way of thinking about rational interpersonal choice. What is basic is the concept of an equilibrium of choices, as developed originally by J. F. Nash, in "Non-cooperative games," Annals of Mathematics, 54 (1951): 286-95. A most useful exposition is to be found in R.D. Luce and H. Raiffa, Games and Decisions (New York: John Wiley, 1958), ch. 4. For a sense of the wide range of variations on, and modifications in, this way of thinking, see in particular J. B. Kadane and P. D. Larkey, "Subjective probability and the theory of games," Management Science, 28 (1982): 113-20; B. D. Bernheim, "Axiomatic characterizations of rational choice in strategic environments," Scandinavian Journal of Economics, 88 (1986): 473-88; and W. Harper, "Ratifiability and refinements (in twoperson noncooperative games)," in M. Bacharach and S. Hurley (eds.), Foundations of Decision Theory (Oxford: Basil Blackwell, 1991), pp. 263-93.1 have tried to explain what I find unconvincing about all of these approaches in "The Theory of Rationality for Ideal Games," Philosophical Studies, 65 (1992): 193-215. Note, however, that it is not a matter of strategic indifference whether you play first rather than simultaneously. The player who goes first, just like one's earlier self, is faced with the task of determining what will maximize her present preferences for outcomes, given that the player who goes second will maximize in an autarkic manner. Once again I have modified the formulation of the relevant separability principle, specifically the one that I employ in "The Theory of Rationality for Ideal Games," so as to leave open the possibility that an agent might just happen (for some non-deliberative reason) to have a preference for coordinating her choice with the other participating agent. It might be objected, of course, that in a game such as a simultaneous choice Prisoner's Dilemma, you will have a quite distinct reason for choosing the non-co-operative strategy, namely, so as to minimize the loss that the other person could impose on you. But this argument cannot be sustained within the context of ideal games played under conditions of common knowledge. Under such conditions, once the separability assumption is replaced by the assumption that rational players will resolutely act so as to secure gains that co-ordination can make possible, each will expect the other to co-operate, and thus the risk factor is eliminated. There is more that needs to be said here, of course, since two individuals each of whom

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Edward F. McClennen is disposed to conditionally co-operate may fail to co-operate even under conditions of common knowledge. The problem is simply that knowing the other to be a conditional co-operator does not ensure that the other will co-operate. There may, in effect, be no decoupling of consequent choice directives from their antecedent conditions. See H. Smith, "Deriving morality from rationality," in P. Vallentyne, ed., Contractarianism and Rational Choice (Cambridge: Cambridge University Press, 1991), pp. 229-53; and P. Danielson, Artificial Morality (London: Routledge, 1992). It might also be objected that in such games there is a distinct argument for taking mutual non-co-operation as the rational outcome, an argument that is based on an appeal to a principle of dominance with respect to outcomes. But dominance considerations, no less than equilibrium considerations, carry little weight in contexts in which a case can be made for a co-ordinated approach to choice. Just how problematic dominance reasoning can be is revealed in N. Howard, The Paradoxes of Rationality (Cambridge: MIT Press, 1971). Howard is forced to admit that his reliance on the dominance principle in his own theory of "meta-games" generates a serious paradox of rationality. I am suggesting, in effect, that we must march in exactly the opposite direction to that taken by virtually the entire discipline in recent years. But, then, radical measures are needed if game theory is to be rescued from the absurdities generated by the standard theory. What the standard theory offers is a marvellous elaboration of the behaviour of "rational fools" (if I may be allowed to borrow that phrase from Sen). In this regard, I have found N. Howard, "A Piagetian approach to decision and game theory," in C. A. Hooker, J. J. Leach, and E. F. McClennen (eds.), Foundations and Applications of Decision Theory (Dordrecht: D. Reidel, 1978), pp. 205-25, most useful. This is, of course, something that is central to the argument in Gauthier's Morals by Agreement. I have tried to say something along similar lines in "Justice and the problem of stability," Philosophy and Public Affairs, 18 (1989): 3-30, and in "Foundational explorations for a normative theory of political economy," Constitutional Political Economy, 1 (1990): 67-99. Space considerations preclude my exploring the relationship between the sort of intentional co-ordination I have in mind here and that which Bratman discusses in "Shared cooperative activity," Philosophical Review, 102 (1993): 327-41. Clearly, however, what I have in mind is only a species of the genus that he delineates. I take this phrasing from Rawls, "Two concepts of rules," p. 24. Bratman would disagree. He levels this charge against resolute choice in "Planning and the stability of intention," Minds and Machines, 2 (1992): 116. See, however, the rejoinder by L. De Helian and E. F. McClennen, "Planning and the stability of intention: A comment," Minds and Machines, 3 (1993): 319-33. The phrase, "a new form of activity" is taken from Rawls, "Two concepts of rules," p. 24.

3 Intention and Deliberation David Gauthier

Successful deliberation about action in the future gives rise to an intention. Michael Bratman expresses the orthodox view about how deliberation, action, and intention are related when he says, "But in deliberation about the future we deliberate about what to do then, not what to intend now, though of course a decision about what to do later leads to an intention now so to act later ... This means that in such deliberation about the future the desire-belief reasons we are to consider are reasons for various ways we might act later" (Bratman 1987, p. 103). I intend to challenge this view, arguing that it rests on an incomplete account of the relations among deliberation, intention, and action. In particular, I want to reject the contrast that Bratman draws between deliberating about what to do in the future, and deliberating about what to intend now. And in rejecting this contrast, I want also to reject his claim that in deliberating about the future we are to consider only the desire-belief reasons that we should expect to have at the time of action. Bratman wants to replace the desire-belief theory of intention in action with what he calls the planning theory. But he does not go far enough; he accepts too much of the desire-belief model to exploit fully the resources that planning offers. But this paper is not directed primarily towards a critique of Bratman. Rather, I intend to use some of his views as a foil for my own position. Of course, I can develop only a small part of a theory of intention and deliberation in this paper. For my present purposes, I intend not to question parts of the desire-belief model that now seem to me to be doubtful. For Bratman, "the agent's desires and beliefs at a certain time provide her with reasons for acting in various ways at that time" (Bratman 1987, p. 15), where desires and beliefs are taken as severally necessary and jointly sufficient to provide these reasons. He accepts desire-belief reasons, although he rejects the simplistic account of their role that the desire-belief model gives. I have come to have serious misgivings about the very existence of desire-belief reasons, but in this paper I propose largely to ignore those misgivings. It would, I now 41

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think, be better to say simply that an agent's beliefs, her representations of how the world would be at a certain time, and how it might be were she to perform the various actions possible for her at that time, give her reasons for and against performing these actions. But nothing in this paper will turn on whether we should employ the desire-belief model or an alternative pure belief model in formulating what I shall call the agent's outcome-oriented reasons for acting. I do not deny that an agent has outcome-oriented reasons. But I do deny both that she has only outcome-oriented reasons, and that in deliberating about the future, she should always consider only the outcome-oriented reasons she would expect to have at the time of action. Of course, deliberation about the future concerns what to do in that future. Successful deliberation about the future typically concludes in a decision now to perform some action then. (The decision may or may not be explicitly conditional, but for present purposes I shall leave conditionality to one side and take successful future-oriented deliberation to conclude simply in a decision.) And, of course, the reasons for acting that one expects to have then must enter into one's deliberation. Indeed, I want to insist that if one deliberates rationally, then what one decides to do must be what one expects to have adequate reason to do. In determining this, one certainly must take into account the outcome-oriented considerations that one expects would be relevant at the time of action. But there are, or may be, other reasons. Suppose there are not. Consider this situation. You offer to help me now provided I help you next week. Or so you may say, but you can't make helping me now literally dependent on my helping you next week. You can, of course, make your helping me now depend on your expectation of my help next week, where you seek to make your expectation depend on whether or not I sincerely agree to help you. You ask me whether I shall return your help, and if I say that indeed I shall, and if you take this to be a commitment, or an expression of firm intention, then you will help me. So now I am faced with deciding what to say, and whether to be sincere in what I say. Let us suppose that I judge that in all likelihood, I shall be better off if you help me now and I return the help next week than if you don't help me now. I think that I have good desire-belief or outcome-oriented reasons to bring it about that you help me, even if I were then to help you. And so, deliberating about what to do now, I think that I should say that I agree to return your help. But let us suppose that, for whatever reason, I think that I should say this only if I am sincere - only, then, if I actually intend to return your help. This leads me to deliberate further - not about what I shall do now, but about what I shall do next week. Unless I decide now that I shall return your help next week, I shall not decide to say that I agree to return your help.

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But the outcome-oriented reasons that I expect to have next week may not favour my then returning your help. For, let us suppose, I am not concerned about our ongoing relationship, or about third-party reputation effects. Although I should like your help, I do not care greatly for you, I do not expect our paths to cross in future, and I do not expect to associate with those persons who would be likely to find out about my not returning your help and think ill of me in consequence. I can see quite clearly that come next week, if you have helped me I shall have gained what I want and nothing, or not enough, in my outcomeoriented reasons, will speak in favour of reciprocating. Faced with the need to decide what to say about what I shall do next week, I have deliberated about what I shall do then. And I have deliberated in terms of the outcome-oriented reasons for the various ways I might then act. Since they are reasons for not reciprocating, that is what I decide to do next week, should you help me now. Hence my intention - if I am rational - must be not to reciprocate. I cannot rationally intend to do what I have decided not to do. And so what I must say now, given that I am for whatever reason not willing to say that I agree to do what I intend not to do, is that I do not agree to return your help. And so you do not help me and we both lose out - each of us would do better from mutual assistance than no assistance. I have deliberated about what to do then, rather than what to intend now, and I have deliberated about what to do then in terms of the outcome-oriented reasons I should expect to have then. I have followed Bratman's lead, and I have paid a high price for it. For if I were to deliberate about what to intend now, then my outcome-oriented reasons would speak in favour of intending to reciprocate. And if I were to form the intention to reciprocate, then I should, of course, say that I agreed so to act. You would help me and - if I then acted on my intention - we should both benefit. Bratman contrasts deliberating about what to do later with deliberating about what to intend now. But this way of contrasting the two modes of deliberation is misleading. For just as deciding what to do in the future leads to an intention now so to act later, so forming an intention about what to do in the future leads to, or is, a decision now so to act later. To form the intention to reciprocate next week is to decide now to reciprocate. And so deliberating about what to intend now is, and must be, deliberating about what to do later. To be sure, there seems to be a difference between deliberating about what to do directly, and deliberating about what to do as a consequence of what to intend. The former proceeds in terms of reasons that one expects to have at the time of performance for acting in one way rather than another, whereas the latter proceeds in terms of reasons that one has at the time of deliberation for intending to act in one way rather than

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another. In many contexts the two sets of reasons coincide. But in my example the outcome-oriented reasons do not coincide, since the intention to reciprocate has effects that actually reciprocating does not have, and these effects manifest themselves in the difference between my outcome-oriented reasons for intending and my outcome-oriented reasons for performing. The contrast that Bratman wanted to draw would seem to relate to this difference, between deliberating on the basis of outcome-oriented or desire-belief reasons for acting, and deliberating on the basis of outcome-oriented or desire-belief reasons for intending to act. But if we draw it in this way, it is surely evident that the latter mode of "deliberation" would be absurd. Suppose, for example, that you will confer some minor favour on me if you expect me to act later in a way that would be very costly to me - the cost far outweighing the benefit from your favour. So I have sufficient outcome-oriented reasons not to perform the costly act, and would have such reasons even if performing the act were the means to gain the favour. Deliberating on the basis of my reasons for acting, I should decide not to perform the costly act. But if I were to form the intention, I should gain the favour, and since intending is one thing and acting is another, it may seem that I have adequate outcome-oriented reasons to form the intention to perform the costly act. But this is to decide to perform the act - which is surely absurd. I agree. To focus deliberation purely on outcome-oriented reasons for intending, where the intention is divorced from the act, would indeed be absurd. But we should not therefore conclude that in deliberating about what to do, considerations relevant to the intention are to be dismissed. For some of these considerations, I shall argue, give the agent reasons for performing the intended act. These are not outcomeoriented, or desire-belief reasons for acting. They may concern the outcome of having the intention, but they do not concern the outcome of performing the intended action. However, I shall argue, they are reasons for acting none the less, and they are relevant to the agent's deliberation about what to do. When they are taken properly into account, deliberating about what to do later, and so what to intend now, and deliberating about what to intend now, and so what to do later, both appeal to reasons, oriented not to the expected outcome of the action alone, but rather to the expected outcome of the intention together with the action. Or so I shall now argue. Let us consider again my deliberation arising out of your conditional offer to assist me. Suppose that I must decide whether to assure you sincerely that I shall reciprocate. But I cannot decide this without deciding whether to reciprocate. Unless 1 intend to reciprocate I cannot sincerely assure you that I shall. A sincere assurance carries with it an

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intention to perform the assured act, and the intention carries with it the decision so to act. Note that I am not claiming that intention is sufficient for assurance. If I intend to go to a film merely because I doubt that there will be anything worth watching on television, I could not sincerely assure you that I should go to the film. Not every intention is strong or firm enough to support an assurance. My claim is only that an intention is necessary support. So deliberating about whether to offer you a sincere assurance that I shall reciprocate involves me in deliberating about whether I shall reciprocate. And now my claim is that I should focus initially, neither on the outcome-oriented reasons I should have for and against reciprocating to the exclusion of all else, nor on the outcome-oriented reasons I should have for and against intending to reciprocate to the exclusion of all else, where the intention is divorced from the act, but rather on the outcome-oriented reasons for the overall course of action that would be involved in sincerely assuring you that I should reciprocate. If I am to give you this assurance, then I must decide to reciprocate; hence I consider my reasons for giving you the assurance and then, should you help me, reciprocating, in relation to my reasons for the other courses of action I might undertake, and that do not include my giving you a sincere assurance. And I conclude that my outcomeoriented reasons best support the first course of action. Note that I may not consider, among the alternatives, giving you a sincere assurance and then not reciprocating. Even if this would be better supported by my reasons for acting, and even though it is possible for me both to give you a sincere assurance and then not to honour it, and indeed to decide to give you a sincere assurance and then later to decide not to honour it, it is not possible for me to make a single decision to do both. For deciding to give you a sincere assurance involves deciding to honour it, so that to decide to give you a sincere assurance and not to honour it would be to decide both to honour and not honour it. Giving you a sincere assurance and not reciprocating is then not a possible course of action, considered as a single subject for deliberation. To this point, my account of deliberation about the future makes reference only to outcome-oriented reasons, but for the entire course of action from now until then, and not merely for the ways an agent might act then. But in deliberating among courses of action, an agent would be mistaken simply to conclude with a decision in favour of that course best supported by these outcome-oriented reasons. A course of action is not decided upon and then carried out as a whole. Insofar as it involves a succession of actions, it involves a succession of choices or possible choices, and these may not be ignored. If 1 am to decide rationally on a course of action, I must expect that I shall have good reason to carry it

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out, and so to choose the successive particular actions that constitute the course. Hence, however desirable it may seem to me to decide on a course of action, I must consider whether it will also seem desirable to perform the various acts it includes, when or if the time comes. This further consideration does not simply reintroduce outcomeoriented reasons for these future acts. Their desirability turns on importantly different factors. But before I explain what these are, I should consider a challenge to the claim that any further consideration is necessary. For if an agent has adopted a course of action, and done so after rational deliberation, then she has reason to carry it out, and so to perform the particular actions it requires, unless she has reason to reconsider. Without reason to reconsider it would be unreasonable for her to ask, when or if the time comes, whether it would seem desirable to perform the various actions required by her course of action. And if it would be unreasonable for her to ask this, then why need she ask at the outset whether it will seem desirable to perform these actions when or if the time comes? Why need she consider more than the overall desirability of the course of action? But what determines whether it is rational for an agent to reconsider? For it may be rational for an agent to reconsider her course of action if she has failed to consider whether it will seem desirable to perform the particular actions it requires when or if the time comes. Now if we suppose the agent's values or concerns are to remain unchanged, then the grounds of rational reconsideration, it may be suggested, must relate to one or both of two possibilities. The first is that the agent has come to recognize that the circumstances in which she adopted her course of action were not what she reasonably took them to be in adopting it, so she should reconsider the rationality of adoption in the light of her new understanding. And the second is that the circumstances in which the agent now finds herself were not those that she reasonably expected in adopting her course of action, so she should reconsider the rationality of continuing in the light of the actual current circumstances. But if these are the only grounds of rational reconsideration, she will have no reason to consider the desirability of performing the various particular actions required by her overall course if she finds herself in any of those circumstances, present or future, that she envisages in adopting it. And so she has no reason to deliberate in advance about the desirability of carrying on with her course of action in the circumstances that she envisages as obtaining or arising. I reject this argument. Grant that it is unreasonable to reconsider one's course of action if one has adopted it as a result of rational deliberation and circumstances that are as one envisaged or expected. Nevertheless, adopting it as a result of rational deliberation requires more

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than considering its overall expected desirability. One must also consider at the outset whether it would seem desirable at the time of performance to choose the various acts it includes - not because one will be rationally required to consider this desirability at the time of performance, but to obviate the need for this further consideration. For suppose there is no such need. Consider this situation. I want to get my way with you in some matter, and I think that the most effective way of doing so would be to threaten you with something pretty nasty if you do not agree. And suppose that for whatever reason I think that only a sincere threat will do - not perhaps because I have any compunction about being insincere but simply because I know from past experience that I am a hopeless bluffer. But if a threat is sincere, then it must involve the intention to carry it out in the event of non-compliance by the threatened party. I cannot decide sincerely to threaten you without deciding to carry out the threat if need be. So I am faced with the prospect of a course of action that includes issuing the threat and, should it fail, carrying it out. Of course, I expect it to succeed, but I must still consider the possibility that it will fail. And now it occurs to me that should it fail, then carrying it out will be something that I shall have good reason not to do. I do not think that carrying it out will enhance my credibility as a threatener to any great extent, so I have nothing to gain from carrying it out, and I should, let us suppose, have to do something that I should find quite costly in order to be nasty to you. Indeed, I think, were you not to comply, then I should be worse off carrying out the threat than if I had never made it. And this leads me to conclude that it would not make sense or be reasonable for me to carry out the threat, and that in fact I would not carry it out. But then I cannot intend or decide to do it - at least not in any direct or straightforward way. If I intend to do something, then I must at least not consciously believe that I shall definitely not do it. To be sure, it would not be rational for me to reconsider whether to carry out a failed threat, provided the circumstances of failure were what I envisaged they would be were the threat unfortunately to fail, if I had rationally issued the threat in the first place. But my point is that it would not be rational to issue the threat, and with it the intention to carry it out if need be, given that should it fail, one would then be worse off carrying it out than had one never issued it. That carrying out a threat would leave one worse off than had one not made it is a reason against carrying it out. It is this feature of the situation in which one would have to carry it out - a feature that one may be in a position to recognize at the outset - that leads one to judge that it would not be rational to carry out the threat, and thereby makes the whole course of action irrational for one to adopt.

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Deliberating rationally about a course of action is a complex matter. An agent must consider both her reasons for choosing the course as a whole in preference to its alternatives, and also her reasons for choosing the particular actions it requires. But the latter need to be set in the context of the former, in a way that I shall now try to explain. In the situation that I have just considered, the problem that I identified in carrying out the threat was not merely that I could do better not to carry it out, but also that I should have done better not to have made it, than I should do to carry it out. If the threat fails, then the situation is no longer one in which my course of action is best. However reasonable my initial expectation may have been, the failure of my threat shows that the course of action consisting of issuing a threat and carrying it out if need be, actually turns out worse for me than an alternative in which no threat is issued. What this suggests is that within the context provided by a course of action, the reasons relevant to performing a particular action focus not on whether the action will yield the best outcome judged from the time of performance, but on whether it will yield a better outcome than the agent could have expected had she not undertaken that course of action. Consider again my deliberation in the light of your offer to help me provided I am willing to return your help next week. I judge that the best course of action that I can adopt is to offer you a sincere assurance to reciprocate, which brings with it my decision now to reciprocate next week. But I then ask myself whether I shall have reason next week to carry this course of action through. And I reflect that, although next week my outcome-oriented reasons would favour not reciprocating, yet I shall still in all likelihood judge the course of action that includes reciprocating as better than any alternative that I might have chosen. (Recall that giving you a sincere assurance and not reciprocating is not a course of action that I can choose.) And this, I claim, gives me sufficient reason to reciprocate. Suppose that I issue a threat and it fails. Then, even though the circumstances I may find myself in are exactly those that I anticipated should the threat fail, and even though I recognized that I might find myself in these circumstances, yet in making the threat, my expectation was that I should benefit thereby, and I now know that expectation to have been mistaken. Suppose on the other hand that I give an assurance and it succeeds -1 agree to reciprocate and you in consequence help me. Then if the circumstances I find myself in are those that I anticipated should my assurance succeed, my expectation that I should benefit thereby has proved correct. And this difference, between the mistaken expectation associated with a failed threat, and the confirmed expectation associated with a successful assurance, is crucial to deciding the

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rationality of continuing one's course of action. It is not rational to continue a course of action if the expectation associated with adopting it has proved mistaken, and if continuing it is not then supported by one's outcome-oriented reasons for acting. It is rational to continue a course of action if the expectation associated with adopting it has proved correct, even if continuing it would not then be supported by one's outcome-oriented reasons for acting. These two theses are at the core of my argument. Let us say that a course of action is confirmed at a given time, if at that time the agent may reasonably expect to do better continuing it than she would have expected to do had she not adopted it. Then to deliberate rationally about adopting a course of action, an agent must consider both whether it is adequately supported by her outcomeoriented reasons at the time of adoption, and whether her expectation is that the course would be fully confirmed - that is, confirmed in each possible situation in which it would require some particular action or decision. Full confirmation constitutes a filter; only courses of action that the agent expects would be fully confirmed are eligible for adoption. Among those she judges fully confirmed, the agent rationally adopts the course best supported by her outcome-oriented reasons for acting, and then does not rationally reconsider unless she becomes aware that circumstances at the time of adoption were in relevant ways not as she envisaged them, or that her present circumstances differ relevantly from what she expected at the time of adoption. If she comes to think that her circumstances at the time of adoption differed from what she then believed, she must ask whether she actually had adequate outcome-oriented reasons to adopt it. If she comes to think that her present circumstances differ from those she expected, she must ask whether in her actual circumstances it remains confirmed. And negative answers to either of these questions require her to reopen deliberation and consider in her present situation what course of action she should follow. This account of deliberation is intended only as a preliminary sketch. One complicating aspect that should be kept in mind is that one course of action may be adopted in the context of a more embracing course, which will serve as a further filter on eligibility. Suppose for example that I have a general policy of giving only sincere assurances; then courses of action involving insincere assurances would not be eligible for deliberative consideration, even if they might be eligible in themselves and best supported by my outcome-oriented reasons for acting. A fully confirmed course of action whose adoption is adequately supported by an agent's outcome-oriented reasons for acting may

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require particular actions not best supported at the time of performance by the agent's outcome-oriented reasons. The agent may know this, but will nevertheless consider herself to have adequate reason to perform such actions. She might express this reason at the time of performance by noting that she expects to do better in terms of her outcome-oriented reasons, than if she had not adopted the course of which the action is part. And done what instead? What alternative does she compare to continuing with her adopted course of action? I suggest as a first approximation that the relevant alternative would be the best course of action that would not have required her to act against the weight of her outcome-oriented reasons at the time of performance. But this is a complex issue that I shall not pursue here. Instead, I shall conclude with two further matters. The first is to note the implications of my account of deliberation for Kavka's toxin puzzle. The second is to state very briefly the rationale for my account - why is deliberation that accords with it rational?

Imagine an individual who is an exceptionally astute judge of the intentions of her fellows, and who - perhaps partly in consequence - is extremely wealthy. Bored with making yet more money, she devotes herself to the experimental study of intention and deliberation. She selects persons in good health, but otherwise at random, and tells each of them that she will deposit $1,000,000 in his bank account at midnight, provided that at that time she believes that he intends, at 8 a.m. on the following morning, to drink a glass of a most unpleasant toxin whose well-known effect is to make the drinker quite violently ill for twenty-four hours, by which time it has passed entirely through his system leaving no after-effects. Her reputation as a judge of intentions is such that you think it very likely that at midnight, she will believe that you intend to drink the toxin if and only if you do then so intend. She puts her offer to you; how do you respond? This is Kavka's puzzle. Kavka - and Bratman - think that it would be irrational for you to drink the toxin. And one can hardly deny that when the time for drinking it is at hand, you would have excellent outcome-oriented reasons not to do so. But let us suppose that you would be willing to drink the toxin in order to gain a million dollars. Suppose, then, that the offer were to put $1,000,000 in your bank account if at 8 a.m. tomorrow you were to drink the toxin. Deliberating now about what to do, you would surely decide to drink tomorrow morning, thus forming the intention to do so. So you can form the intention to drink the toxin. Now return to the actual offer. Since you believe that forming the intention to drink will lead the experimenter to deposit the money in your bank account, then deliberating now about what to do, isn't it rational for you to form the intention to drink the toxin, and so to decide to drink it?

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Bratman agrees that you have good reason now to have the intention to drink the toxin, if it is possible for you to have it without undue cost. But he denies that this reason can affect your deliberation about whether to drink it, since it will not apply tomorrow morning. He insists that you cannot be led to form the intention to drink by deliberating rationally about what to do tomorrow, since your deliberation will turn on what action will then be supported by your outcomeoriented reasons. Bratman thinks that you may have good reason to cause yourself to intend to drink the toxin tomorrow morning, but this way "will not be simply ordinary deliberation" (Bratman 1987, p. 103). Bratman agrees that it would not be rational to reconsider an intention if one has adopted it as a result of rational deliberation and circumstances are as one envisaged or expected. And so if one had acquired the intention to drink the toxin through deliberation, it would indeed be rational to drink it. But he refuses to extend this view of reconsideration to cases in which one has caused oneself to have an intention rather than adopting it through deliberation. Hence, even though he agrees that it would be rational for one to cause oneself before midnight to intend to drink the toxin, he denies that "reasonable habits of reconsideration ... would inhibit reconsideration" (Bratman 1987, p. 106), and claims that reconsideration would lead one to decide not to drink. I can agree with Bratman that if an intention is not acquired through rational deliberation, then it may be rational to reconsider it even if circumstances are as one envisaged. But, of course, I deny that the intention to drink the toxin is not acquired through rational deliberation. Faced with the experimenter's offer, you should consider what course of action to adopt. Outcome-oriented reasons support intending to drink the toxin, even though this carries with it the decision to drink it. You must then consider whether, come tomorrow morning, you will have good reason to implement your decision. And here you do not consider only outcome-oriented reasons. Instead, you ask yourself whether you would expect to be better off carrying it out, than if you had not adopted the course of action that requires you to make it. Had you not adopted the course of action, you would expect to find your bank balance unchanged. Having adopted it, you would expect to find it enhanced by $1,000,000 - well worth one day's illness. And so your course of action would be confirmed. It is rational for you to form the intention to drink the toxin, and to do so deliberatively. And it would not be rational for you to reconsider your decision to drink the toxin tomorrow morning. Bratman thinks it obvious that it would not be rational for you actually to drink the toxin. I think that it would be rational for you to drink it - although I do not claim that this is obvious. Why do I think this? Of

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course, my account of rational deliberation endorses the course of action that embraces intending to drink the toxin and then actually drinking it, but why do I think that my account is correct? A full answer to this question would take me far beyond the confines of this present paper. Part of that answer is implicit in my criticism of focusing exclusively on outcome-oriented reasons in deliberating about what to do, but let me try to make it more explicit. Suppose that, agreeing with Bratman, we accept a planning theory of intention, or perhaps more generally of deliberation. The core of such a theory is the idea that we deliberate, not only to decide on particular actions, but also to decide on courses of actions, or plans. And when an agent deliberatively adopts a plan, she then confines her subsequent deliberation to actions conforming to her plan unless she has adequate reasons to reconsider, where such reasons depend on her recognition that she mistook her initial circumstances, or formed mistaken expectations about the circumstances in which some act is required by the plan. Consider three forms that a planning theory might take. The first is roughly akin to Bratman's, and maintains that a plan is eligible for adoption (henceforth 1-eligible) if and only if each intention that it requires the agent to form would be supported by the outcomeoriented reasons she expects to have in the situation in which she would act on the intention. The second is the one I have proposed, and maintains that a plan is eligible for adoption (2-eligible) if and only if each intention that it requires the agent to form would be confirmed in the situation in which she would act on the intention, in the sense that the agent may reasonably expect to do better in performing the action than she would have done had she not formed the intention, or had not adopted the plan requiring it, but had instead restricted herself to 1-eligible plans. And the third maintains that any plan is eligible for adoption. All three agree that the agent should choose among eligible plans on the basis of her outcome-oriented reasons. These three versions of the planning theory differ on the scope of eligible plans, with the first being the most restrictive. Now one might defend no restrictions, and so the third version of the theory, on the ground that whenever the agent would choose a different plan under the third rather than under either of the other two, it can only be because some plan ineligible under one of the first two versions is better supported by her outcome-oriented reasons than any eligible plan. But against this one may note that according to the third version, it may be irrational for an agent to reconsider a plan even if she recognizes that she would have done better not to have adopted it, and would now do better to abandon it. And if this seems a decisive objection, then we may naturally turn to the second, version, since 2-eligibility excludes a

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plan only if it would require being followed in the face of evident failure. And the superiority of the second to the first version is clear; if an agent would choose different plans under these versions, it can only be because some plan lacking 1-eligibility is better supported by the agent's outcome-oriented reasons both at the time of choice and at the times of execution than any 1-eligible plan. Rational deliberation about plans is responsive to the agent's outcome-oriented reasons for acting. But responsiveness is not a simple matter. The first version of the planning theory affords greater direct responsiveness at the time of performance, but less responsiveness overall, than the second. The third version affords greater direct responsiveness at the time of adoption than the second, but lacks responsiveness at the time of performance. I conclude that the second version mine - best characterizes rational deliberation. What is the relation between intention and deliberation, as it emerges from this discussion? Intentions that an agent has rationally formed, and has no reason to reconsider, constrain her future deliberation; she considers only actions compatible with those intentions. On what grounds are intentions rationally formed? Here there are two different questions. What makes it rational to settle on some intention or other? The benefits of deciding matters in advance, more generally of planning, provide an answer. Given that it is rational to settle on some intention or other, what makes it rational to form a particular intention? The orthodox answer, accepted even by planning theorists such as Bratman, is that it is rational to form a particular intention on the basis of one's expected outcome-oriented reasons for performing the action. And this is frequently the case. But this answer overlooks one important benefit of deciding matters in advance - the effect it may have on the expectations, and consequent actions, of others. This effect may give one outcome-oriented reasons for intending quite unrelated to one's reasons for performing. Deliberation may be concerned not only with what to do then, and so in consequence what to intend now, but also with what to intend now, and so in consequence what to do then. I have tried to accommodate both of these deliberative concerns in a unified account. And the rationale of this account is pragmatic. The person who deliberates about future actions and forms intentions in the manner that I have proposed may expect to do better overall in terms of her outcome-oriented reasons for acting than were she to deliberate about future actions and form intentions solely on the basis of the outcome-oriented reasons that she would expect to have for performing those actions. But, of course, I have not shown that my account can be made fully precise, or that if it can, it must be the best account of intention and deliberation.

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Acknowledgments I am grateful to All Souls College, Oxford, and the John Simon Guggenheim Memorial Foundation, for support at the time this paper was written. References Bratman, Michael E. (1987). Intention, Plans, and Practical Reason. Cambridge, MA: Harvard University Press. All quotations and page numbers are from this book. Kavka, Gregory S. (1983). The toxin puzzle. Analysis, 43 (1983): 33-36.

4

Following Through with One's Plans: Reply to David Gauthier Michael E. Bratman

1. We are planning agents. Planning helps us to realize ends that we value or desire, in part by helping us achieve important kinds of coordination, both intra-personal and social. A theory of practical rationality for agents like us should include a theory of rational planning (Bratman 1987). At least part of such a theory, and the focus of this discussion, will be a theory of instrumentally rational planning agency, planning agency that is rational in the pursuit of basic desires, ends, and values taken as given. A theory of instrumentally rational planning agency puts to one side important questions about the possibility of rational criticism of basic desires and ends. My discussion here of rational planning will be limited in this way throughout, though I will usually take the liberty of speaking simply of rational planning and action to avoid circumlocution. A theory of instrumentally rational planning agency should tell us when it is rational, in deliberation, to settle on a plan for future action. It should also provide a theory of the stability of plans - a theory that tells us when it is rational to reconsider and abandon one's prior plans. Finally, these accounts of deliberation and of plan stability should be linked: I can rationally decide now, on the basis of deliberation, on a plan that calls for my A-ing in certain later circumstances in which I retain rational control over my action, only if I do not now believe that when the time and circumstances for A arrive I will, if rational, reconsider and abandon the intention to A in favour of an intention to perform some alternative to A. We may call this the linking principle.1 In all this I am, I believe, in broad agreement with David Gauthier, both in his "Intention and Deliberation" - to which this essay is intended as a brief and partial response - and in another recent study of his (1996). But there are also disagreements, and some of these will be the focus of this discussion.2 55

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2. Let me begin not with explicit disagreement, but with a difference of emphasis. Gauthier's discussion in "Intention and Deliberation" focuses on certain puzzle cases: (1) Can a rational agent win the money in Kavka's toxin case? (Kavka 1983) (2) Can rational but mutually disinterested agents who will never see each other again (and in the absence of independent moral considerations and reputation effects) co-operate just this once? (3) When can a rational agent offer a credible threat? Elsewhere, Gauthier (1996) has also discussed puzzles posed for a theory of planning by the kind of preference change illustrated by versions of the case of Ulysses and the Sirens. These puzzle cases are fascinating and important; and I will shortly join in the fray. Nevertheless, I think we should be careful not to let these cases dominate our theorizing about planning. It is a striking fact about us that we manage at all - even in the simplest cases which do not raise the cited kinds of puzzles - to organize and co-ordinate our actions over time and socially. Normally, for us to be able to achieve such co-ordinated activity we need to be able reliably to predict what we will do; and we need to be able to do this despite both the complexity of the causes of our behaviour and our cognitive limitations. I see a theory of planning agency as part of an account of how limited agents like us are at least sometimes able, rationally, to do all this. That said, I will focus on the special issues raised by puzzle cases (l)-(3). I will put to one side here issues raised by preference change cases, though I believe that they pose important questions for a full account of rational planning agency.3 Gauthier's lucid discussion saves me from the need to spell out all the details concerning (l)-(3). Suffice it to say that these cases have the following structure: I consider at tl whether I can rationally form an intention to perform a certain action (drink the toxin, help you if you have helped me, retaliate if you do not comply) at t3. I know that my so intending at t1 would or may have certain benefits - my becoming richer at £2; my being aided by you at £2; your compliance at t2. But these benefits, even if realized, would not depend causally on my actually doing at t3 what it is that at t11 would intend to do then; by the time t3 arrives I will either have the benefits or I will not. They are, to use Kavka's term, "autonomous" benefits (Kavka 1987, p. 21). The execution at t3 of my intention would, if called for, only bring with it certain burdens: being sick from the toxin;4 the costs of helping you (if you have helped me); the costs of retaliating (if you have not complied). But at tl I judge that (taking due account of the relevant likelihoods) the expectation of these burdens of execution is outweighed by the expectation of the associated autonomous benefits. And this judgment is based on desires / values which will endure throughout: these are not preference-change cases.

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Given that I judge th at the expected autonomous benefits would outweigh the expected burdens of execution, can I in such cases rationally settle at t1 on a plan that involves (conditionally or unconditionally) so acting at t3l It is not clear that I can. Recall the linking principle. It tells us that rationally to settle on such a plan I cannot judge that I would be rationally required at t3 to reconsider and abandon my intention concerning £3 - the intention to drink, to help (if you have helped me), or to retaliate (if you have not complied). But I know that under the relevant circumstances at t31 would have available a superior alternative option: not drinking, not helping, or not retaliating. These alternatives would be superior from the point of view of the very desires and values on the basis of which I would settle on the plan in the first place. That suggests that I would at t3 be rationally required to abandon that intention. So, given the linking principle, I am not in a position rationally to decide on the plan in the first place. Should we then conclude that, despite their attractions, such plans are not ones on which a rational planning agent can decide on the basis of deliberation? Concerning at least cases (1) and (2) Gauthier thinks that a theory of rational planning should resist this conclusion, and so he seeks a different view about rational reconsideration. It is, of course, sometimes rational to abandon a prior plan if one discovers that in settling on that plan one had relied on a conception of one's present or future situation that was importantly inaccurate. But Gauthier's focus is on cases in which one makes no such discovery. Concerning such cases, Gauthier distinguishes between "three forms that a planning theory might take" (Gauthier 1997, p. 51). Each involves a different view of rational reconsideration. On the first theory, one should reconsider one's prior plan to A at t under circumstance c if one knows one has available at t, given c, an incompatible alternative that is favored over A by one's "outcomeoriented reasons."5 This approach to reconsideration leads, we have seen, to scepticism about whether a rational planner can settle on the cited plans in cases (l)-(3). On the third theory, one should not reconsider if one has discovered no relevant inaccuracy of belief, and the original decision in favour of the plan really was favoured by the expected balance of autonomous benefits and burdens of execution. Gauthier's present proposal falls between these two approaches. Suppose that at the time of action one has discovered no relevant inaccuracy of belief, and the original decision in favour of the plan was favoured by the expected balance of autonomous benefits and burdens of execution. One should at the time of action reconsider this plan if and only if one knows that following through with the plan then would be inferior to what one would have accomplished if one had not settled on this plan in the first place but had instead planned in

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accordance with the constraints of the first theory. Since the first theory highlights a comparison of the planned action with alternatives available at the time of execution of the plan, we may call it the time of execution (TE) view. Gauthier's theory, in contrast, highlights a comparison of the planned action with certain counterfactual alternatives. So let us call it the counterfactual comparisons (CFC) view. Given the linking principle, these different views about rational reconsideration yield different views about which plans are eligible for adoption in the first place. Thus, Gauthier associates the first theory with a plan's being "1-eligible," and his theory with a plan's being "2eligible." Since the third theory provides no further constraint (over and above a concern with the expected balance of autonomous benefits and burdens of execution) on a plan's being eligible for adoption, we may call it the no further constraint (NFC) view. The NFC view is similar to Gauthier's (1984) earlier view about deterrent intentions. On this earlier view, it would be rational to adopt an intention to retaliate if attacked so long as the expected impact of one's settling on this plan was optimal. And if it was rational to adopt the deterrent intention it would be rational to execute it should the occasion arise and one discovers no relevant inaccuracy of belief. An implication was that one could be rational in retaliating even though so acting was completely at odds with one's outcome-oriented reasons for one's various alternatives at the time of the retaliation. Gauthier now rejects this view, replacing it with the CFC view. Very roughly, and ignoring issues about policies to which Gauthier alludes,6 on the CFC view such retaliation may well not be rational. In opting for the deterrent intention one in effect gambled that the deterrence would succeed; now that it has failed one can see that one is worse off, as assessed by one's outcome-oriented reasons, in retaliating than one would have been had one eschewed the deterrent intention in the first place. This approach to deterrent intentions allows Gauthier to drive a wedge between such cases, on the one hand, and cases (1) and (2), on the other. In the toxin case, for example, the intention to drink the toxin can pass the CFC test; for in drinking it one is completing a course of action that is superior, from the perspective of one's outcome-oriented reasons, to what one would have achieved had one earlier decided not to drink it. In contrast, on the standard interpretation of the toxin case, the intention to drink the toxin fails the TE test. At the time of execution one knows one has available a superior alternative to drinking, namely: not drinking. Gauthier's reason for preferring the CFC view to the TE view is pragmatic: "the person who deliberates about future actions and forms intentions in [this] manner ... may expect to do better overall in terms

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of her outcome-oriented reasons" (Gauthier 1997, p. 52-3). In particular, a CFC agent can rationally achieve the benefits at stake in cases (1) and (2). Note the form of argument: A general structure of planning is justified by appeal to its expected long-run impacts; the particular piece of planning is justified insofar as it conforms to this justified general structure. This is a two-tier account, one that seems similar in structure to versions of rule-utilitarianism. That said, we may wonder why Gauthier does not stick with the NFC view; for it seems that that is where a straightforward application of a two-tier pragmatic account will lead. With an eye on this concern, Gauthier writes: "against this one may note that according to the [NFC view] it may be irrational for an agent to reconsider a plan even if she recognizes that she would have done better not to have adopted it, and would now do better to abandon it" (Gauthier 1997, p. 52). But so far this is only a statement of the difference between these two views, not an independent argument. Perhaps the idea is that the two-tiered pragmatic account should be tempered by the intuition that it is rational to reconsider in certain cases of failed deterrence. But then we need to know why an analogous argument cannot be used, against CFC, in favour of TE; for there is also a strong intuition that it would be rational to reconsider the intention to drink the toxin were the time to come.7 So there is reason to worry that CFC may be an unstable compromise between NFC and TE. 3. I agree with Gauthier that a substantial component of a theory of rational reconsideration should have a pragmatic, two-tier structure (Bratman, 1987, ch. 5). But I do not think that such a pragmatic, two-tier approach exhausts the subject. Suppose at ^ I form the intention to A at t2. Suppose that when t2 arrives I discover that I simply do not have it in my power to A then. I will, of course, give up my prior intention; for, baring some very special story, I cannot coherently embark on an effort to execute my prior intention to A once I see that I cannot A. Of course, if I had earlier known that I would not be able so to act I would not have earlier formed the intention to A. But what baffles intention is not the newness of my information, but what information it is. Why am I rationally obliged to give up my prior intention on learning of my inability? Our answer need not appeal solely to the good consequences of a general strategy of giving up one's intention in such cases; though such a strategy no doubt would normally be useful. We can also appeal directly to a kind of incoherence involved in intending and attempting to A while knowing one cannot A. If I am at

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all reflective, I cannot coherently see what I am doing as executing an intention to do what I know I cannot do. Now consider a different case. Earlier I formed the intention to A, and the time has arrived. But I have learned of a specific alternative to A that it is on-balance clearly superior to A as a way of realizing the very same long-standing, stable and coherent set of desires and values which were and continue to be the rational basis for my intention to A. My judgment of the superiority of this alternative has taken into account the various (perhaps substantial) costs of changing my mind - including the need for various forms of re-planning and the impact on my ability to co-ordinate with others. Does this new judgment, like the judgment that I simply cannot A, require me to abandon my prior intention? Though the issues here are complex, it seems to me that the answer is "yes." After all, in the case at issue I suppose that by abandoning my prior intention I best realize the very same long-standing, stable and coherent desires and values which are the basis for that intention in the first place. Now, no plausible theory will deny that many times such a judgment should trigger reconsideration and change of mind. But it might be responded that this is not always the case. Suppose there were a general strategy concerning reconsideration, a strategy that would have superior consequences over the long haul. And suppose that this general strategy supported non-reconsideration in certain special cases in which one has the cited judgment about the superiority of an alternative to A. Would it not be rational of me to endorse such a general strategy and so follow through with my intention to A? Suppose I say to myself: "I would, to be sure, do better this time by abandoning my prior intention - even taking into account the relevant costs of reconsideration, re-planning, and so on. But there is a general strategy of non-reconsideration that has a pragmatic rationale and that enjoins that I nevertheless follow through with my prior intention. So I will stick with my prior intention." But why do I give this general strategy such significance? The answer is: because of its expected impact on realizing long-standing, stable and coherent desires and values. But if that is the source of the significance to me of that strategy, how can I not be moved, on reflection, by the fact that I can even better realize those same desires and values by abandoning my prior intention in this particular case? Following through with my plan is, after all, not like following through with my golf swing: following through with my plan involves the real possibility of changing my mind midstream. Sticking with a general strategy of non-reconsideration in such a case solely on grounds of realizing long-standing, stable and coherent desires and values, while fully aware of a superior way of realizing those very same desires and values on this particular occasion, would

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seem to be a kind of incoherence - as we might say, a kind of "plan worship."8 Or at least this is true for an agent who is reflective - who seeks to understand the rationale both for her general strategies of (non)~ reconsideration and for her particular choices. This suggests that a reasonably reflective planning agent will reconsider and abandon her prior intention to A, in the absence of a change in basic desires and/or values, if she (A)

believes that she cannot A

(B)

believes of a specific alternative to A that it is on balance superior to A as a way of realizing the very same long-standing, stable and coherent desires and values that provide the rational support for the intention to A.

or

Or, at least, she will change her mind in this way if these beliefs are held confidently and without doubts about her judgment in arriving at them. Normally such beliefs will represent new information. But what obliges reconsideration is not the newness of the information, but what the information is. Of course, when the time comes to execute a prior intention one may not have either such belief and yet there may still be an issue about whether to reconsider. Indeed, this is probably the more common case. One may have new information and yet it may not be clear whether or not, in light of this new information, one could do better by changing one's plan. In such cases we frequently depend on various non-deliberative mechanisms of salience and problem-detection: we do not want constantly to be reflecting in a serious way on whether or not to reconsider (Bratman, 1987, ch. 5). Concerning such cases we can take a pragmatic, two-tier stance. We can evaluate relevant mechanisms and strategies of reconsideration in terms of their likely benefits and burdens, with a special eye on their contribution to the characteristic benefits of planning; and we can then see whether particular cases of (non)reconsideration are grounded in suitable mechanisms. Given our limits, the costs of reconsideration and re-planning, and the importance of reliability for a planning agent in a social world, such a pragmatic approach exerts clear pressures in the direction of stability. But this does not override a general demand to reconsider in the face of clear and unwavering beliefs along the lines of (A) or (B). We want mechanisms and strategies of reconsideration that will, in the long run, help us to achieve what we desire and value, given our needs for co-ordination and our cognitive limitations. We can assess

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such strategies and mechanisms in a broadly pragmatic spirit; and we can then go on to assess many particular cases of reconsideration, or its absence, in light of the pragmatic reasonableness of the relevant general mechanisms of reconsideration. But there are also occasions when reconsideration is driven primarily by demands for consistency or coherence. This seems to be true in those cases in which one has beliefs along the lines of (A) or (B). Return now to the toxin puzzle. It follows from what I have been saying that if it is clear to me, when the time comes, that not drinking the toxin is superior, in the relevant sense, to drinking it then I should not drink. And if I know earlier that this will be clear later, at the time of action, then I cannot rationally form earlier, on the basis of deliberation, the intention to drink later. A similar point follows concerning cases (2) and (3): in all three cases rationality seems to stand in the way of the autonomous benefits.9 But how will planning agents achieve the benefits of sequential cooperation? Isn't the support of interpersonal co-operation one of the main jobs of planning? A planning agent will typically be able to profit from the kind of sequential co-operation at stake in case (2) by assuring the other that she will do her part if the time comes. This will work so long as planning agents can be expected to recognize that such assurances typically induce associated obligations. These obligations generate reasons for playing one's part later, after one has been helped by the other. When and why do such assurances generate obligations? A plausible answer might well include an appeal to our planning agency. The ability to settle matters in advance is crucial to planning agents - especially planning agents in a social world. The ability to put ourselves under assurance-based obligations would help us settle certain matters, both for others and for ourselves.10 So perhaps the fact that we are planning agents helps support principles of assurance-based obligation; and we intentionally put ourselves under such obligations in the pursuit of co-operation. In this way our planning agency might help to provide indirect support for co-operation, by way of its support for principles of assurance-based obligation. This would still contrast, however, with an approach like Gauthier's that seeks to explain the rationality of co-operation in such cases by a direct appeal to principles of rational planning. Such a story about assurance can allow that, as Gauthier says, "a sincere assurance carries with it an intention to perform the assured act" (Gauthier 1997, p. 44). It just notes that one's reasons for so acting, and so one's reasons for so intending, can be affected by the assurance itself. My assurance that I will help you if you help me gives me reason

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to help you if the occasion arrives, because it generates an associated obligation which I recognize. If this reason is sufficiently strong it can make it rational for me later to help you, if you have helped me. Knowing this, I can rationally, in giving the assurance, intend to help you (if you help me). My reasons for A-ing that support my intention to A (the intention that qualifies my assurance as sincere) can depend on my assurance that I will A. My assurance can be, in effect, that I will help you (if you help me) in part because of this very assurance. Return again to the toxin case. In posing her challenge to me, the billionaire does not ask me to assure her that I will drink the toxin; she asks, instead, that I just intend to drink it. Indeed, it is an explicit condition for winning the money that I do not support this intention by making a promise to her or someone else that I will drink (Kavka 1983, p. 34). So the toxin case is very different from cases of sequential cooperation in which one achieves the benefits of co-operation by issuing appropriate assurances.11 4. A theory of rational planning should help to explain how limited agents like us manage to organize and co-ordinate so many of our activities over time and with each other. But that does not mean that the accessibility of all forms of valued co-ordination and co-operation can be explained by direct appeal to principles of rational planning. A modest theory of planning would emphasize the central roles of planning and of associated mechanisms of reconsideration. But such a theory would also draw on forms of assurance-based obligation as support for certain kinds of co-operation.12 Plans will normally have a significant degree of stability; intelligent planners will normally follow through with their plans and need not constantly start from scratch. But in versions of cases (l)-(3) in which a reflective agent sees clearly, without the need for further information or reasoning, that, in light of her long-standing, stable and coherent values, she does best by abandoning her prior plan, she should.

Acknowledgments This is a revised and recast version of my 1993 reply to David Gauthier's paper, "Intention and Deliberation." Both Gauthier's paper and my reply (titled "Toward a Modest Theory of Planning: Reply to Gauthier and Dupuy") were presented at the June, 1993 Cerisy, France, conference on "Limitations de la rationalite et constitution du collectif." My original reply, which also includes a response to Jean-Pierre Dupuy's paper "Time and Rationality: The Paradoxes of Backwards Induction," will appear in French translation in the proceedings of that

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conference. My work on this essay was supported in part by the Center for the Study of Language and Information.

Notes 1 The linking principle is concerned with rational decision based on deliberation. It has nothing to say about the possibility of causing oneself to have certain intentions/plans by, say, taking a certain drug. My formulation here has benefited from discussions with Gilbert Harman, John Pollock, and Bryan Skyrms. I discuss the linking principle also in "Planning and temptation" and in "Toxin, temptation and the stability of intention." 2 Other elements of my response to Gauthier's recent essays are in "Toxin, temptation and the stability of intention." 3 I discuss issues raised by preference change cases in "Planning and temptation" and in "Toxin, temptation and the stability of intention." I believe such cases should be given an importantly different treatment than that proposed here for cases (l)-(3). 4 At least, this is how the toxin case is standardly understood. I will proceed on this assumption (which Gauthier shares), but raise a question about it below in note 11. 5 Gauthier uses this terminology in "Intention and Deliberation." I assume that, despite the talk of "outcome," such reasons can involve reference to the past. I assume, for example, that reasons of revenge or gratitude can count as "outcome-oriented." 6 Gauthier develops these remarks about policies in "Assure and threaten." 7 My discussion below, in section 3, offers one such argument; my discussion in "Toxin, temptation and the stability of intention" offers another. 8 I use this terminology to indicate a parallel between what I say here and Smart's (1967) criticism of rule-utilitarianism as supporting unacceptable "rule worship." I discuss this parallel further in "Planning and the stability of intention," where I also discuss related views of Edward McClennan (1990). My remarks here are similar in spirit to those of Pettit and Brennan (1986), p. 445. 9 I reach a similar conclusion, by way of a somewhat different (and more developed) argument, in "Toxin, temptation and the stability of intention." A main concern in that paper is to clarify differences between cases (l)-(3), on the one hand, and certain cases of preference change, or of intransitive preferences, on the other. The latter sorts of cases, as well as constraints imposed by our cognitive limitations, lead to significant divergences from the TE view. 10 See Scanlon (1990), esp. pp. 205-06. Of course, an account along such lines could not simply appeal to the usefulness of an ability to put ourselves under assurance-based obligations. There are many abilities that we do not have even though it would be useful to have them.

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11 There may, however, be a complication. In the science-fiction circumstances of the toxin case, my formation of an intention to drink would be known directly by the billionaire; and I would want the billionaire to come to know this and to act accordingly. All this is common knowledge. It is also common knowledge that the billionaire cares about whether this intention is formed (though she does not care whether I actually drink the toxin) and will act accordingly if it is formed. So in this science-fiction setting it may be that my forming that intention, while not itself a promise, is an assurance to the billionaire that I will drink (in part because of this very assurance). Normally, an assurance to another person requires a public act. But the known science-fiction abilities of the billionaire may make such a public act unnecessary. If my forming the intention is my assuring the billionaire it then becomes an open question whether such an unusual type of assurance induces an obligation to act. If it did turn out that my intention to drink would itself generate an assurance-based obligation, it might then turn out that I could thereby rationally drink the toxin having (earlier) rationally intended to drink. So I might be able, in full rationality, to win the money! Having noted this possible line of argument, I put it aside here. For present purposes it suffices to note that even if we were led to its conclusion it would not be because of Gauthier's CFC view. 12 I believe we will also want to appeal to shared intention. See my "Shared intention."

References Bratman, Michael E. (1987). Intention, Plans, and Practical Reason. Cambridge, MA: Harvard University Press. (1992). Planning and the stability of intention.Minds and Machines, 2:1-16. (1993). Shared intention. Ethics, 104: 97-113. (1995). Planning and temptation. In Larry May, Marilyn Friedman, and Andy Clark (eds.), Mind and Morals (Cambridge, MA: Bradford Books/ MIT Press), pp. 293-310. (forthcoming). Toxin, temptation, and the stability of intention. In Jules Coleman and Christopher Morris (eds.), Rational Commitment and Social Justice (New York: Cambridge University Press). Gauthier, David (1984). Deterrence, maximization, and rationality. Ethics, 94: 474-95. (1994) Assure and threaten. Ethics, 104: 690-721. (1996). Commitment and choice: An essay on the rationality of plans. In Francesco Farina, Frank Harm, and Stefano Vannucci, (eds.), Ethics, Rationality and Economic Behavior (Oxford: Clarendon Press; Oxford University Press), pp. 217-43. (1997) Intention and deliberation. Tn Peter Danielson (ed.), Modeling Rational and Moral Agents (Oxford: Oxford University Press), pp. 40-53.

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Kavka, Gregory (1983). The toxin puzzle. Analysis, 43: 33-36. (1987). Some paradoxes of deterrence. In Gregory S. Kavka,Mora/ Paradoxes of Nuclear Deterrence (Cambridge: Cambridge University Press), pp. 15-32. McClennen, Edward F. (1990). Rationality and Dynamic Choice: Foundational Explorations. Cambridge: Cambridge University Press. Pettit, Philip, and Geoffrey Brennan (1986). Restrictive consequentialism. AMStralasian Journal of Philosophy, 64: 438-55. Scanlon, Thomas (1990). Promises and practices. Philosophy and Public Affairs, 19:199-226. Smart, J. J. C. (1967). Extreme and restricted utilitarianism. In Philippa Foot (ed.), Theories of Ethics (Oxford: Oxford University Press), pp. 171-83.

5

How Braess' Paradox Solves Newcomb's Problem A. D. Irvine

Newcomb's problem is regularly described as a problem arising from equally defensible yet contradictory models of rationality. In contrast, Braess' paradox is regularly described as nothing more than the existence of non-intuitive (but ultimately non-contradictory) equilibrium points within physical networks of various kinds. Yet it can be shown that, from a participant's point of view, Newcomb's problem is structurally identical to Braess' paradox. Both are instances of a well-known result in game theory, namely that equilibria of non-co-operative games are generally Pareto-inefficient. Newcomb's problem is simply a limiting case in which the number of players equals one. Braess' paradox is another limiting case in which the "players" need not be assumed to be discrete individuals. This paper consists of six sections. It begins, in section 1, with a discussion of how paradoxes are typically resolved. Sections 2 and 3 then introduce Braess' paradox and Newcomb's problem respectively. Section 4 summarizes an argument (due originally to Brams and Lewis) which identifies Newcomb's problem with the standard, two-person, Prisoner's Dilemma. Section 5 generalizes this result, first to an M-person Prisoner's Dilemma, then to the Cohen-Kelly queuing paradox and, finally, to Braess' paradox itself. The paper concludes, in section 6, with a discussion of the consequences of these identifications, not just for Newcomb's problem, but for all four of the paradoxes discussed. Newcomb's problem, it turns out, is no more difficult to solve than (the easy-to-solve) Braess' paradox.

1. Resolving Paradoxes Traditionally, a paradox is said to obtain whenever there exist apparently conclusive arguments in favour of contradictory propositions. Equivalently, a paradox is said to obtain whenever there exist apparently conclusive arguments both for accepting, and for rejecting, the same proposition. Yet if either of these definitions is accepted, it follows 67

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that many so-called "paradoxes" - including the Allais paradox, the Banach-Tarski paradox, the paradoxes of special relativity, Gibers' paradox, and others - are not genuine paradoxes at all. They are not paradoxes in this strict sense since they do not entail contradictions. Instead, each is better characterized as merely an unintuitive result, or a surprising consequence, of a particular theory or collection of beliefs. One way of making this observation more explicit is as follows: most arguments are developed relative to a theoretical context or background theory. Typically, the result of each such argument is to supplement this background theory with an additional belief, p. However, in cases where the relevant background theory already includes ~p, this expansion of the background theory results in a contradiction. The contradiction becomes paradoxical only if neither an error in the reasoning which led to p, nor an appropriately modified background theory which excludes ~p, can be found. In contrast, if either of these two conditions is met, the apparent "paradox" is resolved.1 In cases where the background theory is modified to avoid ~p, accepting p may still remain unintuitive or unexpected, but nothing more. Whether the argument which leads to p should be understood as paradoxical is therefore relative to the theoretical context in which the argument appears. Relative to one such context the acceptance of p will be paradoxical; relative to another it will not. Relative to any background theory broad enough to include some large selection of pre-theoretical, "common sense" beliefs, almost any startling or unintuitive result will be judged paradoxical. Yet it is in just these cases that paradoxes are most easily resolved. Consider, as an example, the well-known Allais paradox concerning preference selection under risk. The paradox arises as follows: You are given the option of choosing either of two alternatives, Al and A2, such that Al = 100% chance of receiving $100; A2 = 89% chance of receiving $100 + 10% chance of receiving $500 + 1% chance of receiving $0. At the same time you are also given the option of choosing either of two separate alternatives, A3 and A4, such that A3 = 89% chance of receiving $0 + 11% chance of receiving $100; A4 = 90% chance of receiving $0 + 10% chance of receiving $500.

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It turns out that most people prefer Al to A2 and A4 to A3.2 Yet, to be consistent, an expected utility maximizer who prefers Al to A2 should also prefer A3 to A4. The reason is straightforward. Letting "U" represent an expected utility function and "u" some (arbitrary) unit of utility that is proportional to the value of money, we have U(A1) = (0.89)(100)M + (0.10)(100)M + (0.01)(100)w

and U(A2) = (0.89)(100)M + (0.10)(500)H + (0.01)(0)w. So preferring Al to A2 entails that (0.10)(100)w + (0.01)(100)u > (0.10)(500)w + (0.01)(0)u. At the same time, we have U(A3) = (0.89)(0)w + (.010)(100)w + (0.01)(100)w

and U(A4) = (0.89)(0)w + (0.10)(500)w + (0.01)(0)w. So preferring A4 to A3 entails that (0.10)(500)w + (0.01)(0)« > (0.10)(100)w + (0.01)(100)w. Yet this contradicts the claim that (0.10)(100)w + (0.01)(100)w > (0.10)(500)w + (0.01)(0)w. Resolving this (apparent) contradiction is not difficult. One solution is to conclude that naive expected utility is inappropriate for modeling many cases of rational decision-making - including this one since it is insensitive to factors of marginal utility and risk aversion. A second solution is simply to conclude that people sometimes act - at least in cases such as these - irrationally. Accepting either of these solutions is equivalent to accepting a slight modification to our background theory. In the first case we modify our background theory by rejecting the assumption that the application of an expected utility model will be appropriate in such contexts. In the second case we modify our background theory by rejecting the assumption that observed (real-life) epistemic agents will inevitably have fully consistent contexts of belief. In neither case will we be committed to a genuine

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paradox since in neither case will we be committed to a contradiction. In the first case a contradiction arises only when an admittedly incomplete model of expected utility is employed. In the second case a contradiction arises only as reported from within someone else's (obviously fallible) belief context. Thus, the Allais paradox is a paradox in name only. Relative to some belief contexts there will indeed be a contradiction; but relative to other more carefully considered contexts the contradiction disappears. In this respect, the Allais paradox is not unique. In fact, there are many examples of this relative nature of paradox. Russell's paradox (which concludes that a set S both is and is not a member of itself) arises relative to naive set theory, but not relative to ZFC or any other theory based upon the iterative conception.3 The Banach-Tarski paradox (which concludes that it is both possible and impossible to divide a solid sphere into a finite number of pieces that can be rearranged in such a way as to produce two spheres each exactly the same size as the original) arises relative to pre-Cantorian theories of cardinality (supplemented by the axiom of choice) which disallow the existence of nonmeasurable sets, but not relative to any modern theory.4 Paradoxes of special relativity such as the clock (or twin) paradox (which concludes that two observers in motion relative to each other will each observe the other's clock running more slowly than his or her own, even though at most one can in fact do so) arise only if one retains within relativity theory the (unwarranted, classical) assumption that the rate of each clock is independent of how it is measured.5 Olbers' paradox (which concludes that there both is and is not sufficient extra-galactic background radiation within an isotropic universe to lighten the night sky) arises only relative to a static conception of an infinite universe, and not relative to any modern theory incorporating a finite but expanding model of space-time.6 In each of these cases the purported paradox is resolved by altering the relevant background theory. In some cases these alterations are comparatively minor; in others, they involve a major reworking of the most fundamental aspects of the discipline under investigation. 2. Braess' Paradox Like all of the above paradoxes, Braess' paradox is a paradox in name only. It is a paradox only in the weak sense of describing the existence of non-intuitive equilibrium behaviour within classical networks of various kinds. Braess' original paradox concerned traffic flow. Despite the fact that each driver seeks to minimize his or her travel time across a system of roadways, it turns out that within congested systems the addition of

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extra routes will sometimes decrease (rather than increase) the overall efficiency of the system.7 The result is surprising, since in uncongested systems the addition of new routes can only lower, or at worst not change, the travel time of each driver at equilibrium. In contrast, Braess' paradox shows that within congested systems, the addition of new routes can result in increased mean travel time for all drivers. Since Braess introduced the paradox in 1968, the same non-intuitive equilibrium behaviour has been discovered within a broad range of unrelated physical phenomena, ranging from thermal networks, to electrical circuits, to hydraulic systems.8 Within all such systems it turns out that the introduction of additional components need not increase capacity. To understand the paradox in a concrete case, consider a mechanical apparatus9 in which a mass, m, hangs from a spring, S2, which is suspended by a piece of string of length LI. This string is attached to a second spring, SI, identical to S2, which is fixed to the ceiling (Figure la). Should the connecting string be cut, mass m and the lower spring would fall to the floor. In order to avoid this possibility we add two "safety strings" to the original apparatus, one connecting the top of the lower spring to the ceiling, the other connecting the bottom of the upper spring to the top of mass m (Figure Ib). Assume further that both safety strings are equal in length, i.e., that L2 = L3, and that, by hypothesis, both of these strings will remain limp so long as the original centre string remains taut. Should the centre string be cut, it will be the two safety strings which will take up the tension. The question then arises, if the centre string is cut, will m find its new equilibrium below, above

Figure 1: Braess' paradox

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or identical to its original resting point? In other words, will H2 > HI (Figure 2a), will H2 < HI (Figure 2c), or will H2 = HI (Figure 2b)? Because the two safety strings are each longer than the combined height of the original string together with one of SI or S2, one might at first conclude that H2 > HI. After all, it seems plausible that once the original centre string is cut, m will fall until stopped by the (longer!) safety strings. The new equilibrium will thus be lower than the original resting point and H2 > HI. However, contrary to this reasoning, for many combinations of components, this is not the case. For many combinations of springs, strings, and weights, the equilibrium point, once the supporting string is cut, will in fact be higher than it was originally. In other words, our original (hidden) assumption that SI and S2 will remain constant in extension was mistaken. As an example, consider the case in which LI = %, L2 = L3 = 1, and both SI and S2 have spring constant k. (In other words, the extension, x, of both SI and S2 will be related to the force applied, F, by the formula F = kx.) If we further assume that k = 1, that F = l/i, that the strings are massless and perfectly inelastic, and that the springs are massless and ideally elastic, then the distance from the ceiling to the bottom of LI (or, equivalently, from the top of LI to the weight) will be l /2 + % = %. Since we have assumed that both L2 and L3 (the safety strings) have length 1, it follows that they will initially be limp (Figure Ib), and that the total distance, HI, from the ceiling to m will be l /2 + 3/s + l/i = l3/s. To calculate the equilibrium after LI has been cut, it

h

Figure 2: Predicting the equilibrium point

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is sufficient to note that since each spring now bears only l/i its previous weight, SI and S2 will each have extension ½ x½ = ¼, and the new distance, H2, from the ceiling to the weight will be 1 + 1A = 11A, which is less than the previous distance of 1%. Contrary to our original conclusion, H2 < HI!10 This conclusion becomes paradoxical only if we now fail to abandon our original but unwarranted claim that H2 > HI. (For then it would follow both that H2 > HI and that H2 < HI.) Yet this is surely not the case. Our original implicit assumption that SI and S2 would remain constant in extension was false. If additional evidence is required, one need only construct the appropriate apparatus and make the relevant measurements.11 In other words, Braess' paradox is not a paradox at all. It is a paradox in name only. What is perhaps more surprising is that Braess' paradox is structurally no different from. Newcomb's problem. Both paradoxes are instances of a well-known result in the theory of non-co-operative games, namely that equilibria of non-co-operative games are generally Pareto-inefficient.12 Specifically, Newcomb's problem is a limiting case of the Prisoner's Dilemma, a dilemma which is itself a limiting case of the Cohen-Kelly queuing paradox. This paradox, in turn, is structurally identical to Braess' paradox, a result which has consequences for how we ought to view all four paradoxes.

3. Newcomb's Problem Newcomb's problem is traditionally described as a problem arising from equally defensible yet contradictory models of rationality.13 It arises as follows: consider the case in which you are presented with two opaque boxes, A and B. You may select either the contents of A and B together, or the contents of B alone. You are told that box A contains $10,000. You are also told that if a predictor has predicted that you would select box B alone, then box B contains $100,000. Similarly, if the same predictor has predicted that you would select boxes A and B together, then box B contains $0 (Figure 3). In the past the predictor's predictions have been highly accurate. For the sake of argument, let's assume that these predictions have been correct 70% of the time.14 Given that you wish to maximize your payoff, which is the more rational choice? Should you select the contents of A and B together, or should you select the contents of B alone? Two distinct strategies appear equally rational. On the one hand, dominance suggests that you should select the contents of A and B together. You know that there are only two possibilities: Either there is $100,000 in box B or $0 in box B. Yet in both cases the selection of boxes A and B together dominates the selection of box B alone. In both cases you will be $10,000 richer if you select A and B together.

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Box B contains SO

Box B contains S100.000

Selection A & B

$10,000 - SO

$10.000 t 5100,000

Selection of B

so

S 100.000

Figure 3: Pay-off matrix for Newcomb's problem

On the other hand, utility maximization suggests that you should select B alone, since it is this choice which has the higher expected utility. Simply calculated,15 U(B) = (0.3)(0)w + (0.7)(100,000)« = 70,000« while U(A & B) = (0.7)(10,000 + 0)w + (0.3X10,000 + 100,000)M = 40,OOOM. However, these two strategies clearly conflict. One must either select the contents of A and B together, or the contents of B alone. This is Newcomb's problem. To show that Newcomb's problem is a special case of Braess' paradox, we first consider the relationship between Newcomb's problem and another well-known paradox, the Prisoner's Dilemma. 4. Prisoner's Dilemma Two prisoners - you and I, say - are forced into a situation in which we must either defect or not. (Perhaps defecting consists of agreeing to testify against the other prisoner.) In this context you are told that if you defect you will receive a one-year reduction from a potential maximum eleven-year prison sentence. At the same time, if I fail to defect, then you will receive a (possibly additional) ten-year reduction from the same potential maximum eleven-year sentence. (Perhaps this follows since without my testimony you will be unable to be convicted on anything but a minor charge.) You are also told that I have been given exactly the same information as you, and that I have been offered the same opportunity to defect or not (Figure 4). In the past the actions of two prisoners in similar circumstances have coincided a high percentage of the time. For the sake of argument, let us assume they have coin-

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I defect

I don't defect

You defect

You receive 10 years (I receive 10 years)

You receive 0 years (I receive 11 years)

You don't defect

You receive 11 years (I receive 0 years)

You receive 1 year (I receive 1 year)

Figure 4: Decision matrix for Prisoner's Dilemma

cided 70% of the time.16 Given that you wish to minimize your time in prison, which is the more rational choice? Should you defect or not? As both Brams and Lewis have pointed out, the differences between this problem and Newcomb's problem are differences merely of detail.17 To see this, one need only consider the following slightly altered version of our original Prisoner's Dilemma: Two prisoners you and I, say - are once again forced into a situation in which we must either defect or not. (Again, we can assume for the sake of argument that defecting consists of agreeing to testify against the other prisoner.) In this context you are told that if you defect you will receive a payment of $10,000. At the same time, if I fail to defect, then you will receive a (possibly additional) payment of $100,000. For the sake of argument, we might also assume that defecting in part consists of accepting the contents of a box, A, which is known to contain $10,000. (As Lewis points out, accepting the contents of box A might even be construed as the act of testifying against the other prisoner, just as accepting the Queen's shilling was once construed as an act of enlisting.) The contents of a second box, B - which may or may not contain $100,000 depending upon the choice made by the other prisoner18 will be given to you in any event. You are also told that I have been given exactly the same information as you, and that I have been offered the same opportunity to defect or not (Figure 5). As before we are each facing a potential maximum sentence of eleven years, but in this case it turns out that any monies we obtain can be paid to the court in lieu of time served. Escape from a one-year sentence costs $10,000; escape from a ten-year sentence costs $100,000. As before, the past actions of two prisoners in similar circumstances have coincided 70% of the time. But now, minimizing your time in prison turns out to be equivalent to maximizing your financial pay-off. Should you defect or not?

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I don't defect

You defect

You receive $10,000 (I receive $10,000)

You receive S110,000 (I receive $0)

You don't defect

You receive $0 (i receive$1 10,000)

You receive S100,000 (I receive $100.000)

Figure 5: Revised decision matrix for Prisoner's Dilemma

As with Newcomb's problem, two distinct strategies appear equally rational. On the one hand, dominance suggests that you should defect. You know that there are only two possibilities: Either I have failed to defect (in which case box B will contain $100,000), or I have defected (in which case box B will be empty). In both cases, defecting will give you an additional $10,000. On the other hand, utility maximization suggests that you should not defect, since it is this choice which has the higher expected utility. Simply calculated, U(not defecting) = (0.3)(0)w + (0.7)(100,000)w = 70,000w while U(defecting) = (0.7) (10,000 + 0)u + (0.3)(10,000 + 100,000)tz = 40,000w. However, as in the case of Newcomb's problem, these two strategies clearly conflict. The reason is apparent: from the perspective of either prisoner, his decision matrix is no different from the pay-off matrix of a Newcomb's problem (cf. Figure 3 and Figure 5). The Prisoner's Dilemma is simply two Newcomb's problems run side by side. From the point of view of each individual prisoner, the Prisoner's Dilemma just is Newcomb's problem, albeit in slightly different guise.

5. Generalizing the Prisoner's Dilemma Just as Newcomb's problem can be generalized to form a two-person Prisoner's Dilemma, the two-person dilemma itself can be generalized to form the «-person case.19 To see this, substitute for the second prisoner an array of n —1 prisoners and let the contents of box B become a function, not of the choice of a single second prisoner, but of the choices

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of all n —1 other prisoners. The resulting n-person choice matrix can then be used to characterize a number of well-known phenomena, including several variants of the free-rider problem found in social choice theory.20 The free-rider problem arises whenever co-operative effort produces an improved corporate pay-off but when, at the same time, no single contribution is sufficient in its positive effect to result in compensation for an individual's contribution. In other words, the problem is how to justify, via utility alone, the cost involved in co-operation when it may be argued that, despite the non-negligible benefits of mutual co-operation, each individual's contribution also detracts from his or her personal pay-off.21 As in the original two-person dilemma, only two general conditions need to be met for the n-person version to arise. The first is that defecting - failing to co-operate - must be a dominant strategy for each participant. The second is that universal co-operation must be preferable to universal defection. In other words, although in every case defection will be preferable to the absence of defection, when compared to universal co-operation, universal defection will be worse for some - and typically for all - and better for none. In yet other words, universal defection is Pareto-inefficient when compared to universal co-operation, despite the fact that individual defection is dominant.22 How these conditions affect our particular dilemma can be summarized as follows: as before, you are given the opportunity to defect, in which case you will be awarded $10,000. If some given percentage23 of the other n—\ prisoners fail to defect, then you will receive (a possibly additional) $100,000 (Figure 6). For the sake of argument, let's assume that in the past the choice of (this percentage of) the other prisoners has coincided with your own choice 70% of the time. All prisoners have been given the same information and have been offered the same potential reward for defecting. Provided once again that the probability of your decision coinciding with that of (the required percentage of) the other prisoners is itself above the required threshold, it follows that the resulting n-person choice matrix is simply a generalized version of the two-person Prisoner's Dilemma (cf. Figure 5 and Figure 6). From the point of view of each individual prisoner there simply is no difference. This generalized Prisoner's Dilemma also turns out to be structurally equivalent to yet another paradox, the Cohen-Kelly queuing paradox. To see this, consider a queuing network (Figure 7a) which allows for the transfer of individuals (customers, traffic, messages, etc.) from entrance point A to exit point F.24 Two paths are available for the movement of traffic through the network: ABCF and ADEF. Individual travellers all have knowledge of the mean delays associated with each node in the network but not of instantaneous queue lengths. All arrival

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Collective defection

Collective failure to defect

You defect

$10,000

S110.000

You don't defect

SO

S100.000

Figure 6: Decision Matrix for n-person Prisoner's Dilemma

and exit streams are assumed to be independent Poisson flows. Infinite-Server nodes (IS nodes) represent single-server queues at which individuals are delayed by some random time, x, the average of which is independent of the number of individuals awaiting transit. FirstCome-First-Serve nodes (FCFS nodes) represent single-server queues at which individuals are processed in the order in which they arrive but in which time delays are assumed to be independent exponential random variables based upon the number of individuals arriving at the queue per unit of time. Thus, for a mean number, y, of individuals arriving per unit time, and for a capacity constant, k, with k > y, the mean delay time will be l/(fc - y). System equilibrium is defined as any case in which no individual can lower his or her transit time by altering his or her route whenever all other individuals retain their present routes. It is the goal of each individual to minimize his or her transit time across the network. Typically, in uncongested networks the addition of new routes will decrease, or at least not increase, the transit time of individuals. None the less, it turns out that within (some) congested networks, the introduction of additional capacity will lead to an increase in mean transit time at equilibrium. In other words, as long as individuals choose routes designed to minimize their individual travel time, the mean travel time at equilibrium is higher in the augmented network than in the initial network. To understand why, compare our initial queuing network to an augmented network (Figure 7b). The augmented network includes a new path from B to E which contains an additional IS node, G. In addition, we assume the following: that the mean delay for both C and D is 2 time units, that the mean delay for G is 1 time unit, that the total traffic per unit of time entering at A is n, and that n ^ k — 1 > n/2 > 0

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Figure 7: Cohen-Kelly queuing paradox

Figure 8: Predicting the equilibrium point

(Figure 8). It follows that at equilibrium the mean transit time in the augmented network is 3 time units while the mean transit time in the initial network is strictly less than 3 time units. To check this claim, note that in the case of the initial network, if the Poisson flow from A to B is a, then the mean transit time for route

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ABCF is l/(k - a) + 2 and the mean transit time for route ADEF is l/(fc - (n - a)) + 2. It follows that, at equilibrium, traffic will distribute itself as equally as possible between the two routes in order that a = n - a = n/2. The mean transit time for both routes will then equal l/(k - n/2) + 2. But since k - 1 > n/2 > 0 it follows that 1 > l/(k - n/2) > 0 and, thus, that the mean transit time for both routes at equilibrium is strictly less than 3 time units. However, as in the initial network, individual maximizers in the augmented network will also seek out and use those paths that minimize their travel time (i.e., those paths that maximize their preferences). As a result, individual maximizers will be unable to refrain from selecting route ABGEF despite that fact that this will lead inevitably to an overall increase in mean travel time (i.e., an overall decrease in efficiency) for all travellers. To see this, once again assume that the Poisson flow from A to B is a. If, in addition, we assume that the flow from B to C is b, it then follows that the flow from B to E is 0 — b, that the flow from A to D is n — a, and that the combined flow at E is the combination of that from B to E and from A to D, viz. (n - a) + (a — V) = n — b. The mean transit time for ABCF is then l/(k - a) + 2 while the mean transit time for ABGEF is l/(k - a) + I + l/(k - (n - b)). But if a - b > 0, then the transit times at equilibrium for BCF and BGEF will be equal. Thus, l/(/c - (n - b)) = 1 and n - f c = fc-l.In other words, the mean delay time at E is exactly 1 time unit, the total Poisson flow arriving at E is k - I, and the mean transit time for ADEF is exactly 3 time units. It follows that the mean transit time for ABGEF is also 3 time units. Yet if this is so, then the delay at B must also be 1 and the mean transit time for all three paths will be identical. Thus, for all three paths the mean transit time at equilibrium is exactly 3 time units. If this is so, why use ABGEF at all? The answer is that when a - b = 0, a = n - a (as shown above), and so a = n - b. It then follows that the delay time at E is only I / ( k - a). But since a = n/2 and k - 1 > n/2, it also follows that l/(k ~ a) is strictly less than 1. Thus, for those individuals in the augmented network at node B, delay time will be minimized by avoiding node C. In global terms, what has happened is that the introduction of path BGE allows individuals the opportunity to bypass both of the original IS nodes, C and D, with their mandatory 2 time unit delay. Use of this pathway is therefore inevitable. Despite this, use of path BGE increases the delay at the FCFS node, E, without any compensating reduction in delay time at either of the original IS nodes. The result is an overall increase in mean travel time. Identifying the Cohen-Kelly queuing paradox with an n-person choice matrix comparable to Prisoner's Dilemma is straightforward. Both are instances of non-co-operative pay-off structures in which

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individual defection is dominant for all, but in which universal defection is Pareto-inefficient when compared to universal co-operation. Initially, the only significant difference is that the pay-off function used in the case of Prisoner's Dilemma is typically less fine-grained than that used in the queuing paradox. However, this difference can be eliminated by associating with the queuing network a stepped pay-off function similar to that used in Prisoner's Dilemma. As an example, consider a network similar to the one above, but in which k = 5, the mean delay time for both C and D is 2 hours, the mean delay for G is 1 hour, and n, the total traffic entering at A, is 4. Then M > /c — 1 > n/2 > 0. We then assume that you are one of the travellers and that you must decide whether to take advantage of route ABGEF. As before, you know that either a — b = 0 or a ~~ b > 0. If a - b = 0, then at equilibrium the mean delay time at both B and E will be 1 /(k - a) — II (k -(n- a)) = l/(fc - n/2) = i/(5 - 2) = 20 minutes. Thus the mean travel time for each of ABCF and ADEF will be 2 hours and 20 minutes. In contrast, should a single traveller per unit of time elect to take route ABGEF, then a - b = 1. If, as before, a = n - a = n/2, then the mean delay time at B will be l/(/c - a) = I/(5 — 2) = 20 minutes, while the mean delay time at E will be l/(k - (n - b)) = l/(5 - (4 - 1)) = 30 minutes. Thus, the mean travel time for ABCF will be 2 hours and 20 minutes, the mean travel time for ADEF will be 2 hours and 30 minutes, and the mean travel time for ABGEF will be 1 hour and 50 minutes. As more and more travellers select ABGEF this time will increase until a - b = n = 4 and the mean time becomes l/(5 — 4) + 1 + l/(5 - 4) = 1 + 1 + 1 = 3 hours. Thus, you know that, should you elect to use ABCF, your mean travel time would be I/(5 — 4) + 2 = 3 hours; should you elect to use ADEF, your mean travel time would be 2 + l/(5 — 4) = 3 hours; and should you elect to use ABGEF, your mean travel time would be l/(5 — 4) + l + l / ( 5 - 4 ) = 3 hours. Given these times it might initially appear that it makes no difference which of the three routes you elect to take. Yet this is not so. After all, you also know that if even one of the n — \ travellers other than yourself fails to select route ABGEF, then your most efficient route will be through G. Dominance therefore dictates that you, too, will select ABGEF. To complete the identification of this paradox with the n-person Prisoner's Dilemma, all that remains is to associate the appropriate pay-offs with the required travel times and routes. We do so as follows: if you select route ABGEF (and thereby improve your travel time relative to both routes ABCF and ADEF) you will be awarded $10,000. In contrast, if some sufficient number of other travellers (which we call the "collective") fails to select route ABGEF, you will

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be awarded (a possibly additional) $100,000. The number required to constitute the collective will be a function of the details of the particular queuing network. For the sake of argument let's assume that in the past your choice has coincided with that of the collective 70% of the time. As with the original Prisoner's Dilemma, four distinct pay-offs are then possible (Figure 9). Also as before, one strategy dominates: since nothing you do now can affect the routes taken by other travellers, you know that you will be $10,000 richer if you select route ABGEF. At the same time, you also know that it is much more likely that you will receive the $100,000 pay-off if you avoid route ABGEF. Expected utility therefore suggests that you should avoid route ABGEF. Simply calculated, U(ABGEF) = (0.7)(10,000 + 0)w + (0.3)(10,000 + 100,000)« = 40,000« while U(~ABGEF) = {0.3)(0)w + (0.7)(100,000)w = 70,OOOM. Provided that all travellers have been offered the same potential pay-offs, and that decreased travel time is seen solely as a means of maximizing one's pay-off, it turns out that the choice matrix used in the case of the queuing paradox is indistinguishable from that used in the case of the n-person Prisoner's Dilemma. From the point of view of the individual participant, it is also no different from that used in Newcomb's problem. Identifying the Cohen-Kelly queuing paradox with Braess' paradox is also comparatively straightforward. Provided that similar pay-offs are attached to the various equilibrium points of both networks, the Collective selection of route ABGEF

Collective avoidance of route ABGEF

Your selection of route ABGEF

$10.000

$110.000

Your avoidance of route ABGEF

so

S100.000

Figure 9: Decision matrix for Cohen-Kelly queuing paradox

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two paradoxes can be shown to be structurally equivalent. Once again, both networks are cases in which system-determined equilibria fail to be Pareto-efficient. To understand why the Cohen-Kelly queuing network is structurally equivalent to the mechanical apparatus described in Section 2, simply map the components of the one system directly onto those of the other. Specifically, we begin by letting IS gates correspond to strings, FCFS gates correspond to springs, mean travel time correspond to equilibrium extension, and items of traffic correspond to units of downward force. (In other words, the correlates in the mechanical apparatus of the items of traffic in the queuing network need not be assumed to be discrete individuals. Nevertheless, to motivate the identification, one might imagine the strings and springs as constituting a system of hollow tubes through which "marbles" - units of downward force - flow. Each "marble" is given a "choice" as to which path it will follow. However, unlike the travellers in the queuing network - who are assumed to be rational agents attempting to maximize their individual pay-offs - the "marbles" in this network have their movements decided, not by the laws of rationality, but by the laws of physics.25) It follows that the three pathways in each of the two systems now also correspond. Route ABGEF corresponds to the line of tension, Tv consisting of two springs and the original centre string. Similarly, routes ABCF and ADEF correspond to the two remaining lines of tension, T2 and T3, each consisting of a single spring together with one of the two added safety strings. Just as additional traffic through route ABGEF increases mean travel time throughout the queuing network, increased force along Tj increases the equilibrium extension of the entire mechanical apparatus.26 Finally, we postulate that, just as shorter travel times in the queuing network result in larger pay-offs, shorter equilibrium extensions in our mechanical apparatus do the same. Specifically, for any designated unit of downward force (or marble) there will be a $10,000 pay-off if it follows path Tj. There will also be a (possibly additional) $100,000 payoff if some sufficient number of other units of force (or marbles) fail to follow path Tj. The same in-principle-combinations of pay-offs then appear here as appear in the case of the queuing network (Figure 9). In both cases, too, the likelihood that all travellers (or marbles) will act in concert can truthfully be claimed to be very high! Network equilibrium therefore fails to be Pareto-efficient. Both networks are systems in which individual defection is dominant even though this results in an overall decline in system efficiency. Just as travellers select route ABGEF even though both ABCF and ADEF are available, units of force place tension on Tl even though both T2 and T3 are also available. The

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resulting pay-offs are therefore exactly the same in both cases and the identification between the two networks is complete.

6. Resolving Newcomb's Problem Newcomb's problem, Prisoner's Dilemma, the Cohen-Kelly queuing paradox and Braess' paradox are all instances of the same strategic, non-co-operative "game" (Figure 10). In other words, individual players - whether choosers, prisoners, items of traffic or units of force select one of a variety of alternatives from within a given problem space in an attempt to maximize some goal - be it money, saved time, efficiency, or saved distance. Each alternative (or strategy) results in one of a variety of pay-offs. In some cases pay-off functions are initially assumed to be smooth, but all can be converted to stepped functions of the kind originally found in the two-person Prisoner's Dilemma or Newcomb's problem. In addition, in each case a single alternative dominates; it and it alone serves to maximize individual pay-off regardless of the choices made by other individuals. However, because of the collective effects that individual choices have upon the pay-offs of other individuals, these same actions also act as an equilibrium variable. The result is that user-determined equilibria vary from system-optimal equilibria: in cases where all individuals (or some other threshold percentage of individuals) select the dominant strategy, the resulting equilibrium fails to be system-optimal. It follows that universal defection is Pareto-inefficient when compared to universal co-operation, even Newcomb's problem

Prisoner's dilemma

Queuing paradox

Braess' paradox

Problem space

Payoff Malnx

Decision Matrix

Queuing Network

Classical Mechanics

Players

Chooser

Two (or more) Prisoners

Items of Traffic

Units c! Force

Strategy set

Dominance vs Expected Utility

Dominance vs Expected Utility

Dominance vs Expected Utility

Dominance vs Expected Utility

Payoff

Money

Time

Efficiency

Distance

Equilibrium variable

Predictor's Prediction

Prisoner's Decision

Variations in Traffic

Division of Forces

Figure 10: Hour paradoxes

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though individual defection is dominant. Thus, just as a single Paretoinefficient equilibrium is guaranteed by the laws of physics in the case of Braess' paradox, the same type of Pareto-inefficient equilibrium is guaranteed by the laws of rationality in the Cohen-Kelly queuing paradox, Prisoner's Dilemma, and Newcomb's problem. What Braess' paradox shows is that there is nothing inconsistent in such an outcome. Just as with Braess' paradox, Newcomb's problem is a paradox in name only. Unintuitive it may be, but inconsistent or incoherent it is not. As with the Allais paradox, the Banach-Tarski paradox, Olbers' paradox, the paradoxes of special relativity and others, Newcomb's problem (like Braess' paradox) is therefore best characterized as simply a surprising consequence of a particular set of assumptions, rather than as a genuine contradiction. Yet if this is so, one question still remains: why is it that in the case of Newcomb's problem the argument from expected utility appears persuasive? Put in other words, why is it that it is regularly those players who avoid dominance who obtain the largest pay-offs? The same question also arises for both the Prisoner's Dilemma and the Cohen-Kelly queuing paradox. The argument from expected utility in both of these cases appears to be a strong one. Why is it that it is the travellers and prisoners who fail to defect who typically obtain the optimal outcomes? The answer is a simple one. It is to deny that such a situation in fact obtains. That is, it is to deny that, over the long run, it will continue to be those who avoid dominance, or who fail to defect, who will continue to receive the largest pay-offs. In yet other words, the solution to Newcomb's problem - like the solution to the Allais paradox, Olbers' paradox, the Banach-Tarski paradox, and others - involves an important modification to our original background assumptions. In this case the modification required is straightforward: we simply abandon the (false) assumption that past observed frequency is an infallible guide to probability and, with it, the claim that Newcomb's problem is in any sense a paradox of rationality. Expected utility does not conflict with dominance once it is realized that the appropriate probability assignments for any expected utility calculation will not be based solely upon past relative frequency. Instead, these assignments will be conditional upon both relative frequency and any additional information available. In the case of Newcomb's problem this additional information comes in the form of an argument from dominance. The situation is similar to one in which you know through independent means - e.g., careful physical measurements and a detailed structural analysis - that a tossed coin is fair even though it has landed tails some large number of times in a row. After all, it is perfectly consistent with the coin's being fair that it land tails

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any number of times in a row. It is just that the probability of such an outcome decreases proportionally with the length of the uniformity. The same is true in the case of Newcomb's problem. The solution to the problem is simply to deny that selecting box B alone in any way affects or is indicative of the probability of its containing $100,000. Similarly in the case of the Prisoner's Dilemma, the solution is to deny that a single prisoner's failure to defect in any way affects or is indicative of the likelihood that other prisoners will also defect.27 Similarly in the case of the Cohen-Kelly queuing network, the solution is to deny that a single traveller's failure to select route ABGEF in any way affects or is indicative of the likelihood that other travellers will do the same. Such outcomes are no more likely in the case of Newcomb's problem, Prisoner's Dilemma, or the Cohen-Kelly queuing network than they are in the case of Braess' paradox. Which is to say, they are not likely at all.

Acknowledgments This paper is reprinted with minor revisions from International Studies in the Philosophy of Science, vol. 7, no. 2 (1993), pp. 141-60. Preliminary versions of the paper were read at the annual Dubrovnik Philosophy of Science Conference on 11 April 1992 and at the University of Victoria on 9 March 1993. Thanks go to members of both audiences, as well as to Eric Borm, Joel Cohen, Adam Constabaris, Colin Gartner, Joan Irvine, Kieren MacMillan, Steven Savitt, Howard Sobel, and Jeff Tupper for their helpful comments. In addition, special thanks go to Leslie Burkholder and Louis Marinoff whose many detailed suggestions and ideas were crucial in the writing of this paper.

Notes 1 In some cases both conditions will be met simultaneously. 2 Allais's original 1952 survey concluded that 65% of respondents prefer Al to A2 while at the same time preferring A4 to A3. For example, see Munier (1991), p. 192. 3 For example, see Boolos (1971), pp. 215ff. The same is also true for some non-iterative set theories, including the theory of hypersets. For example, see Aczel (1988). 4 For example, see Jech (1977), p. 351. 5 For example, see Mermin (1969), chs. 16,17 andpassim. Similar "paradoxes" can be constructed with metre sticks or any other type of measuring device. 6 For example, see Irvine (1991), pp. 156-8. The paradox was first proposed in 1720 by Halley and restated more formally by the Swiss astronomer Cheseaux in 1743. The paradox gained its name after being popularized by the German astronomer Heinrich Wilhelm Matthaus Olbers in the 1820s.

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7 See Braess (1986); Dafermos and Nagurney (1984a, 1984b); and Steinberg and Zangwill (1983). Also compare Cohen (1988). 8 For example, in electrical circuit design the addition of extra current-carrying routes can lead to a decrease, rather than an increase, in the flow of current. Similarly, in thermal networks an increased number of paths for heat flow can lead to a drop, rather than a rise, in temperature. See Cohen and Horowitz (1991). 9 The example is from Cohen and Horowitz (1991). The example is a helpful one to consider even though, strictly speaking, Braess' paradox originally concerned only congested transportation networks. The extension of the paradox to physical networks of this kind is due to Cohen and Horowitz. 10 Similar examples can be constructed to show that in some cases H2 > HI and that in other cases H2 = HI. As an example of the former, let LI = Vs. It then follows that HI = 1 Vs and that H2 = 1 %. As an example of the latter, let LI = %. It then follows that HI = H2 = 1V4. 11 Of course, compensation will have to be made for the idealizing assumptions made in the original thought experiment. 12 See Dubey (1986). 13 The problem, due originally to William Newcomb, was first published in Nozick (1969). 14 As is well known, in order for Newcomb's problem to arise, the predictive process need not be 100% accurate; any potentially predictive process above a given accuracy threshold will do. This threshold will be a function of the ratio, r, of the value of the contents of box A to the value of the contents of box B such that r = A / B and the threshold equals (1 + r) / 2. In other words, as the difference between the values of the contents of the two boxes increases, the required accuracy threshold decreases, provided of course that the value of the contents of box A is not greater than the value of the contents of box B. In the above example this threshold will be (1 + 0.1) /2, or 55%, assuming that the value of the contents of both boxes is strictly proportional to the money the boxes contain. 15 As before, we let "U" represent an expected utility function and "u" some (arbitrary but, in this case, positive) unit of utility. 16 As with Newcomb's problem, any percentage of coincidence above a certain threshold will do. In this case the threshold will be a function of the ratio, r, of the value (to you) of your defection, to the value (to you) of my failure to defect, such that the threshold equals (1 + r) / 2. 17 See Brams (1975) and Lewis (1979). For related discussion see Davis (1985); Marinoff (1992), esp. ch. 5; and Sobel (1985). 18 In other words, the role of the predictor is played by the other prisoner. For a real-life example in which an individual's action is believed to be predictive of the choices made by others in similar contexts, Quattrone and

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19

20

21

22

23

A. D. Irvine Tversky present experimental evidence supporting the claim that many voters view their own choices as being diagnostic of the choices of other voters, despite the lack of causal connections. See Quattrone and Tversky (1986). However, it is important to note that, strictly speaking, the decision about whether to place $100,000 in box B need not be viewed as being based upon a prediction at all. Contra Lewis, Sobel and other commentators, any independent process - predictive or otherwise - will do the job. All that is required is that the appropriate coincidence threshold be met and that the contents of box B be determined in a manner that is casually independent of the decision which you make now. As is well known, Prisoner's Dilemma can be generalized in another dimension as well, i.e. the dilemma can be iterated in such a way that each prisoner is required to make the decision about whether to defect, not once, but many times. See Axelrod (1980a, 1980b, 1984); and Danielson (1992). Also called the "tragedy of the commons," the free-rider problem dates back at least to Hume and his meadow-draining project. See Hume (173940), III, ii, vii. As described above, the free-rider problem admits of several variants. Philip Pettit, for example, distinguishes between two types of cases: (1) cases in which a single defection will lower the pay-off for all participants other than the defector, and (2) cases in which a single defection will fail to do so. The former of these two cases, in which even a single defector is capable of altering the collective outcome, Pettit calls the "foul-dealing problem"; the latter, in which each person's individual contribution is insufficient to alter the collective outcome, Pettit calls the "free-rider problem." The distinction arises since in the M-person case, unlike in the twoperson dilemma, there is no uniquely preferred, well-ordering of outcomes. Although the distinction may have strategic consequences, it can be passed over safely in this context since it is the general (n-person) case we are examining. Following Jean Hampton, I shall call both types of dilemma "free-rider" problems since in both cases individuals find it preferable to take advantage of others' contributions to the corporate good without themselves contributing. See Pettit (1986); and Hampton (1987). Compare Hardin (1971). Although other conditions can be used in the construction of w-person choice matrices, these two conditions, and these two alone, are typically taken as standard. For example, see Sen (1969); Taylor (1976); and Pettit (1986). This percentage of the other n—1 prisoners will vary depending upon the details of each specific dilemma. As in both Newcomb's problem and the two-person Prisoner's Dilemma, the probability of your choice coinciding with the relevant "prediction" must be greater than (1 + r ) / 2 where, as

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before, r is the ratio between the value (to you) of your defection and the value (to you) of a non-defecting "prediction". The only difference is that in this case the role of the predictor is played, not by a single second prisoner, but by the collective behaviour of the « — 1 prisoners other than yourself. If we assume that all H — 1 prisoners other than yourself are equally likely to defect, the (minimum) required percentage of agreement among them will be an implicit inverse function of (1 + r ) / 2 . In short, the greater the required accuracy of the prediction (or, equivalently, the smaller the difference in value between the two pay-offs), the lower the percentage of agreement among the n—\ prisoners other than yourself that will be required. Specifically, Louis Marinoff has pointed out to me that iff is the frequency of co-operation on the part of the collective, then we can calculate P(f), the probability with which/exceeds the required threshold, as follows: Assume that the probability, x, of an individual's co-operation is uniform over all n— 1 players other than yourself. Then the probability thatfc other players will co-operate is the product of the number of possible states of k co-operative players and the probability that each such state obtains. Specifically, Similarly, the probability that k + 1 other players will co-operate is 1

Now, if we let k be the smallest integer such that/t > (1 + r) / 2, then it follows that 24 In its essentials, the example is from Cohen and Kelly (1990). 25 Should it be required - for the sake of analogy between Braess' paradox and the other paradoxes discussed - that rational agents be associated with the selection of outcomes, simply associate with each outcome a rational gambler who has bet on that outcome and who will receive the appropriate pay-off. 26 It is worth noting that in both systems there is also the equivalent of a conservation law. In the queuing network (at equilibrium), traffic in equals traffic out at every node, while in the mechanical apparatus (at equilibrium), mechanical force upward equals mechanical force downward at every point in the system. 27 Importantly, it is on just this point that the single-case Prisoner's Dilemma differs from the iterated case. As a result, the iterated case may better be viewed as a type of co-ordination problem. As with co-ordination problems in general, the iterated case then involves more than the simple preference rankings associated with the single case.

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References Aczel, Peter (1988). Non-well-founded Sets. CSLI Lecture Notes No. 14. Stanford, CA: Center for the Study of Language and Information. Axelrod, Robert (1980a). Effective choice in the Prisoner's Dilemma. Journal of Conflict Resolution, 24: 3-25. (1980b). More effective choice in the Prisoner's Dilemma. Journal of Conflict Resolution, 24: 379-403. (1984). The Evolution of Cooperation. New York: Basic Books. Boolos, George (1971). The iterative conception of set. Journal of Philosophy, 68: 215-32. Reprinted in Paul Benacerraf and Hilary Putnam (eds), Philosophy of Mathematics, 2nd ed. (Cambridge: Cambridge University Press, 1983), 486-502. Braess, D. (1968). Uber ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung, 12: 258-68. Brams, S. (1975). Newcomb's problem and Prisoner's Dilemma. Journal of Conflict Resolution, 19: 596-612. Cohen, Joel E. (1988). The counterintuitive in conflict and cooperation. American Scientist, 76: 577-84. Cohen, Joel E., and Paul Horowitz (1991). Paradoxical behaviour of mechanical and electrical networks. Nature, 352 (22 August): 699-701. Cohen, Joel E., and Frank P. Kelly (1990). A paradox of congestion in a queuing network. Journal of Applied Probability, 27: 730-34. Dafermos, S. and A. Nagurney (1984a). On some traffic equilibrium theory paradoxes. Transportation Research: Part B, 18:101-10. (1984b). Sensitivity analysis for the asymmetric network equilibrium problem. Mathematical Programming, 28:174-84. Danielson, Peter (1992). Artificial Morality. London: Routledge. Davis, Lawrence H. (1985). Is the symmetry argument valid? In Richmond Campbell and Lanning Sowden (eds), Paradoxes of Rationality and Cooperation (Vancouver: University of British Columbia Press), pp. 255-63. Dubey, Pradeep (1986). Inefficiency of Nash equilibria. Mathematics of Operations Research, 11:1-8. Hampton, Jean (1987). Free-rider problems in the production of collective goods. Economics and Philosophy, 3: 245-73. Hardin, Russell (1971). Collective action as an agreeable n-Prisoner's Dilemma. Behavioural Science, 16: 472-81. Hume, David (1739-40). A Treatise of Human Nature. London. Irvine, A. D. (1991). Thought experiments in scientific reasoning. In Tamara Horowitz and Gerald J. Massey (eds), Thought Experiments in Science and Philosophy (Savage, MD: Rowman and Littlefield), pp. 149-65. Jech, Thomas J. (1977). About the axiom of choice. In Jon Barwise (ed.), Handbook of Mathematical Logic (Amsterdam: North-Holland), pp. 345-70. Lewis, David (1979). Prisoners' Dilemma is a Newcomb problem. Philosophy

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and Public Affairs, 8: 235-40. Reprinted in Richmond Campbell and Lanning Sowden (eds), Paradoxes of Rationality and Cooperation (Vancouver: University of British Columbia Press, 1985), pp. 251-55. Marinoff, Louis (1992). Strategic Interaction in the Prisoner's Dilemma. Doctoral thesis, Department of the History and Philosophy of Science, University College London. Mermin, N. David (1969). Space and Time in Special Relativity. Prospect Heights, IL: Waveland Press. Munier, Bertrand R. (1991). The many other Allais paradoxes. Journal of Economic Perspectives, 5: 179-99. Nozick, Robert (1969). Newcomb's problem and two principles of choice. In Nicholas Rescher et al (eds), Essays in Honor of Carl G. Hempel (Dordrecht: Reidel), pp. 114-46. Abridged and reprinted in Richmond Campbell and Lanning Sowden (eds), Paradoxes of Rationality and Cooperation (Vancouver: University of British Columbia Press, 1985), pp. 107-33. Pettit, Philip (1986). Free riding and foul dealing. Journal of Philosophy, 83: 361-79. Quattrone, George A. and Amos Tversky (1986). Self-deception and the voter's illusion. In Jon Elster (ed.), The Multiple Self (Cambridge: Cambridge University Press), pp. 35-58. Sen, Amartya (1969). A game-theoretic analysis of theories of collectivism in allocation. In Tapas Majumdar (ed.), Growth and Choice (London: Oxford University Press), pp. 1-17. Sobel, J. Howard (1985). Not every Prisoner's Dilemma is a Newcomb problem. In Richmond Campbell and Lanning Sowden (eds), Paradoxes of Rationality and Cooperation (Vancouver: University of British Columbia Press), pp. 263-74. Steinberg, R., and W. Zangwill (1983). The prevalence of Braess' paradox. Transportation Science, 17: 301-18. Taylor, Michael (1976). Anarchy and Cooperation. London: Wiley.

6

Economics of the Prisoner's Dilemma: A Background Bryan R. Routledge

1. Introduction At the recent conference in Vancouver, scholars in philosophy, biology, economics, cognitive science and other fields gathered to discuss "Modeling Rational and Moral Agents."1 The vast majority of the papers used or referred to the Prisoner's Dilemma (PD). With its stark contrast between individual rationality and collective optimality, it is a natural vehicle to address questions about what is moral and what is rational. In addition, many of the papers presented (and included in this volume) build on ideas of evolutionary biology. Since individual versus species fitness runs parallel to individual and collective rationality, the prisoner's dilemma also serves to link biologists and social scientists.2 This paper offers a brief overview of some of the research related to the PD in economics. Since its introduction in the 1950s,3 the folk story associated with the basic PD has become well known. A clever jailer (DA/Crown Attorney/Sheriff) offers each thief a reduced jail term if he testifies against the other. Since the jail term is lower regardless of what the other chooses to do, both rationally fink on each other. This occurs despite the fact that mutual silence would produce less time in jail than both testifying. Individually rational behaviour leads to collective suboptimality.4 This point is underscored by the fact that staying silent is often called "co-operation" while finking is often referred to as "defection." Surprisingly, many economic and social situations are analogous to that of the two thieves with fate playing the role of the clever jailer. Axelrod (1984), for example, offers many such examples.5 Despite the seeming simplicity, there are many subtle factors which can affect the behaviour of agents in a PD situation. Much of what makes the PD so useful to social scientists comes from these variations. Economic theory, particularly the field of game theory, has addressed many of these subtleties. This paper offers a brief (and far from complete) 6 survey of economic theory and evidence related to 92

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PD situations. For a more comprehensive consideration of strategic situations and game theory see any one of a number of excellent books on the subject.7 The chapter is organized as follows. Section 2 concerns one-shot PD situations. In this section the formal game is introduced and the nature of the equilibrium is discussed. Some of the "solutions" to the dilemma in the one-shot game are considered; including communication, commitment, and altruism. Section 3 considers repeated play of the PD. In repeated play, the strategy space available to the agents is much richer. This richness leads us to issues of sub-game perfection, common knowledge of rationality and renegotiation. Equilibria for both finitely and infinitely (or indefinitely) repeated games are considered. Several modifications of the finitely repeated game which permit co-operation are considered. The final section touches on experimental evidence, evolutionary arguments, and multi-person PDs.

2. One-shot game A. Description of the Game The traditional one-shot game can be represented as a normal form game (as opposed to extensive form games which are mentioned in the next section). A normal form game can be describe by the set G

The game is defined by three elements, a finite set of players (N) and a finite set of strategies (S;) and preferences (ut) for each player. In the PD, there are only two agents or N = {1,2}. S; is the set of available pure strategies to agent z'.8 In the PD, this set is the same for both players and is S2 = S2 = {c,d}. A strategy profile is a combination of strategies for all the agents. The set of all possible strategy profiles, S, is defined as follows. ij

— S\

i^

\^-J

The set of all strategy profiles, or the possible outcomes from the game, in the PD contains only four elements.

where (sl7s2) are the strategies of agents 1 and 2 respectively. The final element in the description of the normal-form game is the agents' preferences. The preferences are over the set of possible strategy profiles. We will assume that these preferences are representable

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by a von Neumann-Morgenstern utility function.9 The utility function maps possible strategy profiles (or outcomes of the game) to real numbers for each agent: U{. S —» IR. It is important to note that while we motivate the PD game with payoffs described by years in jail or money, formally preferences relate to the set of strategy profiles. This point comes up when we consider various resolutions of the PD in part (C) below. A PD game can be completely described by the set GPD as follows.10

However, normal-form games can be more easily summarized in a matrix. The specific PD game used in this paper is presented in Figure 1. Before continuing it is important to note a few of the assumptions that are often implicit in game theory. The first is that agents make choices to maximize their well being. In a narrow sense, agents are selfinterested. However, preferences are defined only on S. The particular reason why an agent prefers (c,c) to (d,d) is not modeled. Utility only reflects the agents' preference over an outcome, whether that utility is derived out of altruism or spite is not important. This topic is addressed again in part (C). The second implicit assumption in the game is that GPD or the matrix in Figure 1 represents everything. In particular, time does not play a role in the game. Agent behaviour is not a reaction to past events nor does it consider the future. We can think, therefore, that the moves are simultaneous. In normal-form games, there is no distinction between behaviour and strategy. To consider other elements (like timing), the normal-form game needs to be changed or, as is often easier, an extensive form game is developed. This is considered further below and in the second portion of the paper on repeated games. Finally, GFD is assumed to be common knowledge. By this, it is assumed that the agents know the structure of the game. In addition, Agent 2 c

A

c

(2,2)

(0,3)

d

(3,0)

(1,1)

Agent 1

Figure 1: The PD game in normal form. The utilities are shown as (agent 1, agent 2).

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agents know that their opponent knows GPD, that their opponent knows that they know GPD, and so on. Due to the nature of the equilibrium, which is discussed next, the issue of common knowledge is not important in the one-shot PD. However, it becomes crucial when considering the repeated PD game, as in Section 3. For more information on the role of common knowledge see Geanakoplos (1992) or Brandenburger (1992).

B. Equilibrium i. Nash Equilibrium The central equilibrium concept in game theory is due to Nash (1951). A Nash equilibrium is defined as a strategy profile in which no agent, taking the strategies of the other agents as given, wishes to change her strategy choice. The strategy profile forms a mutual best response. To state this concept precisely we need to define s\r;. For s G S and rt G S; define

In other words s\rt is a strategy profile identical to s except with agent i playing ri instead of s(. A Nash equilibrium is a strategy profile s such that

for all ri G S, and i G N.11 This definition and an interesting discussion of the concept can be found in Kreps (1987). Nash equilibria have the property that they always exist.12 However, they also raise several problems. In particular they may not be unique. Furthermore there are examples of games where the Nash equilibria lack Schelling's (1960) concept of "focal quality" (see Kreps (1987) for examples). Finally, some Nash equilibria are robust only to single-agent deviations. A group of agents (coalition) may find it profitable to jointly change strategy and improve their welfare. While all these are potential difficulties with the Nash concept, they are not problems in the PD. The reasons why the (d,d) equilibrium is so strong is discussed next. ii. Dominant Strategies It is easy to verify that (d,d) is the unique Nash equilibrium for the PD game.13 A simple check of all the strategy profiles, listed in (3), verifies this. However, the (d,d) equilibrium is stronger. There is no need for a subtle reasoning process in which the agent must conjecture about what the other agent will do. Regardless of what player 2 chooses, player 1 always does better with the choice of d.

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Bryan R. Routledge Formally, for agent /, strategy qt is strictly dominated by r-t if for all s SE S, u,(s\r,) > M,.(sH)

(7)

Note that for each agent d strictly dominates c. The assumption that agents do not play dominated strategies yields the unique (d,d) equilibrium. Note that agents need not assume anything about their opponents. This is why the common knowledge assumption can be substantially relaxed without changing the equilibrium. This contrasts with the finitely repeated game considered latter where agents use the information that their opponent does not play dominated strategies. Hi. What Makes the PD a PD There are two characteristics that make the game in GPD or Figure 1 a prisoner's dilemma. The first is the nature of the equilibrium as one where each player has one strategy which dominates all others (d dominates c). The second is that this unique equilibrium is Pareto-inefficient. Both players are made strictly better off by jointly playing c. The dilemma posed by these two defining characteristics is what makes the PD an interesting test case for rationality and morality. Externalities, like in many economic models, exist in the PD. Agent 1's choice of strategy affects agent 2's utility. Since this impact is external to agent 1's decision problem (i.e., does not affect agent 1's utility), the optimal decision for agent 1 does not coincide with the jointly optimal decision. This form of inefficiency is common in many public policy decisions such as pollution control, common pool resources, employee training, and car pooling. The solution to externality problems often involves public policies (such as taxes) that cause agents to internalize these external effects.14

C. "Solutions" to the Dilemma There are no "solutions" to the dilemma in the normal-form game, which is summarized in Figure 1. Aumann states this strongly as: "People who fail to co-operate for their own mutual benefit are not necessarily foolish or irrational; they may be acting perfectly rationally. The sooner we accept this, the sooner we can take steps to design the terms of social intercourse so as to encourage co-operation."15 Resolutions of the dilemma involve altering the nature of the game. This section discusses some of the more commonly suggested variations. In these cases where the inefficient equilibrium is circumvented, d is no longer a dominant strategy and the new game is no longer a PD. However, the alterations are interesting and their analysis is not trivial, so it is important that they be considered carefully.

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Economics of the Prisoner's Dilemma: A Background

There are two broad ways to alter the PD game; either the strategy space, S,, or the preferences, «;, can be modified. First, I will consider alterations to the strategy space. Second, the notions of altruism and other preference modifications will be considered. i. Changing the Strategy Space

The description of the PD game embodied in GPD (or Figure 1) overlooks some features which are common in many social settings. In particular, this section discusses communication and commitment. Modifying the game to include repeated interactions of the PD where reputation plays a role is left to Section 3. The folk story of the PD often has the two thieves in separate cells unable to communicate. However, simply expanding the strategy space to allow communication will not solve the inefficient equilibrium. For example, consider allowing agents to send m, the message that says "I will co-operate" or n, which can be interpreted as remaining silent. Since there is no truth-telling constraint to the message, defection (md and nd) still dominate co-operation (me and nc). The normal-form matrix is presented in Figure 2. There are multiple equilibria for this game. However, they all exhibit the joint defection of both players.16 The second modification of the strategy space is to allow the agents to commit. The agents would be better off if they could commit to play a dominated strategy. However, for this to be an equilibrium this commitment itself cannot be dominated. For example, allowing our two thieves the opportunity to commit to c or d before conducting their crime does nothing since both thieves will commit to d. Strategy d remains a dominant strategy. The sort of commitment that does solve the dilemma is a conditional commitment. A social mechanism that allows an agent to "co-operate only if other co-operates else defect" (called cc) is not a dominated strategy. This game is shown in Figure 3. Agent 2

Agent 1

me

nc

md

nd

me

(2,2)

(2,2)

(0,3)

(0,3)

nc

(2,2)

(2,2)

(0,3)

(0,3)

md

(3,0)

(3,0)

(1,1)

(1,1)

nd

(3,0)

(3,0)

(1,1)

(1,1)

Figure 2: The PD game with communication. The utilities are shown as (agent 1, agent 2).

Bryan R. Routledge There are two Nash equilibria in the game. The (d,d) equilibrium remains, but now (cc,cc), yielding the Pareto superior-outcome, is also an equilibrium.17 The PD game modified to allow conditional co-operation discussed here is similar to the game used in Danielson (1992) or Gauthier (1986) and used in the evolutionary environment of Danielson (this volume). In addition, the discussion by Frank (1987) that humans have the ability to "read" one another's type can be interpreted as allowing this sort of conditional co-operation strategy.18 ii. Changing the Preferences

The finger is often pointed at self-interest as the cause of the PD. If the two thieves were (completely) altruistic, agent 1 would co-operate (stay silent) hoping that agent 2 would defect (fink) so that agent 2 would be set free. If both acted this way they would achieve the Paretooptimal (c,c) outcome. However, this solution is a bit hollow since it leaves open the possibility that a very clever sheriff (or just bad luck) could, by altering the jail terms, recreate the (d,d) equilibrium.19 Altruism may resolve the dilemma in some situations but not others. The following shows that the amount of altruism required to prevent all PD's is extremely precise. To consider the effect of altruism, consider the following matrix of payoffs in Figure 4. Unlike Figure 1, these payoffs are dollars and not utilities. What preferences over these dollar payoffs will prevent PD situations? Answering this question will help understand the role of altruism in the PD situation. To simplify things, consider a set of agents with identical, monotonically increasing and continuous preferences over the dollar payoffs. For xp x2 £ U, where x1 is the dollar received by the agent and x2 is the dollars received by the other agent, U(xlr x2) will represent the utility to the agent. A strongly self-interested agent puts no weight on the x2 Agent 2

Agent I

c

d

cc

c

(2,2)

(0,3)

(2,2)

A

(3,0)

(14)

•(U)

cc

(2,2)

(14)

(2,2)

Figure 3: The PD game with conditional commitment. The utilities are shown as (agent 1, agent 2).

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dollars in his utility, while a completely altruistic agent puts no weight on xl dollars.20 The following proposition demonstrates that agents must put equal weight on xl and x2 to avoid PD situations.21 Proposition: No PD exists if and only ifU(a,p) = U(j3,a) Ma, /3 e I? Proof: First note, using the Figure 4 dollar payoffs, a PD exists when for some a, /3, y, 17 G IK, U(a,)8) > U(y,-y) > 11(77,17) > U(P,a)

(8)

Sufficiency: l/(a,/3) = U(a,/3) Va, /3 G K implies (8) cannot hold. Necessity: If U(a,f3) > U(f3,a) for some a, /3 G I?, then by the continuity and monotonicity of U, a y and 17 can be found such that (8) holds. In this context, the amount of altruism required is exact. If agents are too selfish or too altruistic, PD situations can arise. For a series of interesting papers on altruism in this and similar contexts, see Stark (1989) and Bernheim and Stark (1988). In particular, these papers consider a different definition of altruism where my utility is a function of your utility rather than your dollars. The latter article considers the "surprising and unfortunate" ways altruism can affect a utility possibility frontier. Finally, Arend (1995) has, in the context of joint ventures, considered the role of a bidding process to eliminate the PD. The bidding process, by auctioning the sum of the dollars in each outcome, effectively creates the equally weighted altruism needed to alleviate the PD. The second variation on preferences is often considered to introduce a preference for "doing the right thing." In the above example, utility depended solely on the dollar payoffs to the agent and her opponent and not on the acts c or A. Clearly introducing preferences directly on the action can eliminate PD situations by ensuring that w;(c,c) > u{(d,c) regardless of the dollar payoffs, years in jail or other such domains for utility. Margolis (1982) and Etzioni (1987), for example, discuss this issue in much more detail. Agent 2

c

d

c

(y,y)

(fc«)

d

(«.&

M

Agent 1

Figure 4: Dollar payoffs for strategy profiles. The dollar payoffs are shown as (agent 1, agent 2) with a, ft, y, r; e K.

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Hi. Institutions

Clearly, if PD situations were indeed common, inefficiency would be chronic. Fortunately, many social institutions have evolved to implement various solutions, or more correctly modifications, to the PD. Some social institutions alter the strategy space while others foster altruism. The most common institutions are those which facilitate repeated interaction where reputation and reciprocity can play a role. Repetition is the topic of the next section. Institutional and transactions-cost economics are beyond the scope of this survey. However, they do address many situations related to the PD. North (1991), for example, discusses the evolution of institutions relating to trade which he models as a PD. Williamson (1985) offers a different perspective using problems similar in character to the PD. 3. Repeated Game The most important modification to the PD to consider is the effect of repeatedly playing the game. When one introduces multi-stage play, notions of reputation and punishment can play a role. For this reason the repeated game is both more complicated and more realistic. This section begins with a few definitions of the repeated game. Next, the nature of the equilibrium is considered in both the finitely repeated game and the infinitely repeated game. The section concludes by considering some interesting modifications to the basic repeated PD. In particular, some of these modifications suggest co-operation is a possible equilibrium even in the finitely repeated game. A. Description of the Repeated PD A repeated game is often best described as an extensive form. The extensive form is a set which includes a description of the players, their utilities, the order of play (often represented as a tree), the information available to the agent each time they are called to move and the actions which are available. However, since the repeated PD game is easily decomposed into a series of periods or stages, a full discussion of the extensive form will be omitted.22 Instead, the repeated PD can be analyzed after considering strategies and preferences in the multi-stage environment. The stages of the game are denoted as t = 0,1,2,... ,T (where T may be finite or infinite). Unlike in the one-shot game (where T = 0), it is important to distinguish between actions (moves) and strategies. At each stage, the agents simultaneously take actions, s,-(f) £ S;. Note that in the PD, the set of available actions, S, = {c,d}, does not depend on time or past moves as it might in a more complex game.23 The pair of actions played by the agents at stage t is s(t) = (s, (t), s2(tj). Each s(f) is a member of S in (3) and contributes to the history of the game. Let h(t) = (s(0),

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s(l),..., s(t-l)) represent the history of the game (where h(0) = 0) and H(t) represent the set of all possible histories to stage t.24 In the repeated PD, when acting at stage t, agents know the history h(t). A strategy, therefore, gives must give a complete contingent plan, which provides an action for all stages and histories. Formally, a strategy for agent i is defined as sequence of maps cr; = (cr • J=0 where each a\: H(t) —> S, map possible histories into actions. Note that in the repeated PD, since S~{c,d] is fixed, we can abbreviate the notation by describing strategies as one (more complicated) function (j;: H —> {c,d}, where H = U/ = 0 H(f) is the set of all histories. The second element that needs consideration in the multi-stage game is agents' preferences. A strategy profile, a = (cr{l M). Finally, the 0 < § < 1 is the discount factor, which plays an important role for infinite games (i.e., where T = °°) and the (1 — 5) is simply a normalization.25

B. Finite Repeated Game - Equilibrium With the exception of the obvious change in notation, the definition of a Nash equilibrium in (6) can be used. The actions induced by an equilibrium strategy profile is called the equilibrium path. In this section, the PD will be repeated a finite number of times. The striking feature of this game is that no matter how large the number of repetitions, as long as T is finite, the equilibrium path consists solely of (d,d). To establish that all equilibrium paths consist only of (d,d), first consider the final stage. In the final stage, T, strategies which call for d strictly dominate those that call for c. Thus, the equilibrium must have (d,d) at stage T. In the second last period (T-1), action d yields a larger stage payoff. In addition, play at stage T-1 does not influence equilibrium play in stage T. Thus, strategies that call for d at T—1 dominate. This logic extends right back to the initial stage. i. Sub-Game Perfection

The above discussion has focused on the actions which occur on the equilibrium path. It is possible that c's occur off the equilibrium path.

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The off-equilibrium path consists of actions that would have been taken if certain histories (not on the equilibrium path) were reached. Since agents' action choice on these branches is payoff irrelevant, Nash equilibrium places little restriction on this behaviour. Thus, there are usually multiple Nash equilibria in repeated games. Selten proposed a refinement of Nash equilibria that requires that strategies form an equilibrium in every possible sub-game. In some games (not the repeated PD) this plays a role since never used "threats" can affect other agents' choices. Sub-game perfection requires these threats to be credible.26 Sub-game perfection in the finitely repeated PD only affects off-equilibrium behaviour. In particular, the only sub-game perfect equilibrium is one where the agents' strategies call for d at every stage and for all histories. However, sub-game perfection arguments are not required to establish that no co-operative play is observed in equilibrium. ii. Dominant Strategies

Despite the similarity in the equilibria, the repeated PD does differ from the one-shot PD. As in the one-shot game, the equilibria in the finitely repeated game are Pareto-inferior. However, in the repeated PD game there is no dominant strategy. For example, suppose agent 1 played a tit-for-tat (TFT) strategy.27 In this case, a strategy that played all-d would do poorly. The best response in the repeated game depends on the opponent's strategy. For this reason, the repeated PD game is not a true PD. In the one-shot game (T = 0), the assumption that agents do not play dominated strategies (i.e., they are rational)28 is sufficient to lead to the (d,d) equilibrium outcome. In a twice repeated game (T = I), this level of rationality predicts (d,d) in the final stage (stage 1). However, if in addition to being rational, agents know that their opponents are rational then they realize that the action at stage 0 does not affect play in stage 1. Realizing this, d at stage 0 is dominant. In a three-stage game (T = 2) an additional level of rationality is required. Both knowing that each is rational gives (d,d) in stages 1 and 2. However, if agents know that their opponent knows they are both rational then at stage 0 both know that future play is (d,d) and d is again dominant. This procedure is called iterated elimination of (strictly) dominated strategies.29 Common knowledge of rationality implies this reasoning. However, for finite games, a finite level of knowledge about knowledge suffices to establish that the equilibrium path consists solely of (d,d). As the length of the game increases, the level (or depth of recursion) of knowledge about mutual rationality increases. This point comes up again in part C below.30

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C. Infinitely Repeated Game - Equilibrium Equilibria in the finitely repeated game depend heavily on the existence of a final stage. In the infinite game, no final stage exists. In fact an alternative interpretation of the game is one of being indefinitely repeated where the game length is finite but stochastic.31 The lack of a known terminal stage, where an agent's optimal action can be determined without regard to history or opponent's strategy, allows the possibility that equilibria exist in which c's are observed. In particular, perfectly co-operative (Pareto-efficient) equilibria may exist in which only (c,c) are observed. Unfortunately, many other different equilibria may also exist. In fact, almost any pattern of behaviour in the infinitely repeated game can form part of an equilibrium. This section addresses equilibria in the infinitely repeated game. Recall that a strategy is a complete specification for all stages and possible histories. In an infinite game, therefore, these strategies can become very complex. For this reason and to avoid supporting cooperation with non-credible threats, the focus of this section is on subgame perfect equilibria. Even with this restriction, one can imagine an equilibrium path which is supported by a complex hierarchy of punishments that induce the agents not to deviate from the equilibrium path and also not to deviate from a punishment should it be necessary. Fortunately, Abreu (1988) demonstrates that complex strategies are unnecessary and attention can be restricted to simple strategies. The strategies are simple in that the same punishment is used after any deviation (including deviations from punishment). As will be seen in the examples below, simple strategies induce a stationarity in the agents' decision problems. To determine if a strategy profile is an equilibrium, only one-period deviations need to be examined.32 i. Some of the Equilibria in the Infinitely Repeated PD

The first equilibrium to note is one where only (d,d) is observed. This equilibrium path is supported by both the agents playing the all-rf strategy (i.e., both play ad(h(t)) = d for all h(t) G H). To verify a strategy profile as an equilibrium, one needs to consider all possible single, period deviations in all possible phases of play. For this strategy profile the only phase of play is (d,d) and the only possible deviation is to play a c. While it is perhaps obvious that this deviation is not profitable this is formally established by calculating the utility playing c and d (using (9) and the stage payoffs in Figure 1) play d

play c

(1-5)1 + SI > (1-5)0+ SI

(10)

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For each of the possible actions, the utility decomposes into the immediate payoff plus the discounted sum of future payoffs under the proposed strategy profile. Since a play of c does not affect future play neither agent has an incentive to deviate and the all-d strategy profile is an equilibrium. The second equilibrium profile to consider is a perfectly co-operative where only (c,c) is observed. Consider the possible grim trigger strategy where each agent plays c so long as only (c,c) is observed otherwise play d. Formally both agents play o-g where:

This strategy profile generates two possible phases; an initial cooperative phase where (c,c) occurs and a punishment phase where (d,d) is observed. First note that if the agents are in the punishment phase neither will deviate since the punishment phase is equivalent to the alld equilibrium discussed above. For an agent not to deviate in the cooperative phase, the following must hold.

Note that when contemplating action d, its effect on future payoffs (i.e., initiating the punishment phase) is reflected. Equation (12) is true so long as 8 > V2. This accords with the intuition that for equilibrium cooperative play to exist the future must matter enough to offset the immediate gain from playing d. ii. Tit-for-Tat Axelrod (1984) focuses heavily on the tit-for-tat (TFT) strategy. The TFT plays c on the initial move and thereafter mimics the opponent's action at the previous stage. While TFT has intuitive appeal, the strategy profile in which both agents use TFT is an example of a strategy profile which can be a Nash equilibrium but not a sub-game perfect equilibrium. The initial phase of the TFT strategy profile produces (c,c). For this to be an equilibrium the following must hold.

The left-hand side of (13) is the immediate benefit of playing a c and the future discount value of remaining (c,c) play. The right-hand side

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reflects the immediate and future value of playing a d. Note that playing a d causes future play to alternate between (d,c) and (c,d). Some algebra shows that as long as 6 2= V2, (13) holds and the strategy profile is a Nash equilibrium. However, for the punishment to be credible, the case that when the TFT strategy calls for a d, it must be in the agent's interest to play a d. For this to be true, the following must hold.

Note that choosing action c instead of a d returns play to the co-operative phase. Since (14) is the mirror image of (13), both cannot be true (except for 8 = V 2 ). If agents are sufficiently patient (d > V2) to play (c,c), then they are too patient to play a d should it be called for. The punishment threat is not credible and, therefore, the strategy profile is not sub-game perfect. It is possible to modify the TFT strategy to yield a sub-game perfect equilibrium. The modified strategy has the agent playing d only if the other agent has played rf (strictly) more times in the past. This modified TFT strategy retains the reciprocity feature of TFT. However, this modified strategy profile avoids the alternating phase. Instead, after an agent is "punished," play returns to the co-operative phase. Hi. The Folk Theorem Two different equilibria for the infinite game have been discussed. The Folk Theorem demonstrates that for sufficiently patient agents (high 8) many other equilibria also exist. To start, note that a strategy profile, a, implies not only a path of play but also a utility profile (Uj(cr), U2(cr)). By considering all possible strategy profiles, the set of feasible utility profiles can be constructed.33 Note that an agent can guarantee that her utility is never below 1 by repeatedly playing d. Thus, utility profiles where Ut(o) ^ 1 for each agent are said to be individually rational. The Folk Theorem states that for every feasible utility profile which is individually rational, a sub-game perfect equilibria exists which provides the agents with that utility (for a high enough 8). Since the minimum guaranteed level of utility happens to coincide with the Nash equilibrium in the (one-shot) stage game, (d,d), it is easy to see how this works. Equilibrium strategies simply revert to the grim (all-d) punishment if either agent deviates from playing according to some sequence which generates the desired utility profile. Since agents are patient (high enough 5), and utility is above the all-d level, neither will deviate. Note that this is quite a large space and equilibrium notions say little about which of the equilibria will occur.34

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iv. Trigger Strategies, Renegotiation, and Noise

In order to support a co-operative equilibria (or any of the many others) agents threatened to play d for a period of time (or forever in the grim strategies). This threat is credible because both agents revert to d after a deviation. Given agent 1 is going to punish by playing d, the best agent 2 can do is to play d and vice versa. However, this mutual support for playing d seems hollow. If the punishment stage actually occurred (for whatever reason), the agents could renegotiate and jointly agree to play c. Both are better off and both should agree (i.e., playing (c,c) is Pareto-superior). However, since players will anticipate that punishment can be escaped through negotiation, agents have no incentive to ever play c. Thus, it might seem that requiring equilibria to be immune from this problem or be "renegotiation-proof" might reduce the set of possible equilibria. However, van Damme (1989) shows that this is not the case. He uses the modified TFT strategy (mentioned above) where the punishment stage ends with repentance. That is, the defecting player must play c when her opponent plays d in order to restart the co-operative phase. In this case both agents cannot mutually agree to forgo punishment (i.e., (c,c) is not Pareto-superior to a (d,c) outcome).35 There is perhaps a second concern with the trigger strategies used to support co-operative play. In particular, if there is some chance that agents can play d by mistake or opponents can misinterpret an opponent's move, then punishments should not be too severe. In these imperfect information games, equilibria can be constructed where trigger strategies are used to "punish" deviations. However, the length of the punishment period is chosen to be as short as possible while still providing agents with the incentive to co-operate. The punishment period is minimized because, unlike the perfect information games considered above, due to noise, in equilibria punishment occurs. (For more on these games see Green and Porter 1984 and Abreu, Pearce, and Stacchetti 1990). Alternatively, Fudenberg and Maskin (1990) use evolutionary arguments to suggest that punishment cannot be too severe.

D. Co-operation in Finite Games In infinitely repeated games, many equilibria including the co-operative one are possible. However, as was discussed above, no matter how long the game, if it is repeated a known finite number of times, no cooperation exists in equilibria. Not only is this result at odds with our intuition, it also seems to be in contrast to experimental results.36 Cooperation in the finitely repeated PD is addressed in this section. As with the one-shot game, various modifications of the repeated game that permit equilibrium co-operation are discussed. First, the role of

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imperfect information of rationality is considered. Second, the stage game is modified so that multiple (Pareto-ranked) equilibria exist. Finally, the effect of strategic complexity and bounded rationality on co-operative behaviour is presented. i. Imperfect Information and Reputation

Kreps, Milgrom, Roberts, and Wilson (1982) consider a finitely repeated PD but relax the assumption regarding the common knowledge of rationality. They assume that there is a very small chance (or e-probability) that agent 2 is "irrational." In particular, an irrational agent plays the TFT strategy. This implies that there is some possibility agent 2 will play c even at the last stage. This could result from true "irrationality" or simply from different (not common knowledge) stage payoffs. The fact that agent 1 does not know agent 2's "type" makes this situation a game of imperfect information. The interesting result of this model is that this small chance of irrationality has a large impact on equilibrium behaviour. In particular, for long but finite games, the e-level of irrationality is enough to dictate co-operation in equilibrium for most of the game. In order to see the effect of this e-irrationality consider why it is the case that in equilibrium (the rational) agent 2 will not play all-d. Given the all-d strategy for agent 2, what would agent 1 believe about agent 2's rationality if, on move zero, he observed a c? Clearly, he would assume that agent 2 is irrational and he should play c (inducing a c from TFT) until the final round where d dominates. Given this response from agent 1, agent 2 has an incentive to mimic a TFT agent and play c on the initial move. In fact what Kreps et al. show is that for a long enough (but finite) game agent 2 mimics the TF T player by playing a c on all but the last few stages. Given this, agent 1 also optimally plays c on all but the last few stages as well. Note that this model captures the intuition that reputation is important. Here the reputation that is fostered is one of a TFT-type agent.37 Recall from the discussion of domination in the finitely repeated game that rationality alone was not sufficient to show that all equilibria consist of (d,d)- Common knowledge of rationality was sufficient so that for any finite T, agents could work back from T, eliminating dominated strategies to conclude that only (d,d) will be played. In the e-irrationality model, rationality is no longer common knowledge. The e-chance that player 2 is not rational acts as if the recursive knowledge about rationality is terminated at some fixed level. Thus, at some stage, I can play c knowing that you know that I know that you know I am rational, but not one level deeper.38 The fact that the irrationality is in the form of a TFT guides the equilibrium to the (c,c) on the initial moves. Fudenberg

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and Maskin (1986) prove a Folk Theorem result that any individually rational payoffs can be achieved for most of the game with the with eirrationality as long as the appropriate form of irrationality is chosen. ii. Pareto-Ranked Equilibria in the Stage Game

In the co-operative equilibrium in the infinitely repeated game, cooperative play is rewarded with future co-operation. In the finitely repeated game, at stage T, such a reward is not possible; play at T cannot depend on the past. However, if one modifies the stage game to allow the players the option of not playing (action n) yielding a payoff of zero for both players (regardless of the other's action) this modifiedstage game now has two equilibria, (d,d) yielding the agents a stage payoff of 1 and (n,n) yielding the agents 0. In this repeated game, at the final stage it is possible for past mutual co-operation to be rewarded by playing d and past defection punished by playing n. Since both of these actions form a Nash equilibrium in the final stage they are credible. Thus, threatening to revert to the "bad" stage-game equilibrium can support co-operation. Hirshleifer and Rasmusen (1989) interpret the n action as ostracism and consider its effect in inducing co-operative societies. Finally, note that Benoit and Krishna (1985), who develop this model, also show that Folk Theorem-like results again hold and such models have a large class of solutions. in. Strategic Complexity Thus far, agents' reasoning ability has implicitly been assumed to be unlimited. Following Simon (1955,1959) some recent research has considered the effects of bounding rationality. Strategic complexity has been developed in game theory to address the effects of bounded rationality. This research defines the complexity of a strategy by the number of states used to describe it. An automata describes strategies with a series of states which dictate an action c or d. In addition, a transition function maps the current state to the next state based on the opponent's last move. For example, the all-d strategy consists of a single d state which, regardless of the other's action, it never leaves. The TFT strategy requires two states (a c and a d state) and transits between them accordingly (e.g., to the c state if opponent plays a c). Formal definitions of automata and complexity can be found in the survey of Kalai (1990). Neyman (1985) notes that in a T-stage game, an automata requires at least T states in order to be able to count the stages and issue a d on the last stage. Thus, if the complexity of agents' automata is restricted (exogenously), then agents may choose co-operative automata in equilibria. If, for example, agent 1 used a grim trigger strategy (two states), agent 2 would like to be able to execute a strategy where she played a c

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in stages 0 to T — 1 and d on the final move. However, since this strategy requires many states, it may not be feasible, in which case agent 2 may optimally select the grim trigger strategy. Essentially, not being able to count the stages transforms the finite game to an infinite game. Zemel (1989) extended these results to note that small-talk (the passing of meaningless messages) can increase co-operation. Zemel's argument is that having to follow meaningless customs (handshakes, thank-you cards) uses up states in the automata and further reduces the number of stages that can be counted.39

4. Other Interesting Things Related to the PD Thus far, one-shot, finitely repeated, and infinitely repeated PDs have been discussed. While this represents a large portion of PD research, it is far from exhaustive. This final section simply touches on some of the major topics which have yet to be addressed in this brief overview. First, a few pointers to the vast body of experimental evidence are presented. Next a mention is made of evolutionary economics related to the PD. Finally, multi-person PD situations are considered.

A. Experimental Evidence Many laboratory experiments have investigated what behaviour is actually observed in PD situations. Roth (1.988) and Dawes (1980) both include surveys of PD and PD-related studies. In general, experiments do report some co-operation in one-shot PD's (about 20% is typical).40 In finitely repeated games again co-operation is observed, however, it tends to decrease towards the end of the game.41 Finally, in an infinitely (or indefinitely) repeated game, co-operative play occurs less often as discount rates decrease.42 Recent studies investigate the Kreps et al. (1982) e-irrationality model. The results are mixed. Camerer and Weigelt (1988) and Andreoni and Miller (1993) find weak evidence that observed patterns of cooperation in finitely repeated games do form the sequential equilibria of Kreps et al. However, in the Camerer and Weigelt paper, which investigates a game similar to the PD, the observed behaviour was most consistent with a sequential equilibria with a higher probability of irrationality than the experimenters induced. Alternatively, Kahn and Murnighan (1993) conclude that uncertainty tends to increase the tendency to play d, which is contrary to the model of Kreps et al. Finally, Cooper et al. (1996) look at both altruism (Is the underlying stage game viewed as a PD?) and Kreps et al. e-irrationality (uncertainty about the common knowledge of the stage game being a PD). They conclude that neither altruism nor reputation are sufficient to understand observed play.

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B. Evolutionary Economics and the PD Evolutionary game theory has developed primarily out of two concerns. First, games (particularly repeated games) often have a large number of equilibria and it is not obvious which will actually occur. Second, the unbounded rationality assumption is empirically false. Evolutionary game theory addresses both of these questions by assuming that agents simply follow rules of thumb which evolve based on past performance or fitness in the population. For introductions to recent symposia on evolutionary game theory see Mailath (1992) or Samuelson (1993). While a comprehensive survey of evolutionary game theory is beyond this paper's scope, there are several evolutionary papers specifically regarding the PD. In the finitely repeated game Nachbar (1992) concludes that, in a randomly matched population, evolutionary dynamics lead to the all-d equilibrium. However, while he demonstrates that the system converges to all-d he notes that populations exhibiting some co-operation can exist for a long time before the all-d takes over. Binmore and Samuelson (1992) investigate the evolution of automata which play the infinitely repeated PD. Following Rubinstein (1986), fitness depends both on payoffs and strategic complexity. They conclude that evolutionary dynamics leads to the co-operative equilibria. However, the automata which evolve are tat-for-tit automata. These strategies play an initial d move and thereafter play co-operatively only if the opponent also began with d. This initial move acts as a handshake and prevents simple (but not stable) all-c strategies from surviving. C. Multi-Person Social Situations Most of the analysis in this paper focuses on two-person games. However, many situations are multi-person. Kandori (1992) and Ellison (1994) investigate co-operation in pairwise infinitely repeated PD games where agents from a population are randomly matched. Both of these papers consider social sanctions or norms which support co-operation under various information assumptions. In particular, Ellison demonstrates that co-operation is possible even if the individual interactions are anonymous, through a punishment mechanism that is contagious. An alternative to multi-person pairwise interaction is the n-person prisoner's dilemma. In these games, an individual agent's utility depends on the action of all the other agents. For example, utility might be increasing in the proportion of agents which play c. However, to maintain the PD nature of these situations, utilities are such that each agent prefers d, regardless of other agents' strategies. For a comprehensive analysis of these situations see Bendor and Mookherjee (1987). Besides, analyzing the repeated version of the n-person game, they

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consider the situation in which other agents' actions are imperfectly observed. When the number of agents in a situation is large, a small imperfection in the ability of the agents to identify an agent's d move has a large impact on the equilibrium.

5. Conclusion This paper has presented a brief overview of economic research related to the well-known prisoner's dilemma. Clearly, much more can be said about the topics introduced here and many interesting lines of research have been omitted. However, the survey has dealt with much of the basic results of the one-shot, finitely repeated, and infinitely repeated PD. It is perhaps surprising that such a simple parable about two thieves and a clever jailer can spawn such a large volume of sophisticated research. However, the struggle to understand what is moral and rational is not easy. They are concepts which are intuitive and familiar from our everyday experience. However, developing their axiomatic properties can be complex. Simple problems like the PD serve as excellent testing ground where the crucial elements can be sifted away from the noise of everyday experiences.

Acknowledgment Thanks to Madan Pillutla and Tom Ross for helpful comments. 1 gratefully acknowledge the support of the Social Sciences and Humanities Research Council of Canada.

Notes 1 Held at Simon Fraser University on February 11 and 12,1994. 2 The Economist (1993) comments on the cross-fertilization of biology and economics using the prisoner's dilemma model. 3 Axelrod (1984) attributes the invention of the game to Merrill Flood and Melvin Dresner. It was formalized by A. W. Tucker shortly after its invention. It is Tucker who receives credit for the invention in both Aumann (1987) and Rapoport (1987) in their contributions to the Palgrave Dictionary of Economics. 4 It is important to note that the sub-optimality is with respect to the two prisoners. Clearly, for the clever jailer this represents the best outcome. 5 I have not compiled a list of economic applications of the PD. The list would be extensive. For a few examples see Kreps (1990) or North (1991). 6 It is difficult to accurately determine just how many papers there are on the PD. Grofman (1975) estimated it at over 2000. Aumann (1987) offers the more conservative lower bound at over 1000. Both of these estimates include the many papers on experiments using the PD from the social psychology literature. These papers are beyond the scope of this survey.

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7 Myerson (1990) or Fudenberg and Tirole (1991), for example. There are of course many other excellent recent textbooks. For an interesting historical overview of game-theory see Aumann (1987). 8 In many games it is important to consider mixed strategies. A mixed strategy, for agent i, is a probability measure over the set of pure strategies, S,, which dictates the probability with which each pure strategy is chosen. Since mixed strategies play no role in the equilibria discussed in this paper, they are not explicitly considered. 9 Typically, preferences are expressed as a binary relationship, R, over some set (S, for example). We say S]Rs2 when outcome s1 is weakly preferred to (as least as good as) S2. To say a utility function, u, represents these preferences means that Vsvs2 E S, w(Sj) s u(s2) SjRsj. Finally, a von Neumann-Morgenstern utility function has the additional property that it is linear in the probabilities over the elements of S. Since mixed strategies are not discussed, this feature does not play a role. These foundations of decision theory are included in Myerson (1990). However, Savage (1954) is the classic reference. 10 The defining characteristics of the PD are discussed below. In addition to the preference orderings indicated in additional conditions are often imposed. The first is symmetry in the utility functions such that M1(s1,s2) = w2(s2,Sj). The second is that u\(c,d) + M t (rf,c) < 2;/1(c,c) and similarly for agent 2. In the one-shot game both of these restrictions are without loss of generality. They do, however, play a minor role in the infinitely repeated game. 11 This definition is actually a Nash equilibrium in pure strategies. The definition is easily extended to mixed strategies where the mixed strategy chosen by each agent is a best response to the other agents' mixed strategies. The randomization agents perform in a mixed strategy profile are assumed to be mutually independent. Aumann (1974) considers the effect of allowing correlation among the randomization devices. However, these issues are not important in the PD game. 12 Nash (1951) proved that in games where the sets of players and strategies are finite, at least one Nash equilibrium exists. However, existence may require the use of mixed strategies. 13 There are no mixed strategy equilibria. 14 Dubey (1986) considers the generic inefficiency of Nash equilibria. He considers models where the strategy space is continuous (i.e., not finite). 15 Aumann (1987), page 468, emphasis included. 16 Note that this description assumes a normal-form game structure. Little is changed if we explicitly consider the timing of the messages. Note that the equilibria differ only in the communication portion of the agents' strategies. In any of the equilibria, agents are indifferent about communicating or not. 17 The Nash equilibrium concept is silent on choosing which equilibria in the game will be played. However, in this case one can note that once agent 2

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realizes that agent 1 will not play c because it is weakly dominated by both d and cc, then cc weakly dominates d in the remaining matrix. This process yields (cc,cc) as the unique equilibrium. However, one must use caution when eliminating weakly dominated strategies iteratively since the order of elimination may matter. The iterated elimination of strictly dominated strategies is considered again in the finitely repeated PD game below. The structure which generates the game in Frank (1987) is quite subtle. Playing A reduces your ability to "look like" a cc player since feelings of guilt cannot be perfectly masked. This technique is often used by villains in the movies who, after getting no reaction from threatening the hero/heroine, turn to threaten the spouse. Using the monotonicity of the utility function, for completely selfinterested agent U(xv x2) > U(x^', x2') x/ > x,'. The formal definition of completely altruistic is analogous. Rusciano (1990) discusses the relationship between Arrow's impossibility theorem and the PD. The author argues that the PD is simply a special case of the Arrow problem. This proposition is in this spirit. See, for example, Kreps and Wilson (1982b) for a formal definition. Note that the notation of s; and S, is consistent with that used in describing the one-shot game. However, it should now be interpreted as actions or moves and not strategies. H(f)=S f , the t-fold Cartesian product of the set of possible outcomes in a stage. Note that if mixed strategies were to be considered, preferences would be taken as expected utility. There are alternative criteria such as the average stage payoffs (or their limit for infinite games). See Myerson (1990) Chapter 7 or Fudenberg and Tirole (1991) Chapter 5 for more discussion. Selten's (1978) Chain Store Paradox is the classic example. Kreps and Wilson (1982a) and Milgrom and Roberts (1982) extend this concept to games where information is not perfect. They not only restrict out-of-equilibrium behaviour but also restrict out-of-equilibrium beliefs on inferences made from the hypothetical actions of others. For more on Nash refinements see Kohlberg and Mertens (1986). The TFT strategy plays c at stage 0 and at stage i plays whatever the opponent played at t—\. See Axelrod (1984) for much more on this strategy which was used in his finitely repeated PD tournament. The fact that this strategy is not part of an equilibrium does not matter when checking for dominance. Recall from (7) that a dominant strategy does better regardless of other agents' strategies. If a player does not play d in the one-shot game we call this "irrational." However, this is simply a convenient way of describing an agent whose preferences (for whatever reason) do not imply d is a dominant strategy.

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29 For more on this issue, see Bernheim (1984) and Pearce (1984) who introduce the concept of rationalizability as an equilibrium concept which begins with the contention that agents will never play strictly dominated strategies. 30 An important issue which is not addressed here is what agents are to believe if they were to observe a c. Is this an indication of irrationality or just a mistake? Reny (1992) addressed this issue. 31 The two interpretations are equivalent. The 8 in (9) can be seen as composed of the agent's patience, A, and the probability that the game ends after any stage, tt, as 8 = (1 — 77)A. Note that 8 can change over time; however, for co-operative equilibria to exist it is necessary that 8 remain large enough. Just how large 5 needs to be is discussed below. 32 This uses fundamental results of dynamic programming. See Stokey and Lucas (1989) for a discussion of dynamic programming. 33 For the specific repeated PD game considered here, this set is the convex hull of {(2,2),(3,0),(0,3),(1,1)|. That is, all points that are convex combinations of the stage payoffs. This is why the (1—8) normalization in (9) is a convenient normalization. Note that a profile,