The Balance of Power: Stability in International Systems

The balance of power: Stability in international systems

The balance of power Stability in international systems

EMERSON M. S. NIOU Duke University

PETER C. ORDESHOOK California Institute of Technology

GREGORY F. ROSE North Texas State University

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CAMBRIDGE UNIVERSITY PRESS Cambridge New York Port Chester Melbourne Sydney

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www. Cambridge. org Information on this title: www.cambridge.org/9780521374712 © Cambridge University Press 1989 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1989 This digitally printed first paperback version 2006 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Niou, Emerson M.S. The balance of power : stability in international systems / Emerson M.S. Niou, Peter C. Ordeshook, Gregory F. Rose. p. cm. Bibliography: p. Includes index. ISBN 0-521-37471-5 1. Balance of power. 2. Political stability. I. Ordeshook, Peter C, 1942- II. Rose, Gregory F. III. Title. JX1318.N56 1990 89-9893 327.1'12-dc20 CIP ISBN-13 978-0-521-37471-2 hardback ISBN-10 0-521-37471-5 hardback ISBN-13 978-0-521-37615-0 paperback ISBN-10 0-521-37615-7 paperback

Contents

Acknowledgments

page vii

1

Introduction 1.1 Perspectives 1.2 The dual nature of instability 1.3 Theory in international relations 1.4 Plan of this volume

1 1 22 29 32

2

Basic elements of a model and definitions of stability 2.1 Preliminary assumptions 2.2 Notation and additional assumptions 2.3 Uncertainty 2.4 Stability 2.5 Enforcement of agreements 2.6 Theoretical domain

36 36 42 56 61 67 72

3

System stability and the balance of power 3.1 Some immediate consequences of our assumptions 3.2 Necessary and sufficient conditions for system stability 3.3 System stability and war 3.4 A note on assumptions 3.5 Appendix: Proof of Theorem 3.2

75 76 85 102 106 111

4

Resource stability and the balance of power 4.1 The relevance of system stability to resource stability 4.2 The nature of resource stability and instability 4.3 A necessary and sufficient condition for resource stability 4.4 General implications 4.5 Some experimental evidence 4.6 Appendix A: Experimental data 4.7 Appendix B: Instructions read to subjects 4.8 Appendix C: Proofs of results

115 115 118 124 127 132 139 142 143

vi

Contents

5

Preventive war 5.1 A dynamic extension 5.2 Preferences 5.3 Equilibria in the first period 5.4 The emergence of a predominant country 5.5 Inessential countries 5.6 The disappearance of stability 5.7 The sources of preventive war 5.8 Appendix: Proofs of results

146 154 160 162 167 174 176 179 182

6

Geography, balancers, and central powers 6.1 Conceptualizing the consequences of geography 6.2 System and resource stability 6.3 Implications: Balancers and central powers 6.4 Balancers and central powers: Some examples 6.5 Appendix: Proofs of results

187 189 197 201 207 212

7

Great-power alliance formation, 1871-1914 7.1 Formulating testable hypotheses 7.2 Methodology 7.3 The League of the Three Emperors 7.4 The Austro-German Alliance 7.5 The Triple Alliance 7.6 The Mediterranean agreements 7.7 The Russo-German Reinsurance Treaty 7.8 The Dual Alliance 7.9 The Entente Cordiale and the Triple Entente 7.10 Balance in the alliance system 7.11 Appendix: Data used in the construction of our index

215 216 223 235 244 248 251 254 255 258 259 267

8

European conflict resolution, 1875-1914 8.1 The Near Eastern Crisis, 1875-8 8.2 The Near Eastern Crisis, 1885-8 8.3 The July Crisis of 1914

271 273 279 288

9

Summary and conclusions 9.1 Balance of power reconsidered 9.2 Contemporary implications 9.3 Conclusion

311 311 318 330

References and selected bibliography on European great-power relations, 1871-1914 Index

333 349

Acknowledgments

Valuable assistance in the preparation of this manuscript and in the refinement of our ideas came from several sources, but we would especially like to acknowledge an intellectual debt owed to R. Harrison Wagner. Although we have approached the analysis differently from the way he might have preferred, the early drafts of Harrison's 1986 World Politics essay, and subsequent discussions with him, profoundly influenced our thinking. If he sees his own ideas in our research without citation, it is only because those ideas have become so much part of our own. Second, we want to thank Roy Bridges, Bruce Bueno de Mesquita, Bruce Cain, Richard Cupitt, Raymond Duval, David Elliot, Jack Levy, Clifton Morgan, Thomas Schwartz, Duncan Snidal, David Sweet, Dina Zinnes, and the staff of the Public Records Office, London, for their suggestions, encouragement, and assistance. We also want to acknowledge the assistance and encouragement of Bruce Russett who, as editor of The Journal of Conflict Resolution, facilitated the publication of the initial manifestations of our research, as well as the editors of Mathematical Modelling and Simulation and Games, who published follow-up research included in this volume. Finally, we would like to acknowledge our gratitude to the University of Texas at Austin and to the support it provided while we engaged in a considerable portion of the research reported in this volume.

CHAPTER 1

Introduction

Among the depressing features of international political studies is the small gain in explanatory power that has come from the large amount of work done in recent decades. Nothing seems to accumulate, not even criticism. Instead, the same sorts of summary and superficial criticisms are made over and over again, and the same sorts of errors are repeated. Kenneth Waltz, Theory of International Politics (1979, p. 18)

Despite the attention of such intellectual giants as Spinoza, Rousseau, Kant, and Clausewitz, we know little more about international conflict today than was known to Thucydides four hundred years before Christ. Bruce Bueno de Mesquita, The War Trap (1981, p. 2)

1.1

Perspectives

Although the causes of international instability and conflict have been the object of intense scholarly concern, the events of our century are not inconsistent with the supposition that we have made little progress in identifying those causes, that, once identified, we have not adequately applied our research to the discovery of correctives, or that those causes have multiplied at a pace that exceeds our abilities of assimilation and analysis. Some scholars amass and analyze vast arrays of data on diplomatic exchanges, military expenditures, economic indicators, the formation of alliances, and the frequency and severity of wars; others apply the mathematics of decision and game theory with varying degrees of sophistication to the description of international relations processes; and still others reason through the meaning and application of concepts and words such as polarity, power, regime, deterrence, neocolonialism, and the balance of power. If this research has uncovered causes and correctives then, for one reason or another, the corresponding scholarly utterances have been less than compelling. If we reflect upon the considerable literature on the causes of war, one of the chief difficulties is that, on the one hand, so many explanations are offered, we are predisposed to discount them all; on the other hand, 1

2

1 Introduction

since it is so easy to find counterexamples to any particular explanation, a great deal of baggage must be carried to ensure a universal understanding. Such difficulties warn us that true scientific explanation is not being achieved, and that rather than isolating a fundamental cause within a logically consistent theoretical framework, this research appears at best to uncover the varying circumstances sufficient to render conflict an understandable choice by key decision makers. Yet, insofar as uncovering fundamental cause is concerned - uncovering the laws that govern the interactions of people in international affairs and the circumstances under which such interactions yield instability and overt conflict - we are no more likely to infer such cause by the massaging of vast data arrays or from a careful compilation and analysis of historical events than we are to infer the physics of falling objects by observing the flight of leaves from trees. We are not so brazen to suppose that the research we report here resolves all issues. Nevertheless, our goals are immodest: to identify the conditions under which international systems are stable as well as the conditions under which they are unstable, the circumstances in which this instability implies conflict, the objectives that that conflict is intended to serve, and the steps we might take in designing domestic and international institutions to avoid such conflict. Our approach is analytical and abstract, and consists of identifying a structure that is fundamental to all international political processes. Our aim is to formalize the ideas offered by the realist view, in which the dominant force directing international processes is the national pursuit of a single-minded self-interest in an otherwise anarchic system. To achieve this formalization, we make a great many compromises with reality, and the theory we offer is but a preliminary step toward a comprehensive, deductive theory. The measure of this volume's success will be the extent to which other researchers follow our lead by refining our model with alternative and more general assumptions. The necessity for and choice of a paradigm So substantial is the literature on international politics and the causes of war, we should believe that it already contains many of the essential components for a general understanding of our subject. If, as Einstein asserted, "The whole of science is nothing more than the refinement of everyday thinking," then it is unreasonable to suppose that the scholarship of centuries is not replete with insights into the fundamental mechanics of international systems. Nevertheless, we must ask why this scholarship has not formed a generally accepted body of theory with corresponding prescriptions for averting or anticipating wars. Our hypothesis is that this

1.1 Perspectives

3

research too often occurs without reference to any basic paradigm of explanation, or to paradigms that are only dimly perceived, poorly specified, and that do not lend themselves to the rigorous derivation of conclusions so that we can see clearly the underlying premises. Without an explicitly defined and formally organized conceptual scheme identifying general concepts and the relationships of these concepts to the observed world, an empirical literature, no matter how well-informed and scholarly, can be little more than an unsystematic mass of insights, conjectures, and stylized facts. Ideas, however insightful, reduce to illdefined constructs held together more by linguistic structure than by a well-understood theory, and understanding and explanation devolve more on familiarity with jargon than on scientific law and generality. A paradigm, or (more properly) a theory based on a paradigm, ensures that our insights and suppositions accumulate, that they can be tested for their generality, and that they can be remembered as part of a coherent structure (Kuhn 1970). Hence, achieving this volume's goal requires that we operate with an explicit paradigm as a guide to our reasoning. A paradigm, however, must do more than merely guide reasoning; it must also organize it in a rigorously deductive way. Human affairs are complex processes, certainly more complex than the situation confronting a natural scientist or engineer who attempts to understand the laws governing airfoils orfluid-flowthermodynamics. Indeed, developing a theory of international processes may entail confronting an even greater complexity than that which confronts us in natural science, which underscores our greater need for theory. Hence, it is essential that we employ a paradigm that permits us to collect our insights in such a way as to ascertain whether seemingly distinct phenomena can be thought of as manifestations of some more general process or whether our intuition requires a fundamental reinterpretation of reality. Our paradigm is that of rational action and methodological individualism, where causal explanation stems from the supposition that all events, save those we attribute to the probabilistic whims of nature, follow from the actions of people pursuing their goals in a world constrained by limited resources, innovative skill, and the actions of others. Explanation, in turn, derives from models in which predicted actions are the only actions that follow logically from hypothesized goals and constraints or that can otherwise be sustained by those goals as equilibria. The mathematics of logical connections - the mechanism we use to ascertain the premises required to sustain our conclusions - are provided by decision and game theory, from which we must recover the appropriate representation of these connections for the substantive task at hand.

4

1 Introduction

The relevance of game theory A fundamental part of our paradigm - the paradigm of rational choice is the supposition that people's preferences over alternative outcomes are "well-defined" and that these preferences can be represented by a measure we call utility. The concept of utility, though, occasions confusing the paradigm and rationality with a mode of decision making in which, after identifying their alternative actions and the likelihood of different consequences given each action, people choose the action that maximizes their utility or expected utility. This model is indeed part of the paradigm, but it is relevant in only highly specialized cases such as voting in mass elections or participating in markets as consumers. Indeed, we must reject the idea that explanation lies in the simple proposition that decision makers choose those actions that maximize their utility or expected utility. Our paradigm admits of more complexity than that, and Jervis (1976, p. 32) identifies the context of that complexity: If he is to decide intelligently how to act, a person must predict how others will behave. If he seeks to influence them, he needs to estimate how they will react to the alternative policies he can adopt. Even if his actions do not affect theirs, he needs to know how they will act in order to tailor his actions accordingly. Snidal (1986, p. 39), in turn, succinctly states the particular problem with which we must deal: [Real international issues lead].. .directly to a strategic rationality which incorporates the realization that pursuit of egoistic interest requires consideration of interactions of one state's choices with other state's choices. No state can choose its best strategy or attain its best outcome independent of the choices made by others... [and] the distinguishing trait of strategic rationality is that actors choose courses of action based on preferences and expectations of how others will behave. International relations, if not most of politics, occurs in an environment of interactive decision making. In this environment, the actions of a single actor determine and are determined by the actions of others. This, of course, seems a trivial proposition, but it leads us to reject approaches that focus exclusively on the motives and beliefs of decision makers taken one at a time. We cannot accept as legitimate propositions such as "actor / chose Y because / preferred Y to Z " unless we are willing to impose some special assumptions. Because a voter's effect on outcomes is so slight in mass elections, it is reasonable to hypothesize that voters ignore interactive effects in deciding how to cast their ballots - each voter ignores the possibility that others will alter their decisions as a function of what the voter in question decides. Hence, we assume that the environment of the voter's decision, including the benefits associated with individual candidates and the competitiveness of the election, is taken as given from

1.1 Perspectives

5

each voter's perspective. Similarly, in microeconomics, when modeling a consumer's decision, we assume that consumers take prices as given, since each consumer's effect on price is imperceptible. For virtually all other social processes, however, we cannot ignore the joint dependency of decisions, in which case our propositions must take the form "/ chose Y because / believed that the other relevant actors, j , k, etc., would choose B, D, etc., and with these other choices, action Y maximizes /'s utility (Y is /'s best response to B, D, and so forth)." Thus, an actor's utility calculations must summarize not merely preferences over outcomes and actions simpliciter, but must also take cognizance of the impact of the actions of others as constraints on that actor's ability to realize his or her objectives. This restatement might seem to represent only a modest revision, or perhaps even just a more complex restatement of the supposition that people act to maximize utility. However, notice that with this perspective we cannot fully explain final outcomes until we utter a parallel statement for person j (as well as for every other relevant decision maker) - that j chose B because, if /, k, etc. choose Y, D, etc., respectively, thenB isj's best response. However, after formulating such sentences, we necessarily confront the problem of higher orders of thinking. If what / chooses is a function of what j , k, etc. choose, and if what j chooses is a function of what /, k, etc. choose, and so on, then how does each decision maker resolve the problem that all decisions are simultaneously determined - that what / does depends on what he believes j believes about him, and so forth? Game theory is the particular branch of decision theory that seeks to disentangle this simultaneity and to discover the logical choices of people in such interdependent contexts; thus, game theory becomes our primary analytic instrument.1 Ours, of course, is not the first study that seeks to apply game theory to the study of international processes. Typically, though, these applications 1

Although we refer to all interdependent decision-making situations as games, game theory identifies three alternative analytic representations of such situations, with each form corresponding to a different degree of abstraction. The first, a game in extensive form, is a representation of the situation that portrays the actions of decision makers, the timing of those actions, and the information of decision makers any time they must act. The key components of a game in extensive form is a list of decision makers, a list of decision nodes, a specification of the decision maker who must act at each node along with the alternatives available at that node, the information about previous choices that that actor possesses at each of his decision nodes, and a list of terminal nodes that specifiesfinaloutcomes and payoffs. The second representation, the normal form, requires a list of decision makers, a specification of the strategies of decision makers - a plan for how to play the extensive-form game that accommodates all possible contingencies - and an outcome function that takes the strategies of all decision makers and specifies a payoff for each decision maker. The final representation is a game in characteristic function form. This representation suppresses the notion of strategy, and focuses instead on the outcomes or

6

1 Introduction

take the form of demonstrating the parallelism between a specific process and a particular game, most notably the Prisoners' Dilemma and Chicken. In their most naive manifestation, we see sentences such as "let China be player 1 with the strategies a and ft,...," and it is difficult to see how such an application can ever lead to a comprehensive theoretical understanding of international processes. Yet, we are not asserting that valuable insights cannot be gained from such an approach. Certainly, our understanding of the possibility of irrational collective action is illuminated by analyses of the Prisoners' Dilemma, the problems of coordination are revealed by Chicken, and recent studies of reputation and deterrence illuminate the profound significance of incomplete information. What we require, though, is a sufficiently general modeling effort that allows for elaboration, admits the possibility of incorporating these earlier studies as subparts, and provides the theoretical structure for an analysis of complete systems at the macro level. The thought to keep in mind is that game theory is but a piece of mathematics and decision theory. Overzealous advocates and unrepentant critics may misinterpret its value and limitations, but game theory in fact offers only the logical connections between abstract concepts. It is not, by itself, a theory of social processes, so that taking a part of it and appending some story to that part does not constitute a theory any more than if we were to append some interpretation to the ;t's and >>'s in the equation (x — y)(*+y) =x2—y2. Knowledge of game theory, however, heightens our sensitivity to the pervasiveness of interactive decision making in politics, and also provides the tools for thinking about such decision making in a logically coherent way. Equilibria as causal explanations Having accepted game theory as the principal analytic component of our paradigm, we are led to our third observation, namely, that concepts of equilibria for both noncooperative and cooperative games are the mechanisms whereby we generate causal explanations. The revision of the notion of utility maximization that requires consideration of best responses is based on the supposition that we can determine the optimal choices for payoffs that can be assured by coordinated action on the part of subsets of decision makers (coalitions). The key elements of such games are again a list of relevant decision makers along with a specification of the "value" of each potential coalition. Ideally, we prefer to model all situations in extensive form, but this is not always possible, and the normal form and the characteristic function form offer representations that are more analytically tractable. For further elaboration of these distinctions and their formal representations, see, for example, Luce and Raiffa (1957), Shubik (1985), and Ordeshook (1986).

1.1 Perspectives

7

people only if we determine all choices simultaneously. And it is the various notions of equilibria in game theory that form the basis for hypothesizing how people solve these simultaneous, interdependent decision problems. Briefly, an equilibrium for a game is a set of actions or outcomes that, given the goals of all decision makers and the structural constraints of the situation being modeled (an identification of relevant decision makers, the actions and information available to each decision maker, and a specification of the relationship between outcomes and joint actions), are the only ones that can endure.2 Hence, we predict certain actions because these actions are in equilibrium. Correspondingly, the cause of a particular outcome is that it follows from actions in equilibrium. This view of causation is important. Cause no longer resides in the mere correlation of events and circumstances or in their temporal ordering. Instead, to specify cause, we must specify the nature of individual goals, the parameters of structural constraints, and the corresponding equilibria. Hence, the attribution of cause to some particular event is moved away from a focus on the events that immediately preceded it; instead, our focus becomes the decision problem confronting all relevant decision makers and the outcomes that follow if those decision makers pursue the goals we assume for them. Insofar as peace and conflict are concerned, we explain one or the other as the consequence of the pursuit of individualistic goals in an environment in which everyone's actions and fates are interdependent; conflict or peace are merely the consequences of a set of individual choices in equilibrium as determined by individual goals and structural parameters. To illustrate, consider the hypothesis that wars are "caused" by a misperception of military capabilities, in particular, by the perceptions of two leaders that they can both win a conflict.3 In this instance, 2

3

Game theory provides a variety of formal definitions of equilibria, so that the choice of a definition depends on the nature of the situation and the particular representation we use as a model. And although a part of contemporary theory seeks to remove the distinction, game theory traditionally divides its subject into two broad categories: noncooperative and cooperative game theory (or, in terms of their analytic representation, games in extensive or normal form versus games in characteristic function form). In noncooperative game theory, we suppose that players are unable to explicitly coordinate their decisions or that, if such coordination is possible, agreements are enforced by wholly endogenous considerations. The names of the equilibrium notions here that are especially relevant to our analysis are Nash equilibria and a refinement called subgame perfect equilibria. In the theory of cooperative games, on the other hand, we submerge the issue of enforcement, assume that coordination is possible, and look for those outcomes to which coordination might lead. In this instance, the relevant equilibrium notions are the core, the von Neumann-Morgenstern solution, the bargaining set, and the competitive solution. We will define each of these notions in this volume as we require them. Although this hypothesis is suggested by Blainey's (1973) argument, it is not intended to be an accurate restatement of that argument (see our discussion in Section 2.3).

8

1 Introduction

however, misperception is merely a precondition, a part of the characterization of the game national leaders play. What remains to be shown (and what is implicit in the hypothesis) is that, given such misperception, one or both parties choosing war describes the equilibrium of that game. This example also serves to illuminate a potential source of confusion in our terminology. Specifically, asserting that an international system is in equilibrium does not mean that there are no conflicts; the equilibrium choices of decision makers may be to war. Indeed, one of the implications of our paradigm is that with an appropriate conceptualization of the game decision makers play, all systems are in equilibrium at all times. It may be that a system is "in transition" - that wars move it from one type of equilibrium state to another - but because the transition path itself is the product of individual "rational" decisions, that path can be described as an equilibrium state. Our task, then, is not to ascertain whether anarchic international systems are in equilibrium; rather, it is to ascertain what kinds of equilibria are attainable under various circumstances. Throughout this book, then, we reserve the word "equilibrium" to denote the actions of decision makers that follow logically from their goals and the other relevant parameters of their environments. We use the word "stability" to describe the international systems in which conflicts of various sorts cannot arise. We should also clarify another source of confusion regarding equilibria. The analysis that follows is in the realist mode to the extent that it focuses on "the struggle for power in an anarchic environment [where] states fend for themselves as they pursue their contradictory interests" (Snidal 1986, p. 39). It is a common mistake to believe, however, that such situations can occasion neither equilibrium nor cooperation. Riker (1980, p. 443), for example, argues that if systems match this conflictual view then equilibria are impossible, in which case "we cannot know much about the future at all." However, in this instance Riker is adopting an especially narrow view of equilibria, one in which a specification of initial conditions determines uniquely the outcome that must prevail. The existence of such equilibria is rare, but the notion of an equilibrium does not refer to a single concept, and game theory offers a variety of definitions of equilibria. Some of these definitions are refinements of others, whereas some treat different classes of games. Although it is true that the conflictual circumstances with which realist thinking is concerned does not typically yield a simple variety of equilibria that occasions unique predictions, prediction in the form of a narrowing of possibilities is still possible. With respect to the issue of cooperation in the conflictual politics envisioned by the realist school, Keohane correctly asserts that

1.1 Perspectives

9

Cooperation takes place only in situations in which actors perceive that their policies are actually or potentially in conflict, not where there is harmony. Cooperation should not be viewed as the absence of conflict, but rather as a reaction to conflict or potential conflict. Without the specter of conflict, there is no need to cooperate (1984, p. 54).

The only qualification we need add is that cooperation is excluded if there are only two parties to the conflict and if the conflict is absolute - if the situation is zero sum. Barring this possibility, which we believe is actually rare in international politics, the various solution hypotheses of game theory (definitions of equilibrium) have as their objective predicting the form of cooperation that ensues. Goals of decision makers We do not suppose that peace requires that international leaders - those who control the destinies of sovereign states - be compelled to act one way or another by idealistic objectives. We cannot assume that a stable or an unstable international system emerges only if decision makers desire such a system. Scenarios such as the Prisoners' Dilemma tell us that even unanimous consent to some objective is no guarantee that people will choose actions that yield the desirable outcome (see, e.g., Olson 1965). Moreover, people may prefer stability to instability or peace to conflict, but these cannot be ultimate objectives. If we wish to assume that nations seek peace or conflict, we must show that such goals are the logical result of the operation of international political systems. Because we do not want to assume our result (that particular systems lead to peace and that others lead to conflict), we must begin with more fundamental objectives and derive the circumstances under which peace or conflict will prevail as the logical consequence of the relationship among states. Ultimate objectives, presumably, might include things such as maintaining one's domestic political power, securing economic benefits for oneself, or even ensuring one's personal survival. Stability (or instability) emerges, then, as a by-product of the pursuit of more fundamental individual objectives. The same argument holds true for balance of power. If a balance of power as we define it later emerges in our analysis, it does so because it can be interpreted as satisfying some more basic objectives of decision makers. It may be true that leaders seek to maintain a balance and that a "balance of power lasts only so long as someone is ready to take risks to maintain it" (Wight 1973, p. 115), but only insofar as a balance serves specific purposes or is the logical result of the pursuit of those purposes. To postulate the pursuit of balance as a primitive objective is to assume what

10

1 Introduction

it is that we are trying to establish: that a balance can emerge in anarchic systems comprised of egoistic decision makers. Domestic versus international politics Theorizing about any phenomenon requires simplification, and in the area of international politics it is often tempting to render our analysis more tractable by ignoring domestic politics. Unfortunately, in specifying the goals of decision makers in international affairs and in identifying their relevant environment, we cannot ignore "mundane" domestic concerns. Certainly, people might be motivated to secure an honorable mention in history's footnotes, and expressions of individual goals may take the form of the lofty rhetoric of international leadership and world peace. However, more commonly, such decision makers are motivated by more immediate concerns such as the establishment or maintenance of a domestic political advantage or, in the case of agents for multinational firms, the pursuit of economic advantage. We need only document those international conflicts that distract attention from domestic problems and thereby undermine domestic political opposition to see how domestic political constraints shape decisions in the international arena. This means that a theoretical dichotomy between international and domestic politics cannot be viable, or at least we must be prepared to formulate an analysis that allows for the eventual bridging of such a dichotomy.4 Rejection of the ultimate viability of such a dichotomy implies that we cannot suppose that conflict is caused by leaders who pursue the "wrong" goals or that peace is secured by more enlightened leaders. Although we cannot discount the influence of a single radical element, the logic of peace and conflict are to be found in the functioning of international systems as leaders react to each other, in the pursuit of their own goals, constrained by domestic politics. We cannot understand the emergence and influence of dominant figures without also exploring the domestic sources of their 4

This is particularly so if the classic grounds for theoretically distinguishing international from domestic politics lies in the "anarchy" of international politics in contradistinction to the "order" of domestic politics. As we argue shortly, the existence or nonexistence of well-defined political rule-systems is a function of the self-interest of political actors. The existence or nonexistence of explicit rule-systems is an endogenous variable, and thus we cannot appeal to the endogeneity of rules to distinguish international from domestic politics. If the international polity is anarchic and the domestic polity archie, it is not because international and domestic politics are categorically different things, but because actors while pursuing essentially equivalent goals in both - generally find strategic advantage in elaborating explicit rule-systems domestically but do not find such advantage in the international arena. Indeed, a study of revolutions and coups d'etat suggests that, in a wide range of cases, domestic politics is as much a Hobbesian state of nature as anything in the international arena.

1.1 Perspectives

11

power and the imperatives of other national leaders with whom they must interact. The problem at hand is raised with particular acuity by Bueno de Mesquita in his discussion of the role of strong leadership in foreignpolicy decision making: We can logically assume, then, that decisions to make war are dominated by a single individual in each state Foreign policies related to war are almost always associated in our minds with individuals such as Richelieu, Metternich, or Bismarck Of course, no leader can afford to ignore completely the desires and interests of those who follow him. Without some support, even the most coercive dictator cannot hope to muster sufficient resources to wage a successful war. Still, it is ultimately the responsibility of a single leader to decide what to do and how to do it (1981a, pp. 27-8).

Although our theoretical analysis, like Bueno de Mesquita's, treats states as unitary actors, the source of our disagreement with this quotation lies in the possibility of confusing institutional arrangements of command and control with the underlying political structure that makes effective leadership possible. The ability of a leader to exercise his institutional authority is determined by a complex set of domestic political games that both constrain and enable such a leader to varying degrees. A model that assumes that such a leader is free to act as he wills cannot explain either why or how a leader acts; it amounts to assuming the explanation. Thus, attention to the constraints imposed by domestic politics is an essential component of any fully comprehensive attempt to predict outcomes in the international system. If domestic politics and these constraints are not incorporated in the analysis, our theory must at least permit us to see how its conclusions are modified when subsidiary analyses identify those constraints. That theory should inform us, moreover, about the forces international politics exert on domestic politics. The issue of the role of domestic politics raises an interesting problem with the "levels-of-analysis" debate in the study of international relations. The paradigm we have embraced provides, if not a complete resolution of the matter, at least a perspective for addressing the issue and organizing our research. Singer poses the problem in the following way: . . . one could, at the systemic level, postulate that when the distribution of power in the international system is highly diffused, it is more stable than when the discernible clustering of well-defined coalitions occurs. And at the subsystemic or national level, the same empirical phenomena would produce this sort of proposition: when a nation's decision makers find it difficult to categorize other nations readily as friend or foe, they tend to behave toward all in a more uniform and moderate fashion. Now, taking these two sets of propositions, how much cumulative usefulness would arise from attempting to merge and codify the systemic proposition from the first illustration with the subsystemic proposition from the second, or vice versa? Representing different levels of analysis and couched in

12

1 Introduction

different frames of reference, they would defy theoretical integration; one may well be a corollary of the other, but they are not immediately combinable. A prior translation from one level to another must take place (1961, p. 92).

Waltz goes further, asserting that systemic and subsystemic explanations are not merely incommensurable, but that systemic explanations alone are the province of the study of international politics and that one cannot expect to obtain precise prediction of specific outcomes through studying international politics: An international political theory does not imply or require a theory of foreign policy any more than a market theory implies or requires a theory of the firm. Systems theories, whether political or economic, are theories that explain how the organization of a realm acts as a constraining and disposing force on the interacting units within it A theory of international politics bears on the foreign policies of nations while claiming to explain only certain aspects of them. It can tell us what international conditions national policies have to cope with. To think that a theory of international politics can in itself say how the coping is likely to be done is the opposite of the reductionist error (1979, p. 72). Waltz, nevertheless, comes closest to our view when he states: Causes at the level of units interact with those at the level of structure, and because they do so, explanation at the unit level alone is bound to be misleading. If an approach allows the consideration of both unit-level and structural-level causes, then it can cope with both the changes and the continuities that occur in a system (1988, p. 618).

The problem for the policy analyst, though, is in the inability to specify precisely the relationship between systemic and subsystemic variables in such a way as to guarantee predictive power over specific outcomes, but a simple example illuminates the contribution our paradigm might make. Assume that three individuals - A, B, and C - bargain, under majority rule, to divide $100. The various approaches in cooperative game theory agree that the structure of this interaction will result in an outcome in which two of the individuals will coalesce to divide the money evenly among themselves, excluding the third participant. In this sense, the structure of the system determines the general character of the outcome - the emergence of a two-person minimum winning coalition that divides the money evenly and excludes the third person. That is, the structure of the system provides a shell that discloses the general form of the outcome. And if we change this structure to, say, one in which persons A and B together can earn $300, A and C can earn $400, and B and C can earn $500, then we can predict that if A and B coalesce, they will divide their winnings $100-200; if A and C coalesce, they will agree to a $100-300 division; and if B and C coalesce, they will split $200-300. In this way, structure sets the constraints on eventual outcomes, and our theory identifies

1.1 Perspectives

13

those constraints. Furthermore, no new theoretical ideas are required to accommodate increases in the number of participants, changes in payoffs, or even alterations in the structure of payoffs from bargaining over dollars to bargaining over, say, policy in some multidimensional issue space. If, on the other hand, we want to predict which particular two-person coalition will emerge in a specific instance - in our example, the alternatives are {A,B}, [A, C}, {B, C] - then we must know a great deal about the details of perception and preference for A, B, and C and the priors they bring to the bargaining process before we can make any accurate prediction. Does A feel more secure dealing with Bl Are there cultural barriers to C s dealing with Al What are the subsidiary games A, B, and C will play that we have thus far ignored in the analysis? Are there domestic political imperatives that we have also ignored that render particular coalitions difficult to enforce? Indeed, it is our experience, after observing hundreds of sessions of undergraduates confronting theoretically equivalent decision tasks in experimental environments, that it may be impossible to determine precisely which coalition will form ultimately. Girlfriends coalesce against boyfriends and classmates against each other; quiet negotiators prevail, as do hard bargainers; and age, sex, and race, if they predict anything at all, predict with minimal significance. Nevertheless, despite the great variety of factors that seem to influence outcomes, the coalitions that emerge and the outcomes that form the basis of agreements nevertheless fit (or closely approximate) the macropattern proscribed by theory. Thus, these and similar considerations pour content into the shell provided by the structural variables our analysis identifies. Prediction of actual coalition behavior in specific instances must be predicated on identifying both the shell and its contents. Our paradigm thus elucidates the relationship between systemic and subsystemic variables in the analysis of international politics. This relationship and the role of the paradigm is illustrated further by an analysis of the general structure of the European great-power alliance system in the last quarter of the nineteenth century. Without attention to the domestic political constraints impinging on international actors, we would be at a loss to predict which nations will ally with which others, that is, we would be unable to predict specific outcomes from either the shell or the content alone. Thus, when examining European great-power alliance formation and conflict resolution, 18711914, in Chapters 7 and 8, we consider some general constraints on action that model the effects of their bargaining in the international arena. However, with Waltz's observation in mind, we also remain sensitive to the modification of domestic constraints by the imperatives of international forces.

14

1 Introduction

Problems uncovered by the theory of social choice Another way in which our paradigm highlights the importance of the interaction of systemic and subsystemic variables in international relations is related to the problems associated with treating aggregates such as the nation-state as individual actors. Any attempt to view nations as unitary actors - as decision makers with well-defined (consistent) goals and preferences - is an abstraction we can apply only with great care. Here we must follow the logic of Kenneth Arrow's (1957) classic theorem, which tells us that regardless of the social and political construction of a state, circumstances may arise in which the actions implemented by political leaders and the heads of multinational firms may not be consistent with any identifiable goal (see Bueno de Mesquita 1981a, pp. 12-18). Although it is convenient from time to time to speak of countries as maximizing their power, resources, and so forth, or offirmsmaximizing market shares and profits, we must be cognizant that such actions are themselves the product of domestic political and economic processes, and that those processes need not be consistent with any identifiable objective for any collectivity. Arrow's theorem and subsidiary results appear to pose a dilemma for any theory that treats collectivities, such as nation-states, as unitary actors. Among the subsidiary results causing the greatest concern is McKelvey's proof (1976,1979) that if decisions are made by majority rule and if we can characterize outcomes as concerning some multidimensional issue space, then even the slightest intransitivity in the social preference order (which occurs whenever every feasible outcome can be upset by some other outcome) requires that the intransitivity encompass the entire issue space. This, in turn, implies that a skillful agenda setter, by the judicious ordering of a sequence of pairwise voting comparisons, has the opportunity to choose from a considerable range of outcomes which outcome will eventually prevail. McKelvey's result might seem to merely open the door to the assumption that any collectivity's preference ought to be reducible, ultimately, to some agenda setter's preference - that an implicit dictatorship cannot be avoided. However, another result of Gibbard (1973) and Satterthwaite (1975) establishes that all procedures of social decision are potentially manipulable. Regardless of the procedure used - be it a simple one such as majority rule with an agenda, or a complex one such as the description we might give to congressional-presidential decision making - that procedure is subject to the strategic revelation of preferences by as few as one and as many as all participants in the decision process. McKelvey's result and Gibbard and Satterthwaite's theorem, taken together, appear to imply the impossibility of imputing a simple discernible preference for any collectivity. Indeed, they suggest that such preferences

1.1 Perspectives

15

emerge only in the context of complex strategic calculation and manipulation. This causes one scholar familiar with this literature to conclude that social decision processes are not merely complex - a fact we already know - but also that imponderable and unpredictable considerations, such as the relative strategic skills of individuals and the precise timing of events, can profoundly affect social decisions and, thus, the implied social preference order over outcomes (Riker 1982). We do not require such theorems to learn that social processes are complex or that the flow of history often turns on critical events. Yet, these theorems are important because they prove and generalize our intuition; and, just as importantly, the research they stimulate yields insight into the potential sources of greater predictability and coherence of social decisions. We have learned, for example, that such coherence does not require even an approximate social consensus over what outcomes ought to be selected. Instead, coherence is supplied in majority-rule systems, for example, if there is unanimity merely over the criteria used to evaluate outcomes (Black 1958) or if the correlation of preferences is "sufficiently great" (Niemi 1969). In this instance, then, we can think of coherence as being supplied by socialization, shared values, and a common perception of the issue under consideration at any point in time. Coherence can also be supplied by various institutional constraints. Although theoretical analysis here has just begun, one apparent implication is if the rules and procedures of social decision are exogenous - if the mechanisms used by a collectivity for aggregating individual preferences and choice into a social choice are not subject simultaneously to reconsideration and change then it is possible to constrain decisions so as to vitiate the implications of intransitive social preferences. Because we cannot survey a vast subfield in a few pages, we cannot delve deeply into the details of abstract theoretical reasoning so as to uncover fully the insights provided by that reasoning. However, the preceding discussion points to two general sources of coherence in social decision: coherence emerging from the constraints of socialization, and coherence emerging from the constraints of institutions when coupled with a consensus on the legitimacy of those institutions. Interestingly, middle-range theory assessing foreign-policy decision making in terms of domestic political variables follows, tacitly, the lines suggested by this analysis of coherence in social decision. The operational-code approach (Leites 1951; George 1969; Holsti 1970; McClellan 1971; Walker 1977; Campbell, Darvich, and Rose 1983) focuses on the consequences of value socialization for preference formation. The bureaucratic politics (Lindblom 1965; Allison 1971; Allison and Halperin 1971) and organization process (Snyder, Bruck, and Sapin 1962; Cyert and March 1963; Allison 1971) approaches focus on the consequences of various institutional arrangements for preference

16

1 Introduction

formation. These results, in turn, permit us to understand more accurately the role of the assumption that we introduce in the next chapter: that nation-states are unitary actors seeking to ensure their continued sovereignty and to maximize their resources. We do not make this assumption because we believe that such a coherent objective necessarily characterizes all the decisions of national leaders. Nor do we make this assumption because we accept Bueno de Mesquita's (1981a) judgment about the dominance of key decision makers in the determination of a country's foreign policy. Rather, we impose it because it is analytically convenient. It permits us to establish the implications for anarchic international systems of states that act, for whatever internal domestic reasons, in a proscribed way. We can then use this model to understand the implications of nationstates that do not act in accordance with our assumption, relating possible violations of our assumptions to the domestic factors suggested by the previous discussion. In this way, we can begin to build into our analysis a consideration of domestic politics and social processes. We emphasize, on this score, that taking cognizance of Arrow's theorem does not require that our models be reducible immediately to an analysis of individual decisions. The assumption of methodological individualism does not require a reductionist approach. Even economists hypothesize that certain collectivities such asfirmsact with a single objective (the maximization of profit), despite the fact that Arrow's theorem applies to any organization and despite the empirical evidence that key decision makers in firms are not always motivated primarily by the profit margins of their employers. What makes this assumption legitimate, however, is the subsequent demonstration that although the goals of the members of a firm need not concern profit, those firms that do not act as unitary agents in the pursuit of profit will disappear, supplanted by those that do. It is the Darwinian force of competitive markets, then, that assures that the consequences of Arrow's theorem will not vitiate the logic of subsequent deductions. Of necessity, then, we will assume that various collectivities in international affairs (specifically, nation-states) act as unitary agents. In doing so, however, we must give good arguments why the process of their internal politics ensures that outcome; or, minimally, why a collectivity that does not act as if it holds a collective goal will either be supplanted by some other form or will otherwise adapt to its exigencies. General adequacy of the paradigm Many students of international affairs might believe that the "rationalchoice" paradigm is inappropriate for their subject. After all, despite

1.1 Perspectives

17

Blainey's (1973, p. 249) conclusion that "no wars are unintended or 'accidental,'" and despite the lessons learned from games such as the Prisoners' Dilemma, wars nevertheless give the appearance of being the consequence of seemingly irrational actions by persons with clear incentives to act rationally. And it is reasonable to suppose that the next great war, if it occurs, will be the penultimate irrational act for our species - in which case, our fear of it seems to be an admission of the possibility of irrational choice. Before turning to our substantive focus, then, we should comment on the adequacy of our paradigm for the study of politics in general and international relations in particular. Such discussions are, of course, already extensively represented in the literature (see, e.g., Harsanyi 1969, and Riker and Ordeshook 1973). Stating our position briefly, the assumption of rational action or the hypothesis that people maximize utility is not subject to direct empirical test. Arguing whether people are rational or not makes little sense, at least within the confines of the paradigm. We judge a person "rational" if that person's decisions are understandable to us or appear that they might be rendered understandable with some attribution of goals and perceptions, imbedded in a tractable model of the decision environment; we judge people "irrational" only if we choose to relegate their actions to the world of the inexplicable. Understanding, on the other hand, is provided by the models we develop for the specific substantive issue before us. Thus, rationality is not an empirical hypothesis like others we might derive from it; rather, its "test" consists of ascertaining whether and to what extent the models based on it (and their corresponding explanations) possess any advantage over other approaches. Naturally, we cannot use the efforts of a single researcher as a basis for any definitive evaluation of the paradigm: it can always be said that we failed to achieve worthwhile insights because of the researcher's deficiencies. Nevertheless, the course of much of the recent theorizing about why people vote gives us some insight into the paradigm's promise as well as the problems we encounter with it. Nearly any representation of the decision to vote suggests that if people are concerned solely with the instrumental benefit of voting versus abstaining - defined in terms of which candidate wins and which loses - then few if any citizens should vote. Thus, it appears that people vote merely because they are socialized to do so, in which case the decision-theoretic structure of our paradigm must be augmented somehow to secure an understanding of this fundamental political act. This does not mean, however, that people are irrational, or that they do not maximize expected utility, or that this application of the paradigm fails to provide valuable insights. Rather, it suggests (but does not prove) that much of the voting decision is determined by considerations exogenous to the particular

18

1 Introduction

election being considered, that expected utility calculations focusing on the parameters of a particular election may be largely irrelevant to the voting act, and that the greater explanatory power in this case is promised by a study of the origins of taste and of the affective (as against instrumental) value people place on the act of voting. This example should also dispel the misperception that our paradigm assumes a world populated by egoists who cannot be motivated by altruism or other "socially conscious" objectives. Admittedly, our analysis assumes an egoism of the sort in which decision makers are unconcerned with the welfares of other states aside from how those welfares affect, via the selection of strategies, the resources and sovereignty of their state. However, this assumption is a hypothesis; it is not a requisite part of the paradigm. The analysis of voting reveals that if we cannot explain the decision to vote by motives that focus on a single voter's influence of the outcome of who wins and who loses, then we must entertain speculations in which the act of voting itself has affective value. Similarly, we may find that people's decisions in contexts that concern us here must also be modified. The point is that we cannot learn whether a modification is required until we explore the implications of a pure form of egoism; furthermore, the possibility of a modification that admits of altruism is not precluded a priori. To be more specific, we could, of course, assume that some national leaders maximize the prospects for world peace, whereas others seek conflict; but because such actions are "close" to the outcomes we want to understand, this attribution of goals threatens a vacuous tautology - one that fits historical facts well but, because of its specificity, yields little generalized predictive power. Thus, we substitute more remote goals instead, in this instance, the desire to maximize a country's relative resources, tempered by the constraint of ensuring its sovereignty. Admittedly, though, even this goal is removed from what geneticists tell us is a more fundamental individual motivation, the maximization of one's "fitness." However, although such a goal promises more generality, the necessity for translating it into predictions about actions requires models that we are not yet equipped to develop. Thus, research within the paradigm necessarily entails a trade-off between generality and tractability, and much of the researcher's art consists of reaching an acceptable balance within the domain of these two dimensions, a balance that must always be prepared to give way to new theoretical developments. The issue as to whether people are "rational" continues nevertheless to arise in political theory, largely because some people believe that simple constructs and words can adequately label and convey the meaning of

1.1 Perspectives

19

complex ideas. Rationality is not a simple elementary concept in our paradigm. It not only embodies a set of axioms about preferences (that they are complete and transitive), but, in addition, much of what game and decision theorists do can be interpreted as the discovery of the meaning of this word in different contexts. Thus, rationality in noninteractive environments may mean choosing actions that can be interpreted as maximizing utility or expected utility; in noncooperative games, it may mean the selection of strategies that are consistent with one definition of equilibrium or another; and in a cooperative game, it may mean the selection of outcomes and the formation of coalitions that are consistent with some solution hypothesis. Indeed, the meaning of rationality is sufficiently complex that very little would be lost if we simply banished the word altogether from our vocabulary. As afinalnote of caution, we should guard against prematurely rejecting our paradigm whenever the complexity of reality seems to defy the development of tractable models based on its precepts. Because we want to use the precise language of mathematics to deduce our hypotheses, thereby ensuring that those hypotheses follow logically from explicitly stated assumptions, we simplify our representations of reality to ensure tractability. This bothers many scholars who see and appreciate politics in all of its complexity. For these scholars, mathematical tractability seems a hinderance to understanding. To some extent this criticism is fair, but only if we abide by a naive view of the role of formal theory. Certainly, it should not be used to judge the paradigm as inappropriate to our subject. The pursuit of scientific understanding is more an art than it is the application of some mechanical formula for proving theorems or conducting empirical research. Because international systems are much too complex to be adequately summarized by any single model, theorem, or set of regression equations, the expression of that art is a coherent combination of whatever theoretical structures we possess, dictated by our general understanding of our subject - an understanding that informs our application of rigorously derived results, which in turn refine that understanding. Applications of the simple Prisoners' Dilemma game to politics provide a useful example. Briefly, such games reveal how sometimes the simple structuring of decisions by circumstance can decisively effect outcomes, in particular, the inability of people to achieve mutually beneficial (Paretooptimal) outcomes. It is hardly profound, however, to assert that little of the real world fits comfortably into the simple structure of a 2 x 2 game. Nevertheless, although we might deem such models imperfect reflections of reality owing to their simplicity, our general understanding of social processes tells us that an important component of those processes is that

20

1 Introduction

politics is a continuing sequence of social interaction and that our actions today are dictated, in part, by how they affect the play of future games. Thus, as we become more technically sophisticated and consider multiperiod versions of the dilemma (Taylor 1976 and Axelrod 1984), or dilemmas in which players possess incomplete information about the game (Kreps and Wilson 1982), our ability to explain additional patterns of action (such as cooperation, threats, and the selection of alternatives that minimize utility in the short run) increases. At this point, however, further analysis requires the mathematical complexity some prefer to avoid. Yet, even at this stage, we can begin to glean important lessons about the precise role of reputation and commitment in circumstances as diverse as terrorist threats, nuclear deterrence, military intervention, and revolution (e.g., Rose 1986; Powell 1987, 1988; Alt, Calvert, and Humes 1988; and Wagner 1988). Even if we stop here, commonalities in empirical phenomena are revealed, so that what was originally seen as diverse phenomena are now known to be manifestations of common theoretical principles. The anarchy of international systems The theory of international systems set forth in this book does not attribute conflict or peace to the presence or absence of various types of international organizations. Indeed, we begin with the supposition that international systems are anarchic in the sense that there exists no exogenously determined organization or structure, save the primitive rule that countries with greater resources can defeat countries with fewer resources. That is, referring to the concept of regimes rather than systems, we follow Keohane (1984, p. 98), citing Young (1979, p. 35), observing that international regimes are decentralized institutions. Decentralization does not imply an absence of mechanisms for compliance, but it does mean that any sanctions for violation of regime principles or rules have to be enacted by the individual members.

The particular issue we address is the emergence of stability, of implicit cooperation, when systems are anarchic. Again, this issue is defined by Keohane: The puzzle of compliance is why governments, seeking to promote their own interests, ever comply with the rules of international regimes when they view these rules as in conflict with what I will call their "myopic self-interest." Myopic selfinterest refers to governments' perception of the relative costs and benefits to them of alternative courses of action with regard to a particular issue, when that issue is considered in isolation from others (1984, p. 99).

Insofar as identifying the specifics of our task, it is tempting to suppose that what differentiates domestic politics from international politics is the

1.1 Perspectives

21

existence in domestic affairs of specific institutional structures that ameliorate the sources of conflict in society, and their absence in international politics. In its simplest form, this argument views the state as an agency authorized to resolve various social Prisoners' Dilemmas such as the provision of public goods, including domestic order and tranquility. Since no such agency exists in international affairs to resolve the dilemmas there for example, arms races that everyone prefers to avoid (see Waltz 1959, Jervis 1978, and Wagner 1983) - then achieving a goal of international peace and tranquility merely requires the establishment of supranational institutions.5 The argument in support of the proposition that peace can be secured via the establishment of "appropriate" supranational organizations supposes, as we do, that the structure of international political systems is anarchic. However, it also assumes, contrary to our approach, that this anarchy is the fundamental cause of conflict. This view is most clearly stated by Riker (1962) in his critique of Kaplan's (1957) analysis of the balance of power. Briefly, on the basis of his application of game theory to the balance-of-power literature, Riker concludes that international systems are inherently unstable, and that there are "only two ways of moderating instability, and only two": the method of moral restraint and the method of institutional constraint (p. 174). Thus, Riker abandons the possibility that simple self-interest can yield stability and argues, instead, that such stability must arise because of exogenous factors. Although our analysis differs significantly from Kaplan's, we cannot accept Riker's conclusion. If stability is to occur at all, it must arise from wholly endogenous considerations of self-interest. We do not dispute that such organizations could play an important role in reducing the costs of information and the likelihood of miscalculation. However, our argument is that an international system is not distinguishable from a domestic system simply by the presence or absence of specific types of institutions, procedural rules, behavioral norms, and the like. Nor do we suppose that supranational organizations per se are a prerequisite 5

As we suggested earlier, the enterprise of explaining the differences between international and domestic political systems in terms of an anarchy/hierarchy dichotomy is questionable. Specifically, it assumes that the ordering principle that determines whether a system is hierarchical or anarchic is exogenous to the system. If, however, the ordering principle is endogenous (i.e., arises from interactions of actors in the system), then differentiating between international and domestic political systems in terms of an ordering principle merely typologizes rather than explains. The endogeneity of rule formation within systems arising from the nonexludability of strategizing is one of the more important implications of our paradigm, and such a result is suggestive of a more fundamental explanation for behavior at both the systemic and subsystemic levels than the anarchy/hierarchy dichotomy provides.

22

1 Introduction

for stability. With respect to the character of domestic versus international systems, we agree that it may be easier to establish mechanisms for the enforcement of agreements in thefirstthan it is in the second, but it is easier only to the extent that decision makers find it in their interest to maintain the features of those institutions that regulate enforcement. We take the view that rules, procedures, and norms are endogenous to political systems: that they are established and maintained only to the extent to which they serve the interests of those who participate in them, and that their existence is governed by the same forces that govern the existence or nonexistence of stability itself. Thus, the objectives of a social or political institution primarily reflect the interests of its more powerful members. When these interests or the relative powers of individuals. . .change, there will be attempts to change the nature of the institution and its objectives in order to reflect significant changes in interests and power (Gilpin 1981, p. xi).

For the specific context of this volume, "if, for example, the equilibrium of the balance of power is maintained through states abiding by rules, then one needs to explain how agreement on the rules is achieved and maintained" (Waltz 1979, p. 121), in which case, for the issue of cooperation, using rational choice theory for the study of international cooperation implies that the relevant decisions of governments, and other actors, about whether to cooperate can be treated as if they were voluntary (Keohane 1984, pp. 70-1).

This argument leads to the view that supranational organizations are merely mechanisms for the implementation of agreements. It is not a gun that kills the victim, nor is it the availability of a gun that "causes" a killing. Rather, it is the person pulling the trigger who commits the homicide, and the gun is merely the mechanism he or she chose to implement an action. Similarly, supranational organizations may facilitate implementation, but we cannot say that they cause peace. They may be facilitating agents, but understanding their function and identifying the benefits that are believed to follow from their formation require that we understand them as the product of more fundamental processes. In short, we cannot use institutions to explain stability or vice versa; both must be understandable from the same source. 1.2

The dual nature of instability

Because this book is fundamentally about stability in international systems, it is essential that we understand this concept's dual nature, for it is that nature that often distinguishes between domestic and international politics. It is also here that we can begin to draw the outlines of the elementary forces that operate in and define international political systems.

1.2 The dual nature of instability

23

To see the dual nature of stability and instability most simply, let us proceed from a rudimentary bargaining game. Imagine that three people, whose initial wealth holdings are $20 each, must use majority rule in deciding how to divide an additional $90. One resolution, suggested by the majority-rule criterion and assuming that each person has an equal vote, is that two people coalesce to divide the sum evenly among themselves, excluding the third from any increment. Thus, persons 1 and 2 might propose to allocate the $90fifty-fifty,and to exclude 3 from any gains. However, person 3 could then offer 2 a "better deal," such as a proposal to divide the amount sixty-forty between 2 and 3, respectively. Person 1 could then respond by offering 3 thefifty-fiftydeal at the expense of 2. Clearly, this cycle continues indefinitely. Notice now that if we repeat this game several times in succession, using an additional $90 each time in the sequence, we could observe any number of possibilities, such as multiperiod coalitions that are periodically broken up by an "irresistible" offer of the excluded player, or the "complete instability" of constantly shifting allegiances and coalitions. An important feature of this scenario to remember, however, is that unless one or more of the participants feels compelled by ethical considerations - if, to properly describe motivations, we must augment the simple objective of maximizing one's wealth with the assumption that people also prefer to avoid "inequitable" distributions of income - the current holdings of a player should be irrelevant to how the next $90 is divided. That is, regardless of the current distribution of income, each person retains his one-third share of the "vote" in deciding how to divide the next $90. Each redistributive opportunity in this sequence corresponds to what game theorists refer to as a simple constant-sum cooperative game that has no core.6 It is constant-sum because our three decision makers cannot 6

Briefly, the core of a cooperative game consists of all outcomes such that, once at one of them, no person or no collection of persons who might coordinate their choices (a coalition) has the desire or the wherewithal to unilaterally move to some other outcome. The definition of the core, then, parallels the definition of (Nash) equilibria for noncooperative games - games that do not permit players to explicitly coordinate choices. Such a noncooperative equilibrium corresponds to a conjunction of strategies, one for each player, such that no player has the incentive to shift to any other available strategy. The core can be viewed as such an equilibrium if we substitute coalitions for players and the outcomes they can secure as strategies. Thus, a cooperative game without a core can, in principle, cycle endlessly as one coalition supplants the outcome proposed by a previous agreement. This does not mean, however, that we cannot predict outcomes if the core is empty. The solution theory of cooperative game theory, although relinquishing the goal of identifying a unique outcome as its prediction, seeks to identify limits on the range of outcomes that can prevail. For example, in a three-person divide-the-dollar game, we can in principle cycle endlessly through all possible divisions, but most variants of solution theory limit the eventual outcome to those three in which two players divide the dollar evenly between themselves.

24

1 Introduction

create or destroy wealth, only redistribute it. It does not have a core because, regardless of what redistribution is proposed, there is another distribution that a majority strictly prefer and that they can secure if they coalesce. Furthermore, if we look at each game in the sequence in isolation from the rest, then the various solution hypotheses treating games without cores all render an identical prediction that accords with our intuition: the formation of a minimal winning majority coalition that evenly divides the $90 among its members. Stated in the formal notation of the game theorist: the value of a coalition C in the first play of this game, v(C), is $20 if C consists of a single member, is $90 plus $20 plus $20 if C contains two members, and equals $90 plus $60 if C contains three members.7 Game theory then predicts that an outcome such as ({1,2}, {3), $65, $65, $20) will prevail in thefirstperiod - that a coalition such as [1,2) will form against {3}, add $45 to the wealth of each of its members, and leave 3 with his initial holding. If the game is repeated a second time with a new $90 subject to redistribution, then f(l) = f(2) = $65, y(3) = $20, v(l, 2) = $220, v(l, 2) = v{29 3) = $175, and u(l, 2,3) = $240. Again, we would predict that some minimal winning coalition will form to divide the next $90 evenly between its members. Notice now that the players in this scenario need not be concerned about the existence of future games in the sequence when deciding how to play a current game. Although the outcome of the current game determines each person's relative wealth, it in no way affects future strategic possibilities. In each game of the sequence, there is precisely $90 to be divided and every person shares one-third of the total vote. In fact, even if we let the current wealth of each person (in addition to the $90) be subject to redistribution, so that the situation is a pure redistributive game, the nature of the instability remains unchanged, as do the strategic considerations of the players. The value of a "single-member coalition" is now zero since it cannot ensure itself anything, the value of a two-member coalition in the first round is $150 since it can reallocate all available wealth, and 7

The notation v(C) is called the characteristic function value of the coalition C. Briefly, v(C) is commonly defined as the set of payoffs the members of C can guarantee for themselves by appropriately coordinating their strategies - the security level of C. There are, however, a variety of alternative definitions of the characteristic function, depending on whether the game concerns a freely transferable resource (games with side payments, such as when some form of exchange is possible), and whether all individual utility functions over this resource are linear (transferable utility, such as when money is equated with utility). If both assumptions are satisfied, then v(C) can be summarized by a single number the amount of resources, money, etc. that C can guarantee for its members and that it can ensure will be at the disposal of its members for distribution. Notice that this definition, because it is based on the notion of a coalition's security value, admits the possibility that the players excluded from C act irrationally in holding C down to its security level. However, in Chapter 4 we modify the definition of v(C) to take account of the rational responses of those countries excluded from the coalition.

1.2 The dual nature of instability

25

a typical predicted outcome is ((1,2), (3);$75, $75, $0). If this game is played a second time, with an additional $90 subject to redistribution,

v(l) = v(2) = v(3) = $0, v(l, 2) = v(l, 3) = v(2,3) = v(l, 2,3) = $240. Although we may witness wider swings in wealth as coalitions form and dissolve in a repeated play of this game, each person "remains in the game," someone to be reckoned with as a potential coalition partner in the game's next redistributive round. The preceding scenario is not much different from the one which Riker (1962) uses to argue that international political systems are inherently unstable. Specifically, if ru r2, and r3 are the resources controlled by persons 1, 2, and 3, respectively, and if K equals the resources added to the current round for redistribution (in addition to the sum R = r{ + r2 + A*3), then v(C) =R+K if the coalition C is any majority; otherwise, if C is losing, i>(C) = 0. In this case, an empty core means that, regardless of what coalition proposes to form and how it intends to distribute its payoff, there always exists some other coalition and proposed redistribution that members of the new coalition can secure and unanimously prefer. Put differently, the set of feasible coalitions and alternative redistributions is an all-encompassing cycle that leads from any distribution to any other distribution, and back again. Now suppose we modify the situation in the following way: although majority rule continues to be the decisive criterion, let each person's "voting weight" be endogenous and equal the resources he or she presently controls relative to all others. If this game is played once, then the players' strategic considerations are not much different from what wefindin a single play of the game in our first scenario. Although voting weights are no longer equal, if there are only three players, if no player's weight is zero, and if no player already controls a majority share of weight, then (as with equal weights) any two players can form a minimal winning coalition that expropriates all available resources. On the other hand, if this game is repeated several times, then the participants' strategic considerations and actions should be quite different from what we anticipate in our previous scenario. Now current holdings play a critically important role since, for example, if any player secures a majority share then he or she becomes dictator in all subsequent games. That is, the existence of future games affects one's thinking about the current game; so, by rendering voting weight endogenous rather than exogenous, the game becomes one of survival in which one or two players can be permanently "retired," rather than a game that is merely concerned with the allocation of money. In summary, then, our two scenarios - thefirstwith exogenous and equal voting weights and the other with endogenous weights - are equivalent if they are played once. However, strategic considerations differ markedly if the players know that the game will be repeated a succession of times.

26

1 Introduction

Thus, combining endogenous weights with repeated play yields a situation that differs significantly from the game Riker uses to analyze international conflict and the balance of power. We draw this distinction for two reasons. First, we want to emphasize the different character of the instabilities in each game and the corresponding differences in the players' strategic considerations. Second, although features of both situations can be found in international political processes, the first situation is more frequently found in domestic politics - at least in reasonably stable democracies - whereas the second is especially germane to the study of international relations. With respect to strategic considerations, notice that in thefirstscenario, players may be quite willing to let a coalition partner secure an advantage since part of the pie is better than none at all. Further, even if everyone knows that the situation will be repeated more than once, each can still concentrate on the current round. Long-term multitrial coalitions are no more stable than single-trial agreements and, because a player retains his or her vote regardless of what coalitions form, agreements in the current round have no bearing on strategic possibilities in future rounds. The second scenario, though, is profoundly different. There participants must guard against agreements in which one person secures a majority of the wealth, since such a person is predominant and can expropriate all available resources. Everyone must guard against having his wealth reduced to zero, since that outcome permanently removes a person from the game. For example, I might agree in the first scenario to let you be the principle beneficiary of a decision to expropriate the resources from a third party. Although I prefer other outcomes, I may feel compelled to this outcome in order to forestall expropriation of my own resources. However, with endogenous voting weights measured by resources, such an agreement may only temporarily postpone my eventual elimination from the game. I will merely become your next victim if I permit you to secure a majority of the resources from the third player. Correspondingly, negotiation strategies and the attention participants pay to multiperiod consequences are likely to be significantly different in the second scenario than in the first. This view of strategic possibilities is perhaps no more forcefully illustrated than by this example from Kennedy (1987, p. 139): The Russian general Kutusov, wishing to halt his army's westward advance in 1812, once the Grand Army had been driven from the homeland, may have spoken for more than himself when he doubted the wisdom of totally destroying Napoleon, since the "succession would not fall to Russia or to any other continental power, but to the power which already commands the sea, and whose domination would be intolerable." Thus, although the first scenario presents us with those instabilities we commonly associate with zero- or constant-sum cooperative games, the

1.2 The dual nature of instability

27

second scenario adds an additional consideration. Not only might we cycle indefinitely as we renegotiate the resources each of us controls, but there is also a potential instability in the number of viable participants. One or more of the players may be so weak - either because of initial inequities in the distribution of resources or because he or she previously agreed to outcomes that were inappropriate to his or her welfare - that he or she may be incapable of securing existence through negotiation and reallocation. Thus, whereas the first scenario offers us a view of what we might term the "microinstability" of shifting alliances and coalitions in constant-sum games, the second scenario adds the possibility of the "macroinstability" of the game's viable participants. The second scenario, moreover, alerts us to the possibility of observing macrostable, but microunstable, systems. The second reason for distinguishing between our two scenarios is the different purposes to which they can be applied in politics. Insofar as domestic politics are concerned, we might use our second scenario to characterize such politics when the forces of economic competition and governmental regulation threaten the economic viability of firms, or when economic status fully measures political influence. For example, if votes are irrelevant because net wealth determines and measures political power, then "voting weights" are endogenous and can be set permanently to zero. This scenario also is obviously relevant to the study of regimes characterized by revolution and armed unrest, where the measure of each competing interest's political influence is its military capability. Of course, the first scenario is also relevant to the study of domestic politics, especially in constitutional democracies with constitutional guarantees to enfranchisement and representation. Legislators may negotiate the reallocation of public resources, but, barring radical political change and subject only to the laws of population variability, legislative districts and the representation of citizens is best treated as exogenous. Despite the fact that we might use both scenarios to model various aspects of domestic politics, only the second seems relevant to the representation of international political processes. Because there are no supranational authorities capable of guaranteeing the representation or sovereignty of countries, "voting weights" are necessarily endogenous. "Power" or economic and military resources, endogenously determined by the vitality of an economy or determined by a country's success at expropriation from others in earlier games, becomes the sole determinant of who can defeat whom and who can expropriate further from whom. It is this endogeneity that gives international politics its special role in human affairs while at the same time rendering organizations such as the League of Nations in the past and the United Nations today largely irrelevant to the central flow of events.

28

1 Introduction

It is the central premise of this volume, then, that the second of our two scenarios describes, albeit in highly simplified and abstract form, a basic process of politics - both domestic and international - and one of our tasks is to elaborate on this scenario so that it becomes a better approximation of reality, an approximation from which we can derive specific and testable hypotheses. At this point, however, we should note the special complications associated with analyzing multiperiod situations modeled after this scenario. In the case of thefirstscenario, an analysis is simplified considerably because we can treat each period as an isolated event. Because the outcome of one period does not prejudice strategic position in the next, we can simply look at optimal or stable decisions that take cognizance only of current factors. However, with the second scenario, the choices of decision makers in one period affect profoundly all future decisions, including even who will be relevant decision makers in the future. Because we do not want to assume initially that decision makers are stupid or naive, we must suppose that they take future conequences into account in deciding what to do now. In the lexicon of game theory, decision makers are playing a potentially unlimited extensive-form game, so strategies must tell them what to do under all possible contingencies, both present and future. Thus, by adopting the second scenario as our basic structural scheme, we must learn to accommodate complex, time-dependent, interactive decision making. Notice also that if we accept the proposition that the second scenario models international political processes, then quite directly, the concept of balance of power becomes the focus of our discussion. Indeed, having accepted the idea that stability, if it arises at all, must arise from wholly endogenous sources, we arrive at a conclusion wholly in accord with Harsanyi's (1969) argument that social processes and their dynamics can be understood only "in terms of the balance of power among the various social groups [nations] pressing for the arrangements most favorable to their own interests" (p. 535). However, in theorizing about international relations, perhaps no idea causes more difficulty or is a greater source of confusion than that of balance of power. Scholars are uncertain about whether this term refers to a theory of conflict and coalitions, to a description of international systems, to the goals of key decision makers, or to a normative prescription about how international systems ought to achieve peace. It should be evident from what we have said thus far, of course, that we reject the notion of balance of power as a goal or as a normative prescription: if balance in the form of system or resource stability prevails, it must do so regardless of whether all, some, or none of the states in the international system consciously aim to maintain a balance or profess balancing as their policy.

1.3 Theory in international relations

29

Nevertheless, even with this narrowing of definitions, the literature's theoretical disarray makes it virtually impossible to identify what ought to be a theoretical primitive and what ought to be a deduction. Should we assume that nations act to oppose other nations that seek to dominate the system, or should such assertions be deduced from more basic assumptions about national interest? What are the indicators of a system out of balance? Is a system unbalanced because the resources of specific countries or alliances do not satisfy certain equations (Zinnes 1967, 1970), or must we observe overt aggressive acts? Indeed, even those who approach the subject from our perspective of nations seeking to advance their own self-interest infer different conclusions as to the meaning of balance of power. To the extent that we rely on our second scenario as a starting point for modeling international relations processes, then quite directly we must formulate our own theory of the balance of power. Despite the confusion in the literature about its precise formulation, it is evident that such a theory ought to be concerned with identifying viable as against unviable alliances (coalitions), with identifying the conditions under which one set of countries will attempt to expropriate resources from another, and the conditions under which one or more countries will be eliminated, or at least rendered irrelevant to the strategic calculations of the remaining countries. However, these are precisely the questions the game theorist tries to answer when studying the games our scenario describes, so a theory of the balance of power becomes our primary product. 1.3

Theory in international relations

We noted earlier that the phenomena that concern social scientists are considerably more complicated than the phenomena treated by natural science, and thus we cannot suppose that a small set of assumptions and mathematical expressions can accommodate anything more than the broad outlines of the forces guiding events. Thus, aside from the obvious fact that our theories must remain abstractions, the models we construct with these theories are of necessity incomplete. For example, despite its continuous development and refinement over two centuries, a review of that most powerful of social theories - classical microeconomics - tells us that our ability to describe and predict even a very special class of social interactions is severely limited. Even today research continues into the causes of market failure, the mechanisms for accommodating uncertainty and incomplete information, and the interface between private and public sectors in areas such as regulation, tax policy, and international trade. Because markets are a component of international affairs, our task entails

30

1 Introduction

the same difficulties (on perhaps an even large scale) as those confronting our colleagues in other branches of the social sciences. We offer these comments so that the reader can properly interpret what follows in this volume. Our assumptions and deductions should not be taken as modeling any specific international process or historical eras, nor should we presume that our analysis is intended to be comprehensive of all possible considerations. Indeed, we acknowledge the fact that the particular assumptions that form the basis of our analysis are chosen not only because we believe that they lead to an understanding of substantively important and general features of international political affairs, but also because they facilitate deduction and analysis. Correspondingly, we do not regard as a criticism the comment that the use of mathematics and explicit assumptions renders the limitations of our analysis apparent. Such comments assure us that we have avoided, at least to some degree, a scientifically unacceptable sloppiness of thinking. As Waltz (1979, p. 11) observes: In international politics, whether because theories are contradictory or weak, discussion and argument about many important matters... are made difficult or useless because the participants are talking about different things while using the same terms for them. Movement towards a remedy is impeded by disinclination to treat the question of meaning as a problem that can be solved only through the articulation and refinement of theories.

To the extent that the forces our analysis uncovers can be interpreted as guiding us to other features of international systems we do not consider, then that fact merely points to the potential profitability of formally incorporating these other features into our analysis and thereby refining further the theory we offer. To illustrate the way in which this incorporation process might proceed, consider once again the "levels-of-analysis" issue. At the first level, we see history driven by the personality or skill of specific individuals. Germany is unified only because of Bismarck, the outbreak of World War II requires a Hitler, Churchill's intransigence keeps Germany from overwhelming Russia, a resurgence of Islam can be led only by a Khomeini, and the Cold War emerges as the result of Stalin's paranoia. At the second level, events are driven by more general features of domestic politics. Great Britain establishes an empire to feed its industrial growth, Germany and Japan's aggression is encouraged by the belief that the United States has not the will to engage its economy in war, and the notion of the nationstate will prove subservient to that of class. At the third and final level, international affairs are dictated by general forces that operate on nationstates. Britain sides with France to block the German threat of preponderance, and the United States seeks a rapprochement with China to forestall a hegemonic coalition between China and the Soviet Union.

1.3 Theory in international relations

31

The theory we offer coincides most closely with the third view, but the potentially devastating error we can make as analysts is to suppose that these views are in conflict - that to adopt one necessarily implies a rejection of the others. Indeed, it is foolhardy to disagree with the proposition that powerful personalities often change the course of history or that domestic politics are relevant to international affairs. Generally, the theory of cooperative games (as we have already noted) merely narrows the range of potentially observable coalitions, so that the specific coalition that forms in some circumstances may be dictated by chance, by the timing of ephemeral events, or by the relative bargaining skills of people. Thus, our "macrotheory" narrows down the range of outcomes, and we must appeal to some "microtheory" to refine those predictions further. Our paradigm, as we have argued, provides a particularly satisfying methodology for uniting these macro- and microtheories. Ideally, our microtheories should employ the same paradigm as our macrotheories. However, in the initial stages of theorizing, which describe the present circumstances of international relations theory, we must often proceed on a more ad hoc basis, and "patch" reality in from time to time. Still, we are confident that the apparently ad hoc flavor of the constraints assumed to operate at the microlevel is, in part, deceptive; they follow from the empirical results of theoretically driven research within the paradigm we have embraced. The fact that our theory is primitive also means that, in reviewing our assumptions, the reader should resist the temptation to begin mapping reality into the analysis, with the search for examples and counterexamples. These comments should not be interpreted as an argument against the empirical application of our theory, since we attempt such an application in the chapters following that theory's development. We are merely offering the caution that in any application or test, we must make a number of compromises. We must temper certain possibilities, for example, with our suppositions about the imperatives of domestic politics, the perceptions and beliefs of key decision makers, and the fact that national leaders are often involved in a complex nexus of games. And to these problems we can add the operationalization or measurement of key concepts. Despite our goal of deductive rigor, this problem remains with our analysis because, rather than grapple with the meaning of such words as power, we merely substitute abstract notation and a different word - resources. We do this, however, with the understanding that such concepts are primitive only at the macrolevel of our analysis and that, depending on substantive context, the analyst has some flexibility in choosing their meaning and measurement. Thus, applying our paradigm and the abstract propositions of our analysis to a complex reality is necessarily more an art than a science. Whether we have executed our art satisfactorily is left to the reader's judgment.

32

1 Introduction

1.4

Plan of this volume

We begin in Chapter 2 with a more extensive discussion of our second scenario, and, in particular, we explore its alternative manifestations and interpretations in politics. Simultaneously, we offer an initial elaboration of this scenario with the politics of international relations specifically in mind. It is here that we present our key assumptions about goals, capabilities, and the alternative actions available to decision makers, and it is here as well that we discuss in detail the specific abstractions we require for tractable analysis. It is also in this chapter that we define more carefully the notions of stability suggested by our second scenario - system and resource stability - and judge the extent to which they capture the meaning of stability in the literature. In Chapters 3 and 4, we turn to an analysis of our scenario, and it is here that we attempt to resolve directly the confusion in the balance-ofpower concept. We do this by reconceptualizing this concept in terms of our two notions of stability. In the case of system stability, we focus on each nation's ability to ensure its survival against every possible contingency threatening its sovereignty. However, even if system stability prevails, this does not mean that some country cannot force others to cede it some of their resources. Hence, a system is resource stable if no one, acting alone or in coalition with others, seeks to upset the current distribution of resources among nations. With these two notions of stability and with the supposition that national leaders are sophisticated decision makers who are cognizant of the ultimate consequences of their actions, we deduce necessary and sufficient conditions in Chapters 3 and 4 for the existence of system stability and resource stability, respectively, in ncountry systems. There are several avenues open for testing the model developed in Chapters 3 and 4, the most obvious being a survey of historical events and the data others use to test similar ideas. However, our theoretical analysis involves a number of assumptions about how people bargain and form coalitions in a specific structured situation. Because other assumptions may be plausible, a focus on the data history alone provides would not allow us to distinguish between a failure of the general theoretical perspective as against the necessity for revising our analysis of bargaining; that is, the general theoretical approach may be correct, but other bargaining assumptions may be more suitable. Thus, Chapter 4 concludes with an experimental test of the predictive accuracy of our assumptions and of the necessary and sufficient conditions for system and resource stability. Our experimental results are encouraging, although of necessity not definitive. Nevertheless, it is evident that regardless of whatever empirical

1.4 Plan of this volume

33

support we can gather for the model in Chapters 3 and 4, there are two essential missing elements. The first such element concerns the subject of preventive wars. We must confront the possibility that neither system nor resource instability is attributable wholly to some current "imbalance" in military position. Rather, such instability may result from the projections of future imbalances and from the corresponding perceived necessity for preventive wars (see Fay 1938; Brodie 1973; Organski and Kugler 1980; and Levy 1987). This dynamical view is reflected also in the debate over domestic versus defense spending and the issue of whether a country should invest resources optimally to maximize economic growth rather than appropriating those resources to maximize current military strength. Indeed, this debate shows why a simple static analysis based on some current distribution of national resources or power cannot capture all relevant aspects of the "balance-of-power" concept in general and the causes of war in particular. In response to such realities, the analysis in Chapter 5 extends the model from Chapters 3 and 4 by adding investment and resource growth as part of the decision criteria of national leaders, who must then judge coalitions and resource reallocations in terms of longer-term as well as immediate consequences. Such an extension, moreover, is natural to our model in that both system and resource stability seem to be precisely the issue involved in the debate over current versus future defense needs. Using these two concepts, the extended model accommodates resource growth as a determinant factor of system or resource stability, and provides an analysis of preventive wars in two ways. First, we let growth be a function of exogenous factors, summarized by a growth rate and an initial distribution of resources. Second, we let national leaders affect growth endogenously. As part of a sequential cooperative and noncooperative game, such leaders can vary the current and future military capabilities of countries by choosing investments into the future, hence the proportion of resources subject to growth. In this way, national leaders do more than simply try to anticipate future events. In choosing their investments and in deciding whether to engage in a preventive war, they must also anticipate the anticipations of others, the actions of others based on those anticipations, and so on, with the understanding that everyone is doing the same. To learn the conditions under which countries can invest and maintain their sovereignty, and when they might contemplate a preventive war because they cannot do so, the analysis in Chapter 5 describes the necessary and sufficient conditions for the existence of system and resource stability in this interactive, dynamic setting. The second element missing from our analysis concerns geography and the asymmetrical role resources play with respect to this factor. History

34

1 Introduction

tells us that, although land armies may be effective for attacking a continguous adversary, they may be useless against one with even a modest water barrier; and a tenet of today's strategic thinking is that modern technology fundamentally changes the implications of such barriers and distance. This argument's implication is that formulations of balance of power in which resources are treated symmetrically, although providing an initial understanding of the problem as well as the basic structure for theoretical refinements, may mislead us as to the necessary and sufficient conditions for stability, and that they may be inappropriate for interpreting historical events and processes. Indeed, profoundly important questions cannot be answered adequately unless geography is taken explicitly into account (cf. Spykman 1944, Dehio 1962). For example, should we attribute England's role as "balancer" in the nineteenth century to the motives and diplomatic skills of key decision makers or to its unique geographical position with respect to the Continental powers? What precisely are the advantages of geographical distance? Should we attribute the unification of China at the end of the Warring States Period in 221 B.C., for example, at the hands of a geographically "peripheral" as against central power as mere happenstance brought about by the fortuitous combination of military and diplomatic skill, or is it reasonable to hypothesize that such peripheral powers have an advantage over central ones? What are the implications for international stability of a decline in geography's importance, as occasioned by advances in the technologies of weapons systems? Can the removal of natural barriers to warfare disadvantage individual states, but make international systems more stable? In Chapter 6, then, but only for the special case of three-country systems, we extend the model that we offer in Chapters 3 and 4 to accommodate geography, and in doing so we are able to specify precisely the roles and special advantages and disadvantages of balancers and central powers. In Chapters 7 and 8, we move toward applying our model to the dynamics of alliance formation and conflict resolution in the classical period of the European balance of power, 1871-1914. Chapter 7 focuses on the dynamics of alliance formation using a valuation of coalitions deduced from our theory and markedly different from one assumed by earlier game-theoretic treatments. Chapter 8 is concerned with conflict resolution in three crises: the two Near Eastern Crises of 1875-8 and 1885-8, and the July Crisis of 1914. Our objective is to show the operation of similar forces in all three crises, but to explain why the third erupted into war. Most interestingly, our treatment of World War I as a preventive war uncovers an important set of factors underlying Russian precipitation of the war that had hitherto escaped analytical notice.

1.4 Plan of this volume

35

Finally, in Chapter 9, we offer a brief review of our analysis, and then we turn to a discussion of the contemporary implications of our analysis. The particular issues that concern us there are the relevance of our theory to a world with nuclear weapons and the theoretical extensions to our analysis that modern technology requires. In this way, we highlight the future research directions to which our analysis points.

CHAPTER 2

Basic elements of a model and definitions of stability International relations continue to be a recurring struggle for wealth and power among independent actors in a state of anarchy. Robert Gilpin, War and Change in World Politics (1981, p. 7)

Classic balance-of-power theory can be interpreted as an hypothesis that in a more-than-two-Power world there are no non-autonomous causes of systemic change;... the purely political aspect of the system's powerpolitical process always tends to produce a stable equilibrium which can be upset, if at all, only by autonomous changes. Arthur L. Burns, Of Powers and their Politics (1968, p. 249)

The balance's underlying principle...was that all the nth disengaged powers would tend to intervene on the side that seemed in danger of losing any ongoing war, to ensure that such a loser was not eliminated from the system and absorbed into an emerging colossus. George H. Quester, Offense and Defense in the International System (1977, p. 64)

2.1

Preliminary assumptions

Our objective in this chapter is to elaborate our second scenario of resource competition and survival - a game wherein the only explicit rule defining its structure is that any player can overcome a weaker adversary so that it more clearly constitutes a model of international conflict and alliances. At the same time, we want to render the concept of equilibrium, implicit in the preceding quotations, analytically precise so that the full implications of egoistic choice in anarchic systems can be uncovered. Before we proceed too far, though, we must be certain that this scenario can model a process as complex as the relations among sovereign states. We have already tried to indicate why we believe that this scenario in particular relates to some fundamental processes of international politics; but even if we accept the proposition that abstract theory can contribute to our understanding of international affairs, it is admittedly difficult to see how a simple three-person bargaining situation in which a single 36

2.1 Preliminary assumptions

37

commodity is the negotiable item can serve as a model of much of anything in politics. Indeed, as we noted earlier, too often the applications of decision and game theory to politics take the form of simple stories or scenarios based on the Prisoners' Dilemma and Chicken games, which, although interesting and informative, fail to provide any general structure for understanding fundamental processes. Hence, before we can become convinced that any one scenario can provide such a model, we must interpret its component parts and structure substantively. First, we must address the issue of the identities of the fundamental decision makers in our analysis. Holding in abeyance our earlier discussion of the implications of Arrow's Impossibility Theorem, our assumption is that countries are unitary actors, or, equivalently and in accordance with Bueno de Mesquita's (1981a) approach, that they are led by a single decision maker. This assumption must confront several objections. Anyone familiar with democratic processes understands the consequences of a Congress with foreign-policy objectives distinct from those of the President, and is thereby cognizant of the fact that the foreign-policy objectives of a state need not be readily identifiable. Indeed, even less democratic governments often speak with a discordant voice, one that is made all the less coherent as policy preferences shift over time and as other elements of the society act in contrary ways. In short, "The existence of transgovernmental policy networks leads to a different interpretation of one of the standard propositions about international politics - that states act in their own interest [T]his conventional wisdom begs two important questions: which self and which interest?" (Keohane and Nye 1977, p. 34). Nevertheless, our assumption plays an important and informative role. Just as we study frictionless planes in physics and the abstraction of competitive markets in economics in order to discern fundamental laws, the assumption of unitary decision makers serves to reduce "friction" in our analysis. We base our assumption of the state as a unitary decision maker on the prior assumption of a relatively complex set of domestic political games the outcome of which is to endow the state, or a single leader of that state, with the effective power of a unitary decision maker. Later, we explore the implications of decision makers who fail to satisfy our assumptions to see what effect such failures might have on our conclusions. Second, we should understand that the identification of key decision makers, and our associating them with nation-states, is more an analytic convenience than it is a theoretical necessity. We do not suppose that nation-states are always the primary actors in international systems, nor do we preclude the possibility of applying our analysis to historical periods in which the concept of the nation-state is irrelevant. Because we are attempting to formulate a general theory of interaction in anarchic

38

2 Basic elements of a model and definitions of stability

political systems, the definition and identification of key decision makers is left to the requirements of the analyst. Thus, although we use the word "country" to identify a key decision-making entity, we could just as easily refer to the city-states of medieval Italy. Indeed, although one of our primary objectives is to study the alliances that form among decision makers, it may be useful from time to time to treat certain alliances as constraints on the system and to suppose that a particular alliance is a unitary actor, especially if that alliance exists more for domestic political reasons that we choose to interpret as exogenous to the immediate study. It may be convenient, for example, to treat Japan, Canada, and the United States as a single entity as long as we focus on immediate military capabilities, as long as Japan and Canada forego the development of independent military capabilities, and as long as Japan in particular continues its indirect support of America's military spending and our corresponding budget deficits. However, if our attention turns to economic conflict and to the competition for control of economic resources, then Japan at least should clearly be entered into the analysis as a differentiated decision-making entity. The point is that the definition or identification of "key decision maker" is for the analyst's analytical convenience. Third, before we can accept the abstraction of assuming that countries are unitary actors, we must satisfactorily specify the objectives of these decision makers. Again, this forces us to confront Arrow's theorem and the possibility that, although countries may act as if they are unitary decision makers, they may also act "incoherently" in the sense of not revealing complete or transitive preferences. In this event, it may be impossible to argue that any collection of persons is acting as if it were pursuing an identifiable goal. Nevertheless, we assume that the unitary decision makers in our model, qua national leaders, act as if they share a readily discernible goal - increasing their countries' resources, provided that the survival of their respective countries is not thereby threatened. We find support for such an egoistic assumption about goals in Kennedy's (1987, p. 73) observation that "The most significant feature of the Great Power scene after 1660 was the maturing of a genuinely multipolar system of European states, each one of which increasingly tended to make decisions about war and peace on the basis of 'national interests' rather than for transnational, religious causes." With a more direct reference to our assumption, we can cite the following observation about the Ch'un Ch'iu period in China: the ultimate aim of political actions was the expression of the sovereignty of the individual states. This became the standard for judging the measures of the rulers and the various officers of the states. The moral code of the former Chou feudal age had meaning only when it added to the power and prestige of the state which

2.1 Preliminary assumptions

39

claimed to adhere to it Thus, major policies were security policies. Security and power were the concern of the day. Such actions as rounding out frontiers, joining leagues, transferring alliances, treaty arrangements, fomenting civil war elsewhere can all be interpreted in terms of the quest for security (Walker 1953, pp. 99-100). Even this quote understates the generality of the assumption of egoism, and of Morgenthau's (1959, p. 5) boldly stated proposition that "statesmen think and act in terms of interest defined as power." Waltz succinctly states the cause for such generality: Because some states may at any time use force, all states must be prepared to do so - or live at the mercy of their militarily more vigorous neighbors. Among states, the state of nature is a state of war. This is meant not in the sense that wars constantly occur but in the sense that, with each state deciding for itself whether or not to use force, war may break out at any time (1979, p. 102). Several important issues, however, arise with our assumption about objectives. Among them is the question of whether the objectives of all nation-states or their agents can be summarized by a single, unidimensional measure. That is, what of the seemingly self-evident fact that Security in traditional terms, is not likely to be the principle issue facing governments. Insofar as military force is ineffective on certain issues, the conventional notion of power lacks precision. In particular, different power resources may be needed to deal with different issues The notion of national interest - the traditionalists' lodestar - becomes increasingly difficult to use effectively. Traditional maxims of international politics - that states will act in their national interest or that they will attempt to maximize their power - become ambiguous (Keohane and Nye 1977, p. 8). In approaching this debate, however, we should keep in mind that the simplification that we adopt from the realist view parallels the simplification that underlies the attractiveness of contemporary economic theory: One important reason for the remarkably great explanatory power of classical economic theory lies precisely in the extreme simplicity of its motivational assumptions, which make economic self-interest virtually the only motivating force of human behavior, at least in economic activities (Harsanyi 1969, p. 518). Morgenthau (1959, p. 12) makes an equivalent argument: if I want to understand "religious man" I must for the time being abstract from the other aspects of human nature and deal with its religious aspect as if it were the only one. Furthermore, I must apply to the religious sphere the standards of thought appropriate to it, always remaining aware of the existence of other standards and their actual influence upon the religious qualities of man. What is true of this facet of human nature is true of all others. No modern economist, for instance, would conceive of his science and its relations to other sciences of man in any other way. It is exactly through such a process of emancipation from other

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2 Basic elements of a model and definitions of stability

standards of thought, and the development of one appropriate to its subject matter, that economics has developed as an autonomous theory of the economic activities of man. To contribute to a similar development in the field of politics is the purpose of political realism.

Correspondingly, such a simplification has proven to be powerful in previous attempts to model diverse political processes at the domestic level, such as in Niskanen's (1971) model of bureaucratic-congressional relations (viz., his assumption that public bureaucrats maximize their budgets), Fiorina (1977) and Mayhew's (1974) studies of congress (viz., their assumption that legislators maximize their probabilities of reelection), in Downs's (1957) seminal study of elections (viz., the assumption that political parties maximize their chances of election), and in microeconomic theory (viz., the assumption that firms maximize profit). Each of these models posits a simple and readily operationalized objective for decision makers with the view that the adequacy of the assumption is tested by observing the adequacy of predictions. If the abstraction leads to a model that predicts well in diverse situations, then we regard it as an acceptable abstraction; otherwise, we must find an alternative specification of objectives or an alternative specification of the system's key actors. If we find, for example, that domestic politics preclude coherent national goals, then we may have to revise our description of international systems so that various domestic actors are explicitly identified and given an appropriate role in the analysis. The potential for disagreement about the identity of key decision makers and their goals characterizes many of the differences in views of international politics. A classical Marxist perspective prescribes class and class leaders as relevant actors, whereas others look at significant economic entities such as multinational corporations, and still others prefer a classic nation-state conceptualization, augmented by the force of powerful personalities. Our approach most closely parallels the traditional realist view, but even acceptance of the criticisms of that view does not vitiate an analysis based on its precepts. If, in fact, "Survival is the primary goal of all states" and "in the worst situations, force is ultimately necessary to guarantee survival" (Keohane and Nye 1977, p. 27), and if "In international affairs, force remains thefinalarbiter" (Waltz 1979, p. 180), then our analysis - like classical microeconomic theory, spatial theories of elections, and analyses of legislatures based on the assumption that legislators maximize the probability of reelection - describes the shell in which all other forces must operate, and provides the broad outlines of international politics that help us discern the meaning and the impact of these other considerations. There remains, nevertheless, one difficulty with applying a perspective in which we hypothesize the goal of maximizing resources or maintenance of sovereignty: no natural unidimensional commodity suggests itself for

2.1 Preliminary assumptions

41

measuring the performance or preference with respect to that goal. We could, for example, try to make our assumption seem more reasonable by substituting the word "power" for resources, but this merely replaces one word with another - one whose definition is even more problematical. What we must do, then, is recognize that the resources to which we refer in our assumption can mean a number of things, depending on context. Thus, at a minimum, we shall use it to refer to territory, at other times to military manpower, and at other times to economic capacity. And we must admit that even these simplifications are often difficult to formulate - witness, for example, the issue of defining and measuring a country's economic capacity in today's complex world market economy. At this point, however, we must emphasize that the assumption that each national leader is motivated to secure resources while guarding his or her country's sovereignty is not necessarily equivalent to the supposition that nations (or their agents) simply pursue the maximization of power or of resources alone. Although the proponents of realpolitik typically equate this goal with "rational" action itself, we must keep in mind our view that international politics does not consist of the single "play" of a simple game; rather, it consists of a successive play of a sequence of games. In this context, if national leaders are concerned with final outcomes or with some view of the future, and hence with what their actions imply about the ultimate, overall distribution of resources, then this interpretation of rationality is not a basic assumption but, at best, a deduction from some model that posits more fundamental goals. If we assume instead that the ultimate goal of national leaders is the survival of their nations as sovereign entities, then those leaders must evaluate a decision to attack another nation, to join a particular coalition, or to cede resources, not in terms of immediate benefits, but in terms of what an action implies ultimately about the likelihood that its survival will be endangered. Echoing Kutusov's strategic dilemma in his pursuit of Napoleon, Kaplan (1979, p. 70) makes a similar observation: "the weakest player, by joining a nearly predominant strong player, only creates a condition in which he will be the next victim." Correspondingly, we can cite Wagner's observation that the basic question that concerns us is whether states will act so as to eliminate other states. If one state is eliminated from a four-actor game, for example, the result is to precipitate a three-actor subgame. If a value can be assigned to such a subgame for each player, it is possible to determine whether any players have an incentive to eliminate other players (1986, p. 551).

So, assuming that international actors are cognizant of the sequential game they are playing, we must check whether immediate objectives (maximizing resources) are consistent with ultimate goals (survival, defined in terms of territorial or economic sovereignty).

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Next, we must specify the rules under which national leaders pursue their objectives - the actions that they may take and the consequences of these actions. Actually, this is a more difficult task than we might otherwise suppose, and there are two alternative approaches. In the terminology of the game theorist, thefirstapproach consists of specifying a game's extensive form. This requires that we model reality by identifying the specific actions available to decision makers, the order in which those actions can be taken, the constraints that specific decisions place on subsequent actions, and the consequences of all possible permutations of actions. In the current context, these actions would include decisions to cede territory or resources to another country, to threaten countries with invasion, to invade, to sue for peace, to negotiate a coalition with someone, to establish mechanisms for enforcing agreements, and decisions about the timing of each of these actions. The advantage of such a method is the empirical verisimilitude of the model it produces. However, the difficulty with such an approach is that it yields a monstrously complex model of reality that defies both specification and analysis. Our solution to this problem is traditional to cooperative game-theoretic analysis: we abstract from the substantive content of actions and focus instead on their consequences. For example, we can look at the decision to transfer resources from one country to another and assess whether such a transfer is dictated by the current situation, ignoring for the time being whether such a transfer occurs via the actual cessation of territory, whether it requires armed conflict, or whether it requires simply that some new tariff be implemented that reorders the economic advantages afforded to the industries of one country versus another. With this approach, we also deliberately ignore the precise sequencing of events, the nuances of bargaining, and the precise mechanisms players might use to enforce agreements. Thus, referring to our first scenario, we can try to decide whether a two-person coalition will form and divide the available resources equally between its members without specifying the details of the negotiation process. In summary, then, our basic scenario consists of a set of decision makers who are confronted with a sequence of situations in which each, without threatening one's future viability, tries to maximize resources via the judicious formation of coalitions as either threats or counterthreats. 2.2

Notation and additional assumptions

The assumptions in the previous section about unitary actors, the identity of those actors, the maximization of resources, and the substantive content of actions permit us to maintain someflexibilityin the substantive interpretation of our formal results. This is important and useful.

2.2 Notation and additional assumptions

43

Although our analysis seeks to be rigorous, we do not want to become constrained by its mathematics when we are required to give it substantive meaning. First, this flexibility permits us to avoid becoming embroiled too quickly in issues of operationalization, such as the definition of sovereign states, measures of relative military capabilities, the specific identities of key decision makers, and definitions of war. Too often discussions of the causes of conflict and the nature of international systems focus prematurely on such issues; but it is precisely these issues that cannot be resolved until general theoretical paradigms and essential concepts are specified. Second, because we are searching for fundamental causes in the structure of international political systems, we prefer to avoid the task of ascertaining the variety of ways in which those causes might become manifest in reality. We cannot know that falling leaves and falling rocks are manifestations of the common phenomena of gravity and friction until the concept of gravity itself is hypothesized. So here, until a general theoretical paradigm is at hand, we cannot be certain in the retelling and comparison of events whether differences are theoretically fundamental or are merely equivalent manifestations of the same phenomena. To elaborate our second scenario, then, we proceed abstractly and as generally as possible by first introducing some formal notation. This notation may be distasteful to many students of international affairs, but it will permit us to state some requisite assumptions rigorously. Keeping in mind that the essential components of that scenario are a specification of the players in the game and an initial endowment of resources for the players that measures their "voting weight," we then require the following notation: S = {1,2,...,«}, the set of all countries in the international system under consideration. C = a subset of S (written C -->rn)> a n fl-tuple of resources, where rt denotes the resources of country /. We assume that A*;>0 for all /, and (for convenience) that the resources of the countries are ordered rx > R = 2?=i /"/, the total resources controlled by all countries in the system under consideration. r(C) — S/ e c f/> the total resources controlled by the collection of countries C. (S; r) = an outcome or a state of the world specifying the countries that constitute the system under consideration and the resources they control individually.

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Merely supposing that we can model international political processes with this notation means that we accept a number of implicit assumptions, including (as we have already stated) that the principal decision makers are unitary actors, that there exists some finite, controllable resource, and that we can specify how much of that resource each country controls. This notation also supposes that we can unambiguously identify the set of countries relevant to the analysis. This, of course, is not always the case since we may be uncertain about which countries are relevant and which countries are mere satellites of another, without true sovereignty in the sense that they cannot express an independent foreign policy. Despite such operationalization problems, we should be prepared to beflexiblein empirical applications of our model. We may prefer, for example, to exclude certain countries from an analysis as a temporary convenience, as when we exclude smaller countries to study superpower relations, or when we ignore the role of the superpowers in the relations among smaller countries in studying, for example, regional matters. Our analysis of European great-power relations, for example, excludes Ottoman Turkey as an actor, since it is more convenient to treat the resources contributed to conflict resolution by that declining empire as merely a means for rendering conflict a nonconstant sum game among the European powers. Alternatively, we may decide that the superpowers have no conflicting interests in a situation, or that their resources are not of a type that impacts on local conflicts, in which case we may choose to apply the analysis only to smaller countries [cf. Walt's (1987) analysis of the Middle East]. Finally, to understand the constraints imposed by domestic politics, we might leave the arena of international affairs and instead associate the members of S with key domestic political interests within a state. Although free to define the set S of relevant countries in a number of ways, depending on research intent, we have already noted that the supposition that a country's resources can be represented by a single number is problematical. Indeed, even if we accept the proposition that nationstates act as unitary decision makers concerned primarily with maintaining their sovereignty, and that military capacity provides the ultimate guarantee, it is clearly the case that a great many things such as geographical advantages, population, size and robustness of an economy, and the degree of domestic political stability are relevant resources; and each of these components even is difficult to measure by a single index. Later, in Chapters 7 and 8, we operationalize this variable in terms of factors especially relevant to the historical period under consideration - namely, a country's military age male population, its ability to mobilize that population for conflict, its production of commodities (e.g., coal, iron, and steel) that are key to any war effort, and its ability to project its military forces

2.2 Notation and additional assumptions

45

offensively and defensively. And although variables such as GNP should correlate with the ones we use, the fact that we do not use economic measures yields a bias in our analysis for measuring directly military capability rather than potential value to an adversary of successful threats. This bias, in turn, reveals an ambiguity in our assumptions. In the analysis that follows, we implicitly equate military capability and the resources a country might lose to an adversary in a conflict. In reality, however, a country with an inferior economy but with a superior armed force might overwhelm its adversary and force the transfer of economic resources that greatly exceed the costs of overcoming the current disparity in strength. Nevertheless, our argument about the conditions for stability and instability can be more clearly developed if we assume that the concept of resources is unidimensional and unemcumbered by complex substantive qualification (for further discussion of this issue, see Section 2.6). This ambiguity in the meaning and operationalization of resources is not much different from the problems we encounter in giving the concept of power theoretical and substantive meaning.1 Of course, these problems are well known, and for those reasons we agree that "power" cannot usefully be a primitive concept in any adequate theory. Nevertheless, because we suppose that "resources" not only measure what it is that countries seek, but that it also measures their abilities to secure additional resources from adversaries, then our use of the word "resources" is synonymous with much of the intuition that underlies the concept of power. To a certain extent, then, we are free to interchange these words, provided that we keep in mind the fact that the notion of resources gains its substantive meaning only through the particular operationalization we supply to it an operationalization we must be prepared to alter, depending on the particular historical period under consideration. Before we can proceed, however, we require two additional assumptions about resources that allow certain mathematical manipulations. The first assumption is that the total of resources held by all countries is constant: Al. R is a constant. 1

For discussion of the plethora of problems associated with operationalizing the concept of "power," see Knorr (1956,1970), Hart (1976), Baldwin (1979), and Organski and Kugler (1980). Despite these problems, however, there is a common theme to attempts at conceptualizing and operationalizing power - namely, the replacement of a complex, multifaceted reality with a simple, unidimensional measure. Our notion of resources seeks this same objective. The resources one country can bring to bear to overcome an opponent or to defend against an adversary are varied and complex, but in order to theorize about fundamental imperatives (as against, say, the specific tactics nations employ) we submerge consideration of the ways in which resources can be combined and transformed, and of the functional relationships between resources of different types and the capacity to resist or engage in aggression.

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An equivalent alternative to Al is the following: Al'. For all / in 5, rt (and thus R) grows at the same rate. Al and Al' are equivalent since rt can be interpreted as country /'s proportion of resources, measured against the total. If a country's resources and R grow at the same rate, then the proportion of those resources to R is a constant. Despite the fact that Al is consistent with the supposition that if resources and power are synonymous then (because power is a relative concept) it ought to be in fixed supply, the most evident limitation of Al, as is made apparent by Al', is that it does not permit us to consider an important class of hypotheses that refer to differential growth rates in national capabilities as a source of conflict in international politics. Put differently, Assumption Al precludes consideration of the causes of preventive wars, those wars that occur in anticipation of some country securing a resource advantage in the future owing to a differential growth rate. That this limitation is serious is underscored by Gilpin's (1981, p. 13) observation that "the most destabilizing factor is the tendency in an international system for the powers of member states to change at different rates because of political, economic, and technological developments." Echoing a consistent theme in the literature, however, Chatterjee states the distinction that outlines our analysis:2 In an international system there are two methods by which stability is sought to be maintained. One method is direct - it involves adjustments of the individual capabilities of the nation... mainly by changes in their armaments levels The other method is the adjustment of alliance configurations among the various nations in the system, including the neutralization of certain actors... (1975, p. 44).

Initially, we concern ourselves with identifying the consequences of the second method - alliance formation and the resource adjustments that the formation of alliances might necessitate - in a world of constant capabilities. We merely postpone until Chapter 5 consideration of the effects of differential growth rates and investment strategies, their importance as a general determinant of stability, and their consequences for alliance formation. Our second assumption is that a country is "eliminated" if its resources equal zero, and that once eliminated it is forever precluded from becoming a member of the international system. That is, 2

Similarly, Waltz (1979, p. 118) remarks that "states, or those who act for them, try in more or less sensible ways to use the means available...[that] fall into two categories: internal efforts (moves to increase economic capability, to increase military strength, to develop clever strategies) and external efforts (moves to strengthen and enlarge one's own alliances or to weaken or shrink an opposing one)."

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47

A2. Country / in S is eliminated if r, = 0, so that in all subsequent states of the world (S'; r'), / cannot be a member of S'. If we interpreted our theory as one that concerns the disappearance of countries, we would have to admit that it is a theory more appropriate to those historical epochs in which we witness the evolution of nationstates from the more basic material of feudal systems than it is to relations among states in an established nation-state system. However, even if states today rarely disappear, assumption A2 admits more general interpretations. First, we can think of the situation in which rt = 0 as corresponding to that in which / has ceased to be an "effective" actor. Thus, if we choose to interpret our model as concerning only "great powers," then elimination can merely refer to the removal of a country from great-power status. Alternatively, elimination can refer to those situations in which, although a country retains its technical sovereignty, it is merely a satellite of some neighboring power, one that controls its foreign policy. These alternative interpretations of the condition rt = 0, then, reflect the flexibility we allow in the specification of 5, the relevant states in the system. For purposes of fully analyzing the nature of conflict resolution and alliance-formation processes, we require one additional assumption about national resources: A3. R is infinitely divisible and freely transferable among the countries in 5. If resources refer to territory, then we can see how A3 might be theoretically viable, but it may not be viable in any practical sense. Although the United States can transfer "ownership" of Hawaii to Japan, of Maine to Canada, and of Texas to Mexico, any elected official held responsible for such a transfer will have served his last term in public office. In short, the transfer of certain types of resources may be impossible owing to domestic political considerations. Alternatively, if resources refer to (say) industrial capacity, it may be technologically impossible to effect certain transfers. We might move a steel mill, but the effective transfer of technological expertise may require the development of entire economic infrastructures. The Soviet Union may be able to steal the technology of microcomputers, but what of that part of the American economy in mail order catalogues, shopping malls, and universities that services these computers with spare parts, software, and programming skills?3 3

Assumption A3 places some technical constraints on resources that warrant emphasis. Aside from the fact that infinite divisibility merely facilitates mathematical manipulation, if we measure r, as a percent of R - that is, if we transform all measures of resources to relative terms - then unless the total resources in the system are constant or unless Al' is satisfied, A3 can be violated. If, for example, relative resources are changed only by a

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Despite its implausibility in certain respects, A3 is nevertheless useful.4 First, we should not suppose that resource transfers require the physical movement of troops, money, or international boundaries. Again, we can be flexible and, thus, we can interpret as a transfer one country ceding some part of its sovereignty to another, as when Poland, acting as a buffer state for the Soviet Union, permits the Kremlin to dictate domestic policy with respect to labor unrest. Second, Assumption A3 is valuable because it corresponds, for us, to our "frictionless model" of international political processes. With it we can understand the types or directions of transfers national leaders might seek. And if such transfers are revealed to be essential for stability, then we can interpret technological or domestic political impediments as impediments to stability, or vice versa. In this way, then, we can begin to understand some of the implications of domestic politics and technological limitations for conflict and stability in international systems. Assumption A3 also permits us to confront the following characterization of balance-of-power theories: the major [adjustment] mechanism... [of balance-of-power theory] is the making and unmaking of alliances. The reason for this dependence on coalitions in order to change the distribution of power is that the power resources of each member of the system are viewed as inelastic. There is no way a nation can increase its own strength very much except by adding its allies' strength to its own, or by decreasing its adversaries' strength by separating it, through persuasion, bribery, or subversion, from its allies (Organski and Kugler 1980, p. 16).

Organski and Kugler offer this characterization as a way to introduce their argument for the necessity of considering endogenous resource growth, but their argument highlights the necessity for Assumption A3, which, by explicitly allowing for an "elastic" shifting of resources across members of the system, brings our analysis in line with previous ideas about the operation of balance-of-power politics.

4

country increasing or decreasing its absolute resources, then any change between two countries necessarily alters the percentage of resources controlled by all other countries and thus free transferability is violated. Any assertion that Assumption A3 diminishes the generality of our analysis must, however, contend with the fact that we have merely made explicit an assumption that is implicit in previous research. Most balance-of-power theorists tacitly make the same assumption (Kaplan 1957, Morgenthau 1959, Claude 1962, Waltz 1979) and their ability to discuss transfers of power and redistributions of capability among national actors rests precisely upon the tacit assumption that power (or resources or capability) is infinitely divisible and perfectly transferable among nations. Moreover, A3 seems more acceptable as an assumption in an era when the measure of military capability can be based more on technology and economic productivity and control than on territory and the counting of bodies available for military service.

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Now we must specify the actions available to the decision makers in our analysis. Because this analysis uses concepts from the theory of ^-person cooperative games, in which a representation of the game that explicitly defines the alternative actions of people (the extensive or the normal form) is reformulated to define a function (the characteristic function) that identifies the relative values of alternative coalitions, of necessity we must abstract away people's specific substantive acts and talk instead of coalition possibilities. Nevertheless, having some sense of available actions is important so that we can more clearly see the substantive implications of our analysis. Briefly, we distinguish these five actions: al: a2: a3: a4: a5:

negotiate to cede resources to other countries, aggressively act to secure resources from other countries, negotiate to secure resources from other countries, aggressively oppose the actions of other countries, act neither to secure nor to cede resources.

We deliberately leave the words "negotiate" and "aggressively" ambiguous. Our purpose is not to distinguish sharply between war and alternative acts of aggression, nor do we want to enter the debate over what constitutes a war and what constitutes an extension of diplomatic posturing for securing a more advantageous bargaining position. Instead, we intend simply to provide a means whereby we might interpret the theoretical results in terms of actions that are more or less peaceful or conflictual. With respect to the preferences of national leaders for alternative allocations of resources, we follow a lead that is implicit in the quotation from Quester (1977) and that also anticipates the logic of our concept of system stability: • A country prefers greater resources to less, unless securing greater resources leads, ultimately, to an outcome in which its resources are reduced to zero, and, in particular, it prefers having some resources to having no resources - survival is preferred to elimination. That is, A4. / prefers the resource distribution r to the distribution r' if rt > r[, provided that / can ensure that no distribution r" results from r in which r" — 0, and, in particular, / prefers any r in which r,- > 0 to a distribution in which its resources are zero. A4 is not inconsistent with the supposition that leaders seek to maximize the resources of their countries, but because we suppose that people look to the future consequences of their actions, it adds a necessary

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qualification: namely, that they seek actions that guarantee the existence of their countries as sovereign entities, or, in the case of "great powers," that guarantee that they remain so. Thus, as we show subsequently, A4 admits the possibility that national leaders might choose to transfer some of their country's resources to another country in the event that such transfers are perceived as necessary to ensure survival. Nevertheless, if a nation's survival is not threatened, its leaders will act, whenever possible, to increase their resources.5 One issue that ought to concern us, however, is that countries do not maximize resources at all - that the goals we attribute to these collectivities are of a different sort. One plausible speculation is that, at least for those countries capable of sustaining prosperity among their inhabitants, maintenance of the status quo is the primary goal and that the maximization of resources or power is merely a by-product or is an objective that is shunned if it interferes with prosperity. This view, summarized by Gilpin (1981, p. 7) as the belief that "peaceful evolution has replaced military conflict as the principle means of adjusting relations among nation states ... [so that] economic and welfare goals have triumphed over the traditional power and security objectives." For example, numerous commentators today characterize the United States as a "status quo power," filling a role once occupied by Britain, and thereby is less interested in the aggrandizement of military power than it is in the maintenance of stable relations among states so that it can pursue purely economic goals. Because we do not suppose, however, that there exists any clean separation between military and economic resources, our assumption matches Gilpin's assertion, as quoted earlier, that "international relations continue to be a recurring struggle for wealth and power," and is reflected in contemporary events by a popular press replete with statements of concern that the strategic position of the United States is threatened by its relative decline in the world economy. Implicit in such concerns, we suspect, is the view that if the maximization of resources or of their proportion - both military and economic - play no role in strategic considerations, then a single country with maximization in mind should find little resistance to its becoming hegemonic. Because hegemonic goals have been pervasive in the past, and because we believe it is only reasonable to anticipate resistance 5

Notice that an apparent alternative to the assumption that resources are maximized is the assumption that national leaders maximize proportion of resources. In fact, to the extent that resources serve the same purpose as power in previous analyses, and since power is (if anything at all) a relative concept, then maximization of proportion seems more nearly consistent with previous research. However, since we assume that total resources in the system are constant, maximization of proportion and maximization of absolute level are equivalent, and we need not concern ourselves with this distinction until we consider, in Chapter 5, the consequences of growth.

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51

to such goals, then maximization must of necessity play a role in everyone's strategic thinking. Turning now to our next assumption, we must proceed with the realization that the processes of international relations are dynamic and the formal representation of these processes is possibly more complicated and laden with strategic possibilities than any game political scientists study. Part of this complexity arises because nations cannot war or transfer resources instantaneously, and decisions to do so leave open the possibility that other nations can respond before final consequences prevail. Unfortunately, formulating an appropriate extensive form so that it is tractable and yields generalizable results is currently beyond our capabilities. Hence, we substitute an analysis in which responses are instantaneous. In particular, we suppose that individually or in coalition, decision makers can choose actions such as a2 and that other countries can be mobilized immediately and their leaders choose al or a4. If these latter countries are defeated, their resources are transferred to the victors unspoiled. The next two assumptions concern the actual play of the game. First, we suppose that: • A coalition of countries can defeat another coalition if and only if the amount of resources held by the first coalition exceeds that of the second coalition. Furthermore, if thefirstcoalition defeats the second, the resources of the second are transferred to the first in a distribution agreed upon by that first coalition. In formal notation, this amounts to the assumption: A5. For the two disjoint collections of countries C and C'^S, C can defeat C" if and only if the resources that C controls, r(C), exceed the resources that C" controls, r(C'), and if no larger coalition attacks C If C defeats C", r(C') is transferred to C as specified by the members of C so that C's resources become r(C) + r(C'). Thus, chance events play no role in determining who can defeat whom. Later, when we interpret historical events in light of our analysis, we must of course make allowances for the uncertainty that confronts decision makers. We discuss this problem further in the next section, although an obviously valuable theoretical extension of the analysis we offer here is to incorporate uncertainty formally. A more serious limitation of our assumption concerns the special role of geography in international affairs. Briefly, our assumption requires that X units of resources be as effective in attacking one country as they are in attacking another, and that those resources can counter an attack by a country possessing X units or less.

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Geography, however, introduces obvious asymmetries in the value of resources. Germany's land army in 1939 may have been just short of adequate for attacking and defeating Russia, but it could not be applied with equal effectiveness against England. This assumption also precludes studying certain kinds of technological change, such as the development of the railroad and the advantages this transportation gave to Germany and Russia. For the present, though, we will ignore this complication, postponing until Chapter 6 a treatment of geography and the asymmetries it occasions. Assumption A5 imposes an additional restriction. By supposing that the resources controlled by C equal r(C) + r(C) after C" is defeated by C, we require that choosing an action such as a2 (as against, say, a3) is costless. It is, of course, silly to suppose that securing resources by aggressive action, including war, takes no more effort than securing those resources by negotiation. We are interested in international instability precisely because of the costs to all of us that wars threaten. Nevertheless, we want to be certain that stability does not prevail in our analysis merely because we have made war and, thus, instability costly. Although this argument (as well as arguments about preserving analytic tractability) rationalizes Assumption A5, Rosecrance's (1986, p. 37) hypothesis that "If war is to be avoided, its cost must be high.. .but it must not be so high as to prevent others from even considering the need to stop an expansive power," forces us to consider the possibility that stability is maximized for some intermediate cost. To see how we might incorporate war costs and thereby determine the logical validity of this hypothesis, an alternative to A5 is to suppose that, after C defeats C , the resources C controls equal [r{C) + r(C')\[\-*r{C')/r{C)]\

(2.1)

that is, the resources destroyed by conflict increase as the strengths of opposing coalitions approach equality. A5 sets a equal to zero. On the other hand, if a = 1 and r(C) = r ( C ) , C is left with nothing - in a contest between equal powers, the resources of both are destroyed. Since even the defeated powers in both World Wars maintained considerable industrial capacity (and ignoring the consequences of nuclear weapons), this possibility seems too extreme, and we might limit a to values between 0 and 1. Throughout most of our analysis, though, we suppose that a = 0, because this allows us to explore balance of power in a "frictionless" world and because such an assumption greatly facilitates the analysis of such issues as preventive war and geography. However, the sensitivity of our conclusion about system stability to the war cost parameter - and, thus, the sensitivity of the fundamentals of balance-of-power politics - are examined in Chapter 3, and are shown not to depend on this assumption.

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53

Regardless of whether we set a equal to zero or not, however, it is still reasonable to suppose that if the same outcome can be obtained through either means, negotiation ought to be preferred to conflict. That is, A6. If securing resources through conflict (a2) and securing resources through negotiation (a3) lead to otherwise identical outcomes, a decision maker strictly prefers a3. Of course, A6 is reasonable only to the extent that the preferences of decision makers somehow match those of the people they purportedly represent. Indeed, A6 precludes the leader who values conflict positively because it draws domestic attention away from his or her own deficiencies and solidifies his or her domestic base of political support. Thus, our analysis may model poorly an Argentine junta. Indeed, Riker (1964) tells us that federalist governments are almost always formed as a response to some external military threat. Hence, if someone sees private political advantage from the formation of such a government, then that person may positively prefer conflict, regardless of casualties or wasted resources. The approach to war and its costs embodied by A6, however, has nevertheless one distinct disadvantage: negotiation and war are barely distinguishable and thus we cannot differentiate sharply between transfers that occur through actions such as negotiation, the mobilization of troops, and skirmishes designed to test or display one's resolve as against those actions in which lives are lost, industrial plants are destroyed, and cities laid waste. If war merely is "diplomacy conducted by other means," then our assumption that wars are costless fails to differentiate among these alternative means. We see later, however, that our analysis of stability in an environment of costless wars applies also to those circumstances in which national leaders and those they lead must confront the costs A6 ignores. Another issue that our assumptions to this point leave unresolved is the consequence of simultaneous moves by several decision makers. In particular, one of the important features of international conflict is the fact that hostilities between any pair of countries leave open the possibility that a third country can enter the fray to its own advantage. Historically, the Soviet Union has been especially attuned to this possibility: Germany attacks Poland, and Stalin reacts to Poland's vulnerability; the United States verges on the unconditional surrender of Japan, and the Soviets declare war on Japan. The next assumption removes some of the ambiguity about how countries can use their resources to attack (a2) or defend (al or a4), and specifies some of the consequences of simultaneous conflicts: • If a coalition attacks and defeats a second, smaller coalition that itself had attacked a third, smaller coalition, then the resources

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of the second coalition are absorbed by the first, but the third coalition is unaffected. That is, A7. If for the three disjoint "coalitions" C, C", and C", C attacks C" and C" attacks C", with r(C) > r(C') > r(C")9 then the larger coalition, C, absorbs C , whereas the target of the attack of C", coalition C", is unaffected. Hence, if r = (150,100,50) describes the initial distribution of resources, and if 1 attacks 2 and 2 attacks 3, then, if no further countermoves arise, (250,0,50) prevails. Actually, A7 is stronger than we require, and any number of assumptions can be substituted provided that they satisfy the following property: If two nations war, then a third more powerful nation can take advantage of the conflict. More formally, for example, we could suppose that C absorbs r(C') as well as some part of r(C")9 or that C absorbs some proportion of r(C') + r(C") and C" absorbs some part of r(C"). Our next assumption merely serves to emphasize the multiperiod context of our second scenario. Briefly, this assumption is driven by the fact that international relations occur in a dynamic setting that cannot be modeled adequately as a "one-shot" game, and that any wholly static description of international political processes abstracts away essential features of reality: • Countries interact over a sequence of periods. The beginning of any period is characterized by a particular distribution of resources over the countries. If at the beginning of any period a country's resources are zero, it is eliminated from interaction for all ensuing periods. Furthermore, the interaction of countries in each period in accordance with assumptions A1-A7 determines the initial resource distribution for the following period. Equivalently: A8. The game consists of a sequence, (1,2, ...,y, ...,m}, of periods. The beginning of period j is characterized by (Sy; r y ). If rj = 0, then / is not a player in the game and i$Sk for all k >y. In each period, countries attempt to redistribute resources in accordance with A1-A7, and the result characterizes the beginning of the next period. The important feature of this assumption is that it implies that decision makers must be concerned with a sequence of plays, rather than with a single negotiation period. Thus, decision makers must look ahead to the

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future consequences of their actions in deciding how to play the game in the current period. One question we should raise with respect to this formulation concerns the calculation of payoffs. Normally, when analyzing a repeated game in which each person receives a stream of payoffs, it is assumed that a person's current payoff equals the discounted value of the stream. However, here payoffs are merely the resources that a country controls: /*/, r/ +1 , etc., and do not necessarily correspond to a recurring stream. Consequently, we interpret Assumption A4 to refer to the resources that a country controls in the next period, and we say that / can ensure its existence if it can pursue a strategy in which no rj in the sequence equals zero. Notice, however, that A8 leaves the details of the game's negotiation process, and thus its full extensive form, deliberately obscured (aside from the requirement that this process consists of the selection of actions al through a5). Rather than specify the process whereby outcomes are negotiated, in the next chapter we introduce a proposition that tells us how resources are actually distributed in a period. Thus, although our analysis up to this point is nearly identical to Wagner's (1986) model of balance of power, it is here that we depart from his approach.6 Wagner proposes to specify an extensive-form negotiation game and to deduce equilibrium bargaining outcomes for that game. Although this is theoretically sound, not only is the analysis of games in this form difficult, but the relationship between cooperative and noncooperative game theory presently is only imperfectly understood. Consequently, general results elude Wagner. The approach we take, which consists of formulating a cooperative solution hypothesis that models the situation at hand without reference to the details of the bargaining process, permits the derivation of general results (but see footnote 8, Chapter 3). Yet, our approach is not without its disadvantages. Like much of cooperative ^-person solution theory and the arguments for the applicability of concepts such as the core and the von Neumann and Morgenstern solution, there is an ad hoc flavor to the analysis that cannot be avoided without resorting to a full extensive-form treatment of the situation.7 Of course, even an extensive-form treatment of a 6

7

One critical difference to this point is that Wagner does not allow countries to transfer resources. As our view of the European great-power relations in Chapters 7 and 8 reveals, however, such transfers, whether implicit or explicit, are an important part of international processes. Moreover, denying the possibility of such transfers, given the flexibility possible in the operationalization of resources, merely denies Blainey's (1973, p. 115) observation that "In peace time the relations between two diplomats are like relations between two merchants. While the merchants trade in copper or transistors, the diplomats' transactions involve boundaries, spheres of influence, commercial concessions and a variety of other issues which they have in common." For a discussion of the core, see footnote 6 in Chapter 1. The von Neumann-Morgenstern solution, on the other hand, can be interpreted as a generalization of the core, and is

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situation must make countless ad hoc assumptions about the sequencing of events and the actions available to decision makers.8 Nevertheless, if we keep our assumptions about bargaining as general as possible, the advantages of a traditional cooperative game-theoretic approach - the ability to deduce general propositions about coalitions and the outcomes they impose - outweigh its disadvantages. 2.3

Uncertainty

Our final assumption concerns the information that decision makers possess about their situation: A9. There is no uncertainty about the resources controlled by each country; and all decision makers have complete information about the game. Game theory admits of two types of uncertainty. The first is statistical and can arise because of the probabilities nature creates, unavoidable measurement error, and ambiguities in the meaning of key concepts. National leaders, for example, may be uncertain about how the various measures of national resources should best be combined into an overall assessment of a country's relative strength and its likelihood of winning a war. Nature may also play a role, as when it dictates the winner of a war as the result of uncontrollable factors such as a storm that sinks an armada or overcast skies that preclude aerial cover of ground troops. Nevertheless, in keeping with our objective of developing an ideal-type model, we assume that this type of uncertainty does not exist. The exclusion of statistical

8

designed, in particular, to treat games for which the core is empty. Briefly, if X is the set of all feasible outcomes then Y, a subset of X, is a von Neumann-Morgenstern solution to the cooperative game over X if Y satisfies two conditions. First, no element of Ydominates another element of Y: if, whenever some coalition has the ability to force a move between elements of 7, the members of that coalition do not unanimously favor such a move; and if they do favor such a move, they do not have the ability to realize it. Second, every element of X not in Y is dominated by some element of Y: if a feasible outcome not in Y is considered, then there exists at least one coalition that has the unanimous preference and the wherewithal to shift the outcome to some element of Y. Notice that in this definition, no allowance is made for the mechanisms whereby coalitional agreements are enforced - the definition of the V set refers merely to what coalitions prefer and can feasibly secure. Thus, the application of such solution notions presupposes the exogenous existence of mechanisms whereby coalitional agreements are enforced. (We emphasize, however, that focusing our discussion at this time on the von Neumann-Morgenstern solution should not be interpreted to mean that we regard it as a wholly satisfactory idea. Indeed, we regard other [albeit related] ideas as more appropriate models of cooperative bargaining and agreements [see, e.g., Ordeshook 1986]. However, we postpone further discussion until Chapter 3, where we must modify solution theory to fit the specific problem at hand.) Aside from the issues of complexity and tractability, the special problem of an extensiveform representation is the timing of moves. The usual structure of an extensive-form

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uncertainty necessarily gives our results a "knife-edged" quality that never exists in reality. Assumption A5, for example, implies that country / can defeat j if /'s resources exceedy's by even the slightest amount. It is never the case, however, that we can assert that one more tank, one more aircraft, or one more dollar of GNP renders one country winning over another. Ideally, we would prefer a model in which we begin with sentences of the form "/'s probability of defeating j is..." and in which we deduce sentences that include phrases such as ".. .makes conflict less likely...," or "... increases the chances that...," or "... leads to conflict with a probability of p, where 0 < p < 1 " Hence, we must learn to temper the interpretations of our abstract results with the realization that reality rarely affords decision makers the luxury of the certainty assumed by our model, and that attitudes toward risk may play a role not found in our analysis (see, e.g., Bueno de Mesquita 1981a). We should not anticipate, however, that the knife-edged quality of our assumptions necessarily implies that our conclusions will be knife-edged as well. Indeed, one of the important conclusions about international stability that we derive from these assumptions is that the necessary and sufficient conditions for system stability at least are not knife-edged, and that the extent to which this form of stability characterizes systems is likely to prove insensitive to the different ways in which we might incorporate statistical uncertainty into the analysis. It is also important to note that the assumption of a certain world does not preclude the study of miscalculation or error as a source of conflict any more than the inclusion of probabilities permits it. Presumably, miscalculation refers to the choice of actions based on factually incorrect beliefs (Jervis 1976), and one can miscalculate in a certain world as well as in a world that one believes is uncertain. Perhaps the presumption that we cannot study miscalculation from a rational-choice perspective derives from the mistaken view of some researchers that "'rational' models of conflict [are] based on the assumption that statesmen accurately perceive external threats and opportunities.. ." (Levy 1983b, p. 76). Yet, there is nothing in our paradigm that game supposes that the sequence with which players implement their decisions, including the possibility of simultaneous decisions, is determined exogenously. Although it is theoretically possible to model sequencing endogenously, this typically yields an extensive-form representation of such complexity that analysis becomes intractable. In most bargaining situations, of course, timing is endogenous, and the only practical solution to this dilemma to date has been to impose ad hoc assumptions on timing and move sequencing. It can be argued, of course, that the theory underlying cooperative solution notions such as the core, the von Neumann-Morgenstern solution, the bargaining set, and the competitive solution merely obscures this issue of timing and that any ultimate theoretical justification for any solution hypothesis must be an extensive-form representation of the moves and countermoves that bargaining allows. Here, though, we must take the tools currently provided by game theory, with the understanding that any analysis based on that theory may have to be modified as our understanding of our paradigm's fundamental elements is refined.

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supposes that individual beliefs, probabilistic or otherwise, are factually correct. We assume simply that people act "optimally," conditional on their beliefs. Thus, as our analysis becomes more sophisticated, we can explore the implications of a world with miscalculation in it. There is, however, a second and potentially more profound type of uncertainty in political processes, which concerns the beliefs of decision makers, and which arises when decision makers are not fully informed about various parameters of the game and when information is asymmetrically distributed. Traditionally, game theory assumes that all players are fully informed about relevant parameters, including the strategies available to an opponent and the opponent's preferences - or at least that they all share the same "imperfect" view - and that each knows that the other knows that the other knows, and so on (this is generally referred to as the "assumption of common knowledge"). Suppose, however, that although person / knows the amount of resources he possesses, he is uncertain about person y's resources, and vice versa. That is to say, / knows y's resources only up to some probability distribution, whereas in fact these resources are determined. Player / could, of course, treat this problem in the usual expected utility way by substituting the expected value of ry into his or her calculations. However, what are we to make of the situation in which j's actions reveal something about ry? In this event, we have a doubly complex strategic situation. If j knows that / may learn something about Tj from j's actions, then j may prefer to take actions that either better reveal or disguise ry-; and if i is aware of j's strategic options, then / must attempt to infer ry-, knowing that j is choosing an action that takes /'s inferences into account. In other words, the usual he-thinks-that-Ithink regress found in noncooperative games extends now to beliefs as well as to strategies. An especially relevant example of the role of such uncertainty includes the analysis of terrorist acts and responses to them. A terrorist, who has seized hostages, may be uncertain about a particular national leader's resolve, just as that leader may be uncertain about the willingness of the terrorist to kill the hostages. If the leader gives in to the terrorist's demands and secures the release of the hostages, this reveals that the leader's resolve is not great, which increases the likelihood of future attacks. If the leader refuses to negotiate, the hostages may die (depending on the terrorist's decision), but the likelihood of future attacks may be reduced. Similarly, if the terrorist kills the hostages whenever his demands are not met, he will certainly die, but he will also strengthen the future viability of this strategy for his compatriots. If he releases the hostages even though his demands are not satisfied, he lives, but this action decreases the viability of this terrorist tactic.

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Presently, we can imagine circumstances in which countries are uncertain not merely about the resources of others, but also about what others believe about capabilities. Thus, actions may be taken to probe the likely responses to others, with the hope that these responses will reveal something about beliefs. Of course, if everyone is aware of everyone else's uncertainty and if everyone knows the purpose of these probes, we are led to formulate a considerably more complicated description of the strategic problem confronting decision makers than the one formulated in this volume. We are only now learning how to treat games with this form of uncertainty and strategic complexity (see, e.g., Kreps and Wilson 1982; Austen-Smith and Riker 1987; Powell 1987,1988; Alt, Calvert, and Humes 1988; Ordeshook and Palfrey 1988), and it is far too complex to incorporate into our analysis, at least until we have identified the basic structure of international relations. We note it here, however, to serve as a warning that a type of uncertainty remains outside of our analysis, and that this type may be especially important for a full understanding of events. Even though we do not consider the strategic imperatives occasioned by this second type of uncertainty, the analysis this volume offers is nevertheless relevant to inferring the consequences of such uncertainty for war. Specifically, a thesis that our analysis supports is that such uncertainty is a necessary condition for war. The argument is as follows: we begin with the first part of Blainey's hypothesis that When nations prepare to fight one another, they have contradictory expectations of the likely duration and outcome of the war. When those predictions, however, cease to be contradictory, the war is almost certain to end (Blainey 1973, pp. 246-7).

Second, such contradictory expectations cannot prevail in the usual setting of game theory - a special form of misperception is required. Finally, because our model precludes this type of misperception, if wars do not arise in our analysis - if a balance of power implies peace - then war requires the misperception that that uncertainty permits, thereby confirming the second part of Blainey's hypothesis. The critical part of this argument is its second step. Suppose, then, that rx and r2 measure the military capabilities of two potential belligerents and that both belligerents, prior to receiving any contemporaneous information about capabilities, share a common subjective prior about the distribution of rx — r2. Of course, if rx — r 2 > 0 then country 1 can defeat country 2, whereas if the inequality is reversed then 2 can defeat 1. Before choosing whether or not to initiate hostilities, however, suppose that each decision maker is privately informed about his own capability, which necessarily causes both decision makers to revise their beliefs that they can win a war. Presumably, the first part of Blainey's hypothesis requires that

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each decision maker believes that he or she can win, but if these revised beliefs are common knowledge - if each person knows the other's belief, if each knows that the other knows, and so on - then probability theory dictates that these posterior beliefs cannot be contradictory (Aumann 1976). Common priors and common knowledge preclude both persons from believing that they can each win against the other. To ascertain the potential sources of such contradictory beliefs, Harsanyi (1968) argues persuasively that the critical condition is the assumption of common knowledge as against the assumption of common priors, since different subjective probabilities ought to be traceable to different experiences - to different observations about the real world. It follows that Blainey's precondition for war requires either a miscalculation of beliefs based on private information or a breakdown of the commonknowledge assumption.9 Game theory, including its application in this volume, generally imposes the common-knowledge assumption (in the form of presuming that all players know the game being played, all know that the others know, etc.), and so, barring miscalculation, the precondition in the second part of Blainey's hypothesis is satisfied by our analysis. Support for the hypothesis is then provided by a balance-of-power analysis that precludes war. If the common-knowledge assumption is critical, then we ought to ask how such knowledge is assured. Knowledge of each other's posterior beliefs is not sufficient - each must also know that the other knows, and so on - and public revelation of one's posteriors is not sufficient either since neither side can be assured that the other is not lying. Indeed, establishing an institutional framework whereby truthful revelation is assured remains one of the difficult design issues of the incentive-compatibility lit9

The common-knowledge assumption does not require merely that each person know the other's posterior beliefs - they must also know that the other knows, etc. - and the validity of the hypothesis that communication facilitates peace may lie here. For those unaccustomed to the jargon of game theory, we offer the following distinctions about the different types of information that players in a game might possess. In a game of perfect and complete information (illustrated by games such as chess and tic-tac-toe), each player is fully aware not only of all aspects of the game, but also of the previous choices of everybody else. In games of imperfect and complete information (that part of game theory made familiar by such expositions as Luce and Raiffa 1957), players are fully aware of all aspects of the game, but the previous choices of some players may be hidden from view, as when decisions are made simultaneously. Finally, in games of incomplete information, players are uninformed about some aspect of the game. In the most interesting case, this information is distributed asymmetrically, as when each person knows his preferences but not the preferences of others. Analyses of games of this last type assume, however, that the basis of preferences is common knowledge. For example, if we suppose that each person's preferences have a random component and that informational asymmetry arises because each person "sees" the result of the draw with respect to their preferences, then it is assumed that the densities governing preferences are common knowledge. Inconsistent beliefs in this instance can arise only if the common-knowledge assumption is violated.

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erature in economics and game theory (for a theoretical discussion, see de Groot 1974 and McKelvey and Page 1986). We can speculate, however, that the appropriate institution is a bargaining format, which leads to the further speculation that the post-World War II argument that peace is fostered by increased communication (by increased trade, cultural exchanges, and the like) is more than simple utopianism. Keohane (1984) restates this argument in a more sophisticated way that hints directly at our argument: regimes are important not because they constitute centralized quasi-governments, but because they.. .increase the symmetry and improve the quality of the information that governments receive. By clustering issues together in the same forums over a long period of time, they help bring governments into continuing interaction with one another, reducing incentives to cheat and enhancing the value of reputation (pp. 244-5). What remains for this volume is establishing the consequences for international stability of "symmetric information" in a form that satisfies the common-knowledge assumption, which is of "high quality" in the sense that there is no statistical uncertainty about capabilities. 2.4

Stability

With our nine assumptions as preliminaries, we must now distinguish between two notions of stability. To see why we need two notions, recall our two scenarios from the previous chapter. In the first scenario, on the one hand, the existence (voting weight) of the players is not an issue. In the event that existence is thus assured exogenously, then the only form of instability that concerns decision makers is instability in the current distribution of resources. In the second scenario, on the other hand, alternative redistributions of resources can threaten existence or at least the opportunity to continue as a relevant participant. In other words, decision makers have a dual (but not independent) concern: securing a more advantageous distribution of resources and securing a distribution that ensures their own survival. Thus, we can speak of two forms of stability. The first notion concerns the instabilities in the distribution of resources among countries that threaten the sovereignty of some countries. In this instance, stability is of a particularly profound sort and means that no country can be eliminated from the game. The second notion concerns instabilities in the distribution of resources that do not threaten sovereignty. Morgenthau (1959), in his description of the operation of balance of power, suggests this distinction: So long as the balance of power operates successfully... [i]t creates a precarious stability in the relations between the respective nations, a stability that is always in danger of being disturbed and, therefore, is always in need of being reestablished.

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This is, however, the only stability obtainable under the assumed conditions of the power pattern. For we are here in the presence of an inevitable inner contradiction of the balance of power [B]alance of power is supposed to [establish].. .stability in the power relations among nations; yet these relations are...by their very nature subject to continuous change. They are essentially unstable. Since the weights that determine the relative position of the scales have a tendency to change continuously by growing either heavier or lighter, whatever stability the balance of power may achieve must be precarious and subject to perpetual adjustment in conformity with intervening changes (pp. 161-2).

Similarly, although not abiding by our convention of distinguishing between the concepts of equilibrium and stability, Gilpin (1981) offers this hint at the necessity for identifying two separate notions of stability: In every international system there are continual occurrences of political, economic, and technological changes that promise gains or threaten losses for one or another actor. In most cases these potential gains and losses are minor, and only incremental adjustments are necessary in order to take account of them. Such changes take place within the existing international system, producing a condition of homeostatic equilibrium. The relative stability of the system is, in fact, largely determined by its capacity to adjust to the demands of actors affected by changing political and environmental conditions. In every system, therefore, a process of disequilibrium and adjustment is constantly taking place. In the absence of large potential net benefits from change, the system continues to remain in a state of equilibrium (p. 13).

Distinguishing between two forms of stability is important not merely because our scenarios suggest it, but also because it clarifies much of the confusion in the balance-of-power literature. In our view, balance of power is neither a goal of decision makers nor a necessary property of international systems, but is instead a property that results when conditions lead decision makers to conclude that certain actions but not others are in their interests. Briefly, notice that our second scenario, as elaborated by Assumptions Al through A9, satisfy Waltz's conditions for balanceof-power politics: that "the [international] order be anarchic and that it be populated by units wishing to survive" (1979, p. 121). Given an anarchic system - a system in which no organizations or constraints on action are imposed exogenously - composed of decision makers whose principal objective is to maximize their resources, provided that their existence is not threatened, then our two notions of stability taken together define a balance of power.10 Later, we provide formal definitions in terms of some game-theoretic concepts, but for now, informal definitions of our two types of stability 10

Of the quotations that introduce this chapter, Quester clearly has system stability in mind when referring to balance of power; Burns, however, is more ambiguous, but we can read into it the necessity for considering both forms of stability.

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will suffice. First, if in fact "Whenever conflict between organized groups occurs, some question of sovereignty is involved" (Walker 1953) and if "there seems to be little disagreement.. .that a fundamental objective of creating a system of balance of power is to protect the security and independence of the participant nations" (Chatterjee 1975), then the following definition of the first type of stability - system stability - is central: System stability. The set S of countries in conjunction with the resource distribution r, where rt is strictly greater than 0 for all / in S, is system stable if no country or coalition of countries will be eliminated by some other country or coalition of countries. Notice that this definition avoids any reference to the "ability" of countries or of alliances to eliminate others in terms of specific equations or inequalities relating the resources of one set of players to that of another set. Insofar as ability is concerned, simple algebra tells us that if no country controls a majority of resources then, in any three-country system, for example, some two-country alliance controls more resources than the third. So some coalition always has the ability to eliminate someone else.11 However, until we formally link objectives, actions, and consequences, we cannot suppose that the mere fact that any two countries outweigh the third country implies that some pair of countries will in fact act to eliminate the third. We avoid specific algebraic relationships in the preceding definition, because such relationships ought to be deductions from the model as necessary and sufficient conditions for the selection of certain actions as against others, and not as theoretically primitive propositions. System stability focuses on final consequences and actions, and not on theoretical capabilities, and our second notion of stability - resource stability - takes a similar view. Later, we restate the definition of this second type of stability in terms of a particular solution concept for a cooperative ^-player game in characteristic function form called the core: the set of 11

Bull (1977, p. 113) states the situation correctly: "To speak of a complex or multiple balance among... three or four powers is not to imply that they are equal in strength. Whereas in a system dominated by two powers a situation of balance... can be achieved only if there is some rough parity of strength between the powers concerned, in a system of three or more powers balance can be achieved without a relationship of equality among the powers concerned because of the possibility of combination of the lesser against the greater." As simple as the algebra of three powers is concerned, however, its consequences have escaped universal notice. Specifically, if stability requires "a delicate balance among the relative capabilities of competing, fluctuating coalitions" (attributed to Claude 1962 by Wallace 1973), then three-country systems cannot be stable except in the unlikely event that one power controls precisely half the resources. Hence, even if it were true, as Jervis (1986, p. 60) asserts, that a balance of power requires that "there must be several actors of relatively equal power," or [as Blainey (1973, p. 110) states in his survey of the literature] that balance "now usually signifies equality and equilibrium," equality cannot ensure balance among alliances.

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feasible outcomes that no coalition has the means or the unanimous inclination to upset in favor of some other outcome.12 However, referring to the possibility that countries will not act to redistribute resources to one country at the expense of another, the following definition is adequate for now: Resource stability. The system (S; r), with r, strictly greater than 0 for all / in 5, is resource stable if for no coalition C^S will the members of C redistribute resources to some new distribution r', where r\ is strictly greater than rt for all / in C. Thus, a resource-stable system is one in which no individual country or coalition of countries has an incentive and the means to alter the current distribution of resources. There is, of course, a close relationship between both types of stability. First, notice that if (S; r) is resource stable, then it is system stable: countries can be eliminated only by transferring resources. However, the opposite need not be true. As we show later, although countries may choose to force a redistribution of resources, the corresponding system may nevertheless be system stable in that no coalition will seek a redistribution in which some subset of countries is eliminated. Second, we can distinguish one form of stability from the other by noticing that a system is system stable if no redistribution of resources yields rt = 0 for any /, whereas resource stability implies that no redistribution whatsoever occurs. Despite the simplicity of this distinction, the two notions of stability are often confused in the literature, or the possibility or the necessity for a distinction goes unnoticed. For example, Rosecrance (1986, p. 45) argues that "A power relationship determined at a prior period may be undermined if a rising state transforms its position through industrialization. Then the balance of... responsibilities and rewards.. .will no longer correspond to power realities, and the new challenger will seek a redistribution of the spoils, usually through war." What this argument fails to specify, however, is the circumstances under which the instability takes the extreme form of system instability in which survival is threatened, as against the circumstances in which a more modest redistribution results in which no country's position of sovereignty is altered. 12

If a game has a core, then that core is necessarily a subset of any von Neumann-Morgenstern solution. A core is especially attractive as a solution hypothesis because of the type of stability its definition captures - once at the core, no coalition has the unilateral incentive or means to force any other outcome. Generally, however, most n-person («>3) games of political interest do not have cores, such as all games in which the players must divide a finite, fixed amount of some commodity or resource. It is, of course, this fact that Riker (1962) uses to assert his conclusion that balance-of-power politics is necessarily unstable.

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For another example, Claude (1967, pp. 49-50) catalogues those scholars who define balance specifically in terms of the equality of power or resources. However, this equating perhaps best illustrates the confusion that our two definitions try to resolve. Is equality a necessary condition for, say, the absence of war? Is it a sufficient condition for peace? Is equality a definition of "balance" or is it the product of a balance of power and an indicator that some sort of "balance" prevails? The definitions of system and resource stability do not provide answers to such questions directly, since answers must be deduced from assumptions and from the interpretations we give words such as "negotiate" and "aggressive." Yet, the distinction between the two forms of stability permit us to avoid certain ambiguities. Because (S;r) can be system stable without being resource stable, then, unless proper distinctions are made, we are likely to confuse one sort of instability with another.13 With respect to war, for example, if (S; r) is not system stable, and if we interpret the choice of action a2 to imply the overt use of force, then some form of international conflict seems inevitable - some country will be absorbed or have its status as a great power altered. An important question, then, is: What distributions of resources imply system stability? Regardless of how we interpret various words, system stability does not mean that countries will not seek reallocations of resources by some means. Indeed, if (S;r) is system but not resource stable, reallocations are predicted. Those reallocations can occur in many ways, including the renegotiation of fishing rights and the actual transfer of territory from one nation to another. Thus, system stability does not presuppose that wars cannot occur, since it does not specify how resources might be transferred. Presumably, though, wars will take a limited form if system stability prevails, since each country, by definition, has a strategy that precludes its complete elimination from the game. Indeed, we can imagine 13

We note in passing that Organski's and Kugler's critique of balance-of-power theory (1980, pp. 14-17) rests on such a confusion. The problem is compounded by a non sequitur arising from their attempt to use a citation from Organski's (1968) work on world politics to show that "the 'balance' must represent an equal distribution of power" (p. 16). The passage in question simply asserts: "This is how the power system is supposed to work:... Given large numbers of nations with varying amounts of power each one striving to maximize its own power, there is a tendency of the entire system to be in balance. That is to say that the various nations group themselves together in such a way that no single nation or group of nations is strong enough to overwhelm the others, for its power is balanced by that of some opposing group. As long as the balance can be maintained, there is peace and the independence of small nations is maintained" (p. 274). As we see shortly, the situation Organski describes is realized under both equal and unequal distributions of power among nations; how he moves from the relative equality of potential coalitions to an equal distribution of power among national actors, we cannot fathom except insofar as he conflates system and resource stability.

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circumstances in which countries might "voluntarily" agree to a transfer of resources so as to ensure system stability, or at least their survival in a system, and one of the issues that we will study in this volume is an elucidation of the circumstance under which such transfers might occur. Resource stability, on the other hand, implies not only that the system is stable, but also that no subsequent reallocations of resources are possible. Resource stability implies the unanimous choice of alternative a5 and, certainly, the absence of wars. It remains to be seen, though, whether equality of resources is either a necessary or a sufficient condition for such stability. In any event, we should anticipate that the circumstances under which a system can be resource stable are likely to be exceedingly fragile and easily upset by changes in technology and changes in the capabilities of countries through the natural evolution of domestic markets. A number of theoretical issues arise immediately with these two definitions of stability. First and foremost, of course, is a specification of the necessary and sufficient conditions for either form of stability to prevail. Burns (1968), Kaplan (1957), and Wright (1942), for example, argue that systems with fewer states are more stable than systems with a larger number of states, whereas Waltz (1979) argues the opposite. Are large systems favored over smaller ones? The contradictory and at times confusing assertions about the relationship between stability and the distribution of resources occasion a second important question: Must resources be distributed uniformly, and, if not, how great can the asymmetry in the distribution become before instability prevails? Are there nontrivial conditions of a country's resources that guarantee its survival? What types of coalitions and alliances are we likely to see? Will the ebb and flow of events more nearly correspond to balance as against bandwagoning behavior (cf. Waltz 1979 and Walt 1987)? That is, will smaller states join the smaller alliance to keep a large alliance or country from becoming predominant, or will they seek alliance with the larger member, hoping to gain some share of the spoils? If a system is not system stable, can it become stable, and, if so, what will the configuration of countries look like? If a system is system but not resource stable, can it ever become wholly stable, and, if not, what can we say about the likely pattern of redistributions? When we allow for growth, are states with the greatest growth rate advantaged? Do coalitions that initiate a preventive war necessarily exclude the threatening country? What patterns of investment and of alliances are we likely to see when resources grow at exogenously determined rates? And when we allow for the asymmetries occasioned by geography, are peripheral or central powers advantaged? Are system and resource stability more or less difficult to achieve when all countries are contiguous or when one country is central and others are peripheral? Is it necessarily

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true, as Waltz (1979) maintains, that "the notion of a balancer is more a historical generalization than a theoretical concept" (p. 164)? That is, can we generalize the role of, say, Britain in the nineteenth century to its special geographic character relative to continental powers? 2.5

Enforcement of agreements

Seeking answers to the preceding questions is our preoccupation in the next four chapters. Before beginning, though, we should comment on the important issue of the logic of cooperative action and the mechanisms countries might use for enforcing agreements. Understanding cooperative action and the approaches game theory offers for modeling such action requires, first, that we abandon viewing simplistic games as models of political processes. For example, because international systems are anarchic, it is commonplace to believe that their processes occasion Prisoners' Dilemmas that cannot be resolved or that can be resolved only with great difficulty. Simple analogies appealing to the Prisoners' Dilemma or to a single play of any simple 2x2 game, however, are problematical. Political processes are a never-ending sequence of games, and even so simple a game as this one, if played sequentially, offers quite different strategic incentives than the dilemma played once. For example, Wagner (1983) shows how the imperatives of the dilemma are changed if we merely give players a chance to detect defections from cooperation, as well as the opportunity to respond, and if we make these facts common knowledge. Similarly, Prisoners' Dilemmas, such as those that occasion various economic market failures, pervade our existence, and their resolution with and without appeals to exogenous forces, when the stakes become sufficiently great, is not uncommon. The history of human affairs attests to the considerable degree of imagination and invention - including the invention of the state itself - that has occurred in response to the problems such dilemmas occasion. There is no reason to suppose a priori, unless other factors operate, that with the stakes as great as they are today in international affairs, invention cannot occur here as well. Merely acknowledging the complexity of political processes, however, sidesteps a complex facet of cooperation, which we can best consider in the context of a simple two-person Prisoners' Dilemma. The usual context for the dilemma is to suppose that, regardless of whatever communication occurs beforehand, each player must choose his or her strategy without knowing the strategy selected by the other, in which case the joint selection of dominant actions for all participants leads to an outcome they unanimously prefer to avoid. Alternatively, suppose this game is played in a cooperative context that allows the players a completely free hand in determining the outcome, so that the issue of whether choices must be

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made simultaneously, or whether binding contracts between the players are feasible, is wholly endogenous to the situation. At this point, we have two alternative ways in which we can proceed with our analysis: (1) model the complex game that unrestricted negotiation allows, including the mechanisms that might be employed to enforce contracts; or (2) simply ascertain what contracts we are likely to observe if binding agreements are possible, ignoring the mechanisms whereby enforcement of agreements is assured. Because modeling endogenously determined games, as well as the mechanisms players might implement, is exceedingly difficult, cooperative game theory traditionally focuses on the second mode of analysis. Hence, if confronted with a Prisoners' Dilemma, cooperative game theory, without specifying precisely how agreements are implemented, simply predicts that a Pareto-optimal outcome will prevail, as against the Pareto-inefficient outcome that results from noncooperative (simultaneous and uncoordinated) choice. In particular, if lotteries are precluded, the prediction is the outcome that obtains when both persons cooperate, because, in the terminology of game theory, this outcome is the unique element of the core. This is the approach we take in this volume. Thus, our analysis seeks to specify the agreements and outcomes that prevail if decision makers have at their disposal mechanisms for the enforcement of agreements: for committing to one alliance rather than another, and for implementing the actual tranfer of resources. (However, see footnote 8 in Chapter 3 for the outline of an extensive-form model that rationalizes our analysis as a noncooperative equilibrium and that thereby assures the endogenous enforcement of the agreements our analysis otherwise circumvents.) Questions nevertheless remain as to whether we can design such mechanisms, whether they are likely to arise "naturally" in anarchic systems, and whether feasible mechanisms admit the sorts of agreements that are part of system-stable systems. There are at present two seemingly distinct, but in fact closely related approaches to this issue, both of which take cognizance of the fact that any such mechanism must correspond to a noncooperative game (in order that we not solve the problem of cooperation by assuming it away) and in which the actions sufficient to yield the prespecified cooperative outcomes are dominant choices or, minimally, equilibrium choices.14 The 14

There is a third approach, introduced in population genetics and recently popularized in political science (Axelrod, 1984). Using the notion of evolutionary stable strategies (ESS) - strategies that cannot be "invaded" by mutant strategies (see, especially, Maynard-Smith 1974) - the approach here is to establish various strategies that yield cooperative outcomes as ESS. One difficulty with this approach, however, is illustrated by the fact that the commonly cited source of cooperation in repeated Prisoners' Dilemmas, tit-for-tat, is not ESS (Boyd and Lorderbaum 1987).

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first approach seeks to identify the specific institutional structures that guide and constrain individual choices and that are thereby sufficient to yield the desired outcomes as the consequence of dominant or equilibrium choices. Although research to date has focused on mechanisms that use some type of taxation device for solving a market failure, there is no reason to suppose that this approach cannot have a broader application. The theoretical difficulty with implementation of cooperation in anarchic international systems by relying only on institutional design, however, is that institutions survive only if they serve the purposes of those whose actions they seek to regulate. If they are to serve the purpose of inducing cooperation then, excepting the existence of some exogenous "higher" authority that enforces their rules, they themselves must ultimately be viewed as the product of noncooperative choices in anarchic systems. To reiterate a point made earlier, if "the equilibrium of a balance is maintained through states abiding by rules, then one needs an explanation of how agreement on the rules is achieved and maintained" (Waltz 1979, p. 121). This view compels us, then, to regard institutions more as facilitating agents, and less as the fundamental source of cooperation or the primary explanation for cooperation.15 The second approach takes cognizance of the fact that the "games" nations and national leaders play entail sequences of moves - indeed, even repetitions of the same game - and it explores the circumstances under which such repetition solves the problem of cooperation that recourse to institutional design alone cannot solve. In particular, what repetition allows is the possibility that players can retaliate subsequent to an "uncooperative" choice, and that retaliation renders cooperative action an equilibrium. For example, one mechanism for inducing such action in the Prisoners' Dilemma is to require that the two participants play the game repeatedly, without knowing which game in the sequence will be the last. In this instance, a Pareto-efficient, cooperative outcome (via the joint selection of a strategy such as tit for tat) corresponds to an equilibrium (Taylor 1976). Thus, it is tempting to assert that this view of politics as a repeated game rationalizes the existence and durability of fundamental political institutions such as constitutions, and indeed this hypothesis about the source of cooperation and the enforcement of contracts is especially appealing in international systems.16 We can discern, for example, the incentives to abide by one's treaty obligations in the fact that by pursuing such a strategy, one secures a reputation that becomes an asset in 15

16

A debate of the issues surrounding this view is offered, for example, in Shepsle and Weingast's (1984) response to McKelvey and Ordeshook (1984). For a discussion of these issues in the context of international relations theory, see Snyder and Diesing (1977), Axelrod and Keohane (1985), and Oye (1986). The citations in Oye's essay, in particular, are a good starting point for review of the relevant literature.

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subsequent negotiations - in subsequent repetitions of the "international politics game." Similarly, institutions themselves are rendered viable because they become part of the technology for playing a repeated game. At this point, however, game theory poses an especially perplexing problem. Regardless of the particular definition adopted, infinitely repeated games generally possess a great many equilibria, and a variety of "folk theorems" establish that nearly any outcome, including Pareto-inefficient ones, can be sustained by some equilibrium of strategies.17 In the infinitely repeated Prisoners' Dilemma, for example, both players always defecting remains an equilibrium: anyone playing a strategy that cooperates some of the time against someone who always defects would do better by defecting all of the time as well. This multiplicity of equilibria occasions two difficulties. First, it leaves unanswered the question of why Paretoefficient (cooperative) as against -inefficient outcomes prevail. What precisely is the process that allows players to implicitly coordinate in order to achieve efficient (cooperative) as against inefficient (noncooperative) equilibria? Second, if (#, b) and (c, d) are both equilibrium strategy pairs then it is not necessarily the case (unless the game is two-person and zerosum) that (#, d) or (c, b) are equilibria, in which case we must still ascertain how players implicitly coordinate to achieve any equilibrium in the fundamental noncooperative game we employ to explain cooperation. There is a second problem. Our analysis assumes not only perfect information (and thus the absence of uncertainty), but also complete information (and thus common knowledge of all aspects of the game). However, games of incomplete and asymmetric information, like their infinitely repeated counterparts, also typically possess multiple equilibria, and the nature of these equilibria is sensitive not only to the initial beliefs of participants but also to the institutional details governing the sequential revelation of information. Even seemingly innocuous features of a situation, such as a nonbinding straw poll taken prior to a formal vote, can profoundly affect outcomes. Unfortunately, we have only just begun to understand the analysis of such games, and we are no better here at resolving the problems occasioned by multiple equilibria than we are in repeated games in general. 17

See, for example, Fudenberg and Maskin (1986) and the literature they cite. To appreciate the consternation that multiplicity of equilibria occasion, see also Harsanyi and Selten (1988). One solution to this problem, which we adopt in the model outlined in footnote 8 of Chapter 3, is to assume that strategies are stationary - that players make the same choices whenever they find themselves in identical circumstances, regardless of whatever events have transpired (assuming that those events do not alter beliefs or capabilities). Such an assumption greatly facilitates analysis and reduces the number of equilibria with which we must contend. However, aside from appeals to cognitive capabilities, stationarity is itself an ad hoc hypothesis.

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Thus, whether through an explicit institutional structure, through the strategic imperatives of repeated play, or through combination of the two, ensuring cooperative action and the enforcement of agreements is not simple. It is probably the case, of course, that by looking for the sources of cooperation in systems that are anarchic "in the extreme," we have pushed the problem to too fundamental a level: no event or negotiation occurs in an environment in which there does not exist some institutional structure and some historically based expectations about what others will do. Although the meaning of words such as "culture" and "tradition" await definition and operationalization in these formal analyses, it is perhaps here, in the context of the issue of equilibrium selection, that we see a role for such notions. In any event, it is the recognition of problems of this sort that helps explain the difficulty nations confront in resolving the details of agreements, which, although appearing inconsequential when compared to the scope of the general issues under consideration, can nevertheless undermine their ability to ensure that abiding by the agreement is in the complete interest of both persons. And to the extent that countries cannot implement agreements in a timely fashion, conflict can occur even if all of our assumptions are otherwise satisfied. This fact not only makes the issue of implementation and the study of the sources of cooperative action a primary concern, it also provides, if not an argument against the theory this volume offers, then an argument for requiring that we regard the conditions we subsequently establish for system and resource stability as merely necessary but not sufficient. Our analysis assumes the feasibility of implementing agreements, but if technology or the details of international processes preclude this, then self-interest may imply instability. There is, then, no obligatory conflict between realist arguments and those regarding institutions as essential to the maintenance of stability and peace. On the one hand, we must accept the fact that, unless there exist mechanisms for the implementation and enforcement of agreements, the forces for stability described in this volume can be ineffectual. On the other hand, mechanisms alone cannot ensure stability unless there are incentives for their maintenance and their use, incentives such as the ones this volume seeks to describe in terms of necessary and sufficient conditions for stability. Thus, it is prudent to maintain the hypothesis that Nonhegemonic cooperation is difficult, since it must take place among independent states that are motivated more by their own conceptions of self-interest than by a devotion to the common good The importance of regimes for cooperation supports the Institutionalist claim... that international institutions help to realize common interests in world politics. An argument for this view has been made here... by relying on Realist assumptions that states are egoistic, rational

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actors operating on the basis of their own self interest. Institutions are necessary, even on these restrictive premises, in order to achieve state purposes Realism should not be discarded, since its insights are fundamental to an understanding of world politics... but it does need to be reformulated to reflect the impact of information-gathering institutions on state behavior, even when rational egoism persists (Keohane 1984, pp. 245-6).

2.6

Theoretical domain

The preceding discussion helps identify the context and limitations of our analysis. Within this context, Assumptions Al through A9 define our theory's domain: the class of phenomena to which it can be legitimately applied. Not all assumptions, however, perform this task equally well. The assumption that decision makers are rational merely places our theory in a particular paradigm - in this instance, a paradigm that is not intended to study only part of social reality but all of it. Other assumptions, such as that information is complete, are intended to facilitate initial formulations, and we can only hope that they will be weakened if the theory appears promising. However, our assumptions about resources, Al, A3, A4, and A5, together define the domain of our theory, if not all balanceof-power theories. Assumption A4 requires that national leaders maximize /•/, which to this point we have ambiguously referred to as country /'s national resources, whereas A5 requires that whatever rt measures also determines a state's capability of securing rt in conflict, its military capability. Furthermore, A4 requires that all countries share a common maximand, and A5 requires that the capability of countries is also measured by a common variable. Finally, Al and A3 suppose that whatever r measures is transferable across countries on a one-to-one basis and in finite supply. The consequence of these assumptions, then, is that our theory concerns those international systems in which readily identifiable sovereign entities (countries) maximize the same 'thing,9 which is in limited but transferable supply, and which translates directly into a measure of a country's capacity for securing its ends and maintaining its security. It is tempting in this context to relabel rt as country /'s power and to assert that countries maximize power. After all, whatever power is, there is likely to be some consensus that it is relative and therefore must be in finite supply at least to the extent that an increase in one person's power decreases someone else's; and assuming that decision makers maximize power seems only reasonable if the alternative is to suppose that they minimize it or deem it irrelevant to their goals. In theories of rational choice, however, the concept of power is afictionas a theoretically primitive idea it is at best a derivative notion (e.g., market power in microeconomics) -

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and relabeling rt as /'s power merely replaces notation with a theoretically undefined word. In any event, such a relabeling takes us no closer to an operational definition of rt than does the notation itself, and theories of politics that pretend otherwise are doomed to ambiguity. If, on the other hand, we operationalize resources less ambiguously, as Organski and Kugler (1980) do by basing their analysis on measures of GNP or GNP per capita, the empirical limitations of the consequence of our three assumptions become more apparent. Assuming that all countries maximize some measure of their per-capita wealth is not an unreasonable basis for theorizing about international politics. Such an assumption is not inconsistent with a variety of views about international politics ranging from a belief that private entrepreneurs in search of profits direct the destinies of states to the hypothesis that national leaders are motivated simply by the median welfare of their populations. However, to assert that GNP also translates directly into an ability to wrest such resources from another sovereign state (i.e., a military capability) is problematical. Organski and Kugler argue that GNP and military capability are closely related - that GNP correlates highly with the measures of capability that others offer - but clearly such a relationship is at best imprecise. Even if we could convert every component of GNP into wealth with which we might purchase military capability, and even if every aspect of military capability had a monetary antecedent, time lags in the conversion of one resource into the other would render the correspondence imperfect. Evaluating any particular operationalization of the thing we call resources, then, reveals that even if we sidestep the social preference problem defined by Arrow, the things countries maximize may not be immediately convertible into a capacity for resisting aggression or for otherwise securing those ends. Thus, it is also tempting to view the consequence of Al, A3, A4, and A5 as a limitation of the theory that some theoretical refinement ought to rectify. Whether the theory requires rectification, however, is a matter of judgment based on an empirical assessment of the limits of the theory's domain relative to the variety of domains offered by nature. Clearly, some restriction on our theory is required. For example, suppose national leaders and the populations they represent maximize "spiritual contentment" as measured by the logical consistency of a common religious dogma. Because the means to such an end will not set nations in conflict - indeed, it may render the issue of sovereignty itself moot - the theorems we offer in subsequent chapters will be irrelevant to an understanding of international relations in such a system. If only international systems of this type exist, then our theory would be little more than an exercise in logical thinking. Alternatively, if a national leader seeks to maximize colonial territory and if the ability to secure and defend that

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territory and one's sovereignty depends on a wholly unrelated military capability, then once again our theory will prove irrelevant (although we suspect that such a leader will enjoy a brief tenure). To the extent, though, that territory and military capability correlate, then the relevance of our theory increases and we should begin to detect the operation of the forces our theory describes. And if that correlation becomes 1, then the forces our theory describes should be the only forces that operate. The particular difficulty with which we must contend, then, is the possibility that this correlation is low and that the discrepancies between the two aspects of resources - goals and means - occasion forces that render the balance of power an insignificant or otherwise minor consideration in the relations of states. We ought not to try to resolve this empirical issue, however, without reference to the empirical verisimilitude of the theory's implications. Perceptions about motives can be deceptive, and the relationship between means and ends - between, say, a country's economic status, the living standard of its inhabitants, and its ability to influence (if not militarily overcome) other countries - may be obscure even to astute observers; witness the contemporary debate over the implications of deficit spending and balance of payments. Detailed diplomatic histories, for example, might provide valuable insights into the motives and perceptions of key decision makers, but we cannot take such histories as definitive: if events (e.g., the pattern of alliances, the outbreak of war, and changes in sovereignty) corroborate the theory, but if motives as revealed by public utterances, private correspondences, and public and secret treaties are distinct from the motives required to apply the theory, then we ought to suspect the interpretation of those histories. On the other hand, if events fail to corroborate the theory's predictions, then we should use those histories (as well as any other data at our disposal) to infer the motives of decision makers and to guide the development of an appropriate theory.

CHAPTER 3

System stability and the balance of power

The balance of power is a system designed to maintain a continuous conviction in every state that if it attempted aggression, it would encounter an invincible combination of the others. Quincy Wright, A Study of War (1942, p. 254)

The aspiration of power on the part of several nations, each trying either to maintain or overthrow the status quo, leads necessarily to a configuration that is called the balance of power and to policies that aim at preserving it The balance of power and policies aimed at its preservation are not only inevitable but are an essential stabilizing factor in a society of sovereign nations. Hans Morgenthau, Politics Among Nations (1973, p. 161)

In this chapter and the next, we analyze the consequences of the assumptions offered in Chapter 2 by ascertaining the conditions under which both system and resource stability prevail in ^-country systems. In doing so, we are able to resolve much of the confusion between assumptions and conclusions found in earlier attempts to theorize about balance of power. And although our model differs significantly in form from the verbal models that others before us formulate, our principal conclusion accords with the scholarly intuition underlying balance-of-power theories: stability of both sorts is possible. We contest Riker's (1962) hypothesis, then, that international systems are characterized by the instabilities of cooperative zero-sum games. Self-interest, we show, does not lead necessarily to the formation of minimal winning coalitions that expropriate all from the losers. These conclusions suggest that wars, especially those that yield significant changes in great-power resources, are occasioned by matters we shall consider in Chapters 5 and 6 (differential growth rates of resources and the asymmetries occasioned by geography) or by the "frictions" we have not explicitly operationalized as part of our analysis (such as those arising from the constraints imposed by domestic politics, or because of the technological infeasibility of actually transferring resources from one country to another). 75

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3.1

Some immediate consequences of our assumptions

Although a number of other assumptions might be employed to constrain and define an international system, Al to A9 not only yield a set of consequences that are consonant with much of the scholarly thinking about balance of power, but also many of these consequences often have been taken as theoretically primitive assumptions in other balance-of-power theories or as descriptions of systems "in or out of balance." We begin with one consequence, which, although a trivial implication of our assumptions, is nevertheless an important component of our analysis: • If any single country controls a majority of the resources available to all countries in the system, then that country will eliminate all others and expropriate all the available resources. Formally: Remark 3.1. If rt>R/2 for some / in S, then / will eliminate all others, and ({/};/?) prevails as the final outcome. This consequence is obvious, given the assumptions, and it asserts that any state securing hegemony becomes a dictator.1 This remark, however, once again underscores the "knife-edged" nature of our assumptions. Because our model does not consider uncertainty and because it, of necessity, supposes that imprecise concepts can be represented precisely, the results depend on the specific values of parameters. That is, if a country secures more than exactly half the resources, it becomes dominant, with certainty. In reality, of course, and as we noted earlier in our discussion of uncertainty, we can never say that a particular dollar added to the GNP or a resource unit added to some index renders a country dominant and capable of eliminating the sovereignty of the rest. We are especially interested in learning, then, whether the conditions for stability in international systems require certainty or uncertainty. If these conditions are similarly knife-edged, then we can infer that stability, if it exists at all in reality, can exist only if constraints exogenous to our analysis impinge on international affairs or if the statistical uncertainty associated with the measurement of resources and military capabilities, coupled with assumptions about attitudes toward risk of key decision makers, preclude the 1

The literature offers a variety of definitions of a hegemon (Aron 1973, Keohane and Nye 1977, Gilpin 1981, Modelski 1983, and Keohane 1984), but throughout this text, a hegemon or (equivalently) a predominant country simply refers to a country that controls more than half of the system's resources. A near-predominant country is one that controls precisely half the resources. In Chapter 6, we generalize these definitions to accommodate the discounting of resources occasioned by geography.

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choice of aggressive acts. However, if these conditions are not knife-edged (which they are not), then our confidence that such stability can prevail in reality is increased. Aside from this issue, the maxim embodied in Remark 3.1 nevertheless retains a considerable degree of empirical plausibility: namely, that states (or, more properly, their policy makers) are sensitive to resource thresholds that enable or constrain potential opponents from achieving hegemony. The resources in question may take the form of political prestige and military might, or, more straightforwardly, economic forms. In the former case, and although the example refers to an alliance rather than a single country, we find it difficult to explain the vigor with which the Quay d'Orsai sought in the early 1890s to dislodge Czarist Russia from its alliance with Wilhelmine Germany except in terms of the strategic perception that the resources amassed by a German/Austro-Hungarian/Russian coalition would establish it as hegemonic on the European continent, and thereby threaten not only the ability of France to amass resources from colonial possessions and protectorates in the Near East and South Asia, but its eventual sovereignty as well. Indeed, the same concerns preoccupy the Triple Entente with respect to potential Germanic hegemony over the Balkans and the Straits. The diplomatic archives are replete with documentation of this shared perception that the perseverance of this alliance would permit its members to breach some resource threshold and achieve eventual hegemony. The classic statement of this perception is Sir Eyre Crowe's 1907 memorandum in which he observed: History shows that the danger threatening the independence of this or that nation has generally arisen, at least in part, out of the monetary predominance of a neighboring State at once militarily powerful, economically efficient, and ambitious to extend its frontiers or spread its influence, the danger being directly proportionate to the degree of its power and efficiency, and to the spontaneity or "inevitableness" of its ambitions. The only check on the abuse of political predominance derived from such a position has always consisted in the opposition of an equally formidable rival, or of a combination of several countries forming leagues of defence. The equilibrium established by such a grouping of forces is technically known as the balance of power, and it has become almost an historical truism to identify England's secular policy with the maintenance of this balance by throwing her weight in this scale and now in that, but ever on the side opposed to the political dictatorship of the strongest single State or group at a given time (Gooch and Temperley 1926, pp. iii, 403). Note also a little later in the same document where he suggests as an explanation of Germany's weltpolitik that Germany is definitely aiming at a general political hegemony and maritime ascendancy, threatening the independence of her neighbors and ultimately the existence of England... (Gooch and Temperley 1926, pp. iii, 417; emphasis added).

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Similar views, albeit focused on Russia and Great Britain, were current in German military and diplomatic circles (Conrad 1922, pp. iii, 597; and Fischer 1961, pp. 59ff). In another case, wefindconsiderable evidence that Imperial Japanese planners from the late 1920s onward perceived United States policy in Asia and the Pacific as aimed at diminishing Japanese access to economic resources - particularly vital raw materials - so as to prevent the resource threshold necessary for the Asian hegemony Japan envisioned in its Greater East Asian Co-Prosperity Sphere. Although Remark 3.1 may state the maxim with a "knife-edged" precision that only imperfectly obtains in empirical cases, it merely renders explicit an intuition upon which statecraft has been predicated since the emergence of states. Indeed, if one understands the "elimination" of a nation to consist not in its destruction and the extermination of its people, but rather in deprivation of its freedom of political, military, and economic action, in its subsumption by the political dictatorship of another state, the vassal or satellite of which it becomes while technically retaining national existence, then Remark 3.1 - even with its knife-edged precision - captures an essential element of both the strategic thinking of statesmen and the theorizing of balance-of-power theorists. Another consequence of Remark 3.1 bears directly on the debate over the greater stability of bipolar versus multipolar systems, since it asserts that the only system- and resource-stable two-country (bipolar) system is one in which resources are exactly evenly divided. Again, the empirical meaning of such a consequence must be tempered by those features of reality we do not consider. It nevertheless would follow straightforwardly from almost any reasonable probabilistic adaptation of our model that "as the expected values of the resources of two countries in a two-country system become equal, the probability that one country will succeed in any attempt to become a hegemon will decrease." For example, suppose rx and r2 are both random variables, so that we must speak of the probability that one exceeds the other and vice versa. Notice, however, that as random variables, they have two interpretations. First, we might be referring to the decision maker's uncertainty about parameter values; second, we might be referring to our (the analyst's) uncertainty as to what a decision maker will believe about parameter values. This is an important distinction. Consider the first possibility, and let p denote the probability that /*! exceeds r2, which is a probability we can deduce directly from the presumed joint distribution of rx and r2. If p also measures a decision maker's perception of the probability with which an attack by 1 on 2 will be successful, then 1 need only perform the appropriate expected utility calculation in deciding what to do. That is, attack if pu + (l— p)u'> w",

3.1 Some immediate consequences of our assumptions

79

f

where u is the utility of success, u is the utility of failure, and u" is the utility of the status quo; otherwise do not attack. Here again, the analysis will be knife-edged in that if the expected utility of an attack exceeds u" by however small an amount, then we will predict an attack; otherwise not. Still, even if reality is knife-edged, we are not likely to be able to measure the parameters of decision criteria with perfect accuracy, in which case the uncertainty is ours as analysts. In this event, we can only utter predictions of the form "an attack by 1 on 2 is more likely as rx increases relative to r2."2 The implications of Remark 3.1 should affect profoundly the decisions of national leaders when they choose whether or not to form alliances. If, in an /7-country system, the elimination of (say) country j leads to the eventual elimination of country / in subsequent plays of any of the n-1, n-2, « - 3 , etc., country games, then / will act to block y's elimination (if such action is feasible). Hence: • If the resources of each country in an international system are less than or equal to half the resources available to all countries in the system, no country will act to increase the resources of any other to a majority of the available resources. That is: Remark 3.2. If r^R/2 for all /, no country will act to increase the resources of another country in excess of R/2. By permitting another country to control over half the resources, each country knows that the only stable system and stable resource distribution that can prevail ultimately is the unique outcome in which the dominant country eliminates all others. Thus, Remark 3.2 helps "fill in" each country's preference order over alternative outcomes this way: / strictly prefers any 77-tuple of resources r in which rt > 0 and r7 < R/2 for all jeS to any ^-tuple r' in which rj>R/2, j^i; that is, every country prefers a resource distribution that prevents the preponderance of any other. 2

Remark 3.1 also reveals one possibility that our model does not incorporate: the breakup of sovereign entities into two or more independent decision makers, that is, civil wars. Clearly, we could use our analysis to model internal wars simply by reconceptualizing the elements of S as, say, the significant factions in a country. Throughout our analysis, though, we suppose that once / "absorbs" j , i can be forced to cede resources to another country only by considerations that are external to it. Further, once the number of countries reduces to n, it cannot increase later to n +1 or more owing to the division of a country. If such a division occurs, we must assume that it does so because of circumstances endogenous to that country. We could certainly analyze the resulting (n — l)-country system (assuming that our results are general and do not depend on the number of countries in a system), but we cannot predict its emergence.

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Our third remark takes us closer to establishing a necessary and sufficient condition for system stability: Remark 3.3. If rt = R/2 for some country / in S (if some country is near-predominant), then no member of S—{/} will attack another member of S— {/}. If CcS—[i] attacks some C'CS— {/}, then, by Assumption A7, / will attack C and absorb all of its resources, in which case, by Remark 3.1, the final outcome will be ({/};/?). That is, aggressive choices among the members of S— [i] create opportunities for the near-predominant country, /, to become predominant. Remark 3.3 demonsrates the critical nature of Assumption A7 for our analysis. With it, countries must guard against the possibility that third parties can take advantage of "localized" conflicts. This has the important consequence of keeping such conflicts from occurring, especially if a third country is close to becoming hegemonic. Without this assumption, on the other hand, we might anticipate that even if r; = R/2 for some /, that some members of S— [i] could attack other members of this set. Yet, this could only occur if all the members of S— {/} agree to "terminate hostilities" the minute / attacks one of them, and it is indeed difficult to envision such an agreement among warring parties. We can find no such case of prior agreement to terminate hostilities in the eventuality of attack by a third party in recorded history; however, some examples of such agreements after a third party's attack can be found - for example, Constantius' agreement to Sassanid Persia's peace terms to permit redeployment of the Eastern Roman armies against the then-usurper Julian's eastward march on Constantinople from Gaul in A.D. 361. However, even here, despite the Persians' clear preference for the relatively ineffectual Constantius as opposed to the more militarily gifted Julian as Roman Emperor, the Sassanid Shah imposed draconian terms - terms that undermined Constantius' support among his generals - on the very party whose victory in the battle for the throne he sought to ensure. This example suggests that the existence of sufficient self-interest to impel countries into war militates against sufficient prescience to prepare before the fact for peace so as to face other threatening contingencies. Nevertheless, such agreements may be theoretically possible, and, as such, they lead to a violation of our assumption, in which case Remark 3.3 is no longer valid. Nonetheless, we find little reason to dispute the general plausibility of our assertion that states with sufficient self-interest to form coalitions against other (potentially hegemonic) states will be unlikely to wage war within that coalition for fear of enabling their common enemy's predominance.

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Remark 3.4. If r; = R/2, then / has no means for increasing its resources. Country / can increase its resources in only two ways (keeping in mind that our assumptions do not yet permit differential internal resource growth across countries): by attacking or threatening to attack a weaker country or coalition, or by forming a coalition with other countries to attack countries not in the coalition. By Remark 3.3, no country in S— [i] will attack another country in S— {/}. Thus, all countries in S— [i] are free to attack / either individually or collectively, but if / attacks some country or coalition and wins, it becomes dominant. Thus, all members of S— [/}, whose total resources necessarily equal rh have action a4 as a dominant choice, which in this instance is "attack / if / attacks anyone in S— [/}." Hence, / cannot increase its resources by attacking. Similarly, since no country has an incentive to relinquish any resources to / as a consequence of any coalition, / knows that no coalition is possible in which it can secure additional resources. An important consequence of this discussion, then, is this: Remark 3.5. If r( = R/2 for some / in S, then (S;r) is system stable and resource stable. From Remark 3.3, no member of S— [i] will attack another member of S— {/}; from Remark 3.4, / has no means of increasing its resources. Thus, no country's resources will be redistributed to other countries and no country will be eliminated. Remark 3.5, then, provides a sufficient condition for system and resource stability. Notice also that if we keep Remarks 3.3 and 3.4 in mind, then Remark 3.5 summarizes the intuition that a balance-of-power system protects the survival of small states (see Morgenthau 1973 and Hertz 1959). In Remark 3.3, such states are protected from larger but not predominant ones by the threat that warfare poses to all but the largest state, whereas in Remark 3.4, smaller states are protected from attack by a predominant one because no one can permit the largest state to become predominant. 3 Herein, then, is the basis for Crowe's observation on the July 1914 crisis that Our interests are tied up with those of France and Russia in this struggle, which is not for the possession of Serbia, but one between Germany aiming at a political dictatorship in Europe and the Powers who desire to retain individual freedom (Gooch and Temperley 1926, pp. iv, 101). Herein also is the basis for Morgenthau's (1973, p. 167) insistence that 3

Following both Morgenthau and Herz, we are tacitly assuming here that smaller states are less resource-abundant than larger states. Empirical reality may violate this assumption under a number of conditions, including differential distribution of technologies over states.

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the international balance of power is only a particular manifestation of a general social principle to which all societies composed of a number of autonomous units owe the autonomy of their component parts; that the balance of power and policies aimed at its preservation are not only inevitable but are an essential stabilizing factor in a society of sovereign nations...

and for Waltz's characterization of the balance of power as a "self-help system" (1979, p. 118). Remarks 3.3, 3.4, and 3.5, then, reveal a profoundly important feature of the balance of power, which students of international affairs such as Morgenthau and Waltz appreciate; namely, the balance of power is not merely the equating of one combination of resources against another combination. Rather, the balance of power necessarily embodies, implicitly or explicitly, a nexus of "mutual security" relationships among states, enforced by the understanding that hegemonic dominance threatens more than a single state. To reiterate an earlier point, notice that Remark 3.5 depends critically on Assumption A7 (by way of its dependence on Remark 3.3). If countries can somehow agree to localize their conflicts with the guarantee that they will jointly move against the intervention of external parties, then the condition that rt = R/2 is not sufficient to ensure either system or resource stability. However, with such a guarantee, the countries party to it simply become a new subsystem, and Remarks 3.1 through 3.5 apply to this subsystem. This leads to an interesting speculative question: Is (5; r) system stable whenever all of its subsystems are system stable? And, conversely, if all subsystems are system stable, is (S;r) system stable? The hint that such relationships need not exist, but that any such relationship is important, is offered by Bull (1977, pp. 116-17) in his summary of the meaning of balance of power: First, the general balance of power serves to prevent the system of states from being transformed by conquest into a universal empire Second, local balances of power - where they exist - serve to protect the independence of states in particular areas from absorption or domination by a locally preponderant power Third, both the general balance of power, and such local balances as exist at present, help to provide the conditions in which other institutions on which international order depends are able to operate.

We cannot answer our two queries until we identify the necessary and sufficient conditions for system stability. However, notice that all subsystems are system stable if and only if every two-country subsystem is system stable. It follows from Remark 3.1, then, that every subsystem can be system stable only if resources are uniformly distributed across all n countries in S. And, as we see shortly, this is sufficient to ensure that (S; r) is itself system stable. The converse of this proposition, however, is not true. Suppose in particular that 5= {1,2,3} and that rx = R/2>r2>r3.

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Remark 3.5 tells us that the system is system and resource stable. However, subsystem ([2,3 j ; (r2, r3)) is not stable since 2 is predominant in it. Our next remark, which follows directly from the preceding remarks, moves us closer to a condition that is both necessary and sufficient for system stability: Remark 3.6. If rt Tj and rt + rj > R/2, then one or more of the remaining countries should coalesce withy to defend j ; otherwise / will dominate all countries. On the one hand, since /*;R/2 and rt r(C'k), the resources of the members of coalition Cj are greater than those of the members of coalition Cjc; iii. rj = 0 for all / e C'k, the resource allocation for all countries in coalition Ck will be zero if the threat is not countered; iv. r'j > rj for all j e Cj, the resource allocation for each country in coalition Cj will be greater if the threat is carried out. We say that (r',C) is a threat against / in particular if / is a member of the threatened set of countries, Ck. Surveying this definition, condition (i) requires that Cj and Ck both be disjoint coalitions in the coalition structure C . This is only reasonable, of course, since if i attacks j9 we can hardly say that / and j have coalesced to coordinate their strategies. Admittedly, though, this condition ignores the sort of complexity that characterizes the role of different states in many conflicts, as illustrated by the United States' role in mediating between the disputes of its allies or those it seeks as allies (e.g., Britain versus Argentina, Israel versus Egypt).

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Condition (ii) is borrowed from the presumption that countries will attack others only if they anticipate the possibility of victory, hence the requirement that the resources of Cj exceed the resources of C'k. Condition (iii) requires that a threat is a proposal to eliminate attacked countries. Generally, of course, threats are aimed merely at securing some portion of a country's resources, but as we have just noted, this extreme form of a threat is part and parcel of our current focus on system stability (as against resource stability, in which threats can have more limited aims). Finally, condition (iv) states that the members of Cj will coalesce to attack others only if, individually, each anticipates some immediate gain in terms of increased resources from such an act. The threat, in other words, must be profitable for all participating countries. Clearly, only near-predominant or predominant countries - countries controlling half or more of the resources in a system - are immune to threats. Thus, system stability prevails only if all countries also possess counterthreats - threats against one or more of those who are making the original threat or a proposal that induces some of those countries into joining a new coalition. This view, then, merely reiterates the previously cited observation of Waltz (1979, p. 102) that all states must at all times be prepared to use force, if only to counter the ambitions of others. Of course, just as threats need not be explicit hostile acts, counterthreats can similarly exist merely in the minds of decision makers: Counterthreat. (r",C") is a counterthreat to (r',C) by the collection of countries K^ CfkDC^ if i. either Cj g C£ or Cj DCj^^Q, where C^ and C"h are both elements of the coalition structure C"; ii. (r", C") is a threat to C"h by C£; iii. r'( > r\ for all i in C£, and r'( > rt for all / in K if C"h * 0. Thus, a counterthreat by the collection K is, according to conditions (i) and (ii), a proposal in which K is in both Cjc (the coalition which is being attacked) and C^ (the coalition that is formulating the counter), as well as a proposal that either threatens all the members of Cj (the originally threatening coalition) or coopts one or more members of Cj. In addition, condition (iii) requires that all countries in the countercoalition C^ prefer the counter to the original threat. Notice now that condition (iii) is the feature of counterthreats that differs from the notion of a counter offered in bargaining set theory. Because bargaining set theory asks whether a particular distribution is "stable," those making the counter must prefer it to the original proposal. Here, though, we allow two types of counters. First, one or more countries can transfer resources directly to another so as to render it near-predominant

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(in which case C# is empty). Such a counter reduces the threatened players' resources below what they have in r; if successful, such a ploy, in accord with Remarks 3.3 to 3.6, "freezes" the system and thereby secures their sovereignty. We justify this modification of the original idea offered by Aumann and Maschler (1964) because we are merely interested in whether countries can counter to ensure their existence. The second type of counter allows one or more of the threatened countries to formulate an alternative threat directed at members of the originally threatening coalition. However, because the new threat does not freeze the system, these members must be concerned that their resources will be whittled away in repeated action; so, in this instance, we require that members of the new threat who were targets of the old threat defend, as in the bargaining set, their original payoffs. Counterthreats are available to nearly all players for the game we are presently considering. Yet, for a smaller country, say /, counters may require the assistance of others and we must be certain that these other countries have some incentive to join / - that they have some incentive, in Quester's words cited earlier, "to ensure that... a loser was not eliminated from the system and absorbed..." (1977, p. 64). Thus, we require an additional refinement, which we call a viable counterthreat.6 Viable Counterthreat. The counterthreat (r", C") is viable for / in K if there is no subset Co of C^ that excludes / (i.e., no Co rj.

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component of stability in international politics, and formally incorporates the notion of collective security as a central component of our analysis. We emphasize that our introduction of the notions of threat and counterthreat admits of an important dynamic in balance-of-power systems. As will become clear from our application of the model to the case of the European great powers, 1871-1914, in Chapters 7 and 8, it is not always the formal existence of alliances that drives the balance of power or maintains system stability, but the potential alliances embodied in threats and counterthreats. For example, from 1873 to 1892, the actual alliances observable within the system were: Austria-Hungary, Germany, and Russia (1873-87); Austria-Hungary and Germany (1879-1918); Austria-Hungary, Italy, and Germany (1882-1914); and Germany and Russia (1887-90). However, the real balancing dynamic from the early 1870s onward was between the potential threat of a Russo-French alliance (which did not actually materialize until 1892-4) and the potential counterthreat of an Austro-Italo-German alliance (which was not actually consummated until 1882). Indeed, the potential threat of the Russo-French alliance did not become actual until Russia was left with no better option than the alliance with France. The particular hypothesis we offer now about the properties of outcomes to our game is the following: A10. Country / in S will not be eliminated if and only if it possesses a viable counterthreat to every threat. Implicit in this assumption, of course, is the supposition that if / is vulnerable - if there exists some threat it cannot counter - then that fact will be known to all other participants whether or not the threat is explicitly made, and they will take advantage of the situation to eliminate i.1 Thus, our next remark follows immediately from Assumption A10: Remark 3.7. (5; r) is system stable if and only if, for all i in S and for every threat against /, / has a viable counterthreat. The coalition Cj might pose a threat to C'k because it outweighs C'k, so threatened countries must devise viable counterthreats in order to survive. The definition of a counterthreat is weak and excludes few countries from 7

By basing our formulation of balance of power on the notion of threats and counterthreats, it might seem that our analysis conflicts with Walt (1987), who argues for a distinction between a balance of power and a balance of threat. Walt, however, is primarily interested in explaining observed alliances and thus, consistent with our assumptions, he asserts that alliances will necessarily be responses to perceived or actual threats. Thus, his analysis is compatible with ours. Because we are concerned with the general issue of survival, we do not want to place any ad hoc constraints on potential threats, and thus we assume that survival is ensured if and only if all threats can be effectively countered.

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having one in any specific contingency. Most of the work is done by the notion of a viable counter. Briefly, a counter is viable for / in K if /'s partners in the counter have some incentive to coalesce with /, if C^ — {/} cannot also threaten CjJ (or C# joined by /) so that they prefer this counterthreat to the counter that includes /.8 8

We prefer deducing A10 and its primary implication (Theorem 3.2) from more primitive assumptions, since we can often rationalize cooperative predictions as noncooperative equilibria (Roth 1985, Binmore and Dasgupta 1987, Harsanyi and Selten 1988). To see how we can do this in the present context, let nature choose country / to make a threat (failing to threaten or the refusal of /'s partners to unanimously approve the threat leads to the selection of another player). If /'s threat of r' by C against C is approved by C— {/], nature chooses a y e C to counter. If the counter, (r", C"), is not unanimously accepted by the new proposed coalition, nature chooses another member of C , and so on until all members of C have the chance to counter. If no counter is approved, the threat is implemented; if a counter is a mutually agreed-to transfer, the transfer is implemented; if an approved counter involves involuntary transfers, it becomes the new threat, etc. Nodes in which a player receives R/2 resources are terminal. But because some branches repeat earlier nodes, we can have infinite play, and the method of solution is to posit continuation values, v(Tr>), for the subgames beginning with each threat and to assume "stationary strategies" that posit the same action whenever a node is repeated, where strategies and continuation values are simultaneously consistent. To illustrate the analysis of this game if all players are essential and can buy stability, let r = (120,100,80), so no sequence of decisions can yield less than 70 to 1 and 2 and 50 to 3 since each can avoid lower payoffs by buying stability. Simplifying the discussion, consider only the threats (150,150,0), (0,150,150), and (150,0,150), and the transfer outcomes (150,70,80), (150,100,50), and (70,150,80). Letting t;(r (150 , 0 , iso)) = (150,70,80), v(T{0> 150,150)) = (70,150,80), and v(T{i50,150,0)) = ^(r(i5o,0,150))/2 + t;(F (0)150) 150))/2 = (110,110,80), suppose 1 threatens (150,0,150) and let 3 accept. If 2 counters with (0,150, 150) then 3 rejects, because the original threat is worth more to 3 than v(F(o, 150, iso))- Similarly, 1 rejects a counter of (150,150,0); so 2 must transfer to 1, which is consistent with f(r(i5o,o,i5O))- On the other hand, if 1 or 2 threatens (150,150,0) then 3 can counter with (150,0,150) or (0,150,150). Both counters are certain to be accepted, but since 3 is indifferent between them, let 3 choose each with equal probability, which is consistent with the value f(F(i5o, i5o,o)) w e previously posited. Notice now that if we posit an equilibrium in which no country participates in an initial threat if it does not gain from doing so, then no threats are made, whereas if we suppose that countries participate in initial threats if they do not lose, then either 1 transfers to 2 or 2 transfers to 1. In either case, system stability prevails. To see the general structure of this example, notice that with the presumed continuation values for (150,0,150) and (0,150,150), neither threat is a counter to the other, but one or the other serves as a counter to any other threat if we posit continuation values similar to the one we choose for (150,150,0). This construction, in turn, implies that the presumed continuation values are consistent with subgame perfect strategies. Hence, although our game is noncooperative, these continuation values make {(150,150,0), (0,150,150)] look like a core: neither distribution "dominates" the other in that neither is a counter to the other, all other threats are "dominated" by (150,0,150) or (0,150,150) since one or the other counters any other threat, and no other threat "dominates" (150,150,0) or (0,150,150). For four-country systems, if all countries are essential and can buy stability, as with r = (110,80,60,50), the analysis proceeds as in the three-country case except that the particular threats that have "corelike" stability are (150,85,65,0), (150,85,0,65), (150,0,75,75),

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To illustrate A10 and Remark 3.7 in combination with earlier assumptions, consider again the three-country example with the resource distribution (140,100,60), and suppose that countries 1 and 2 propose to coalesce in order to attack 3. We noted earlier that 3 might try to oppose this threat by offering to transfer 10 units of resources to country 1, which yields a distribution both 1 and 3 strictly prefer to anything {1,2} might agree to. To see how this proposalfitsour definitions, notice that {1,3] can be interpreted as a coalition that threatens C#. Thus, condition (ii) in the definition of a counter is satisfied. Condition (i) is satisfied since C\ DC* = {l,2}n{l,3j = {l}, which is not empty, as is condition (iii), since 1 prefers securing 150 units of resources "peacefully" rather than aggressively (Assumption A6), and since 3 prefers any resource level in which it is not eliminated to one in which it is (Assumption A4). The counter is viable for 3, because player 1 cannot secure a better outcome than "150 units without aggression" (Assumption A6 and Remark 3.6). and (0,150,75,75), where y(r(i50,85,65,0)) = (150, 80,60,10), u(r(i5Of 85,o,65)) = (150,80, 20, 50), y(r(150, o, 75,75)) = (150, 40, 60, 50), and y(r(0,150,75,75)) = (40,150, 60, 50). So suppose 4 is inessential as with r = (110,90,70,30), and let 1, 2, or 3 threaten (120,100, 80,0). If 4 merely offers any two countries more than 10 units each, keeping part of r$ for itself, then even if its offer is accepted, 4 is merely postponing its day of reckoning and will eventually have its resources reduced to zero. So suppose instead that 4 counters with (150,150-e,0,e), which, if accepted, is countered by 3 with either (150,0,150,0) or (0,150,150,0). If the continuation values are as in the three-country case with zero payoffs for 4, then 2 must transfer to 1 or 1 to 2; and if 3 chooses each of its counters with equal probability then 1 and 2 prefer avoiding this lottery by rejecting 4's counter in favor of (120,100,80,0), in which case 4 is eliminated. Similarly, if 4 counters with (150,0, 150 — e, e) then 2 must transfer to 1; but then 3 is left with a payoff of 70, so 3 rejects 4's counter and again 4 is eliminated. In this way an inessential country cannot maintain its sovereignty. Concluding that this noncooperative model fully rationalizes our cooperative approach requires a general analysis, which follows our treatment of three-country systems. First, letting max[C] denote the largest member of C, we identify type-1 threats as those in which a winning coalition C threatens S— C, where S—C can render max[C] near-predominant. Second, we isolate a particular set 7"of such threats, where every country that might be the largest member of a winning coalition is represented by some threat in T, and where the members of Tall satisfy two properties: (1) for no two coalitions Cand C" represented by threats in T'\s it the case that max[C] and max[C"] = CflC"; (2) there does not exist a CoeWsuch that C o flC = max[Co] ^max[C]. Threats in Tare assigned a continuation value in which the largest member of the coalition becomes near-predominant and its partners maintain their current allocation; all other threats are assigned values in which no one becomes near-predominant and all but the largest coalition members are confronted with the possibility of a resource loss. In certain circumstances (as when resources are not uniformly distributed and no country can buy stability), we add the assumption that countries threaten to cede their resources to the largest country when confronted with elimination (because such threats preclude others from taking actions that lead to such possibilities), and we suppose that, ceterisparibus, threatened countries prefer to coalesce with themselves in formulating counterthreats. These values and such assumptions give Tits "corelike" stability, render those values consistent, and imply that a stationary equilibrium supports AlO's implications (Niou and Ordeshook 1989c).

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This argument is not a formal proof, but it suggests why A10, together with the previous assumptions, implies the following result (which is a corollary of the central theorem, Theorem 3.2). Theorem 3.1. All three-country systems are system stable provided that rt < R/2 for / = 1,2, 3; and only those distributions in which rt = R/2 for some / are resource stable. Although we generally relegate the proofs of our results to appendices, we discuss the logic of this theorem here so as to illuminate the role of our assumptions. We begin with the observation that, in any three-country system in which 0 < rt < R/2 for all / in 5 (in which no country is predominant), for any pair of countries / and j in S it must be the case that r{ + rj exceeds or equals R/2 (otherwise, rk>R/2). Hence, rt>R/2 — rj. This inequality means, however, that /'s resources, rh necessarily exceed the amount of resources required to render j near-predominant, R/2 — rj, in which case (by Remark 3.5) the system becomes both system and resource stable. At the very worst, then, i can seek to counter any threat to its existence and freeze the system by transferring R/2 — rj of its resources to j . Such a counter is clearly viable since it thwarts the elimination of /, and since, by Remark 3.6, j knows that it can never hope to form a coalition that promises it more than R/2, and since, by Assumption A6, j prefers a voluntary transfer that secures R/2 to a coalition with k to eliminate / for the same end. Thus, the system is system stable. To show that the system is resource stable if and only if rt = R/2 for some country /, Remark 3.3 establishes that rt = R/2 is sufficient for resource stability. Necessity is established by observing that if rt < R/2 for all /, then for any / and j in S it must be the case that rz + rj > R/2, so that / and j together can threaten the third country and force it to cede resources. Despite the fact that this result deals with only the special three-country case, a number of substantively important conclusions follow from it. First, with regard to the relative stability of two- versus three-country systems, although the only two-country system that is system or resource stable is one in which both countries have identically equal resources, all three-country systems without a predominant country are system stable, even ones in which the distribution of resources is "inequitable " As long as decision makers maximize relative resources, and as long as our other assumptions are satisfied, instability is inevitable in two-country systems unless perfect equality of resources is maintained; but it is anything but inevitable in three-country systems. This does not mean that we should interpret Theorem 3.1 as implying that wars cannot occur in such systems, since we have shown that not all three-country systems are resource stable and since we have not specified the mechanisms nations use to effect a

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transfer of resources. Yet, the key feature of a three-country system is that no nation will be eliminated. This conclusion differs markedly from Waltz's that "smaller systems are more stable" (1979, p. 136) and, specifically, that bipolar systems are more stable than multipolar ones (1979, pp. 170-6). However, Waltz's conclusions do not follow in any rigorous way from his assumptions. Although it may be true that "the costs of bargaining increase at an accelerating rate as the number of parties becomes larger" or that "the diversity of parties increases the difficulty of reaching agreements" (1979, pp. 1356), the opportunities for stabilizing counterproposals also increases as number and "diversity" increase. Indeed, if a system is truly bipolar, so that the resources of all other countries are irrelevant, then arguments about the costs of negotiation, diversity, and the like are irrelevant. All that is relevant is who can defeat whom and at what cost. In a three-country system, on the other hand, everyone can be defeated by some combination of resources, in which case either this mutual interdependence is recognized and system stability prevails, or someone errs. Whether error is more likely as the number of countries increase, we cannot say, but Waltz's conclusion about relative stability does not follow because he ignores or misconstrues the full consequences of his prior observations about the self-help nature of international systems. Theorem 3.1, moreover, disconfirms the supposition that equality of resources is required to achieve balance, and thus supports Morgenthau's (1959) belief that exact parity is not a necessary condition for peace {qua stability). However, excepting the circumstance in which some country's resources exceed R/2, it fails to support his corresponding supposition that rough equality is necessary. Theorem 3.1 also contradicts the supposition that ensuring (system) stability requires precise measurement of capabilities. Such a supposition is correct in two-country systems, but not in three-country systems. It is perhaps not surprising, then, that the literature seems to lead to an ambiguous conclusion about the relationship between power inequities and the frequency of war.9 In any event, this feature of three-country versus two-country systems suggests an issue that we elaborate on in Chapter 9, namely, that negotiations for disarmament are more likely to be successful when there are three powers rather than when there are only two. In bipolar systems, negotiators must be especially 9

Ferris (1973), Sabrosky (1975), and Siverson and Tennefoss (1984), for example, argue that equality correlates with peace; Garnham (1976) finds the opposite; Singer, Bremer, and Stuckey (1972) argue that the relationship depends on the characteristics of states; and Healey and Stein (1973), Karsten, Howell, and Allen (1984), Maoz (1982), and Wolf (1978) find no relationship whatsoever. Although our analysis has not yet taken rates of resource growth and geography into consideration, these studies taken as a whole can be interpreted as supporting the implication of Theorem 3.1: there is no relationship.

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careful that agreements do not inadvertently give a slight advantage to one side or the other, whereas in three-country systems, the consequences of miscalculation and error are less severe - all three countries will remain necessarily essential. A further implication of Theorem 3.1 is that, although three-country systems may see shifting alliances as resources change, we need not witness meaningful permanent coalitions. If alliances are "commitments.. .which bind two or more states to concert their policies in time of crisis" (Singer and Small 1966, p. 5), we must reject the supposition that alliances are the sole measure of the operation of balance of power. Indeed, it is not merely coalitions that prevent wars, but also the threat of coalitions that performs this function. At the distribution (140,100,60), the coalition of 2 and 3 should never actually form to eliminate 1. If such a coalition intends to eliminate 1, 1 can forestall this possibility by relinquishing resources to 2. The threat of 1 and 2 coalescing does not yield a war of total absorption either, since 3 can cede 10 units to 1. Thus, with three countries, we do not predict merely that we should observe "minimal winning coalitions," but instead we should allow for the possibility that no explicit coalitions will be observed, and that the implicit threats and counterthreats of coalitions may be as important to the dynamics of the system as the explicit coalitions revealed to us in the form offormal treaties and agreements. This is not to say that we would not want to understand those treaties and agreements in the context of balance-of-power politics, but only that implicit as well as explicit possibilities should be considered. Consider now the presumption that Theorem 3.1 and its consequences are known to all decision makers, or at least that its logic is implicitly understood by them, in which case the leaders of nations who might otherwise seek to be predominant know that they cannot succeed and therefore that it is fruitless to try (at least with the instruments we have thus far allowed). We infer from these comments, then, that those who view the explicit formation and dissolution of alliances as an essential part of a balance-of-power system (Wright 1942, Claude 1962, Organski 1968) generally have something other than system stability in mind (or that they are merely failing to distinguish between system and resource stability). Insofar as system stability is concerned, the threat of alliances and not their actual formation is sufficient for a three-country system to be system stable. Finally, and despite the fact that it deals with a special case, perhaps the most profound implication of Theorem 3.1 is that it establishes the possibility of system stability without appeal to exogenous institutions or constraints, or to an analysis that posits balance as a goal. Balance in the form of system stability emerges wholly endogenously as the consequence of self-interest and a recognition of the interdependence that describes the

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system. This theorem, then, is a counterexample not only to Riker's view of balance of power, but also to Utopian schemes that see world government (however established) as the sole guarantee of stability. Self-interest alone provides such a guarantee in anarchic three-country systems, and Utopian schemes are merely accessories. The importance of what we have just said is greatly diminished, of course, if we cannot generalize Theorem 3.1 to larger systems in the form of necessary and sufficient conditions for system stability. Before we proceed in this direction, however, we must consider a potentially confounding strategic possibility arising in systems with more than three countries. Specifically, because the possibility cannot arise in three-country systems, we have not ruled out a strategy whereby inessential countries are eliminated and their resources absorbed in such a way as to render a previously essential country inessential. In fact, though, if an inessential country is eliminated, the survivability of essential countries is not threatened in subsequent games, as indicated by the following result that simplifies considerably the strategic planning of countries contemplating the possibility of expropriating resources from someone else. Remark 3.8. If any or all members of S—E (the set of countries that are not essential) are eliminated then, regardless of how the resources of S—E are reallocated, all members of E remain essential. Suppose, as a worst case, that j receives no resources from the eliminated countries. Since j is essential before some or all of the inessential countries (those in the set S-E) are eliminated, there is by definition a minimum winning coalition C such that C— {j} is no longer winning. But C augmented by all inessential countries, the coalition CU(S-E), is also winning (although not minimal winning), and, since S—E is not essential, CU(S—E— {j}) is not minimal winning either.10 Therefore, even if C-[j] absorbs any or all of S-E's resources, j remains essential to C and therefore is essential in the new game.11 10

11

This assertion requires that, if 5—E is the set of all countries that cannot individually render a losing coalition winning, then S—ii combined into a single entity cannot do so either. To see that this is true, suppose the contrary, that C is losing but that C plus the members of S-E are winning. In this case, we should be able to delete each member of S—E one at a time from C plus S—E until what remains is losing, in which case that member is essential - a contradiction. Note, however, that this result is true only when the internal rate of resource growth for all essential countries in the system is held constant; as we shall see in Chapter 5, treating differential internal rates of resource growth as endogenous to the model makes the absorption of the resources of inessential countries very troublesome to essential countries.

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Remark 3.8 has an important substantive implication that permits us to make some predictions about superpower relations. Namely, a country should be unconcerned about the attempts of other countries to absorb the resources of inessential countries unless, for reasons exogenous to the present analysis (e.g., differential rates of internal resource growth among the essential countries), such an absorption is part of a strategy by the absorbing country to become predominant. Perhaps we should not be surprised, then, to find the major Western powers agreeing to the absorption of a Latvia or an Estonia. The usual argument for explaining acquiesence to this absorption is that countries like Latvia and Estonia are within the "sphere of influence" of the Soviets or that they have a "historical attachment" to Russia. Yet, this leaves open the question of how spheres of influence are arrived at in specific cases and defined in general. Indeed, such spheres are artificial concepts and commonly are proved to be ephemeral in the face of dominant military capabilities. In the terms of our analysis, countries are allowed spheres of influence since the explicit addition of resources of entities within those spheres does not materially affect the stability of the international system. Similarly, although each superpower can make useful propaganda out of the other's military involvements in Vietnam and Afghanistan, neither truly perceives the outcomes there as threats to international stability. The United States, however, should take a far different view of a Soviet invasion of, say, the People's Republic of China, just as the Soviets might react to an American decision to occupy by force the oilfieldsof the Middle East. In such cases, it is precisely the changes in resources implied by such acquisitions that make them matters of considerable concern to the parties involved. Despite their substantive import, Theorem 3.1 and Remark 3.8 are nevertheless but preliminaries to the central result of this chapter, which provides the necessary and sufficient condition for system stability. Theorem 3.2. (S; r) is system stable if and only if all countries in 5 are essential (i.e., if and only if S = E). We relegate the formal proof of this result to an appendix, but the principles that underlie it are straightforward. Each country knows that everyone else will oppose it if it threatens to become predominant, and therefore that the best outcome it can secure is control of precisely half of the system's resources. The condition that S equals E merely guarantees that every country possesses, either individually or in combination with some other set of similarly vulnerable countries, enough resources to render someone near-predominant. Since every country prefers to become nearpredominant in this way, the recipient of such an offer has a clear incentive to accept it, which, in accordance with our earlier remarks, freezes the system. The formal proof of Theorem 3.2, then, merely ensures that

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the result is consistent with our assumptions and that all essential countries can meet every threat with a viable counterthreat. As long as we maintain the assumption that countries maximize "power" in the form of maximizing their proportion of resources in the system (subject to the constraint, of course, that their sovereignty is not threatened), it is straightforward to see, moreover, that this theorem does not depend on our assumption that wars are costless, where a costly war or threat means that resources for the loser as well as perhaps the winner are lost in the conflict [as expression (2.1) in Chapter 2 illustrates]. First, suppose that all countries can form viable counterthreats by transferring resources to the largest country that might threaten them so as to render that country near-predominant. The same must be true, of course, if wars are costly, since that cost is irrelevant to the transfer. Now suppose that all countries are essential, but that some cannot individually buy stability, and, in particular, suppose that the target C" of a threat made by C does not have enough resources to buy stability. Notice that we can suppose that C=S—C; even if C does not explicitly include all of S—C\ the members of S— C — C are implicitly parties to the threat since, with the destruction of any of the resources of C, their share will increase. And because every / in C" is essential, it must be the case that some Kri.n Finally, if C" is inessential, its "coalition partners" can always increase their proportion of resources by dropping C" from the counter - even, perhaps, adding C" to a counterthreat's target - in which case an inessential C" cannot form a viable counter. Returning to the assumption that wars are costless, to illustrate the logic of Theorem 3.2 further, especially the operation of threats and counters, we consider the following five-country examples:13 12

13

Suppose, for example, that r = (100, 80,60,40,20) and suppose that (1,2) threatens 4, in which case 3 and 5 also gain in proportion from any destroyed resources on the part of 1, 2, and 4. But 4 can counter with, say, {2, 3,4] threatening (1, 5 j . Since the proportion that {2, 3,4] controls after 1 and 5's resources are reduced to zero also equals 100 percent, and since 4's partners are fewer in number than the number of members of the original threat, those partners can be assured of gaining more, proportionately, in the counter than in the threat. An interesting sidelight to this theorem is its relationship to the Shapley-Shubik power index (1954). Originally designed to measure a person's "power" in formal voting systems (see footnote 5), notice that a country is essential if and only if, after interpreting the r/'s as voting weights, the power index is nonzero. Hence, we can interpret our analysis as showing that only those countries with positive power indexes can survive.

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Example 3.1. Let r = (120, 50, 50,40,40), which corresponds to Case 1 in the theorem's proof. Although there are a variety of potential threats against different countries, even the smallest (4 or 5) has enough resources to "buy" stability when threatened. At the very worst, 5 can form a viable counterthreat by offering to transfer 30 units to 1. In this event, 5's coalition partner can do no better than have its total resources increased to 150, and the distribution (150,50,50,40,10) serves as the basis for a viable counter. Example 3.2. Let r = (100, 80,50,35,35), in which case countries 4 and 5 cannot individually form viable counters by "buying off" the largest player; hence, this illustrates the second case in the proof. If 1 and 2 threaten 5, notice that the maximum gain to C= [1,2} is 35, so country 2 can at best realize the resource level 115. But if 5 coalesces with 2 to attack 1, 2 can be made strictly better off (even securing 150 units of resources), but 2 cannot do better by acting alone. It is interesting to note that, except for the number of countries, the strategic considerations in this example are identical to those that prevailed in Europe between 1871 and 1914. At the time of the second Near Eastern Crisis in 1885, for example, resources were distributed on a percentage basis approximately (5,35,14,21,6,19) across Austria-Hungary, Great Britain, France, Germany, Italy, and Russia, respectively (see our discussion of resources in Chapter 7). To see that all countries are essential, if we ignore the geographical imperatives of the period, notice that although Austria-Hungary, Italy or France did not possess sufficient resources to render Britain near-predominant, any two of them together could do so in the form of a minimum winning coalition. Hence, all countries are essential. Example 3.3. Let r = (110,80,50,50,10), in which case country 5 cannot "buy off" any other country, nor is it essential to any coalition for the implementation of a threat. For instance, the coalition {2,3,4} can attack 5, and although 5 might counter with, say, {2,5} threatening 4, 2 does not require 5 to make such a threat. Hence, the counter is not viable. Example 3.4. Let r = (60,60,60,30,30), and suppose that {1,2, 3} threatens 4 and 5. Notice that, although countries 4 and 5 are essential, and although they have enough resources to ensure their sovereignty by setting one country's resources equal to half the total, such a move reduces their resource holding to zero. Thus, it appears that countries 4 and 5 can, paradoxically, secure

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survival only by eliminating themselves. However, 4 and 5 can coalesce with 2 and 3 so that the coalition's gain is also 60. Since 4 and 5, by the definition of a counterthreat, are not required to gain more than what they have in the original distribution, 2 and 3 can divide the gain in such a way that they are better off in the counter than in the threat. These examples reveal an important component of balance-of-power politics that is not always fully appreciated. In Examples 3.1, 3.2, and 3.4, in particular, the country in question secures its existence by forming a viable counterthreat that transfers resources to someone else so as to render the target of the transfer near-predominant (thereby freezing the system in accordance with Remark 3.3). Thus, although students of international affairs such as Morgenthau and Waltz, as we noted earlier, make us sensitive to the mutual-security aspects of balance-of-power politics, we see here that the threat of counteralliances is only part of the process whereby system stability is assured - the other part being the ability to transfer resources (territory, spheres of influence, money, etc.).14 Theorem 3.2 and its proof confirm the intuition expressed by Gilpin that "Territorial, political, and economic adjustments among states in response to conflicting interests and shifting power relationships function to relieve pressure on the system, thereby preserving it intact" (1981, p. 46). Although it is far from clear that Gilpin has in mind the resource transfers we formalize as part of viable counterthreats, the logic of Theorem 3.2 quite directly shows how the possibility of such adjustments secures system stability. It is interesting to speculate, then, why theorists pay such modest heed to this feature of counterthreats as a mechanism for ensuring balance (Claude 1962 and Organski 1968). However, aside from observing that resource transfers are a feature of international politics (witness Franco-British and Russo-British negotiations prior to the formation of the Entente Cordiale as well as the resolutions of the Near Eastern Crises of 1875-7 and 1885-8) and aside from noting the subtlety possible in a transfer (granting of trade concessions, replacing a hostile minister with one who is likely to be subservient to the threatening power), such speculation is premature. Just as our theory does not require the actual implementation of a threat and counter, our theory does not suppose that such transfers invariably occur so as to secure system stability. Rather, it is the potential for such transfers that renders a system system stable. Predicting 14

Wagner (1986) argues that such transfers are not essential to system stability and that it is unclear how such transfers are enforced, although he implicitly admits such transfers by allowing countries to absorb the resources of others. As we note earlier, the issue of enforcement is a legitimate concern, but what is unclear at this point is whether transfers are more difficult to enforce than other types of agreements.

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the transfers that we might actually observe requires the analysis we offer in the next chapter on resource stability. 3.3

System stability and war

If there is a serious ambiguity in our analysis, it is the one between reallocations that might occur through the choice of some aggressive act (a2) and reallocations that, although requiring coercion, occur through negotiation (a3). Assumption A6 presumes the feasibility of making this distinction, but such assumptions must eventually give way to ones in which countries are presented with a greater diversity of alternative actions, where each one imposes different costs on decision makers. Although the present theoretical structure gives us a clear sense of empirical possibilities, it is probably impossible to classify acts into two such abstract categories. Of course, there is always a gap between abstract concepts and their empirical operationalization. What we are suggesting here, then, is that because the primary outcomes that concern us - peace and war - manifest themselves in varying degrees, it would ultimately be useful to extend our analysis so that it better incorporates this variety. It should also be apparent that the failure of the analysis in this chapter to consider several factors limits its ability to discuss alternative sources of conflict. Consider the following remark, which merely lays out formally our earlier discussion of subsystem stability: Remark 3.9. If S' is a subset of S and if r' is the distribution of resources within S', then even if (5;r) is system stable, (S";r') may or may not be system stable. For example, (S; r) = ({1,2,3,4, 5,6], (100, 80, 30, 30, 30, 30)) is a systemstable system, but (S';r') = ({2, 3,4), (80,30,30)) is not. Thus, if - for reasons presently exogenous to the analysis, such as geography - countries 1,5, and 6 will not attack those in S\ then we might anticipate a war in which 2 attacks and absorbs 3 and 4. Hence, the formal results of this analysis do not accommodate "local conflicts" such as ones between Iran and Iraq or between Great Britain and Argentina. To summarize, then, the particular import of Theorem 3.2 is that it reaffirms much of what we said about Theorem 3.1. In particular, we find that there is no magic number for achieving a balance of power, nor is there any magic formula relating the distribution of resources to system stability. Nevertheless, because in a bipolar (two-country) system, system stability requires a precise equality of resources, and because system stability in multicountry systems exists under a broad range of resource distributions, we can only conclude that such stability is much more likely

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to be retained in multipolar systems. Thus, Waltz (1979) is probably correct to argue that miscalculation presents special dangers in bipolar systems. We do not concur, however, with his assertion that "systems of three [countries] have distinctive and unfortunate characteristics," as compared to multipolar systems because "two of the powers can easily gang up on the third" (p. 163). Theorems 3.1 and 3.2 offer no reason for supposing that three is less secure than four, or that five is "the lowest number that promises stability." In addition to the empirical counterexample that, for 200 years, seven conflictual states co-existed in ancient China (the Warring States Period, 453-221 B.C.), we find no theoretical basis for Waltz's assertions. If the simple counting of powers cannot explain the likelihood of war, then we should ask: What are the sources of conflict and war in our analysis? Despite the relatively broad circumstances for stability identified by Theorem 3.2, wars do occur, including wars between great powers that in no way can be conceptualized as inessential. The opponents of those who have attempted to secure hegemony - of the Habsburg realms under Charles V and Phillip II, France under Louis XIV and Napoleon, and Germany under the Kaiser and Hitler (Dehio 1962) - were clearly not inessential, as evidenced by the outcomes of the wars themselves (Wright 1942 and Dehio 1962). And if a hegemonic war is one that involves "a direct contest between the dominant power or powers in an international system" and in which "the fundamental issue at stake is the nature and governance of the system" and in which nations use "unlimited means" to secure their ends (Gilpin 1981, p. 199; see also Levy 1984), then Theorem 3.2 cannot provide an answer to our question. A partial answer is provided, however, by Blainey's (1973, pp. 111-12) observation that we possess not an axiom for peace but an axiom for national independence. And that in fact was the main virtue of a balance of power in the eyes of those who originally practiced it Gulick.. .was adamant that its clearest theorists and practitioners - the Metternichs and Castlereaghs - "all thought of war as an instrument to preserve or restore a balance of power." In essence, a balance of power was simply a formula designed to prevent the rise of a nation to world dominance. It merely masqueraded as a formula for peace. Although we would add that a balance is also a formula designed to prevent the elimination of essential countries in addition to the rise of a hegemon, the point of this argument is that because system-stable systems are not necessarily resource stable, we cannot preclude the possibility that resource instability implies armed conflict that stops short of the elimination of essential actors. We should not preclude the possibility, however, that conflict nevertheless occurs also because one or more of the assumptions of our model are

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violated. Ignoring for the moment the issues of resource growth and decline and of geography, notice first that our model presupposes that all countries possess complete freedom to bargain and to threaten, and that counterthreats to transfer resources are believable. Clearly, then, we can hypothesize that in systems described by constant resources for all countries, system instability and the wars they occasion are caused by inflexibilities and constraints on bargaining. Inflexibility can arise from a great many sources, and this is perhaps the reason why scholars can so readily cite different reasons for a particular war or for all wars. Domestic politics provides perhaps the most evident constraint on a national leader's flexibility. Few if any such leaders, except those who are absolute dictators and who rule by military force, are afforded the luxury of being able to transfer resources to another country in the form, say, of territory. If particular allocations are regarded as infeasible, then Theorem 3.2 no longer applies, system stability may no longer hold, and even essential countries may be unable to invent the viable counterthreats that secure their existence. However, following the dictates of the literature as well as common sense, we postulate survival as a supreme goal, which suggests that inflexibilities that are allowed to threaten sovereignty must have profound sources and the inflexibilities themselves must be of an extreme sort. Simple disputes over, say, colonial possessions, tariffs, and the like should fall before the goal of survival. Thus, impediments to formulating viable counters to a threat may be a contributory factor to war, but wars to eliminate essential countries are likely to have a different source. As we suggest in Chapter 1, that source is most likely uncertainty and misperception. Even if domestic politics plays no role, resources cannot be readily measured and transferred. Hence, even reasonable negotiators can fail to agree on what specific transfers (such as the assignment of spheres of influence) will ensure their sovereignty and thus maintain balance. Naturally, the uncertainty and misperception that our assumptions exclude can play critical roles as well. The impact of uncertainty, in fact, can be profound, at least for the determination of final outcomes. Without it, essential countries can ensure, with certainty, that no country becomes predominant. Yet with uncertainty, the "luck of the draw" may yield such predominance. Also, essential countries can in our model ensure against the possibility that they in particular become inessential. Still, again, probabilistic events may dictate otherwise, regardless of what steps are taken by decision makers beforehand. Thus, even though uncertainty is precluded by our assumptions, breakdowns of system stability and outbreaks of war need not be taken always as refutations of the forces our analysis uncovers; such outbreaks can be the result of chance events. This, of

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course, requires that in any empirical test or application of our analysis, we distinguish between those conflicts that arise because of chance events, those that arise because leaders have misjudged each other's capabilities, and those that arise because our model is simply in error. An additional source of difficulty, discussed in Chapter 2 but warranting reemphasis, concerns the enforcement of agreements. Throughout this analysis, we have assumed that countries need merely to share some mutual interest to render coordination toward the attainment of that interest feasible. Diplomats and foreign-policy analysts alike, however, know that the sticky details of an agreement - designed to assure that those who are party to it abide by its terms - often derail otherwise mutually beneficial outcomes. The perception that an agreement in one coalition is more readily enforced than an agreement in another can, of course, introduce important asymmetries. Particular threats or counters may be unbelievable owing to factors such as ideology and domestic politics, whereas other threats and counters may play a more significant role because factors we do not consider reenforce mutual self-interests. Finally, we should not ignore two major empirical violations of our initial assumptions as a likely cause of war: the absence of resource growth among countries and the failure to discount offensive capabilities for geography. As we attempt to make clear in Chapter 5, where we explicitly add the dimension of differential rates of growth among countries, and in Chapter 8, where we discuss the conflict-resolution implications of our model, much of the observed conflict between essential countries arises from differential rates of resource growth. Similarly, as we show in Chapter 6, where we add geographical considerations, and in Chapter 7 as well as 8, where we illustrate how geography shapes the strategic environment, wefindthat geography can place differential burdens on the resources required to render a country essential. At this point, we should not belabor particularistic sources of instability and conflict, since what we want to emphasize is that, although results such as Theorem 3.2 point the way to abstract principles and to the conclusion that the anarchic order of international relations does not necessarily yield instability and conflict, these results are not an end to theorizing. We ought not to interpret the analysis offered in this chapter as providing any definitive conclusion about whether the realist perspective necessitates the conclusion that balance (system stability) precludes war. Indeed, we have not yet considered a second type of stability, resource stability; to better understand the possible sources of conflict in international systems, as well as the necessity for distinguishing between system and resource stability, in the next chapter we shall focus on whatever reallocations are likely to arise when no country's existence or sovereignty is threatened.

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3.4

A note on assumptions

We can debate any assumption ad infinitum without necessarily resolving whether it is reasonable or unreasonable; the most valuable means of assessing an assumption is by gaining an understanding of its critical function in the analysis. Aside from the model of bargaining (which we offer in this chapter in the form of A10), two assumptions introduced earlier are especially critical to this chapter's conclusions: Assumptions A6 and A7. First, Assumption A6 states that countries prefer to secure additional resources without conflict. Intended to model the costs of wars without introducing new parameters, this assumption plays a profound role in the specification of necessary and sufficient conditions for system stability. Indeed, without it or some equivalent assumption, only two-country systems are system stable. To see why this assertion is true, consider the threecountry system (120,100,80). In our analysis, country 3 protects its sovereignty by the threat that it can transfer 30 units to 1. Assumption A6, in turn, ensures that 1 prefers to accept this transfer over any forced feasible reallocation of resources. But without A6, a proposal by 1 and 2 to attack 3 so as to secure the distribution (150,150,0) cannot be countered by 3. At first glance, it might seem that we should add a stronger assumption to accommodate the costs of wars and, in particular, to accommodate the fact that conflicts destroy resources. Nevertheless, our assumption that wars are costless to the extent that they do not destroy resources does not affect our primary result about system stability (Theorem 3.2): if countries maximize proportion of resources (power), then S = E remains a necessary and sufficient condition for system stability. Thus, if potential combatants maximize proportion - if strategic planners in the United States and the Soviet Union view a conflict in which the United States retains a population of 25 million compared to 10 million for Russia as a "victory" for the United States - it is perhaps somewhat comforting to learn that balance-of-power politics, although not aiding inessential countries, will nevertheless ensure the sovereignty of essential ones. Although this result is consistent with Organski and Kugler's (1980) assertion that "nuclear weapons have not modified [the rules of international political processes], they have instead reinforced them" (p. 217), we might have conjectured that war costs would make system stability more pervasive by rendering otherwise inessential countries essential. This intuition, though, seems to have in mind decision makers who maximize the absolute level of resources rather than their proportion, or at least decision makers who perceive a tradeoff between "power" and absolute resource levels.15 15

Another way in which such costs can induce stability, of course, is if the technology of modern warfare (nuclear weapons and their delivery systems in particular) assures that

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We suspect, then, that the issue of war costs as a deterrent to conflict arises whenever national leaders are implicitly assumed not to be concerned merely with proportion of resources, but with the absolute level of resources as well. To see this, suppose that (conditional on maintaining their sovereignty) countries maximize absolute level, and that absolute level and proportion are decoupled because threats, if implemented, destroy resources. If we otherwise maintain A6 and if all countries are not merely essential but can individually "buy" stability with a resource transfer to the largest threatening country, then the system remains system stable since, as before, all countries can freeze the system with such a transfer. What changes here, however, are the conditions under which stability can be bought. To see this, suppose war costs are described by expression (2.1) - suppose that the resources of C after implementing an uncountered threat against C" equal [r(C) + r(C')][l-ar{C')/r(C)] in which case ({1,2}; (200,100)) is system stable whenever a > 2/3, since any attack by 1 against 2 leaves 1 with fewer than 200 units of resources. This example, in fact, seems to correspond to the balance of power that existed between the United States and the Soviets. Despite the considerable resource advantage enjoyed by the United States and its allies over the Soviet bloc, the destructiveness of nuclear weapons allowed an otherwise unstable system to remain stable. However, before we infer from this example that war costs are stabilizing, notice that if (in a three-country system) r = (120,100,80) then country 3 cannot buy stability unless it cedes all of its resources to 1 - our analysis of the two-country system shows that 1 can hold up to 200 units of resources before it dominates all others. Indeed, some simple algebra establishes that country 3 is inessential in this instance - it cannot formulate a viable counterthreat against a threat by 1 and 2.16 We should not interpret this example to imply that war costs destabilize otherwise stable systems or that there is any simple monotonic relationship between war costs and system stability whenever national leaders maximize absolute resources. For example, in the system ({1,2, 3,4]; (90,90,90,30)), if a = 0 then country 4 is not essential; if a = 1/2 then 4 is essential; and if a — 2/3 then 4 is inessential once again.

16

any two coalitions are effectively of equal size - in which case, with a = r(C)/r(C')> expression (2.1) implies that any two antagonists destroy each other totally. In such a world, of course, any reasonable attribution of preferences to a world in which r = (0,0,0,...) is necessarily the consequence of any threat would imply that no coalition will ever threaten another. For additional discussion of war costs in the context of the present model, especially as the effect of those costs are mediated by alternative national goals, see Niou and Ordeshook (1989b). By using expression (2.1), the gain of (1,2) attacking 3 is positive, whereas the gains of counters by 3 (coalescing with either 1 or 2 to attack the remaining member of the system) are not positive. Thus, 3 has no viable counterthreat either in the form of a direct resource transfer to render one country near-predominant or in the form of an attacking counter.

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The general implication of this discussion, then, is that the relationship between war costs and stability are mediated by goals. If leaders pursue power, defined in relative terms, then such costs are irrelevant to stability. On the other hand, if those leaders maximize absolute levels of resources, then war costs can either stabilize or destabilize systems - and the precise nature of the consequence of such costs depends on their magnitude (as well as on the functional relationship between those costs and the resources of potential combatants). The impact of war costs on the stability of international systems, then, may depend profoundly on domestic political matters that influence the weight given by national leaders to relative versus absolute resource levels. A concern with relative resources is relevant to decision makers who are concerned primarily with international issues and state sovereignty; a concern with absolute resources is perhaps more relevant to decision makers concerned about their domestic bases of support and with the survival of their regimes. Hence, the analysis in this chapter merely opens the door to a more comprehensive theoretical treatment of the relationship among stability, war costs, and the tradeoffs national leaders perceive between goals. Assumption A7 is no less important for our analysis, its critical feature being the supposition that any larger country or coalition of countries not a party to some conflict can secure resources from that conflict. Consider again the distribution (120,100,80), and consider country 3's reasoning as it contemplates a possible attack by 1 or 2. Everyone knows that in the event of such an attack, 3 could offer to transfer 30 units to 1 and 1 would be predisposed to accept the transfer. But moving the system to the distribution (150,100,50) does 3 little good if it is then attacked by 2. What keeps this attack from materializing, though, is Assumption A7: country 2 know that if it attacks 3 then 1 can profit from the conflict, securing the extra resources or proportion of resources required to become predominant. Thus, reasoning backwards, everyone knows that if 1 controls exactly half the resources then 2 cannot attack 3, in which case everyone also knows that a 30-unit transfer by 3 to 1 establishes system stability. Again, reality need not correspond to A7. This is most easily seen by noting that A7 can imply that third parties to a conflct can secure small gains from a conflict among other states. Clearly, geography and other impediments to effective military mobilization lead to violations of this presumption, and if third parties cannot anticipate gains that outweigh costs then regional conflicts precluding the emergence of system stability are possible. Thus, multicountry systems are system stable only if regional conflicts are "policed" in the international system by the threat that larger countries will take advantage of such conflict. If this threat is not present then the stability of subsystems must be analyzed independently.

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Our last critical assumption is A10, which models the bargaining that occurs among countries insofar as that bargaining pertains to the ability of countries to survive. Readers familiar with game theory might wonder why we do not directly graft to our analysis the reasoning that defines any one of the several cooperative solution hypotheses offered by game theory. Indeed, applying the reasoning of the V-set or the bargaining set yields quite different predictions than the ones produced by A10. Consider the bargaining set, the motivation of which closely parallels our assumption, and consider the distribution (150,150,0), which we suppose is the result of a coalition of 1 and 2 against 3. Bargaining set theory asks whether this distribution is "stable" in the sense that it can endure once arrived at. To see that this distribution is in fact stable according to that theory's logic, suppose 1 objects with the proposal (150,100, 50), which is to say that 1 demands receiving 150 units through negotiation rather than through conflict. This is a legitimate objection since, by Assumption A6,1 prefers his payoff in (150,100,50) to his payoff in (150,150,0). Further, only objections of this form are possible, since objections in which 1 or 2 receives more than 150 units are implausible; everyone knows that no coalition partner would agree to such an arrangement. However, 2 can now counter with (0,150,150). Indeed, there is a counter objection to every objection, and (150,150,0) with the coalition structure ({1,2), {3}) is in the bargaining set. Conversely, the distribution that results whenever 3 implements a transfer so as to secure its existence, (150,100,50), is not in this set. Presumably, this distribution should be associated with the coalition structure ({1,3}, {2}), so consider an objection by 3 against 1 in the form of the proposal (0,150,150), which is to say that 3 threatens a coalition with 2 in which 2 and 3 attack 1. But A6 implies that 1 has no counterobjection. A counter, according to the definitions of bargaining set theory, must be a distribution that 1 likes at least as much as (150,100,50) and that its coalition partner(s) like as much as (0,150,150). That is, with the counter, country 1 must be able to "defend" whatever payoff is received in the particular proposal under scrutiny, (150,100,50) in this instance. Since this distribution is the best possible outcome for 1 (the maximum possible resources secured through negotiation), any other distribution in which 2 or 3 secures as much as it is offered in the objection, 150, cannot be obtained without conflict if 1 also gets 150, in which case 1 likes this 150 less than the 150 in (150,100,50). Since 1 thus cannot defend the 150 units it gets in (150,100, 50), this distribution is not in the bargaining set. Indeed, in the present three-person situation, bargaining set theory predicts that only outcomes of the form (150,150,0), (0,150,150), and (150,0,150) prevail - one of the three countries will be eliminated.

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The application of traditional solution hypotheses to our problem, then, predicts that all systems ultimately reduce to two-country systems. Yet such applications are ill-suited to our purposes. Traditionally, these hypotheses are designed to be applied to games "in characteristic function form," which represents the advantages and disadvantages of players in terms of what outcomes can be enforced by the different coalitions. These applications presume, in effect, that aside from the constraints represented by the characteristic function form, bargaining occurs in a "state of nature" - there are no additional constraints on what is possible. Abstractly, all advantages and disadvantages, if they exist, derive from individual preferences and the structure of the bargaining situation. The application we have just reviewed takes account of preference, but it ignores structure. We have not tried to model the structure of bargaining among countries, but notice that the purpose of A10 is not to render a prediction about payoffs. Rather, A10 is designed to aid in predicting which countries can and which countries cannot survive given a current distribution of resources. Thus, the status quo [the distribution (120,100, 80) in the example] plays an especially important role here, and all objections that propose a change must be objections to it. With this in mind, A10 proposes some simple modifications of the objection-counterobjection process detailed in the bargaining set literature. For example, we do not require that those countries being objected to recover to their original position. Bargaining set theory imposes this constraint on counter objections because it seeks to predict payoffs; but here we are trying to predict survival, and survival merely requires that a country is assured of retaining some resources. However, one might object to this rationale for A10 with the argument that the initial resource distribution serves merely to identify winning coalitions, and that bargainers can otherwise ignore it. However, there is another view that justifies A10. Suppose countries threatened with attack can make "one last move" to transfer resources. Hence, if everyone knows that an attack on an essential country can be thwarted by a transfer, then the possibility of such a transfer should be taken as a constraint on whatever bargains are reached. Bargaining environments that support A10, then, are those in which negotiators have some certainty that transfers can be implemented as a blocking response to the decision to attack and to eliminate countries (see Footnote 8). As with our other assumptions, though, this justification for A10 suggests how its violation yields a second class of environments in which conflict and the eventual emergence of a bipolar system might become a reasonable prediction. If, in our example, 3 misreads 1 and 2's intentions and thereby fails to initiate the transfer to 1, then the attack will occur and

3.5 Appendix: Proof of Theorem 3.2

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3 will be eliminated. In this instance, imperfect information and miscalculation preclude the certainty of "last moves," in which case we should not presuppose that countries can guarantee their survival with resource transfers. There is, then, a second class of bargaining environments, in which there is no assurance that responses of a particular type are possible. Although it is uncertain what solution is appropriate in such a context (bargaining set theory seems irrelevant since it presumes, in effect, that players negotiate with the assumption that counter objections can always be implemented as a last move), hypotheses such as the main simple V-set (which completely forgoes a bargaining rationale) predict minimal majority coalitions that eliminate maximal minority coalitions. Which class better models international systems is subject to speculation and the character of the particular system under discussion. If the system is characterized by imperfect information, by barriers to communication, and by domestic politics that inhibit the timely implementation of appropriate counterproposals, then the second class seems more appropriate. Yet, if there is a time lag between the decision to war and the actual first attack, and if there are no impediments to the implementation of an appropriate response, then we ought to use A10. Minimally, then, our analysis identifies conditions appropriate for A10 as sufficient for multicountry system stability, and the appropriateness of classical solution hypotheses as sufficient for predicting the eventual emergence of bipolar systems.

3.5

Appendix: Proof of Theorem 3.2

To establish sufficiency, we will show that (S; r) is system stable if / is essential for all i 6 5. Remark 3.5 shows that if rt = R/2 for some / in S then (S; r) is system stable. Thus, suppose that rt < R/2 for all / in S. First, we let g(C'h Ck) = r(Ck) denote the resources that C\ can secure from Ck if it defeats Ck, assuming, of course, that Q g S - C / . Let (r',C) be a threat against Ck by C'h ieC'k9 and let y = arg max [rk], keS-{i]

which means that j is the country with the greatest resources in 5— {/} (if j is not unique, then we can arbitrarily choose one such j). There are now two cases to be considered. Case 1: ieE

and rt>—2

max [rj\. jeS-U)

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3 System stability and the balance of power

Suppose that jeCj, so j is the country with the greatest resources in S— {/} and j is part of the threatening coalition. By assumption, / has enough resources to transfer the amount R/2 — Tj to j , thereby setting j's resources equal to R/2, From A6 and Remark 3.6, R/2 is the maximum most preferred outcome possible for j . And since / prefers any outcome in which rt > 0 to one in which rt = 0, then / and j both prefer the transfer to the outcome of the attack. Hence, letting C"m= [ij] and Q' = 0, K = {i} has a counter to the threat (r', C ) . The counter is viable since, again from Remark 3.6 and A6, j most prefers the transfer to any other feasible outcome, and therefore j cannot be better off by excluding / from [/, j}. Now suppose that j $ Cj, and let C^ = {/, j} and C/J = Cj. We know that rt + rj>R/2, so (r",C") is a threat against Q' by C^, which also implies that r">ri. By Remark 3.6, j can secure at most R/2 — rj additional resources. Hence, (r", C") is a counter to (r', C ) by K= {/}. And, as before, the counter is viable, since j cannot do better than secure R/2 by acting alone. Case 2: ieE

but

/*/ rf for all jeC0 is impossible. Accordingly, (r", C") is a viable counter for /. The second possibility is that the inequality in expression (3.A1) becomes " /*/, for all jeC0. However, since Coe W*, r(Co) > r(CJU(/)), which implies that (r0, Co) is a threat to Q'U {/}. Moreover, g(C0, C^U{/}) > g(C^, C£), and since resources are infinitely divisible and transferable, all members of Co can make themselves better off in (ro, Co) than in (r", C"), which contradicts the conditions of a viable counterthreat. Q.E.D.

CHAPTER 4

Resource stability and the balance of power

... disagreements about how benefits should be distributed permeate the relations among actors and persist because bargains are never permanently valid Furthermore, this struggle to make others adjust is played repeatedly. Apparent victory can be illusory or defeat ephemeral, for political bargaining and maneuver result not in definitive choices conferring power on some people rather than others, but in agreements that may in the future be reversed or in discord that signals a continuation of bargaining and maneuver. Robert O. Keohane, After Hegemony (1984, p. 18)

This discovery of instability. ..points up sharply the contrast between economic activity, most of the models for which are self-equilibrating or assume some kind of "dynamic" equilibrium, and political activity, where fundamental instability seems inherent and ineradicable. William H. Riker, The Theory of Political Coalitions (1962, pp. 173-4)

4.1

The relevance of system stability to resource stability

Hitherto, our analysis has focused simply on the first part of the game, which concerns international actors. This focus is especially important because it deals with the fundamental issue of sovereignty and the survival of regimes, and it shows how such survival is ensured in anarchic systems. The central thesis of this volume, however, is that there are two types of instability, and we should also be concerned about the possibility that the second type - resource instability - can also upset systems and lead to conflict. Echoing the observation from Keohane, Gilpin (1981, p. 13) hints not only at the necessity for distinguishing between system and resource stability, but also at the pervasiveness of resource instability in systems that are otherwise stable: In every international system there are continual occurrences of political, economic, and technological changes that promise gains and threaten losses for one or another actor. In most cases these potential gains and losses are minor, and only incremental adjustments are necessary in order to take account of them. Such changes take place within the existing international system, producing a condition 115

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4 Resource stability and the balance of power

of homeostatic equilibrium. The relative stability of the system is, in fact, largely determined by its capacity to adjust to the demands of actors affected by changing political and environmental conditions. In every system, therefore, a process of disequilibrium and adjustment is constantly taking place. In the absence of large potential net benefits from change, the system continues to remain in a state of equilibrium.

In a similar vein, Kennedy (1987, p. 73), referring to the period 1519-1659 as one of "short-term, shifting alliances," suggests the "microinstability," encapsulated by our notion of resource instability, which seems an inevitable part of "macrostable" systems, that is, system-stable systems. Despite such tantalizing hints in the literature, system and resource stability, by and large, have not been analytically distinguished. Consequently, some scholars believe that cooperative game theory, whenever applied to situations in which the total of resources isfixed,yields a constant-sum game in which no balance of power is possible, and in which dominant alliances war on and absorb the rest. This supposition not only runs contrary to our argument that the forces of self-interest can ensure essential countries against absorption by other essential countries and, hence, elimination, but if the argument presented in the previous chapter is genuinely compelling then our attention must be drawn to a further implication: the dynamics of system stability affect the set of feasible redistributions. To see what we mean by this, notice that our analysis thus far can be interpreted as modeling the contemplations of actors prior to anyone actually making a threat or initiating a transfer of resources. Although it does not tell us what alliances, if any, will form, or what specific reallocations of resources will be implemented by the various coalitions, if everyone has complete information about the situation, so that everyone knows that essential countries cannot be eliminated, then this act should be reflected in any subsequent negotiations and resource transfers. Thus, the baseline of negotiations should not be the elimination of essential countries; that is, it should not include setting rt = 0 for any country / in E. Instead, that baseline should be the amount of resources countries can ensure for themselves, given that they will not be eliminated. To illustrate this implication using some elementary notation from cooperative game theory, if the initial distribution of resources is $50 to each of three players, and if all resources are subject to expropriation and redistribution, then the "value" of a single player, v(i) - /'s security level if it joins no coalition - equals zero since an individual can guarantee nothing under simple majority rule, whereas the value of any winning coalition is $150, the total of all available resources. Thus, the "negotiable range" - the set of feasible payoffs to any individual player - varies theoretically from zero to $150. In this instance, we can appeal to any

4.1 The relevance of system stability to resource stability

117

number of solution hypotheses to predict that a minimal winning coalition will form and "split the difference": its two members will divide the $150 evenly (assuming that we can equate the measure of a player's utility with the measure of how much money he or she possesses). Of course, if the distribution (75,75,0) appears imminent then player 3 could propose (0,80,70), which 2 and 3 unanimously prefer to the original proposal. However, now 1 could counter with (75,0,75), to which 2 counters with (0,70,80), and so on, ad infinitum. Thus, in this instance, the game is both system and resource unstable: players are threatened with elimination, and no resource distribution is immune from challenge. However, the assignment of "value" to alliances in this example, the corresponding prediction about coalitions, and the arguments about the game's inherent instabilities are legitimate only if players can secure more than half the resources and if the players are indifferent about how they secure their resources. If, as in our model, a player prefers to receive resources through negotiation rather than through conflict, and if everyone acts on the presumption that no one can secure more than half the total, then if any player agrees to cede half of his or her initial endowment of $50 to another, the game ends in the sense that no further reallocations are possible. Hence, rather than setting the security value of /, v(i), equal to 0, we should set this value equal to $25, which is what any player will be left with in the event that it must cede half of his or her initial endowment to counter a threat and forestall elimination. This change in the representation of the implicit and explicit bargaining game that affects nations is fundamental to our understanding of international political processes, since it is no longer evident that such a game is necessarily resource unstable. Instead, we must analyze this new game to learn its properties with respect to resource stability. Hence, in the next section we discuss situations that are system stable but that may not be resource stable: systems in which the distribution of resources remains the subject of negotiation, but in which each country is initially essential and can thereby ensure its own survival. Our analysis will establish the necessary and sufficient conditions for resource stability and predict what distribution of resources will arise from an initial distribution. As we note in the previous chapter, researchers offer a variety of speculations about the details of balance-of-power systems, including assertions that relate to the relative stability of large versus small systems, and the question of whether equality or inequality of resources among countries best fosters stability. Chapter 3 addresses such issues in the context of system stability, and we find no necessary relationship between the number of countries and system stability, or between the precise distribution of resources and this type of stability. This chapter, then, will also address

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4 Resource stability and the balance of power

these same issues in the context of resource stability, and we seek answers to the following questions: Are three-country systems more resource stable than ^-country systems? Is there any "magic" number of countries that ensures resource stability? Does such stability require equality or near equality in the distribution of resources? Finally, to be certain that our model makes sense in the context of cooperative bargaining processes, we shall conclude with an experimental test of the ideas presented in this chapter and Chapter 3. 4.2

The nature of resource stability and instability

Suppose that all nations are essential, so that the survival of regimes or countries is not an issue and the purpose of alliances is simply to coerce a more advantageous position with respect to the overall distribution of resources. What we now wish to consider is the bargaining that might occur within an alliance if it seeks to extract resources from some other set of countries. The literature on bargaining and experimental games contains a great many ad hoc hypotheses about how an alliance's members might divide their "winnings," such as that those winnings accrue to the largest member of the alliance, that members divide winnings in proportion to their relative weight, that these winnings are divided on the basis of a player's likelihood of being the pivotal member of a winning coalition, or that equitable (i.e., equal division) outcomes prevail. Here, however, we take a more game-theoretic view, and we suppose instead that those resources are redistributed among the alliance's members on the basis of what each member might have received if some other coalition had formed. To transform this somewhat imprecisely stated idea into a formal analysis, there are at least four important theoretical questions a game theorist would ask about the situation. First, how in general can we represent cooperative games that take account of the analysis given in our previous chapter? That is, how do we formally represent a cooperative game among essential countries that reflects Assumptions Al through A10? Second, will such games have cores? This question is merely a technical way of asking whether there are any equilibrium resource distributions - resource distributions that naturally suggest themselves to negotiators and that, once arrived at, cannot be upset by any other proposal. Negotiations in cooperative games without cores, though, can cycle endlessly through an indefinite (possibly finite, possibly infinite) series of proposals as each proposal on the table is upset successively by some coalition with a new proposal. Third, if no core exists, but if alternative distributions are arrived at by some sequential recontracting process, then what are the end

4.2 The nature of resource stability and instability

119

points, if any, of this process? That is, if there is no natural "sticking point" to which negotiations among essential countries might converge, will negotiations in fact proceed endlessly or will certain types of outcomes emerge? Finally, might essential countries become inessential in the process of negotiation? To illustrate how we can go about answering these questions, suppose that the set of countries is S= {1,2,3 j and that the initial distribution is r = (120,100,80). In accordance with Theorem 3.1, this system is system stable. Suppose now that the coalition {1,2} tentatively forms with the intention of threatening 3 with elimination or at least in order to coerce 3 out of some of its resources. Notice that one possible response is that 3 offers to cede 30 units to 1, at which point a resource-stable distribution (150,100,50) results. However, we certainly should take some account of the fact that country 2 could form an alliance with 3 instead to expropriate 50 units from 1, in which case 1 can respond by noting that it can coalesce with 3 to expropriate 30 units from 2. Thus, each country has bargaining advantages and disadvantages of different sorts. Country 3 is disadvantaged in that none of these coalitions permit it to secure additional resources. Yet, if 3 is excluded from any coalition, so that only {1,2} is feasible, then country 2 cannot secure additional resources either. Thus, 3 (being essential) can place some demands on 1 or 2 in the event either approaches it to form an alliance. Although it is still too early to suggest that the situation we model here is precisely reflected in the empirical world, what we seek to model in this situation are the systemic bargaining implications of the bargaining position of a weaker (here, the weakest) but essential power in an international system. Abstracting for a moment from the number of essential members of such a system, the situation we have in mind is similar to that of, say, Austria-Hungary with regard to the other great powers (particularly Germany) in the last quarter of the nineteenth century, or England in its dealings with Spain, Portugal, France, and the Holy Roman Empire at the dawn of the sixteenth century. The three-country situation we have described corresponds to a threeperson cooperative bargaining game that we can model in the usual characteristic function form representation of ^-person cooperative games. However, for reasons we have already noted, some care must be taken in defining that function. Traditionally, a coalition's value is set equal to whatever it receives if the excluded players act in the coalition's worst interests, even if that means that these players act irrationally. If, for example, coalition {2,3} forms and 1 must play alone as a "single-member" coalition, then 1 cannot ensure against the possibility of being eliminated by {2,3), in which case it seems appropriate that we set v(l) = 0. Yet, we

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4 Resource stability and the balance of power

already know that in this circumstance, country 1 should offer to cede 50 units of resources to country 2. Country 2 now has a dominant choice accept the offer. This strategy ensures l's continued sovereignty and gives 2 its most preferred feasible outcome. This argument leads to the conclusion that we should set v(\) equal to 120, less the 50 units it may be required to cede in order to exist; that is, v(l) = 70. Continuing with this definition of the characteristic function, we set v(2) = 70 (acting alone, it must cede 30 units to 1), and v(3) = 50 (acting alone, it must cede 30 units to 1). With respect to coalition [1,2), although it has the ability to absorb 3, in evaluating ^(1,2) we should be cognizant that 3 can disrupt the coalition by ceding 30 units to 1. Thus, if {1,2} assumes that 3 responds rationally to its formation (and recall that responses can be made instantaneously in our model), its members know that the coalition can expropriate at most 30 units from 3. Thus, v(l, 2) = 220 (the resources it controls originally) plus 30 (the resources it can secure from 3), which totals 250. By a similar argument, we should set v(l, 3) and v(2,3) each equal to 230. Finally, we set v(l, 2,3) equal to 300, since the system's resources are constant and we assume that resources cannot be destroyed (or, equivalently, that countries are concerned solely with proportion of resources controlled). It is evident that the corresponding characteristic function, based on the idea that countries respond rationally to offers and counteroffers, yields a zero-sum cooperative game that has no core. As before, outcomes can cycle endlessly. If the distribution (150,100,50) is proposed, 2 can offer 3 (70,150, 80) instead, to which 1 can counter with (150,70,80), to which 3 might demand (50,100,150), and so on. Notice, however, that this instability is of a different sort than what Riker (1962) asserts characterizes international systems. Specifically, it does not support the conclusion that balance of power is a fiction, at least insofar as the existence of sovereign countries is concerned, since the survival of countries is not an issue in this game. That is, the results of our previous chapter justify the assumption that essential countries will not be eliminated, and this fact has been incorporated into the definition of v(C). Instead, we are now focusing on ascertaining final distributions of resources that can be arrived at through negotiation and through the "threat" of war as against its actual implementation. With respect to the general idea of international instability and conflict, then, the direct manifestations of resource instability are likely to correspond less to overt conflict than will manifestations of system instability. System instability threatens sovereignty; resource instability merely implies that all countries are positioned to lose as well as win resources from each other. This does not mean, though, that resource instability is not dangerous to peace. On the contrary, the tactics whereby countries seek

4.2 The nature of resource stability and instability

121

to extract economic and military advantage from others are the threats and counterthreats defined in the previous chapter, and thus, in a resourceunstable world, countries are necessarily in a state of muted conflict and competition. The complexity this instability occasions is well articulated by Waltz (1979, pp. 165-6): With more than two states, the politics of power turn on the diplomacy by which alliances are made, maintained, and disrupted. Flexibility of alignment means both that the country one is wooing may prefer another suitor and that one's present alliance partner may defect. Flexibility of alignment narrows one's choice of policies. A state's strategy must please a potential or satisfy a present partnerf,] ... strategy is at least partly made for the sake of attracting and holding allies... [and] if pressures are strong enough, a state will deal with almost anyone. In the quest for security, alliances have to be made... by states that have some but not all of their interests in common. The common interest is ordinarily a negative one: fear of other states. To ascertain which specific alliances and distributions might prevail in this complex and competitive environment, we must turn to the various solution concepts offered by game theorists to solve cooperative games in characteristic function form. However, rather than review these hypotheses here as they might pertain to the specific problem at hand, it is merely sufficient to note that each of them takes the view, first, that a unique outcome cannot be predicted. These hypotheses, however, do offer a response to Waltz's rhetorical question: "The three-body problem has yet to be solved by physicists. Can political scientists or policy makers hope to do better in charting the course of three or more interacting states? " (1979, pp. 192-3). The answer is "yes" at least insofar as we can narrow the set of feasible possibilities to a significantly smaller set. This narrowing occurs through the imposition of restrictions on predicted outcomes that seem feasible (e.g., no outcome should prevail in which all of the members of the coalition that bring it about unanimously prefer some other outcome; or, if two coalitions are possible, then those players who pivot between them should not all strictly prefer the proposal of one coalition to the other). Fortunately, despite a seeming plethora of competing hypotheses (e.g., main-simple V-set, strong bargaining set, competitive solution, aspiration bargaining set), each of these solution concepts predicts for the preceding example that one of the following three distributions will prevail: (125,70,105) via coalition {1,3}; (125,125,50) via coalition {1,2); (70,125,105) via coalition {2,3}. As is usual in cooperative games without an equilibrium such as the one defined by the concept of the core, our predictions are not unique,

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4 Resource stability and the balance of power

but instead take the form "if the alliance C forms, it will establish the new resource distribution..., if the alliance C" forms, it will establish..." and so on. More importantly, notice that, after reflecting the relative bargaining positions of the players, we do not predict that one country will achieve half the total of resources. Instead, we predict that if, say, {1,3} forms then 2 loses the amount it would have to cede to 1 to end the game. However, these 30 units do not all accrue to player 1, because player 2 also has a claim to receiving some resources (such as the threat that {2,3) forms against 1). Hence, 1 and 3 divide these 30 units 5 to 25. Similarly, if 1 and 2 coalesce then 3 loses the 30 units it must cede to ensure survival, but now 1 and 2 divide this amount 5 to 25. Finally, if {2,3} forms then 1 must give up 50 units, which 2 and 3 divide evenly. Notice, however, that these predictions may be valid only in the "short run." For example, if (125,70,105) prevails, then renegotiation can reoccur with a new characteristic function that reflects the new distribution resulting from the previous bargaining round. Based on our previous arguments, then, the corresponding characteristic function is v(l, 2) = 220,

v(l, 3) = 255,

v(2,3) = 220;

That is, country 1 must now cede 45 units to 3 in order to ensure survival, which means that 1 can guarantee itself 80; country 2 must cede 25 to 1, so it can guarantee itself 45; and country 3 must cede 25 to 1, so it is assured of receiving no less than 80. Applying any of the cooperative solution concepts mentioned earlier to this representation of the game yields the following three distributions as predictions: (127.5,92.5,80) via {1,2}; (127.5,45,127.5) via {1,3}; (80,92.5,127.5) via {2,3}. This example leads to several important observations that warrant generalization. First, it follows from a well-known result in game theory about the nonexistence of cores for zero- or constant-sum ^-person cooperative games that since we assume that resources can be neither destroyed nor created, there is no distribution that cannot be upset by some alternative proposal (provided, as in our example, that no country controls precisely half the resources). Put differently, and in accord with the intuition that Waltz reveals in the previously cited quotation: if the survival of all countries is assured, then regardless of which alliance is proposed, there always exists another proposal that the members of some other alliance all prefer and that they can secure.

4.2 The nature of resource stability and instability

123

Second, unless one country can somehow secure one-half the resources in our example, this process of negotiation, dissolution of alliances, and renegotiation will continually change the distribution of resources. The only distribution corresponding here to a resource-stable system is one in which rt = R/2 for some /, so "rt = R/2 for some /" is both a necessary and a sufficient condition for resource stability. That is, in accord with the quotation from Keohane that introduces this chapter, if the existence of all countries is assured, then no alliance is permanent. Instead, we should observe the constant shifting of alliances as countries jockey for more advantageous positions. As Claude observed, "for balance of power, competitive struggle is the general condition, to be dealt with by the realization of cooperative arrangements within limited groups" (1962, p. 145). Indeed, it is precisely this dynamic that led Bismarck to summarize the implications of a balance-of-power system in the dictum that "all politics reduces itself to this formula: Try to be a trois as long as the world is governed by the unstable equilibrium of five Great Powers" (quoted in Joll 1984, p. 37). As well, this remark accounts considerably for the fluidity of the late nineteenth- and early twentieth-century European alliance system. It is exactly this constant shifting and the threats to security it engendered that led to British involvement in "entente" rather than alliance with France and Russia and that reduced Italian participation in the Triple Alliance to "ententelike" behavior. One caveat we offer about this implication is that, because time is not an explicit variable in our analysis, we cannot say anything about the temporal duration of alliances. In reality, of course, they may endure long enough so that technology and the natural growth rates of countries alter the distribution of resources in a way that essentially "resets" the system. Certainly, the economic growth of West Germany, Japan, Korea, and Taiwan - in conjunction with a stagnant Soviet economy and nearly stagnant Eastern European economies - has altered the relative resources of whatever alliances emerged from World War II. The consequences of this change are now being felt as the Soviets try not only to revise the structure of their economy, but also to negotiate a position for themselves that allows them to recoup their lost position. Finally, notice that a fully resource-stable system - one in which one country controls precisely half the total resources - can emerge only in the limit of an infinite sequence of alliances and negotiations. That is, barring differential domestic resources growth (and error on the part of negotiators), in the negotiations for resources among countries, no country emerges as predominant or even near-predominant. Hence, this feature of our example, if it holds true generally, implies that leaders who seek to render their regimes or countries predominant have but two alternative

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4 Resource stability and the balance of power

courses of action. If they believe that the intelligence of their opponents is flawed, then they can try to secure their objective by nearly any means at their disposal, including the negotiation of successively more advantageous resource allocations; but if stupidity and misinformation do not clear their path, they must rely instead on internal sources of growth - on the dynamism of their economies, on their willingness to sacrifice consumption for military power, and, perhaps, on the inevitability of certain sociopolitical imperatives in which they believe. 4.3

A necessary and sufficient condition for resource stability

To establish the generality of these assertions, which are based on the analysis of a single example, we must first formulate the general form of the resource allocation game's characteristic function: the value of each alternative coalition relative to all others. Letting r be a vector that denotes the initial distribution, we assume first that all countries are essential. That is, we suppose that inessential countries have been eliminated and that their resources have been reallocated to yield r. Letting C be a winning coalition (r(C) > R/2), and letting e be an arbitrarily small positive number, if we take Theorem 3.2 as an operating constraint (assuming that no country in S can have its resources set equal to zero) then we have two possibilities. First, if S—C controls enough resources to render the largest member of C near-predominant, then C cannot guarantee that it receives more than the difference between R/2 and the resources of this member. And since S—C can always "freeze" the system with such a transfer, we set v(S-C) equal to r(S-C) minus the resources it must cede. Second, if S- C hasn't enough resources to render the largest member of C near-predominant, then one or more members of S—C can counter a threat from C only by attracting one or more of the members of C to some new coalition. However, all we know at this point is that, because everyone is essential, no member of S— Cs resources will be set equal to zero (Theorem 3.2). Thus, we set v(S—C) = e and v(C) = R — e since one or more members of S—C may find it necessary to cede an unspecified amount of its resources in forming the counter. Restating this argument in terms of our notation, we have: if ieC

R—e r(S-C)-

r(S-C)>~-msix[ri]i

/

*

ieC

otherwise; 'R \ ( —-max[rt])

if r(S-C)>

/

otherwise;

(4.1) R — -max[rj, 2

ieC

(4.2)

4.3 A necessary and sufficient condition for resource stability

125

v(S) = R; and if C is blocking - if C controls exactly half the resources then S- C is blocking and v(C) = v(S- C) = r(C). Thus, if country / is large enough to render the largest member of S— {/} near-predominant, then v(i) equals rt minus the minimum resources it must cede to render someone near-predominant; on the other hand, if / cannot by itself render anyone near-predominant, then v(i) = e. Notice, moreover, that if we set e = 0 but no longer interpret rt = 0 as implying the elimination of /, then the preceding characteristic function is identical to the usual one used to model a simple majority-rule game, but with the important exception that v(S- C) > 0 if S - C is large enough to "buy" stability. We proceed now by establishing three results about games with this representation of coalitional value. The first result, which follows trivially from expressions (4.1) and (4.2), states simply that the corresponding game is an essential constant-sum game - that v(C) + v(S— C) equals a constant for all C and there is at least one pair of coalitions that gains something by joining. Games of this sort cannot have a core, and because this property of essential is satisfied whenever rt < R/2 for all / (whenever no player controls half or more of the resources), then r{ < R/2 for all / in S implies that (S;r) cannot be resource stable. The second result states that negotiations will necessarily change the distribution of resources unless one country controls exactly half the resources. Thus, rt-=R/2 for some / is a necessary and sufficient condition for resource stability. The last result states that if the agreed-to reallocations of resources correspond to the predictions of any one of several solution hypotheses offered by game theorists about such situations then, regardless of what negotiated outcome prevails, if no country controls half the resources when negotiations begin, no country will control half the resources when negotiations end. Thus, barring changes that are exogenous to the analysis, resource instability is a permanent feature of international political systems. Theorem 4.1. If rt < R/2 for all / e S, then the cooperative «-person game among the essential countries described by expressions (4.1) and (4.2) is an essential constant-sum game, and thus has no core. Theorem 4.1 corresponds to Riker's assertion that "no rules of balance can be formulated for (essential) ^-person zero-sum games" (1962, p. 169) and it reasserts his conclusion about instability cited in the introduction to this chapter. However, it is imperative that we keep in mind that Theorem 3.2 implies that this conclusion does not apply to system stability. That is, resource instability does not imply system instability. Also, Riker fails to consider the possibility that the game among nations becomes inessential whenever one nation is nearly predominant. That is, if one country controls exactly half the resources then no threats are possible, and

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4 Resource stability and the balance of power

thus no reallocations of resources can occur: systems with a nearly predominant country are both system and resource stable (notice that with a nearly predominant country, all countries are essential). Stated as a formal result with r' denoting the negotiated outcome: Theorem 4.2. For the game in characteristic function form described by expressions (4.1) and (4.2), r] ^ ry- for two or more j e S unless rt = R/2 for some /. Next, we address the issue of whether the condition for resource stability, that rt = R/2 for some /, can ever emerge in a system that is not initially resource stable. In the statement of our next result, Theorem 4.3, we focus on the main-simple V-set as a vehicle for predicting payoffs, since such payoffs correspond to both the competitive solution and to the strong bargaining set (McKelvey, Ordeshook, and Winer 1978).! Theorem 4.3. If r is the initial distribution of resources, if rt< R/2 for all ie S, if r' is the negotiated outcome, and if r' is an element of a main-simple V-set, then r\ 2, a uniform resource distribution is not sufficient for resource stability or necessary for system stability, nor does the emergence of either form of stability produce such uniformity. If we apply any of the usual game-theoretic solution hypotheses to the characteristic functions of our examples, a uniform distribution of resources does not emerge as a predicted outcome. Moreover, even if such a distribution emerges as the result of, say, the dynamics of domestic economic growth, it cannot correspond to a resource-stable outcome. If our assumptions have any empirical validity then coalitions can extract resources from excluded countries, and we should anticipate that unless the costs of international conflict are sufficiently high (and decision makers are concerned with absolute as well as relative resources), countries will maneuver for advantage. A third conclusion we can draw from the analysis is that neither system nor resource stability precludes the possibility that no nation or combination of nations has the ability to defeat some other nation. Clearly, the implied algebraic identity of such a requirement cannot be satisfied in any constant-sum game if the number n of countries in S is greater than or equal to 3. Rather, balance is achieved by a nation's ability either to "buy off" a key member of a threatening coalition or to propose an alternative coalition that makes its members better off than they would otherwise be 3

Alternatively, we might interpret Waltz's hypothesis to mean that the "extent" of the instabilities is less with broadly asymmetric distributions of resources. Of course, the meaning of the word "extent" is ambiguous, but we can give it a formal interpretation. Consider the distribution r and suppose that from any other distribution TQ there exists a path (ri» T2,..., rm) such that there is a winning coalition that prefers ri to ro, a winning coalition that prefers r2 to r i , . . . , and a winning coalition that prefers ro to rm. That is, regardless of where we begin, we can find a way to reach ro. If this is true for any possible resource distribution in the system, then we will say that the "extent of the instability" is total, whereas systems with a smaller set of points that satisfies this property is "less unstable." Unfortunately, it is easy to see that the initial distribution of resources is irrelevant to this degree of instability - if we can get to any distribution from any starting point then any distribution can become a new status quo point. Indeed, as long as no country controls half or more of the resources, then it is a simple corollary of McKelvey's (1979) result, cited in Chaper 1, that all systems are equally unstable: every distribution can be reached from any starting point.

4.4 General implications

129

if the original threat succeeded. To this extent, then, the analysis agrees with the view that system stability is maintained primarily by the skillful formation of alliances and counteralliances, or (more precisely) by the threat of such alliances. Organski and Kugler are thus correct to observe that "the major mechanism through which the balance-of-power system is maintained is the making and unmaking of alliances" (1980, p. 16). However, as we noted in Chapter 2, they offer the wrong explanation for this fact: The reason for this dependence on coalitions in order to change the distribution of power is that the power resources of each member of the system are viewed as inelastic. There is no way a nation can increase its own strength very much except by adding its allies' strength to its own, or by decreasing its adversaries' strength by separating it, through persuasion, bribery, or subversion, from its allies (1980, p. 16). Further, The balance of power...models argue [that]...the units of the system do not change (at least not much): they simply combine in different ways, and different distributions are the result of such combinations (1980, p. 24).

If our theory were thus restricted then Organski and Kugler would be correct in their critique; but, if anything, our analysis allows too much flexibility in the transfer of resources among countries. In any event, allowing such transfers makes the formation of coalitions more, not less, profitable and likely. The preceding discussion thus serves to emphasize that our analysis presumes that all decision makers are cognizant of the game they are playing, including the strategies available to all other decision makers, and that they anticipate and take account of the feasible responses of others to any challenge. Agreements reached among nations should reflect this knowledge, and presumably "skillful negotiation" entails making certain that all other participants are aware of one's advantages and options. Keep in mind, moreover, that our analysis presents decision makers with the simplest environment. If uncertainty takes the form of a lack of perfect information about the situation - about the preferences and perceptions of others - then, as we noted earlier, strategic complexity is increased considerably. This, of course, gives an even greater premium to strategic skills. At this point, it is worth commenting on Morgenthau's (1959, p. 171) observation that "The opposition of two alliances, one or both pursuing imperialistic goals and defending the independence of their members against the imperialistic aspirations of the other coalition, is the most frequent configuration within the system of the balance of power." Morgenthau subsequently qualifies this assertion to include a role for a third,

130

4 Resource stability and the balance of power

balancing country, and indeed we do not find within our analysis reasons for supposing that bipolarity necessarily prevails. Certainly, the various cooperative solution hypotheses to which we have appealed predict that a zero-sum redistribution game, if played once, yields a minimal winning coalition expropriating from those who have been excluded. This establishes at least one force for bipolarity. On the other hand, we prefer to be somewhat more ambiguous in our predictions since the game nations play in system-stable systems is a recurring process of negotiation and renegotiation. Indeed, in this fluid environment it may be difficult to say precisely who is a member of what coalition at any specific point in time, and our impression is that this is often a more accurate characterization of international affairs than one that sees countries cleanly aligned into two opposing alliances. Fourth, if the number of countries exceeds 2, then a uniform distribution implies only system but not resource stability. System stability requires simply that no nation be so weak that it is inessential - a condition that is satisfied if resources are uniformly distributed - whereas a system is both system and resource stable if and only if one country controls exactly half the resources. Hence, aggressive choices are precluded only if r; = R/2 for one country. Otherwise, a system will be either system or resource unstable, in which case wars remain a possibility unless nations can cede resources without armed conflict. This conclusion, then, undermines the equating of the absence of wars with uniform distributions of power (see, e.g., Claude 1962). It is certainly not the case that the balanceof-power theory offered in this volume "suggests that when power is more or less equally distributed among great powers or members of major alliances [that] peace will ensue" (Organski and Kugler 1980, p. 14). Aside from the fact that uniform distributions are sufficient for system but not resource stability, our analysis shows that the precise nature of the resource distribution is largely irrelevant to whether a system is system or resource stable, and thereby it is consistent with the inference Organski and Kugler draw from their data that "power distributions are... not a predicator of the coming of war" (1980, p. 49). Fifth, no "magic" number of countries is required for the attainment of system or resource stability. It is the traditional wisdom that in threecountry systems, two of the powers can easily ally, threaten, and even conquer the third, divide the spoils, and drive the system back to bipolarity. Thus, this argument implies that systems must contain at least four countries to be stable. Yet, our analysis shows that as long as resources are distributed to satisfy the conditions of Theorem 4.2, three-country systems are no more or less resource stable than ^-country systems. What this wisdom seems to have forgotten is that if the third country is capable

4.4 General implications

131

of anticipating threats, it can propose counter alliances to any threat or, at the very worst, offer to cede resources to one of the countries posing a threat. Certainly, the Soviets can anticipate the consequences of an SinoAmerican alliance, just as the United States can anticipate the implications of Soviet hegemony over all of Asia. Just as the United States may be willing to grant various economic benefits to China in order to forestall its moving too closely to the Kremlin, the Kremlin can seek to negotiate its border disputes with China. China, being the willing recipient of this largess, need only be concerned that an American-Soviet rapproachement weaken its pivotal position so as to provide the Soviets with an excuse to reassert their territorial claims and to reduce the United States' desire to aid in China's industrial advancement. We should not, of course, carry the implications of our analysis beyond its limitations. Certainly, as the number of relevant essential countries increases, the transaction costs of negotiating and enforcing agreements will increase as well; and the increasing complexity will make miscalculation more likely. Thus, in reality, smaller systems may have an advantage over larger ones with respect to the ease with which system-preserving negotiations can proceed, which is to say that the intuition that small systems are more stable than larger ones may indeed be correct. Our analysis merely asserts that this differential stability must be explained by considerations that are currently exogenous to our analysis and that our analysis reveals the feasibility of stable systems of any size. Hence, schemes for inducing peace need not rely on the prior existence of any magical number of primary states. Before we review the results of an experimental test of our theoretical structure, we should comment on one aspect of system-stable, resourceunstable systems. It should be evident from what we have said thus far that we regard the outcomes associated with resource instability to be less traumatic from the perspective of wars and armed conflict than those we associate with system instability. This should not be interpreted to mean, however, that resource instability is not dangerous to peace. Threats and counterthreats as a means of securing advantage continue to play a role in resource-unstable systems even if all contries are essential. Although in the perfect and complete information context of our theory these threats will not result in the elimination of such countries, we should anticipate that those informational imperfections that are likely to characterize reality, as well as the constraints on action imposed by domestic political considerations, can upset our calculations. Hence, reality may magnify the consequences of the resource-reallocation game allowed by resource instability and can lead, as a result, to conflict that was otherwise unintended.

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4 Resource stability and the balance of power

4.5

Some experimental evidence

The distortions of reality occasioned by our assumptions in this and in the previous chapters are apparent. The one assumption that is especially difficult to evaluate, however, is A10 - the imperatives of bargaining that we assume. This is not an assumption that can be readily studied by looking at actual international relations processes. Although evaluating our analysis's performance with respect to the data generated by such processes provides an indirect test, such testing does not permit us to focus on any such specific assumption - the model may fail because other assumptions are inappropriate. And to this point, our justification for A10 is that it is a reasonable extension to the particular problem at hand of bargaining process theory encapsulated by bargaining set theory for cooperative Ai-person games in a context (transferable utility games) in which that theory yields predictions corresponding to the predictions of competing ideas (the main-simple V-set and the competitive solution). What we want to do in this section is provide, via some laboratory experimentation, a bit more confidence in our assumption; thus, we turn to an experimental test of the necessary and sufficient conditions for system and resource stability.4 Here we describe the design of an experimental "balance-of-power" game that we use to test three specific hypotheses about our theory. Section 4.6 reports our data and Section 4.7 offers the instructions read to subjects. Before proceeding, however, a note of caution is in order about how we interpret these experiments. We emphasize that they are not simulations nor are they designed to test the theory, although extreme departures from predictions would certainly be cause for alarm. Our experimental design is not equipped, for example, to accommodate the fact that countries are collectives we have abstractly chosen to treat as a single decision-making entity, because in the experiments we report, the role of countries isfilledby single individuals. Thus, we cannot test the veracity of this anthropomorphic assumption or the effects on our predictions of national decisions being controlled by a collectivity governed by rules of their own creation. Instead, these experiments are designed merely to see if we can have any confidence in a small part of our analysis - Assumption A10 and its implications for bargaining. The experiments use undergraduates at the University of Texas at Austin as subjects. Each experiment begins by randomly assigning numbers to the subjects after seating them face to face at a large conference table. 4

This section and the experimental results we report are taken from Niou and Ordeshook (1988).

4.5 Some experimental evidence

133

The rules of the experiment are as follows: the experimenter begins by revealing to all subjects how 300 divisible and transferable points are initially divided among them, where this division is determined beforehand by the random assignment. Subjects are told that an experiment ends when there is unanimous agreement on the part of the active players (those with positive points) to do so; hence, the duration of an experiment is endogenous (each lasted between one and two hours). 5 Each subject's final dollar payoff from participating is determined by the points he or she controls at the termination of the experiment. To ensure that subjects are motivated solely by the resources they control rather than by, say, the desire to extract as much money as possible from the experimenter, at the end of the experiment each subject converts the points he or she controls at that time into money according to an exchange rate specified in a sealed envelope. Exchange rates differ among subjects, and although everyone is aware of this fact, no one knows his or her own exchange rate or the rate of anyone else. Thus, the dollar value of 10 points depends on which subject controls those points. Also, to minimize the possibility that equity considerations distort the preferences we seek to induce, no discussion of schemes for dividing money after the termination of the experiment is allowed. Each subject is paid in secret, so there is no need for one subject to ever learn how much others earned. The average net payoff to each subject in an experiment was six dollars (no subject was allowed to lose money). An especially critical issue concerns how subjects secure points from others, since here experimental methodologies are fraught with difficulty. On the one hand, we must ensure that subjects play the game addressed by our theory. On the other hand, there is the danger that subjects are so constrained that it is impossible for them to act contrary to the hypotheses under consideration. Since this is the first test of this particular model, we are concerned more with the first consideration than the second, and we proceed under the supposition that if subjects do not act as predicted when constrained, it is unlikely that they would act in accordance with the model when the various rules are wholly endogenous. First, then, in order to secure points from another subject, we require for the allocation resources that one or more of the subjects make a threat that proposes to take all or part of the points held by one or more other players. Those who attempt to expropriate points are called the Attack Group; those who are their target, the Target Group. A threat must satisfy three 5

Although an experiment must end, in actual political processes, of course, there is no end to the game. We are uncertain about the effects of this feature of the experiment, and follow-up research might consider a procedure in which the experiment is terminated at a time unknown to the subjects.

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4 Resource stability and the balance of power

conditions: (1) it must specify how many points are to be taken away and from whom (the Target Group); (2) it must specify how the points are to be distributed among members of the Attack Group; and (3) the Attack Group must outweigh the Target Group. Our theory presumes that the Target Group can thwart an attack if it can formulate an appropriate counterthreat. In accordance with our definitions, our experimental procedures must allow for two types of counterthreats: Type 1: A proposal by one or more members of the Target Group to give points to some other player or players who are not in the Target Group and that was agreed to by those player (s). Type 2: A new threat, by any subset of the Target Group, that either (i) threatens all members of the original Attack Group, or (ii) includes one or more members of the original Attack Group in the new Attack Group. A Type 2 counterthreat replaces the old threat as the current threat and current threats that are not countered are implemented. The actual bargaining in each experiment uses both informal and formal negotiation procedures. Subjects can freely discuss alternative possibilities among themselves, including the possibility of formulating threats and counters. However, before any reallocation of resources is realized before a player is forced to relinquish points or is eliminated from the game - a threat (or counterthreat) must be made formally byfillingout a 3x5 card that records the distribution of resources among the members of the relevant coalition and that contains the signatures of the members of that coalition. To accommodate our assumption that countries prefer receiving points freely rather than by initiating an attack (Assumption A6), subjects are told that if they receive points because a threat is not countered then those in the Attack Group are charged a fee of two dollars. Also, to accommodate the assumption that if one coalition attacks another then those who are not party to the attack can secure resources from the Attack Group as well (Assumption A7), subjects are informed that members of the Attack Group will be awarded 80% of the points in their proposal (at a fee of two dollars for each member in the group), whereas subjects who are neither in the Attack nor in the Target Group will divide the remaining 20% of the points in the proposal, in proportion to their weight. Subjects who are left with no points are removed from the game. To illustrate these rules with a numerical example, suppose the initial distribution is (100,80,60,30,30). Players 2 and 3 could then threaten players 4 and 5, proposing to divide their gains evenly. If there are no

4.5 Some experimental evidence

135

counters to this threat, then 2 and 3 are awarded 80% of 4 and 5's total 24 points each if their agreement is to divide the spoils evenly - and player 1 is awarded 20%, or 12 points. The result is that players 4 and 5 are removed from the game, the new distribution (112,104,84) is established, and players 2 and 3 are each charged a fee of two dollars. Alternatively, players 4 and 5 might counter with the proposal that they each give 10 points to player 1. If this offer is accepted, then no fees are charged and the distribution (120,80,60,20,20) results. Or, 4 and 5 might counter with a proposal that they, in coalition with 1 or 3, attack 2. If 1 or 3 agrees to this proposal, then it becomes the new current threat. Finally, suppose 1 and 2 attack 3. Notice that 3 cannot counter with an attack on 4 since this leaves the coalition of 1 and 2 unchallenged. But 3 could propose that he and 1 or he and 2 attack someone else. We report here on a total of 15 experiments: seven 5-person games and eight 6-person games. The data generated by these experiments are presented in Section 4.6. Notice that if a subject proposes a Type 1 counter, or if a threat is uncountered, then the initial distribution of points changes and subsequent negotiations must take this new distribution as the status quo (until the distribution is changed again). Each such new distribution, then, can be thought of as a starting point for a new experiment. Hence, Section 4.6 describes the exact sequence of threats, counterthreats, and new starting points for each experiment. Also, "*" denotes a player who has been eliminated, and the points held by inessential players are identified in italic. Aside from the prediction that no player should become predominant, our theory yields several hypotheses with respect to this data, which we can summarize as follows: HI. H2. H3. H4.

No essential player should be eliminated. No essential player should be rendered inessential. Inessential players should be eliminated. No inessential player should become essential.

Looking first at the six experiments with no inessential players (experiments 1, 4, 5, 7, 9, and 11), notice that, except for player 4 in game 5, no player is eliminated. Indeed, if we take the 18 initial or new distributions in which no player is inessential, this is the sole instance of an essential player's elimination. This elimination can be attributed to that subject's refusal to counter a threat against her. When asked afterwards why she did this, she reported that she was angered by the attitude of several of the male subjects and decided to quit playing the game. In fact, if we look at all 31 initial or new distributions, this is the sole instance of the elimination of an essential player. Hypothesis HI, then, appears to be

136

4 Resource stability and the balance of power

well supported by the data. Similarly, Hypothesis H2 is also well supported: although there were proposals to the contrary, no essential player was ever rendered inessential. The support for Hypothesis H3, on the other hand, is more ambiguous. Looking at the experiments with at least one inessential player in the initial distribution (experiments 2, 3, 6, 8, 10, 12, 13, 14, and 15), only 7 of the 13 such players were ultimately eliminated. And if we count each new distribution as a separate experiment, then only 7 of 17 inessential players were eliminated (alternatively, we could say that of 16 initial or new distributions, 5 yielded the elimination of an inessential player). Of course, we could defend Hypothesis H3 with the argument that more inessential players would have been eliminated if the experiments had been allowed to continue, and indeed this possibility is suggested by the striking fact that no inessential player ever gained points. Aside from a simple counting of eliminations, however, the comparison of essential and inessential players is interesting. We might speculate that the seven eliminations of inessential players can be explained more simply by factors not captured by our theory and by the fact that such players had so few points. Notice, though, that subjects with low points are not necessarily inessential, and that when they are essential they are not eliminated. For example, the initial distribution in experiments 9 and 11 is (70,10,10,70,70,70), so that players 2 and 3 (who were essential) have only 10 points each. Experiment 14 offers a similar example. The initial distribution there is (70,15,70,70, 5,70); nevertheless, player 2 with 15 points is essential. Yet, in experiment 9, players 2 and 3 are included twice in a counterthreat; in experiment 11, three times; and player 2 in experiment 14 is included in a counter three times, which suggests that these players are not eliminated because, in accordance with A10, they are valuable in the threat-counterthreat negotiation process. Ambiguity also pervades the data with respect to Hypothesis H4: Of the 17 players who are inessential in an initial or new distribution, 5 such players become essential (experiments 2, 3, and 6). What is interesting to note, however, is that in every such instance, the player is rendered essential while losing points because some essential player gains sufficiently to put them at or near 150. Indeed, it is not altogether clear whether H4 is in fact implied by our theory. This hypothesis is based on the supposition that inessential players will be eliminated first, but our theory does not specify when an essential player will counter a threat with a new threat or will counter by rendering another country near-predominant, thereby freezing the system. And in the event that one country is rendered nearpredominant, all countries, including previously inessential ones, become essential. Our experimental data, then, suggest an important empirical

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137

possibility that does not appear to be without historical precedent: the competition among "major" powers leads to the emergence of a nearpredominant power, in which case "marginal" powers become relevant to balancing the threat of the potential hegemon. Regardless of how we interpret the data with respect to H4, it is important to keep in mind that we should not anticipate a perfect match between experimental data and theoretical predictions. Our subjects are "inexperienced players," and may be more prone to strategic error and to missed opportunities than their counterparts in ongoing political systems. The deviations from H3 and H4 merely alert us to the fact that this volume's theorems and remarks are not "iron-hand, immutable natural laws" - the uncertainties that pervade our understanding of individual decision making, as well as our measurement of preferences and perceptions, can redirect events away from theoretical predictions. The point that warrants emphasis, though, is that we can reject the supposition that the final outcomes will entail a minimum winning coalition expropriating all from the losers. The option of a minimal majority expropriating everything from a maximal minority is available to our subjects, and indeed such proposals permeate our experiments (see, e.g., Game 1, in which every threat is such a proposal). Nevertheless, our data demonstrate clearly that if subjects are cognizant of a dominant player's ability to expropriate all resources, then system stability prevails. Indeed, not only are essential players not eliminated, but six experiments end not with the maximal expropriation from a maximal minority but rather with a resource transfer rendering one subject near-predominant (experiments 1, 2, 3, 4, 7, and 11). For further insight into the mechanics of this equilibration process, notice that all experiments ended with one or more persons transferring points to others so as to reach a unanimous agreement to terminate. Thus, in effect experiments ended with Type 1 counters as against Type 2 counters, which a myopic constant-sum cooperative game model of international processes implies. Notice also that if a country is essential and if it confronts a threat, then we anticipate observing it countering by ceding resources to some country or by proposing a viable counter threat. On the other hand, if a country is inessential then it does not have enough resources to avert elimination, nor is its inclusion in any coalition essential to the formation of a viable counterthreat. With this in mind, consider Table 4.1, which shows the number and types of threats that inessential and essential players confront, the number of times they are included in a Type 1 or Type 2 counter, and the number of eliminations. Briefly, inessential players successfully countered threats 13 times, but in no instance was a Type 2 counter employed - counters always required that the inessential

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4 Resource stability/ and the balance of power

Table 4.1.

Threat Type 1 Counter Type 2 Counter Elimination

Inessential

Essential

30 13 0 7

76 23 52 1

player cede points. Thus, it appears that subjects recognized that inessential players did not have enough points to ensure stability via participation in viable counterthreats. It also appears that essential players are content to slowly "bleed" an inessential participant - the fact that they could not participate in a Type 2 counterthreat explains why no inessential player ever gained any points. Essential players, however, use Type 2 counterthreats more frequently than Type 1 counters. In the case of essential players, our theory does not state which type of counter should be observed more frequently. The "first instinct" of subjects, however, is to counter a threat with a new threat, and a Type 1 counter seems to be an action "of last resort," an action taken when no other players are willing to join in the formation of a Type 2 counter. Thus, it is not surprising to learn that Type 2 counters are more prevalent than Type 1 counters, which serves to emphasize the argument thatflexibilityin alliance systems is an essential component of the operation of balance-of-power processes. (Generally, a Type 1 counter is used by an essential player to ensure stability when a threat to divide all the points evenly by a two-member attacking group was initiated or when it seemed the easiest route to ending the experiment.) In summary, a static one-period view of anarchic international processes that are characterized by a transferable measure of resources and value implies that system stability is impossible, because some minimal winning coalition will form to eliminate all other players. However, if we take a more dynamical view and if we assume that each country seeks to maximize its resources or proportion thereof, subject to the constraint that its survival is not threatened in the future, then system stability is possible and the resource instability that characterizes constant-sum nperson games takes a more limited form. Our experimental results support this argument, and in particular they support the bargaining assumptions summarized by A10. This does not prove, of course, that the processes of real international systems correspond to our model or that we cannot design experiments to support some alternative bargaining assumption.

4.6 Appendix A: Experimental data

139

Decision makers in international affairs are much more sophisticated than the subjects of our experiments, and reality is far more complicated than the scenario confronted by our subjects. Yet the intent of these experiments is not to show that our model describes reality, but only that if the conditions presumed to operate in our model do in fact operate then anarchic international systems do not necessarily lead to instability and the elimination of essential players. And if our subjects can respond to the strategic imperatives our model purports to represent, and if our model captures an essential part of reality, then it is reasonable to suppose that more sophisticated decision makers respond to those imperatives as well.

4.6

Appendix A: Experimental data Player

Game 1

New

End

Game 2 End

Game 3

New New

End

Game 4 New New

End

6

Description

1

2

3

4

5

Type of Counter

60 100 0 40 0 80 0 23

60 0 90 100 130 0 150 150

60 0 90 60 90 120 150 60

60 100 0 40 80 0 0 23

60 100 120 60 0 100 0 43

Initial distribution 1,4,5 threaten 2,3 2,3,5 threaten 1,4 1,4 cede 40 points to 2 2,3,4 threaten 1,5 1,3,5 threaten 2,4 2,3 threaten 1,4,5 1,4,5 cede 50 points to 2

20 0 10

60 0 40

60 150 60

60 0 40

100 150 150

Initial distribution 3,5 threaten 1,2,4 1,2,4 cede 50 points to 5

IT 1

30 50 0 20 0 15 50 0 100 0 12

30 50 0 20 0 15 50 0 100 0 13

80 0 150 120 150 140 0 60 100 150 150

80 104 0 60 0 50 90 120 0 0 45

80 96 150 80 150 80 110 120 0 150 80

Initial distribution 1,2,4 threaten 3 3,5 threaten 1,2,4 1,2,4 cede 40 points to 3 3,5 threaten 1,2,4 1,2,4 cede 20 points to 3 1,2,4,5 threaten 3 3,4,5 threaten 1,2 1,2,3 threaten 4,5 3,5 threaten 1,2,4 1,2,4 cede 10 points to 3

IT 2 1 IT 1 IT 2 2 2 1

60 0 50 90 50 82 106 0 40

60 0 50 90 50 82 0 0 40

60 100 60 0 35 68 0 75 35

60 100 60 0 35 68 52 75 35

60 100 80 120 130 0 142 150 150

Initial distribution 3,4,5 threaten 1,2 1,2 cede 20 points to 5 1,2,5 threaten 3,4 3,4 cede 50 points to 5 1,2,3,4 threaten 5 1,5 threaten 2,3 3,4,5 threaten 1,2 1,2 cede 20 points to 5

IT 1 IT 1 IT 2 2 1

initial threat (IT) 2 1 IT 2 2 2

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4 Resource stability and the balance of power

Appendix A (cont.) Player

Game 5

End

Game 6 End

Game 7

New

New End

Game 8

New New

End

Game 9

New End

Game 10

Description

Type of Counter

60 0 0 66

Initial distribution 1,2,4 threaten 3,5 1,3,4 threaten 2,5 2,3 threaten 4, 4 quits

IT 2 1

80 0 58

30 0 11

Initial distribution 1,3 threaten 2,4,5 2,4,5 cede 60 points to 3

IT 1

60 78 0 80 0 60 69 66 100 0 30 0 25

60 72 100 140 150 60 69 66 100 0 30 0 25

60 0 100 0 75 61 108 0 100 100 61 0 40

Initial distribution 2,3 threaten 5 1,4,5 threaten 2,3 2,3,4 threaten 1,5 1,4,5 threaten 2,3 2 cedes 1 point to 5 3,4,5 threaten 1,2 1,2 threaten 5 3,4,5 threaten 1,2 1,2,5 threaten 3,4 3,4 cede 60 points to 2 1,2 threaten 3,4,5 3,4,5 cede 31 points to 2

IT 2 2 2 1 IT 2 2 2 1 IT 1

60 100 0 80 0 68 60 61 100 150 61

60 100 150 0 75 61 70 71 0 75 71

60 0 90 120 0 61 60 61 100 0 56

40 0 60 100 150 42 40 46 0 75 56

20 0 0 0 0 0 10 0 * * *

Initial distribution 1,2,3 threaten 4,5,6 3,4,5 threaten 1,2,6 2,4,5 threaten 1,3,6 1,3,5 threaten 2,4,6 1,2 threaten 6 6 cedes 10 points to 3 5 threatens 6, 6 quits 1,2,4 threaten 3,5 2,3,5 threaten 1,4 1,4 cede 10 points to 5

IT 2 2 2 2 1 IT IT 2 1

70 0 100 0 75 150 0 75 96 70 120 90

10 15 0 20 0 25 50 75 0 8 30 20

10 15 0 20 0 25 38 75 0 8 30 20

70 75 100 0 75 0 106 0 75 70 120 90

70 75 100 120 0 100 0 75 74 70 0 40

70 105 0 140 150 0 106 0 75 74 0 40

Initial distribution 2,6 threaten 1 1,4,5 threaten 2,3,6 2,3,5,6 threaten 1,4 1,4,6 threaten 2,3,5 1,2,3,5 threaten 4,6 2,3,4,6 threaten 1,5 1,2,3,5 threaten 4,6 1,4,6 threaten 2,3 2,3 cede 4 points to 6 1,2,3,4 threaten 5,6 5,6 cede 64 points to 1,2,3,4

IT 2 2 2 2 2 2 2 1 IT 1

60 75 120 0

40 0 60 100

60 75 120 0

20 0 0 0

60 75 0 100

60 75 0 100

Initial distribution 1,3,5,6 threaten 2,4 1,2,3 threaten 4,5,6 2,5,6 threaten 1,3,4

IT 2 2

1

2

3

4

5

60 100 120 66

60 100 0 84

60 0 60 84

60 100 120 0

80 150 80

30 0 11

80 150 140

60 72 100 0 75 60 54 109 0 100 60 150 60

60 78 0 80 0 59 0 59 0 100 119 150 150

60 100 0 0 75 68 60 61 100 0 56

6

4.6 Appendix A: Experimental data

141

Appendix A (cont.) Player

New New End

Game 11

End

Game 12 New End

Game 13 New

End

Game 14

End

Game 15 New New

End

5

6

Description

Type of Counter

0 0 18 0 10 0

150 0 55 56 55 56

0 60 60 61 60 61

1,3,5 threaten 2,4,6 1,3,6 threaten 2,4,5 2,4,5 cede 10 points to 3 2 threatens 4 4 cedes 8 points to 1 2 threatens 4, 4 quits

2 2 1 IT 1 IT

10 0 21 0 40 0 35 10

70 75 0 100 130 0 80 70

70 75 86 100 0 100 0 30

70 75 86 0 90 100 0 30

Initial distribution 1,4,5,6 threaten 2,3 1,2,3,5,6 threaten 4 1,4,5 threaten 2,3,6 2,3,4,6 threaten 1,5 1,5,6 threaten 2,3,4 1,2,3,4 threaten 5,6 5,6 cede 80 points to 1

IT 2 2 2 2 2 1

20 0 * *

20 0 * *

80 100 0 80

20 0 * *

80 100 150 110

Initial distribution 1,4,6 threaten 2,3,5 1,6 threaten 4 4 cedes 20 points to 1,6

IT 2 1

60 61 100 0 80 0 100 63

40 56 100 0 80 0 100 58

60 61 100 150 0 75 100 63

60 61 0 75 140 150 0 58

20 0 * * * * *

60 61 0 75 0 75 0 58

Initial distribution 2 threatens 5, 5 quits 1,2,3 threaten 4,6 3,4,6 threaten 1,2 1,2,4 threaten 3,6 3,4,6 threaten 1,2 1,2,3 threaten 4,5,6 4,6 cede 4 points to 1,2,3

IT IT 2 2 2 2 1

70 75 100 120 0 100 0 0 50 100 0 50

15 0 100 0 60 100 0 50 100 150 0 8

70 75 100 0 100 0 75 150 0 0 100 80

70 75 0 90 140 0 75 100 150 0 100 80

5 0 0 0 0 0 0 0 0 0 0 2

70 75 0 90 0 100 150 0 0 50 100 80

Initial distribution 1,3,4,6 threaten 2,5 1,2,3 threaten 4,5,6 1,4,6 threaten 2,3,5 2,3,4 threaten 1,5,6 1,2,6 threaten 3,4,5 3,4,6 threaten 1,2,5 2,3,4 threaten 1,5,6 1,2,4 threaten 3,5,6 1,2,6 threaten 3,4,5 3,4,6 threaten 1,2,5 1,2,5 cede 90 points to 3,4,6

IT 2 2 2 2 2 2 2 2 2 1

60 62 70 71 0 60 150 71

60 61 60 61 80 120 0 60

60 61 60 61 80 120 0 60

50 58 50 51 70 0 75 52

50 58 50 56 70 0 75 57

20 0 10 0 * * * *

Initial distribution 4,5 threaten 6 6 cedes 10 points to 1 5 threatens 6, 6 quits 2,3,4,5 threaten 1 1,2,3 threaten 4,5 1,4,5 threaten 2,3 2,3 cede 2 points to 4,5

IT 1 IT IT 2 2 1

1

2

3

75 120 60 61 68 69

0 0 37 51 37 43

75 120 70 71 70 71

70 75 86 100 0 100 150 150

10 0 21 0 40 0 35 10

80 100 150 110

4

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4 Resource stability and the balance of power

4.7

Appendix B: Instructions read to subjects

This experiment is designed to model various aspects of conflict, and the amount of money you earn here depends on how carefully you listen to instructions and how well you play the game. The experiment begins by dividing 300 points among you. The precise nature of this division has been predetermined, and it is not necessarily an equal division. Some of you will start with more points than others. At the end of the experiment, each of you can convert the points you have at that time into money according to an exchange rate specified in the sealed envelope before you. Like the initial distribution of points, these rates vary from player to player. Although you cannot open your envelopes until the experiment ends, let me assure that these rates are positive: Thefinalmonetary payoff each of you receives increases as the number of points that you control increases. Notice that although each of your exchange rates is unknown to you, everyone will always know the current distribution of points among you. An important rule concerns how points are secured from other players, which is the only way to increase the points in your possession. First, one or more of you must propose a threat that takes all or part of the points held by one or more other players. I will call those who are attempting to expropriate points the Attacking Group and those who are their target the Target group. I will consider giving those points to the Attacking Group provided that the threat satisfies two conditions: First, the Attacking Group must identify itself and indicate on one of the slips of paper in front of you: (i) how many points are being taken and from whom, and (ii) how those points will be distributed after they are taken. Second, the total points presently held by the Attacking Group must exceed the total number of points held by the Target Group. That is, only bigger coalitions can attack smaller coalitions. Calling this the Current Threat, I will then entertain counteroffers. There are two types of counters. The first is a proposal by one or more members of the target group to voluntarily give points to players who are not being attacked. These other players may or may not include members of the Attacking Group. In the event of such a counter, I will exchange the points as agreed to, recompute the distribution of points among you, and then let negotiations continue by opening the floor to new threats and counters. The second type of counter is a new threat by a new Attacking Group that includes some, but not necessarily all, of the old Target Group. This new Attack Group must either threaten the members of the old Attack Group or it must induce at least one such member to join the new Attack Group. In this event, the counter will replace the old threat as the Current Threat, and I will entertain counters to it.

4.8 Appendix C: Proofs of results

143

Notice that by these rules, if anyone secures over half the points, it is impossible to counter any threat that this person might make. So such a player can secure all the points in the game. Now if neither type of counter is made or otherwise agreed to in response to a threat, I will award the points to the Attacking Group as specified in the Current Threat, but with two provisos: (1) Those who are neither in the Attacking Group nor in the Target Group will divide, according to their current relative points, 20 percent of the points being expropriated from the Target Group. Those who have made the uncountered threat will divide the remaining 80 percent as specified in their agreement; and (2) I will charge each person receiving points a fee of $2.00. At this stage, I will again recompute the distribution of points among you and if any of you are left with no points, you will be removed from the game. Those so removed will be paid $2.00 for their participation. The experiment will end when there is unanimous consent to terminate, and you will then be permitted to open your envelopes to see the exchange rate converting points into money. Each of you will be paid in cash the appropriate amount, subtracting any necessary fees for your actions. 4.8

Appendix C: Proofs of results Theorem 4.1

Consider CeW, so that S—C$W. From the definition of v,

v(C) + v(S- C) = r(C) + ( ^ \

2

ieC

ieC

or v(C) + v(S— C) = R — e + e = R> which defines a constant-sum transferable utility game in characteristic function form. It is well known that such games have no cores if they are essential - if there exist two disjoint subsets of S, say C" and C", such that v(C') + v(C") < v(C'UC"). It is straightforward to see from the definition of v(C) that this game is essential (namely, let C" and C" be losing coalitions, but let C'UC" be winning). Note, incidentally, that if rt = R/2, the game is still constant-sum, but it is now inessential and thus it has a core - the original distribution of resources. Q.E.D. Theorem 4.2 Theorem 4.2 implies that winning coalitions will change the resource distribution unless one country controls R/2, in which case no such coalition

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will form. Suppose to the contrary that rj = r7- for all j . If the winning coalition C forms then - from the definition of v(C) and the fact that regardless of whether we use a solution concept such as the main-simple Vset, the competitive solution, or the strong bargaining set - r' must be Pareto optimal and feasible for C: jeC

But since rj = rj for all j , this equality becomes R r . — = max[r / ], which is impossible since rt y . Suppose that the maximum payoff to each partner j of i*eC is rj. But again, from the properties of a main-simple V-set, j must receive rj from every coalition in which /* is a partner. Consider now all those minimal winning coalitions that exclude /* and suppose that the maximum payoff to j when it is a member of any one of these coalitions is rj*. Again, though, rj* = rj. But this means that if any coalition C** forms that excludes /*, it cannot get more for its members than they can get individually if they join /* to yield a resource distribution in V. Thus, the previous inequality implies that

4.8 Appendix C: Proofs of results

145

However, except for the case in which /* has R/2 resources to begin with, v(C**) > /*(C**), contradicting the fact that if r** is in a main-simple Vset then it must be Pareto optimal for C**. Hence, if /* begins with less than R/2, no resource distribution in the main-simple V-set gives /* the amount R/2. Q.E.D.

CHAPTER 5

Preventive war

What made the [Peloponnesian] war inevitable was the growth of Athenian power and the fear which this caused in Sparta. Thucydides, The Peloponnesian War (trans. Warner 1954, p. 49)

The great wars of history - we have had a world war about every hundred years for the last four centuries - are the outcome, direct or indirect, of the unequal growth of nations. Halford J. Mackinder, Democratic Ideals and Reality (1962, pp. 1-2)

Whether to invest in additional power resources or to consume some of those that have been accumulated is a perennial issue of foreign policy. Many of the most important choices governments face have to do with the relative weight given to consumption.. .versus investment, and with devising strategies for action that are both viable in the short run and capable of achieving wealth and power objectives in the long run. Robert O. Keohane, After Hegemony (1984, p. 24)

Intuition and a considerable theoretical and empirical literature concur with Thucydides' assertion that the cause of international conflict lies as much with differential growth rates in resources as with the relative power or resources countries enjoy at any specific point in time. Hence, conflicts might not be attributable wholly to some current "imbalance" in military position, but also to the projections of future imbalances and to a corresponding perception by some that only a preventive war can guarantee their security.1 Indeed, so compelling is the threat of imbalance owing to the growth of some potential adversary, that even domestic constraints on aggressive action and sentiments for peace can be quickly swept aside. As forcefully argued by McNeill (1982, p. 309): Presumably, as long as sovereign states exist, whenever one of them seems to be growing so powerful as to threaten the continued independence of the others, 1

A great number of sources can be cited for the hypothesis of preventive war, but for a comprehensive general survey, see Levy (1987). 146

5 Preventive war

147

everything tending to encourage hostility to the potential hegemonical power finds congenial conditions within the states that feel threatened. Rapid changes of mood and popular sympathy can and do occur under such circumstances, forming and dissolving alliances in a matter of a few weeks or months. Contrary intentions and conflicting ideals prevailed only when no pressing external threats provoked balance-of-power behavior. This view of international conflict is reflected also in contemporary public debates over, say, domestic versus defense spending and the issue of whether a country should invest resources to maximize economic growth rather than appropriating those resources to maximize current military strength. Kennedy (1987, pp. 539-40) identifies the dilemma clearly in his commentary on contemporary international affairs: To be a Great Power.. .demands a flourishing economic base Yet by going to war, or by devouring a large share of the nation's "manufacturing power" to expenditures upon "unproductive" armaments, one runs the risk of eroding the national economic base, especially vis-a-vis states which are concentrating a greater share of their income upon productive investment for long-term growth Ideally, of course, "profit" and "power" should go hand in hand. Far too often, however, statesmen found themselves confronted with the usual dilemma: between buying military security, at a time of real or perceived danger... or keeping defense expenditures low, but finding one's interests sometimes threatened by the actions of other states The present large Powers... are thus compelled to grapple .. .with the uneven pattern of economic growth, which causes some of them to become wealthier (and usually stronger), relative to others; and... [with the necessity] to choose between a more immediate military security and a longer-term economic security. This debate and contemporary scholarship concerning the issue of "consumption now" versus "greater consumption later" show why a simple static analysis based on some current distribution of national resources or power cannot capture all relevant aspects of balance-of-power reasoning in particular and the causes of war in general. In response, this chapter extends our analysis to include the possibility that the resources that measure a country's capabilities are not static. In so doing, we contest the supposition that a conceptualization of international processes based on "balance of power" is necessarily distinct from one based on "power transition." Indeed, the extension we seek is natural to our analysis, because we assume that decision makers judge coalitions and resource reallocations in terms of future as well as immediate consequences. After all, it is presumably the anticipation of some type of future imbalance and the decisions it implies that leads to preventive war, and we already have adopted the view that national leaders are forwardlooking, that they choose actions in anticipation of what others will do in response to their current decisions. Moreover, the concept of system

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stability seems to define precisely the imbalance that should concern decision makers. The literature recognizes, of course, that the relative resources or power of nations change for various reasons, and that even if we focus exclusively on those resources relevant to the measure of military power, such as technological sophistication and agricultural production, some countries may simply possess advantages of climate, geography, and the like that yield greater growth rates than other countries. To the extent, though, that the same literature offers national leaders only the options of warring or acquiescing to a dominant power, it is inconsistent with the idea that they can also respond to power imbalances by trying to augment their resources. The possibility revealed in the quotation from Keohane that introduces this chapter means that growth rates may vary for endogenous reasons: countries may simply choose to invest their resources differently, with some preferring to invest in the future via additional expenditures on research, education, and a general strengthening of the economy, whereas others act to increase current resources at the expense of the future by way of contemporaneous military hardware expenditures. Spending billions on the procurement of, say, a particular aircraft increases current military preparedness, but at the expense of future resources to the extent that this procurement detracts from expenditures on research and development or on the general strengthening of the economy. Hence, we extend our model in two ways. Taking the idea that growth is a function of both exogenous factors (geography, climate, availability of natural resources) and endogenous factors (vitality of the economy, trade, share of the budget directed to military procurement), we begin by supposing that each country is confronted with an exogenously determined growth rate, which it cannot control. However, we also assume that this rate applies only to those resources countries invest toward the future. As part of a sequential cooperative and noncooperative game, they can vary their current and future military capabilities by choosing the amount of resources to be invested. Since only the proportion invested is subject to growth at the exogenously determined rate, decision makers thereby determine the overall rate at which resources grow. In our extended model, then, national leaders do more than simply try to anticipate future events and choose whether or not to engage in a preventive war. Through alternative investment decisions, they also affect the future. Naturally, in choosing their investments and in deciding whether to engage in a preventive war, they must also anticipate the anticipations of others, the actions of others based on those anticipations, and so on, with the understanding that everyone is doing the same. To learn the con-

5 Preventive war 149 ditions under which countries can invest and maintain their sovereignty, and when they might contemplate a preventive war because they cannot do so, our analysis describes the necessary and sufficient conditions for the existence of system and resource stability in this interactive, dynamical setting. Our objective, however, is not merely to extend a particular abstract formulation. It is to answer some specific substantive questions about the preconditions for and nature of preventive wars. The literature on such wars, beginning with Thucydides' observation, provides the perspective for the questions we raise. With respect to definitions, Brodie (1973) defines a preventive war as one in which a country undertakes "to destroy an already strong rival whose power one fears may grow faster than one's own." Fay's (1938) definition is "the waging of war upon a neighbor while he is still weak... to prevent his growing stronger " Implicit in all such definitions is the assumption that preventive wars are not waged simply because some country or coalition acts to take advantage of a current imbalance in the distribution of resources. Rather, preventive wars are designed to overcome a perceived future resource deficit. This view that preventive wars are initiated by a stronger state against an adversary that threatens to grow stronger, however, is not universally accepted. For example, Organski and Kugler (1980), citing Germany and Japan, speculate that the rising power initiates the war. More generally, they observe that "before the nuclear age the great powers, although far stronger than the middle and small powers and very much involved in the diplomatic interchanges that preceded wars, were not the initiators of major wars. They fought when others decided to fight" (p. 56). This dispute reveals the theoretical vacuum that characterizes the debate. It is probably rare for the economic and military capabilities of all countries to grow in wholly parallel fashion, and national capabilities, however measured, almost certainly intersect frequently whenever graphed on a time line (for the case of the great powers of Europe between 1871 and 1914, see Figure 7.1 in Chapter 7). Yet, not all of these positional changes yield preventive wars. Correspondingly, Organski and Kugler suggest that we ought to look solely at the two major contenders: "The fundamental problem that sets the whole system sliding almost irretrievably toward war is the differences in rates of growth among the great powers and, of particular importance, the differences in rates between the dominant nation and the challenger that permit the latter to overtake the former in power" (1980, p. 61). Earlier, however, they observe that "If one looks at the point at which conflicts begin in relation to the point at which the trajectories of the challenger and the dominant nation intersect, it is

150 5 Preventive war clear that conflicts occur after the intersection when the two nations fight alone... but before the coalition of the challenger overtakes the coalition of the dominant country... [which] seems a curious pattern" (p. 60). The notion of system stability, however, offers a view that makes this fact less curious: it is not relative position per se, but whether growth or decline undermines system stability by rendering countries inessential. Quite naturally, then, our theory forces us away from analyzing the situation from the perspective of two countries in isolation, and toward viewing matters from the perspective of complete international systems. Similarly, with respect to the issue of who attacks first and to whether distinctions between preventive wars and preemptive attacks are readily discerned, suppose that there is no uncertainty and thus no possibility of miscalculation or accident; additionally, suppose that all aspects of the situation - growth rates, current resources, etc. - are common knowledge (everyone knows these parameters, everyone knows that everyone knows them, etc.). In this instance, not only is the "declining power" (say A) capable of foreseeing the consequences of its decline if it fails to respond, but those consequences are known to the rising power (say B) as well; and if A concludes that its dominant strategy is to initiate a preventive war, then B also knows this and can respond before A acts - either by offering to bargain away some part of its future advantage so as to dissuade A from attack, by establishing alliances that render it secure until its resources have grown sufficiently, or by launching a "preemptive preventive war" on its own. Thus, even at this level of the analysis, we cannot say who will attack first or, indeed, whether there will be an attack. However, matters can be even more complicated still. If B prefers a preemptive preventive war, then A should know this and can respond with a "preventive preemptive preventive" war; however, B should know that A knows this, and so forth and so on, so that the spiral of conjectural variation continues backward forever. Unless we supply additional structure in the form, for example, of assumptions about bargaining opportunities, about the advantages of different types of attacks, about uncertainty, or about violations of the common-knowledge assumption, we can rationalize any proposition as to who attacks first merely by terminating our conjectural thinking at an appropriate point. That people can anticipate the actions of others, that they will anticipate the anticipation of others, and so on, blurs the theoretical distinctions between preventive and preemptive conflict, so that, regardless of the labels we give to players who choose particular acts, preventive and preemptive attacks must both be understood in the context of the same theoretical structure. Furthermore, unless we elaborate our analysis to include such matters as a specification of the strategic complexity decision makers can

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151

accommodate, the relative advantages of offensive versus defensive capabilities, and the advantages of first versus second moves, then it is unlikely that we can resolve unambiguously whether the presumed conditions for preventive war imply a preventive strike by a declining power or a preemptive attack by the rising power. In this context, then, we find the source of much of the confusion in the identity of the aggressor in some wars. 2 Was Germany the culprit in World War I with its seeming overzealousness in military expenditures, or was Russia the instigator with its early mobilization? Later, in Chapters 6 and 8, we suggest that Russia's actions and Germany's preemptive thrust into Belgium followed a clear logic dictated by the dynamics of the general resources of the European powers. However, with national leaders able to anticipate the anticipatory moves of others in a game we can only imperfectly model, it can never be clear who has the last move and who has the first. Thus, the distinction between preventive and preemptive actions is more a verbal convenience that helps us summarize general strategic considerations than it is a rigorous theoretical construct. This discussion of the possibility that people might anticipate the actions of others raises additional issues, including the possibility that the growing power, B, might choose to negotiate away its potential advantage so as to avert a war. Levy (1987, p. 96) mentions this strategy, and brings to the forefront the potential for bargaining in the circumstances of threatened war: One might argue that... the problem be conceived as one of bargaining. There is some level of concessions that the challenger would prefer to grant rather than to fight Similarly, there is some level of concessions that the declining state would prefer to accept from the challenger rather than to initiate a war. Levy sets this possibility aside, however, with the conclusion that "It would not be technically incorrect to introduce the defender's calculations... but this would probably add far more complexity than explanatory power" (p. 97). To the contrary, it is impossible to reach definitive conclusions without an analysis that accommodates the reactions and counterreactions of all affected parties, including not only aggressive preemptive moves but less militaristic moves also. To do otherwise relegates the subject of preventive war to a theoretical backwater, and precludes any possibility of ascertaining the circumstances under which countries might seek some 2

See, for example, Allison's (1971, pp. 14-15) discussion and citation of Schmitt's (1930) lament about the disagreements that persist among diplomatic historians over precisely this issue. The problem is that such histories proceed without a well-structured theoretical paradigm, and instead focus on a "cause" that is particular to the circumstance. Such a cause, of course, cannot exist.

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accommodation to differential resource growth other than war.3 Indeed, Levy's argument precludes meeting his goal of identifying "the conditions under which declining states choose to provide for their future security through alliances, an expanded armaments program, or other internal changes rather than through preventive military action" (p. 107). One response to our simple scenario might be to reject it as too abstract and to argue that we ought to appeal to the data. However, although there is ample empirical evidence to support the hypothesis that a necessary condition for wars that we call preventive is differential growth in resources that yields the displacement of one leading power by another, we also know that this condition is not sufficient - witness the failures of Britain to thwart its displacement by the United States, of the Soviets to block the rise of China, of the United States after 1945 to block the Soviet's military expansion (Lebow 1981). Unless we choose to argue that Germany did not begin World War II, we have the additional uncomfortable example of Britain's, France's, and Russia's failure to block the rise of Germany under Hitler. Thus, what we require is an analysis that does more than merely point to circumstances propitious for preventive wars - we should direct our research toward identifying a necessary and sufficient condition. Deducing a necessary and sufficient condition is important because we want to learn whether the cause of preventive war lies in our model, or whether other circumstances must be met as well. Knowing only that differential growth is necessary does not preclude the possibility that these other factors are a contributing source to conflict and that conflict can be avoided if we direct our attention there. Indeed, elaborating on his argument that bargaining is precluded as an alternative, Levy (see also Levy 1983b) hints at the existence of other factors (notably uncertainty): The issue is not a conflict of concrete interests in which each party can easily calculate its gains and losses from different levels of concessions and negotiate accordingly, but a question of future influence over a range of diverse and partly 3

To reiterate our earlier point that we must consider the complete international system before any definitive conclusions can be drawn, it is in fact difficult to see how any of the standard conclusions about preventive wars survive even if we admit the reactions of a second country. If one country is about to be surpassed by another, and if no other countries are relevant so that the country in decline is about to be rendered inessential and its antagonist predominant, then preventive action should have been instigated by the country in decline at an earlier time under the label of a war of conquest. If geography had previously precluded such a war, then it will preclude a successful preventive war as well until the time the country in ascendancy gains predominance. Whether or not wars of conquest occur in this two-country system is not our concern here, but it should be evident that a preventive war is an impossibility. Hence, this chapter only examines systems with three or more countries.

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unpredictable issues that cannot be calculated with any degree of precision and that are not easily amenable to negotiation. Moreover, the very intensity of the conflict tends to generate different perceptions by the adversaries of the stakes, threats, legitimate aims, and acceptable alternatives to war (1987, p. 96).

Thus, one hypothesis we ought to examine is whether factors such as uncertainty must be added to identify sufficient conditions for a preventive conflict as against the hypothesis that our model already contains the seed of such conflict. To these broad issues, we should also add a series of more specific questions. For example, are preventive wars limited necessarily to two adversaries and are such wars initiated by a single large country? What of the possibility that preventive wars are initiated by coalitions to keep some large adversary from becoming predominant? These questions are suggested by the debate over whether it is the declining or rising power who initiates a preventive war. Organski and Kugler (1980) are forced to admit that preventive action is indeed dangerous for the rising power. However, they may be retreating too readily: if several powers are rising or declining simultaneously, those of the same type may coalesce for preventive action (or we may even see mixed coalitions). The debate over who initiates, in fact, seems overly preoccupied with two-country scenarios, and the countries Organski and Kugler cite in support of their proposition hardly corresponded to a unified entity (Germany, Italy, and Japan in World War II). Later in our analysis, then, we distinguish between those situations in which a single country threatens to become predominant and to render all others inessential, and those situations in which no country will become predominant but some subset of countries are threatened nevertheless with becoming inessential. Additional questions that will concern us include: Are preventive wars initiated only when one country grows at a faster rate than someone else or are such wars possible also when all countries grow at the same rate? Are preventive wars predetermined in the sense that, given initial growth rates and resources, there is only one possible resolution to instability? Equivalently, is there a unique and predictable response to a perceived future imbalance in the distribution of resources among countries, or is there a variety of possible responses? Also, despite our treatment of countries as unitary actors and the distribution of resources and their growth as the fundamental determinants of conflict, we must be cognizant of the hypothesis that domestic politics plays an important role in determining whether preventive wars are undertaken (Lebow 1981, 1984). Although we incorporate domestic issues in a limited way by allowing decision makers to determine what share of their countries' resources is to be invested into the future and what share is to be used for strategic maneuver in the current period, because we assume

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that decisions are unconstrained except by their countries' resources and by the imperatives of international politics, we do not take full account of the domestic bases of foreign policy. Nevertheless, to the extent that we define what unconstrained decision makers ought to do, we provide a basis for speculating about the consequences for preventive war of constraints such as those imposed by domestic politics. 5.1

A dynamic extension

To this point, national leaders are forward-looking only in the sense that they will not form coalitions that, although perhaps temporarily useful, disadvantage them in the future by threatening their sovereignty. Now, however, we assume a two-period structure that admits differential resource growth rates among countries and that also permits decision makers to choose the amount of resources they will invest into the second period. Specifically, we suppose that country /'s resources at the beginning of period 1 is rh that / can invest any part of its resources, say pt (0< Pi < A*/), and that the rate of return on an investment is wh so that this investment grows to the amount (W/ + \)pt in period 2, where wt > 0 for all countries / in S. That is, after making this investment, and if nothing else occurs, / will have rt—pt resources available in period 1 and, ceterisparibus, ri-pi-\-{wi-\-\)pi = ri-\-wiplresources in period 2. Of course, we are not asserting with this formulation that a model with only two periods is wholly general. Time is continuous and the temporal horizons of decision makers may vary. If, for example, politicians in democracies "plan only until the next election," then perhaps their opponents in dictatorships plan for what might occur after that election. That is, domestic factors affect time horizons and cause them to vary across decision makers. Nevertheless, a two-period model captures the essential issues confronting people in making any investment decision - the weighing of current against future benefits. Returning to our notation, we emphasize that we are not predicting that country /, after it invests ph realizes AV+W/JP,- resources in period 2. Such a prediction ignores the assumption that everyone's information is complete, and thus the fact that others can anticipate this investment with preemptive moves. To see more clearly what this view implies, we can think of the two-period situation in terms of four stages, where period 1 has three stages and period 2 has a single stage. Assuming that (S;r) emerges as the prevailing state from whatever has transpired beforehand, preventive wars (if they occur at all) occur in the first stage, as do resource reallocations induced by other means in anticipation of the future.

5.1 A dynamic extension

155

Investment decisions are made in the second stage, and in the third stage, with the state being (S;r'), countries can be eliminated unless they are capable of defending their sovereignty. Period 2 consists of a single stage in which the prevailing state is (S'; r"), after which coalition formation, redistribution, and the like reoccur. Summarizing: Period 1 Stage 1: With (5; r) the status quo, and in anticipation of what is to follow, countries form alliances, redistribute resources, or launch a preventive war. Stage 2: Countries choose their investment strategies. Stage 3: With (S;vf) the status quo, countries seek to eliminate inessential players. Period 2 Stage 1: With (S';r") the status quo, countries form alliances, etc., in accordance with the model described in Chapters 3 and 4. Notice now that we cannot predict that r" ultimately equals (rx + wxpl,..., rn + wnpn), since the actual distribution of resources countries use as the basis of their investment decisions may be modified in the first and third stages of period 1. Indeed, even if (ri + w^Pi,..., rn+wnpn) were to prevail ultimately, it would be changed in period 2 as countries begin the negotiation and redistributive process implied by any absence of resource stability. Our objective is not to predict thefinaldistribution of resources per se. Rather, it is to measure the anticipations of decision makers in order to identify the root causes of conflict and, hence, the forces that operate to yield one coalition rather than another in the first period. Consequently, rather than build coalitional considerations directly into the determination of ultimate resource distributions, we initially ignore such considerations, taking r" as the basis whereby countries judge the implications of their investment decisions and the necessity for a preventive war, ceteris paribus. Coalitions and preventive wars may arise to redistribute resources before r" prevails, and it is almost certainly true that the game will continue should r" prevail so as to change this distribution. Yet, it is the anticipation of this particular postinvestment vector, in conjunction with strategic options afforded by the initial distribution of resources, that dictates events. In summary, we suppose that, beginning with (5; r), countries must decide what investment decisions to make under the presumption that there will be two rounds of maneuver, thefirstwith the system (S;r') = (S; (rl— P\,...,rn— pn)) and, if no one is eliminated in the first period, the second with (S;r") = (S; (rx + wxpx, ...,rn+wnpn)).

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Aside from this delineation of a decision problem, we also must add some assumptions about preferences for alternative investment decisions that reflect the decision problem at hand. Because we are interested in the conditions for the emergence of preventive wars and because we suppose that the survival of a country depends on its resources relative to other countries, the following assumption seems a logical one: conditional on being able to defend one's sovereignty in period 1, / maximizes its proportion of resources in period 2, (^• + w / /? / )/2 / 6 5 (^ + wy/?/). Notice that if countries are unconcerned about the possibility of elimination in period 1, such as when investments at the expense of current capabilities can be undertaken in weapons systems that will not become immediately operational because current vulnerabilities can be disguised from an adversary, and if all growth rates are positive, then countries should invest all of their resources. However, the qualification this assumption makes about maintaining the ability to defend one's sovereignty in period 1 implies that, in deciding how much to invest, each country must consider the possible investment decisions of other countries, keeping the following considerations in mind: (1) it cannot invest "too much," or (given the decisions of others) it mightfinditself eliminated in the third stage of period 1, in which case its current and future resources are divided among the remaining countries; and (2) it cannot invest "too little," since that may make it vulnerable to being eliminated in period 2. These two strategic considerations are, of course, the two sides of modern strategic planning and weapons procurement, and they give rise to two important questions. First, under what conditions (in terms of the initial distribution of resources r and the growth rates of countries w) does each country possess an investment strategy that guarantees it will not be eliminated in both periods? Indeed, do such strategies exist? Second, under what conditions will a country find it impossible to ensure being essential in period 2, and thus will prefer - either alone or in concert with others - to initiate a preventive war in period 1? To see why we must ask and answer the first question, notice that the two-period scenario, beginning with stage 2, is in part an ^-person noncooperative game in which each nation's strategy is its investment decision and in which its payoff is 0 if it is eliminated in either stage, and equals the proportion of resources it controls in period 2 otherwise. To illustrate, let the initial distribution be r = (120, 80,60,40) and let the vector of growth rates for this four-country system be w = (1,1,0.5,0.5). Suppose country 1 invests 100 units. Notice that, with 20 units of resources remaining, this country cannot be rendered inessential in the third stage of the first period unless the actions of one or two of the other countries render the remaining country predominant (e.g., countries 3 and 4 invest

5.1 A dynamic extension

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40 and 20 units, respectively, and country 2 invests nothing). Given current growth rates, this means that in the second period, l's resources total 20 (the amount withheld in the first period) plus 2 times 100, which totals 220. But if country 2 believes that investing 100 units is indeed l's strategy, then it can invest 70 units without fearing the possibility that it will be rendered inessential, short of "irrational actions" by countries 3 and 4. However, if 1 believes that 2 intends to invest 70 units, then it should invest nothing instead and become predominant in the first period. This reasoning corresponds to the familiar "he-thinks-that-I-think" regress, and one of our tasks is to ascertain the conditions under which such regresses terminate because the countries possess dominant strategies or because the game possesses a pure strategy Nash equilibrium - an investment decision for each country such that, even after all choices are revealed, no country unilaterally prefers a different decision.4 We must be cautious, nevertheless, in trying to resolve such regresses, because we must first decide how to accommodate one important feature of reality - namely, that real investment decisions are not made at a specific instant. Rather, investment strategies evolve gradually, during which time other countries can observe decisions and adjust accordingly. The United States, for example, cannot act to shift resources from, say, procurement of weapons (the expenditure of current resources) to research and development (an investment) without the Soviet Union perceiving this shift and adjusting to it, and without the United States anticipating the possibility of making a readjustment, long before any funds are actually spent. Because we have rendered decision points discrete, even a threestage conceptualization of period 1 abstracts from this reality. Unfortunately, a model allowing for the possibility of continuous adjustments and readjustments seems too complex to specify or to analyze. Perhaps a differential game analysis would be appropriate, or a very elaborate 4

By a dominant strategy we mean an investment decision that is best, regardless of what others do. A Nash equilibrium is, as we stated earlier, a set of investment decisions - one for each country - such that no country prefers to change its strategy after all others reveal theirs. Game theory admits the notion of a mixed strategy - a strategy wherein a player chooses his or her investment decision by a lottery - but in the present context such strategies seem unreasonable (for a discussion of why we might prefer to exclude such strategies in other contexts, see Ordeshook 1986, pp. 180-3). Notice, nevertheless, that noncooperative games can have more than one Nash equilibrium (e.g., the game of Chicken), in which case it may be difficult to rationalize the prediction that equilibria prevail as outcomes without some form of preplay communication and coordination. Indeed, the proliferation of equilibria in extensive-form games that attempt to rationalize cooperative action as a noncooperative equilibrium remains as a serious theoretical roadblock to the ultimate integration of noncooperative and cooperative game theory. Thus, throughout this analysis, we will be especially sensitive to the possibility of identifying the circumstances under which equilibria are unique.

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description of an extensive form, but analysis would become unduly heavy without any guarantee that we have achieved generality. Hence, to accommodate the possibilities of secondary, tertiary, etc. adjustments within period 1, we implement a different and simpler approach that uses the notion of a strategy's security value. Game theory traditionally defines the security value of a strategy such as an investment decision in terms of what it guarantees a person should everyone else make decisions in that person's worst interests. Calculated in this way, a security value represents a pessimal view of possibilities. However, suppose instead that we calculate security values under the presumption that other people respond to a particular choice in their own best interest, which presumably should occur if they choose after the person in question acts, or if they are allowed an opportunity to readjust their decisions. To be more specific, consider the following definition of the security value of the period-1 postinvestment resource level for country /, r/, which defines the security value of a strategy after considering certain responses and adjustments by other decision makers. Letting E(r) denote the set of countries in S that are essential when the distribution of resources is r, then: The security value of r/, denoted S;(r-)9 equals 1 if and only if / e E(r° = (rf,..., rl,..., r°n)) for all r° satisfying: for no jeS, j*i9 is it the case that j $ E(r°) but j e E(r*), where r% = r°k for all k ^ j and r} > rf > rf. Otherwise, st(r/) = 0. As formidable as this notation might appear, if we keep in mind that we are only talking about the first period, it is merely the formal representation of some logical ideas. To see what this definition entails and to understand how we can use it to accommodate investment adjustments within a period, notice first that our definition set 5/(r/) = l if, regardless of the investment decisions of others, / is essential. For example, the largest country has at least one strategy with a security value of 1, namely, the strategy of not investing anything. But if country / is inessential at rj, given the particular distribution r, there are two possibilities: everyone else is essential or someone else is inessential. If everyone else is essential, then if attacked, / cannot suppose that its sovereignty will be "protected" by other countries defending themselves because (from the definition of inessential) no other country requires /'s support or existence to ensure its own survival. Hence, in this instance, our definition of security value sets Sj(r-) equal to 0. That is, the security value of r\ is zero if there exists some pattern of investments among the remaining countries that renders / uniquely inessential. For the final possibility, which is the one that our notion of security value is intended explicitly to accommodate, we do not require that / be

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essential with r' in order to set S;(A*/) equal to 1. Suppose that / is not alone in being inessential, and suppose that whenever one (or more) of these countries responds to make itself essential by reducing its investment, such a move necessarily renders / essential as well. Then we set s^r-) = 1. The rationale for this attribution of security levels is this: if / and j are inessential, but if each is made essential when the other one acts to become essential, then everyone knows that one or both countries cannot be eliminated from the game. Owing to the interdependence of their security, if one country, say j , is attacked, then the other, /, must respond to make itself essential - otherwise / will become the next victim, because without the possibility of any rational response from j , / cannot hope to become essential and maintain its sovereignty. So, to accommodate collapsing a dynamic investment-adjustment process into a static one-shot decision stage, we suppose that countries, if faced with an attack because they are inessential, attempt to adjust their investments so as to render themselves essential, and we further suppose that all countries, in evaluating the security levels of alternative strategies, assume that other countries respond rationally if someone attempts to take advantage of a temporary vulnerability. This definition of security value, then, reformulates Bueno de Mesquita's (1981b, p. 543) assumption that "States act to prevent other states from gaining a preponderance of power" (with which he characterizes Gulick 1955, Kaplan 1957, Claude 1962, Morgenthau 1973, Organski and Kugler 1980, and Singer 1980) to mean that states formulate their strategic plans under the presumption that other states will act to avert their own elimination. Hence, to the extent that this notion is critical to our analysis of systems with dynamical resources (which it is), we see quite directly how the idea of collective security is an essential part of the balance-ofpower theory. Morgenthau (1959, p. 175) sought to distinguish balance of power from the notion of collective security: Balance of power alliances are formed by certain individual nations... on the basis of what those individual nations regard as their separate national interests. The organizing principle of collective security is the respect for the moral and legal obligation to consider an attack by any nation upon any member of the alliance as an attack upon all members of the alliance. Consequently, collective security is supposed to operate automatically; that is, aggression calls the counter-alliance into operation at once and, therefore, protects peace and security with the greatest possible efficiency. Our formulation, though, argues that this distinction is meaningless, and it presumes that whatever "moral and legal" imperatives exist, their basis lies in the egoistic motives of individual states. Put simply, our definition of security value presumes that countries invest even if such action renders

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them vulnerable in the short run, provided that others have the incentive and the means, however indirect, to render them essential should the need arise for them to do so.5 Aside from rendering the notion of collective security explicit within a balance-of-power framework, our concept of security value has the technical advantage of associating a single number with each strategy that is independent of the actions of all others. If we are able then to justify the supposition that this number indexes preferences (i.e., denotes the utility that a person associates with each strategy), then we can reduce a gametheoretic problem to a simple decision-theoretic one in which people simply choose strategies that maximize their security value. In the theory of two-person, noncooperative games, that justification is provided by showing that if both persons maximize their security values then their choices are necessarily in equilibrium - neither person has any incentive to alter his or her choice unilaterally after the fact. This justification does not extend to all games in general, but in the next section we use it to model preferences over alternative investment decisions.6 5.2

Preferences

We are now in a position to specify the investment game extension of our basic model. Recall that a noncooperative game has three characteristics: the set of players, the strategies available to each player, and a payoff 5

6

At this point, however, we should emphasize that our analysis makes critical use of the preceding definition of security value, and thus, if it is invalid, then our conclusions about stability and preventive war may be invalid as well. The critical aspect of this definition, clearly, is the presumption that the short-term viability of some countries is established by a mutual vulnerability and, subsequently, by the assumption that everyone acts in their own self-interest. If, however, uncertainty and misperception cloud the picture and if others cannot be relied upon to act in their own self-interest (defined, in part, by the assumption that everyone else is doing the same), then investment poses a risk that national leaders may rationally choose to avoid. Before we use this notion of security value to define preferences over alternative investment decisions, we must first consider the possibility that everyone computes the security value to them of their strategy in r' to be 1, but if everyone adopts the strategy specified by r' then a subset of S finds itself inessential and unable, owing to the strategies of the remaining countries, to extricate itself from this situation. If this possibility arises then there may be no basis for supposing, as we do shortly, that countries maximize their investments subject to the constraint that, whenever possible, they not choose strategies with a security value of 0. To see, then, that this situation cannot arise, notice that / alone cannot be inessential; otherwise, / cannot be certain that, if attacked, no one offers assistance out of their own self-interest, in which case s,(r/) = 0. So suppose that all countries / e XC S are inessential with r', that all countries / e S—Xarc essential, and that £,(/"/) = 1 for all / e S. That the strategies of the countries in S—Xcannot "trap" two or more countries in X, say the subset Y, into being inessential follows because, by setting 5/(r/) = l,

5.2 Preferences 161 function, which either specifies the utility to / from each vector of joint strategy choices or which specifies ordinal preferences over these vectors. The set of players is 5, and z's strategy is the amount of resources it fails to invest, rft = ri—ph so its strategy space is Rt = {r- |0 rk and j is inessential in

5.8 Appendix: Proofs of results

185

(S; r ) , then k is inessential in (S; r ) . If k is essential, then by definition there

exists a CeW* such that keC, r(C-{k}) + rk>R/2 and r(C)-rk< R/2. Because rj>rk, it must be the case that r(C-{k}) + rj>R/2 and r(C) — Tj < R/2. So j is essential, which contradicts the assumption of the lemma that j is inessential. Turning to the proof of the remark, it must be true that ry < R/2, for all j ^ /. It follows from the lemma just stated that / can invest to reduce its resources in period 1 to r2, the second largest country: If / becomes inessential, all countries are inessential, which is impossible, so by All, the unique period-1 equilibrium is an r' in which / invests at least rt-r2. Q.E.D. Remark 5.6 From Remark 3.5, if rt = R/2 for some / e S, then (S; r) is resource stable and thus system stable. By Remark 5.5, no country can invest except /. But if w > 0 , then r" > 2 " = I A / / 2 in period 2, which is to say that / becomes predominant. Hence, S— {/} can threaten / in stage 3 of period 1 if / invests. Since r(S— {/}) = rt > /*/, S— {/} can force some reallocation, but in no case will / be allowed to secure more than half the resources in period 2. Since / is assured of controlling half by not investing, it prefers not to invest, in which case (S; r) describes period 2. Q.E.D. Remark 5.7 To prove sufficiency, we must specify the range of w such that expressions (5.1) and (5.2) in footnote 13 are satisfied: (i) If N< 0 (which means that the members of C except / invest more than half of the total investment), which implies that K< 0, and if M > 0, then to satisfy (5.1) and (5.2), the range of w is -L/K<w< -M/N; (ii) if N= 0, and (1) if K< 0, to be essential, w>-L/K, (2) if K > 0, L < 0, then w < -L/K, (3) \fK=0, L < 0 , then w > - l ; and (iii)if A/>0, and (1) if K< 0, then w > -L/K and w > -M/N, (2) if K>0, L < 0 , then -M/N<w - l . To prove necessity, notice that when the condition on w is not satisfied, ceterisparibus, conditions (5.1) and (5.2) are not satisfied, and / is not essential. Next consider the cases that are not included in the previous: (i) If N (R + wI)/2.

In other

186 5 Preventive war words, coalition C— {/} possesses more than or exactly half of the total resources at period 1, and invests more than half of the total investment. Hence, Ce Wbut C$ W*, and / is not essential; (ii) if Af>0, K>0, and L > 0, then r(C) + wp(C) < (R + wI)/2. In other words, coalition C possesses exactly or less than half of the total resources at period 1 and invests exactly or less than half of the total investment, which implies that C$W; hence, / is not essential. Q.E.D. Remark 5.8 Because rt < R/2 for all /, any three-person coalition is winning. Consider now the allocation (R/3, R/3, R/3,0) in the presumed set of core points, Co. From Remark 5.1, all three countries invest everything to yield the postinvestment outcome ((l+w)R/39 (l+w)R/3, (l+w)R/3,0). Because the equivalent outcomes in which we permute the identity of the countries do not dominate each other, to prove that such a set is a core we must show that no alternative outcome dominates any outcome in Co. We have three cases: (i) (1,4} is winning and they divide all resources evenly between themselves (since an uneven distribution renders one of them predominant). Since 1 and 4 cannot invest in this event, such an allocation dominates what results from r in Co if and only if both countries prefer R/2 to what they get from r, and, in particular, if one of them prefers R/2 to (l + w)R/3. But this inequality requires that w0.5; and (iii) consider any allocation of the form r' = (r{, A*2, /"3,0). Once again, we need only show that 2 prefers r to rr. As before, 2's maximum postinvestment resources are, from Remark 5.1, r{ + w(r2 + A*3 - r[) = r{+w(R -2r[), the derivative of which is negative as r[ increases if w > 0.5. Q.E.D.

CHAPTER 6

Geography, balancers, and central powers

The geographic location of a state in the world is of basic importance in defining its problems of security. It conditions and influences all other factors... [and] regional location defines potential enemies and allies and perhaps even the limits of a state's role as a participant in a system of collective security. Nicholas J. Spykman, The Geography of the Peace (1944, pp. 22-3)

If a Soviet strategic planner could be granted one wish, it should be to move his country somewhere else. Stephen M. Walt, The Origins of Alliances (1987, p. 277)

A fundamental difficulty with formulating a fully comprehensive theory of stability in anarchic international systems lies in conceptualizing a country's resources so that we adequately summarize the strategic imperatives of a decision maker seeking to ensure a country or a regime's sovereignty. In his early work on coalitions, the size principle, and the application of cooperative game theory to a formulation of the concept of balance of power, for example, Riker (1962) assumes that such systems are constantsum games in which winning coalitions are those that control a majority of resources. Much of the formal theorizing in the balance-of-power literature, including our analysis thus far, follows Riker's lead of assuming that resources are additive across a coalition's members and that any coalition can defeat a coalition with fewer resources. This formulation supposes implicitly that resources to attack one country can be used equally well to defend against an attack, or that those resources can be directed with equal effectiveness at any country. Yet, such assumptions are not wholly general. History tells us that even inferior land armies may be effective for defending against a more powerful adversary if that adversary has not the technology for supply far from its home base, and that even if an army can defeat a contiguous adversary, it may be useless against one with even a modest water barrier. Although the United States had perhaps the most technologically advanced and powerful army in the world during the Civil War, we can generally ignore its influence in nineteenth 187

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century European great-power diplomacy simply because of its geographic remoteness. Indeed, although a tenet of today's strategic thinking is that modern technology fundamentally changes the implications of such barriers and distance, Morganthau's (1959) assertion that "[t]he most stable factor upon which the power of a nation depends is obviously geography" (p. 102) and his more specific argument that the geographical location of the United States remains a fundamental factor of permanent importance which the foreign policies of all nations must take into account, however different its bearing upon political decisions might be today from what it was in other periods of history (p. 103) finds contemporary support in Pearson's (1974, p. 455) reasoning that "even in the post-World-War-II technologically advanced environment, geographic distance seems to represent a cost in the movement of troops which all but the most powerful states are unable or unwilling to pay," and in Bueno de Mesquita's (1981b, p. 44) assertion that modern technology does not eliminate the necessity for discounting capabilities among even today's great powers. This argument's implication is that formulations of balance of power in which resources are treated symmetrically, although valuable for an initial understanding of the problem, may mislead us about the necessary and sufficient conditions for stability, and may be inappropriate for interpreting a broad range of historical events and processes. Indeed, profoundly important questions cannot be answered adequately unless geography is taken explicitly into account. Should we, for example, attribute Britain's role as "balancer" in the nineteenth century to the motives and diplomatic skills of key decision makers or to its unique geographical position with respect to the Continental powers? Waltz (1979, p. 164) correctly observes that "the notion of a balancer is more a historical generalization than a theoretical concept," but he goes further to assert that "balance of power theory cannot incorporate the role of balancer because the playing of the role depends on such narrowly defined and historically unlikely conditions" (1979, p. 164). Is this view justified, or is there a general theoretical explanation for, in Churchill's words, "the wonderful unconscious tradition of British foreign policy"? What precisely are the advantages and disadvantages of geographical distance? For example, should we attribute the unification of China at the end of the Warring States Period in 221 B.C. at the hands of a geographically "peripheral," as against central, power as mere happenstance brought about by the fortuitous combination of military and diplomatic skill, or is it reasonable to hypothesize that such peripheral powers have an advantage over central ones? Can that part of Germany's special, threatening role in European power

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politics from 1871 to 1945, which is seen as a consequence of its geographically central position, be understood theoretically and generally? What are the implications for international stability of any decline in geography's importance, as occasioned by advances in the technologies of weapons systems? Can the removal of natural barriers to warfare disadvantage individual states, but nevertheless make international systems more stable? Finally, to the extent that nuclear weapons neutralize the great powers and render them unable or unwilling to play anything but a peripheral role in regional conflicts, what are the strategic imperatives in particular regions such as the Middle East, where conventional weapons and resources, and thus geography, follow a classical logic? To answer such questions, this chapter extends our earlier analysis to include the asymmetries occasioned by geography in the use of resources. In formulating this extension, however, we must make certain compromises. First, our analysis focuses on the model that Chapters 3 and 4 detail, and it forgoes any treatment of differential resource growth and preventive wars. Second, we only treat three-country systems. Naturally, we prefer a general ^-country model, but even the analysis of a special case is enlightening, since many of our results are unanticipated. We identify, for example, the circumstances under which one country is uniquely positioned to play the role of "balancer." We find that "central powers" countries that are contiguous to two noncontiguous powers - are at a disadvantage in terms of the resources they must possess to be essential and to maintain system stability. We also find that if geography discounts the offensive capabilities of all countries (but not "too much"), then no threecountry system can be system stable. Thus, although we might disagree as to whether war is made less likely by modern technology because that technology increases the costs of conflict to mutually unacceptable levels, our analysis shows that even with this consideration aside, those features of modern weapons technology that have the effect of removing the barriers of geography among all great powers simultaneously make international systems more stable. 6.1

Conceptualizing the consequences of geography

The essential feature of geography that we seek to capture in our analysis is that resources may be especially advantageous for defense as against offense, and they may be especially advantageous or disadvantageous when used to attack one country as compared to another. Germany's army in 1940 might be well suited for overwhelming France but not England, even though France certainly had a more powerful army than England. Yet,

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even if ill-suited for attacking across the Channel, that army was sufficient for repelling a counterinvasion by England alone. Spain once held an advantage over England in colonizing the world and even dominated the Continent, but it could not extend its power to invading England; and a relatively minor "power" in the New World, led by an oft-defeated general, could secure its independence from a state that was almost certainly the most powerful in Europe. On the other hand, if we conceptualize resources not in terms of direct military capabilities, but instead as economic control, this special asymmetry between offense and defense remains relevant. Aside from the obvious advantages of language and familiarity with legal institutions, countries can erect barriers to investment by foreigners so that domestic entrepreneurs can more readily secure economic control of industrial capacity within their own countries than they can of capacity elsewhere. In whatever way we choose to conceptualize and measure resources, then, it seems only reasonable to suppose that defense has an advantage over offense - that a "unit" of resource mobilized for defense can at least counter a unit of resource mobilized by a second country for offense against the first. Hence, to accommodate the imperatives of geography, we first reinterpret the resource vector r a s a summary of the defensive capabilities of countries. Next, we suppose that a country's resources, say country /, can be applied offensively against another country, say j , but only after they are discounted by some parameter dtj, where 0 < dtj < 1. Thus, we let r/rf/y denote /'s offensive capabilities with respect to j , and as a modification of Assumption A6, which posits complete symmetry in offensive and defensive capability, we assume instead that 1 A6'. Country / can defeat j (/ can unilaterally absorb all of y's resources but not vice versa) if and only if r/rf/y- > rj9 and j can defeat / if and only if rjdjt > rh 1

Assumption A6' differs from the way in which Wright (1942, pp. 1389-91) tries to incorporate distance. Wright assumes that the ability to resist aggression equals the "degree of separation" minus the disparity in power. Although /'s "ability to resist aggression" on the part of country j is not a theoretical primitive in our analysis, we can roughly take it to be measured by /•,• — ry djt, so that (in Wright's notation) ry djt = 5// — Pi. Boulding (1962) assumes that offensive capabilities decline monotonically with distance, as does Bueno de Mesquita (1981b); however, because we treat countries as discrete and stationary entities, all such accommodations for distance are equivalent as long as we assume that geography discounts offensive but not defensive capability. Admittedly, though, our assumption is a problematical abstraction for a country such as Russia in which the discounting that ought to be applied with respect to other European powers should differ from the one that should be applied even today for the Asian powers of China and Japan. For additional discussion of the results this chapter reports, see Niou and Ordeshook (1989a).

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191

Notice that even if resources as represented by the r's are not equal, neither country may be able to defeat the other if the d's are sufficiently small. An immediate implication of this fact concerns the number and nature of winning coalitions. Recall that if all d's equal 1 in a three-country system, then either one country is predominant (if it controls more than half the resources) or all two-country coalitions are winning in the sense that they can overwhelm the third. With discounting for geography, however, we must generalize the definition of the set W* of two-country winning coalitions as W*=\

CCS, |C| = 2, 2 ieC

So if the d's are sufficiently small, then W* is empty, whereas with an appropriate selection of discount factors, only one or two coalitions may be winning. The comparison of the results in this chapter with those we offer in Chapters 3 and 4 is especially important. For example, we can interpret du = dji = 1 as meaning that / and j are contiguous, at least from the perspective of the technology of offensive resources. If resources can be converted into weapons systems that can render geographical barriers irrelevant, then even if countries are not geographically contiguous, they ought to be treated as contiguous. Theorem 3.1 tells us that if all d's equal 1, then all three-country systems are system stable. Hence, the results we obtain in this chapter tell us what effect such modern technologies have on international stability. To deduce results formally so that we can make such comparisons, however, we must impose a simplifying assumption about a particular type of asymmetry in the discount variables that reduces the complexity of the conditions for stability that we offer later. Specifically, any asymmetry in the d's of the form d;j > djt means that / has a technological advantage over j - that /'s resources are more effective for attacking j than are j's resources for attacking / (viz., the United States' short-lived advantage over the Soviet Union following World War II). If no such asymmetric superiority exists, then A12. dtj = dji for all / and j in S. Keeping in mind that A12 implies nothing about the relation between dtj and dik or between d^ and djk, we use this assumption to facilitate analysis, although it is not required by all of our results. We should emphasize that we do not suppose that this assumption models all historical circumstances. Certainly, asymmetric investments in navies generated asymmetric offensive and defensive capabilities among European powers throughout

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modern history, just as the Mongol hordes and barbarians of the Steppes could invade and replace established empires because of the special advantages of mobility afforded them by their horsemanship. To illustrate now how our analyses in Chapter 3 and 4 are altered with the preceding reformulation of resources, consider the three-country system with the initial resource distribution (140,120,40). If all rf's equal 1, then this system is system but not resource stable. Suppose, however, that dn-1> ^13 = ^23 = 0-2. Then 2 cannot absorb 3, nor can 1 absorb 3. But 1 can attack 2 without 3 being able to assist 2 in a viable counter - 3 cannot transfer sufficient resources to 2 so that 2 can defend against 1 nor can 3 divert sufficient resources from 1 by attacking it. Thus, 1 absorbs 2, and, with 260 units of resources, it subsequently absorbs 3. Hence, even though it initially controls less than half the resources, country 1 is in fact predominant. Alternatively, (160,80,60) is wholly stable if dl2 = dn = 2/3 and d23 = 0.7. To see this, notice that if 1 threatens to absorb 2 and become predominant, then 3 can join 2 in a viable counter by attacking 1. If 3 attacks 1 then 1 must hold r3rf31= 40 resources in reserve to counter 3's attack, leaving it with insufficient resources to threaten 2. And because 2 cannot threaten 3, (160,80,60) is resource and thus system stable. This simple example, then, illustrates a fact that we might have suspected was true, namely, that with geographical considerations taken into account, countries can become predominant even if they do not initially control a majority of resources, and countries with a majority of resources need not be predominant. Proceeding further to generalize the implications of this example requires that we choose between two alternative assumptions about how countries can use the resources of other countries. Previously, we assume that if country / defeats country j and absorbs all of y's resources (Assumption A5), then / can target rt + rj resources at k. However, now we must choose between these two assumptions: A13. If / absorbs y, then / can target (rt + rj)dik resources at k, and A13'. If / absorbs y, then / can target ridik + rjdjk resources at k. Choosing between these two assumptions entails asking whether, in the case of A13', it is merely the sovereignty of resources that are transferred (in the same way as the United States used its strategic position following World War II to target the resources of both Japan and Germany at the Soviets) or whether, in the case of A13, / absorbs y's resources into its own territory (in much the same way as the Soviet Union absorbed

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193 2

Germany's captured industrial plant after that same war). If resources refer to territory and the population it holds, then the second assumption is appropriate. A13', for example, fits the scenarios of the early development of empires in which, because of the poor technology of transport, armies had to rely for their food, and even for their personnel, on indigenous, conquered populations. If, on the other hand, resources are readily transportable (such as when the resource is wealth), then we should impose the first assumption. Thus, both assumptions are plausible. A13, though, seems more consonant with an analysis that conceptualizes resources and international conflict in terms of economic wealth and control of industrial capabilities. Thus, all of the results we offer in this text employ A13. However, when we come to the central result about stability, notably Theorem 6.1, we indicate a parallel condition that assumes A13'. Before proceeding further with the clarification of some definitions and with our formal results, let us preview the broad outline of the results we establish. In general, we learn that, at least for three-country systems, geography profoundly affects the circumstances under which system stability prevails. We know from Theorem 3.2, of course, that if dtj — dik = djk = 1, then ({/, j , k]; r) is system stable. Theorem 6.1, on the other hand, establishes that if none of these d's is equal to 1 - if no two countries are contiguous - then the system cannot be system stable. Stability requires either that two countries be both contiguous to a third or that two countries be contiguous to each other. The possibility that two countries are contiguous to a third but not to each other gives rise to the notion of a "central power," and our analysis shows that central powers are disadvantaged if the discounting of resources between the two "peripheral powers" is sufficiently great - the central power must control more resources than either peripheral power in order to ensure its sovereignty. The possibility that two countries are contiguous but not to a third, on the other hand, gives rise to the notion of a balancer - the sole peripheral power - who, as we shall see, enjoys the advantage of being able to ensure that it is near-predominant. We cannot establish these substantively tantalizing results, however, unless we render certain ideas more precise. In particular, we turn now to the issue of when a country can become predominant - when it can threaten the sovereignty of others without confronting any viable counterthreats. Without considering geography, the corollary to Theorem 3.1 tells us that in three-country systems, any attack by one country on a 2

A third possibility allows a country to choose the portion of its resources to target via djk and djk. This possibility yields a somewhat more complicated analysis, but the conclusions we offer do not appear to be qualitatively altered.

194

6 Geography, balancers, and central powers

smaller one threatens the third (unless the attacking country already controls over half the resources in the system), and, correspondingly, any attack by a country on another can be countered by a viable counter. However, with geographical considerations, a country, say /, can become predominant, first, only if, upon the absorption of, say y's resources, it can overcome the third, k. Second, predominance requires that k cannot effectively assist j in formulating a viable counterthreat by transferring resources to j. The final requirement for the predominance of / is that k cannot participate in a viable counterthreat by attacking / directly (thereby causing / to divert some share to defending against k while attacking j). Formally, this yields the following revised definition of predominant: Country / is predominant if and only if neither j nor k, acting alone or in concert, can formulate a viable counterthreat to a threat by / that renders / predominant - if and only if there exists a jeS-{i] such that: (1) (ri + rj)dik>rk (i.e., if / absorbs j9 i can defeat k); (2) ridij>rkdkj + rj (i.e., / can defeat j even if k transfers all of its resources to j); and (3) r/C^- > rkdkidiJ + rJ (i.e., / can defeat j even if k tries to aid j by attacking / directly). Correspondingly, recall that, without any consideration given to geography, country / is near-predominant if rt = R/2 - if the addition of e resources, however small e might be, renders i predominant. This idea can be generalized by removing the reference to the specific number R/2 and by saying simply that / is near-predominant with rh first, if / is not predominant, and, second, if it is rendered predominant whenever its resources are increased to rt + e, where e is any number that exceeds 0.3 Before proceeding further, it is important that we first verify that a two-country (bipolar) system can be system and resource stable. For the model offered in Chapters 2 and 3, two-country systems are system stable if and only if both countries control an equal share of resources. As we have previously noted, however, such a "knife-edged" stability condition is the by-product of our mathematics - of the fact that we have not incorporated the momentum of existing alliances and the constraints of domestic politics, and of our failure to consider the uncertainty confronting real decision makers in their assessments of resources and their attitudes toward risk in their assessments of strategic options. We should make 3

Under A13', we say that / is predominant if and only if there exists a j in S- [i j such that (1) ndik + rjdjk >rk- that is, after / eliminates and absorbs all of j , k can be eliminated; (2) ndy > rj + rkdjk - that is, / can defeat j even if k aids j by transferring resources to j ; and (3) (ri — rkdkj)dij > rj - that is, / can defeat j even if k aids j by attacking / directly. The definition of near-predominance is unchanged with this alternative assumption.

6.1 Conceptualizing the consequences of geography

195

certain, however, that incorporating geography does not destroy so fragile a condition, and, in the process, check whether our formulation of geography renders bipolar systems more stable. The following remark, which does not require that dtj = djh answers our query: Remark 6.1. In two-country systems, either one country is predominant or the system is both system stable and resource stable. Neither country is predominant if and only if 1/(1 + dtj) > rt/R >

This remark follows from our assumptions and some simple algebra. First, suppose that country / has no threat against j (i cannot defeat j ) , in which case we must have rtdij Rdjt/{\ + tfy7), which implies the inequality in the remark. To see how this result relates to our analyses in Chapters 3 and 4, notice that if dij = dji = 1, then the inequality becomes 1/2 > rt/R > 1/2, or simply rt = R/2. More generally, the inequalities in Remark 6.1 can be satisfied by an appropriate choice of rt only if the first term is at least as great as the third; otherwise we have a contradiction and system stability is impossible. Some simple algebraic manipulation shows, however, that l/(l + djj)>dji/(l + dji) if and only if l>rfy7d/y-, which is an inequality that is necessarily satisfied, given the constraints on the rf's that they not exceed 1. Thus, system stability is possible in two-country systems (as is resource stability since both forms of stability are equivalent in such systems). And not only is system stability possible in two-country systems after we take geography into consideration, but Remark 6.1 establishes that stability no longer requires a "knife-edged" equality of resources. For example, even if country 1 has twice the resources of country 2, the system ({1,2}; (200,100)) is stable and country 2 is essential as long as tf12 rk, (b) du = djk, and (c) either d ij = djk = \ or dik = \. A straightforward corollary to this theorem, then, is as follows: Corollary. If [i,j], {j,k}, and {/,/:} are all winning coalitions, then (5; r) is system stable if and only if all the tf's equal 1 (if all countries are contiguous). One way to interpret this theorem is to consider the following result about systems that violate condition (3): Remark 6.2. If there are only two winning coalitions, say {/,yj and {j, k] in a three-country system, if dtj > djk, and if j is not predominant when it absorbs either / or k, then the system cannot be system stable. In particular, a stable two-country system will emerge without /. To see the logic of this remark, suppose j is not predominant over / even if j controls ry -I- rk resources and that it is not predominant over k 7

Under assumption A13', condition (3) becomes simply: if every country is in some minimum winning coalition. This follows since if {/, k) is winning, then by the result in footnote 4, a = gi({ij}) rki and by the result in footnote 5, k does not control sufficient resources to render / nearpredominant.

6.2 System and resource stability

199

8

even if j controls rt + rj resources. Notice from Lemma 6.1 and from the definition of gj that j gains something from each winning coalition. Since j is the sole pivot between minimal winning coalitions, it follows that there cannot be any viable counter to a threat by {/, j) to eliminate k or by {j, k] to eliminate /. It follows from Lemma 6.1 that #,•({/, j})< gj({j, k}) if and only if djk < dij9 so, in particular, j prefers a coalition with k if and only if djk 0.8(100)+ 80-46.66. Indeed, we see here that 0.8(166.66) exceeds x( 100) + 80-46.66 for any value of x less than 1 - for any degree of noncontiguity between 2 and 3.

The conditions of Theorem 6.1 are important since they imply substantively meaningful results about the special role of countries that balance in international systems and countries that we call central powers. However, postponing the interpretation of this theorem for the moment, let us first use it to define, in conjunction with the definitions of predominant and near-predominant, a reformulation of the characteristic function for the cooperative three-person game among countries; thus: v(S) = R and v(i, j) = rt + rj and v(k) = rk if W* is empty or if / or j are near-predominant.

(6.1)

Clearly, if W* is empty, no threats are possible, so, trivially, there is a valid counter to every threat in which no country has to transfer resources to others. If some / is near-predominant, then, as in our earlier model, j and k cannot threaten each other for fear that / can become predominant (Assumption A6), nor will / join in a coalition to threaten a third country since it knows that it will not be allowed to gain any resources in that coalition. Next, and if either / or j is near-predominant if / or j controls 6 additional resources, d," and / becomes predominant, which means that it is near-predominant without e. Finally, suppose / threatens j alone. If dkJ = 1, then rj + rkdkj = rj + rk = rjdjj, k can join j in a counter by transferring resources to y, and thus / is not predominant. But if e is added to /'s resources, the last equality becomes" < " and k cannot join j in a viable counter by attacking / or by transferring

6.5 Appendix: Proofs of results

213

resources to j . An attack yields a viable counter only if (r, + e — rkdki)dij < r7, or, equivalently, (rt + e)dij- 0 , this requires that rt djj < rj + rk dik dtj. We know that dik dtj < 1, so this inequality cannot hold. Nor can k form a counter by transferring its resources to j . Such a transfer yields a viable counter only if ry + rk > r^dy + edtj. Since e > 0, this requires that ry + A^ > rt dtj, which violates the assumption of the lemma. So again, / is near-predominant. Suppose, on the other hand, that dki = dtj - 1. Then, if / threatens j , k can join j in a viable counter by attacking /. Country / must target more than /y of its resources at j and at least rk at k. If rt = rj + rk, this is impossible, whereas if e is added to fs resources then / has sufficient resources to defeat j and k. To prove necessity, we already know by definition that if r^dy > rj + rk, then / is predominant; so suppose that " < " holds. By Lemma 6.2, g;(C) > 0 for some C in W*9 i e C. But / cannot subsequently gain enough to eliminate its coaliion partner, since such a coalition agreement violates our model's rationality assumptions. Suppose then that " = " holds. We are left with two cases. First, if dkJ< 1 andd k i rj + rkdkj and rt dtj > /y 4- rkdki dtj, in which case / is predominant. Second, if dkj < 1 and djjrj + rkdkj and rjdij> rj + rkdkidu. Q.E.D. Theorem 6.1 It is evident that conditions (1) and (2) are sufficient. To show that (3) is sufficient, suppose (rr, ({j, k], {/))) is a threat against /. From the definition of a threat, this requires that {j,k}eW*9 rjdjj + rkdki>rh r- = 0, and r]>rj, and r'k>rk. But if, as (3) assumes, /, when threatened by (rr, ([j9k], {/})), is in some winning coalition, say {/,./}, such that the conditions of (3) are satisfied, then / can transfer gy({/, j}) resources toy. In this case, by Lemma 6.2, j becomes near-predominant and the transfer is a counter. That / can transfer gy({/, j}) resources to j without setting its resources to 0 follows because (r;+rj)djk > rk, which implies that gjdjj k})< ri - otherwise, j is predominant when it secures gj([j9 k}), in contradiction of Lemma 6.1. It is a viable counter since, by A6, j prefers the transfer to securing an equivalent amount of resources by eliminating /, and since, by Lemma 6.1, j cannot gain more than gj([i9 j}). To prove that the conditions of the theorem are necessary, suppose (5;r) is not system stable, W* is not empty, and no country in the system is nearpredominant. Then, if dtj ^ djk9 or if dik ^ 1 and dtj or djk ^ 1, by Lemma 6.2, / cannot render j near-predominant. Finally, if (rt + rj)djk v(i) + v(j) = ri + rj-2b. But if W* is not empty and no / is near-predominant, there exists a b > 0 that renders some / predominant, so the game is essential. It follows, because only inessential constant-sum ^-person games have nonempty cores, that the core is empty for situations described by the characteristic function in (6.2), and that the system is not resource stable. Q.E.D. Theorem 6.3 For sufficiency, by Theorem 6.1 and the definition of a balancer, conditions (1), (2), and (4) imply that / is a balancer and the system is system stable, whereas (3) implies that neither j nor k can be balancers. To prove necessity, if (4) is violated, then {/, j] and {/, k] cannot be winning and / cannot be a balancer. If d(rj + rk) > rh then / is not the unique balancer. Specifically, {j, i] and {j, k) are winning: {j, k] is winning from the assumption that (3) is violated; and condition (4) implies that /y +tfr,-> rki so {j, /} is winning. Third, if (2) is violated and dtj = dik = 1, then, ceteris paribus, by (3), rj-\-rk GR > AG = GI. 20

The interests of Russia in the Balkans and the Ottoman Empire generally related directly to its ability to maintain its rate of domestic resource growth and its rate of growth relative to the other great powers. The Austro-Hungarian need to resist such Russian ambitions accounts for the collapse of the League. However, this argument in no way mitigates Russia's general preference for a coalition with Germany to a coalition with France in the event of Austria-Hungary's nonavailability - especially at the point in which the League becomes irrelevant (1887), and a Russian-German coalition becomes winning (see Figure 7.4). Although Russia was able to credibly threaten a potential coalition with France against German interests in an effort to preempt German support for AustriaHungary, the underlying validity of our analysis is suggested by the Russo-German Reinsurance Treaty of 1887 and the stillborn 1905 Treaty of Bjoerko.

245

7.4 The Austro-German Alliance

1871 1875 1879 1883 1887 1891 1895 1899 1903 1907 1911

BR

BF

AB

BG~ABG~ABGI •

BFR

-B-

ABI

Figure 7.10 q(C)/r(C) for Britain, and FG, AR, and GR infeasible.

Thus, the apparent German preference for an alliance with Austria-Hungary (as evidenced by its formal treaty obligations) over alliances with Russia and Great Britain is unwarranted if we assume that size dictates the preferences of national leaders for coalition partners. However, a more detailed examination serves to emphasize that we cannot explain individual pairings within an alliance without reference to the alliance structure that characterizes the full system. First, to ascertain whether a German alliance with Austria over one with Britain violates our model, we ought to keep in mind that alliances must be mutually beneficial. Germany, as we have already seen, prefers a coalition with Austria, Italy, and Russia, or Austria and Italy as long as Britain remains neutral to any Continental conflicts. A preference for a coalition with Britain emerges only in the event of the unavailability of AGIR or of British willingness to coalesce with France or Russia against Germany. If we suppose, though, that a Franco-German as well as Austro-Russian rapprochement is impossible, and that by coalescing with Austria, Germany signals to Britain its unwillingness to side with Russia, thereby precluding a Russo-German threat to British interests, Britain's optimal strategy, as Figure 7.10 shows, is to remain unaligned, at least up to 1899. It also is apparent that by 1886, Anglo-German competition for economic markets had emerged as an important political-economic dynamic that characterized the period

246

7 Great-power alliance formation, 1871-1914

through 1914.21 The repeated attempts of Great Britain and Germany to reach a rapprochement and their repeated failures to do so suggests that we are dealing with marginal considerations, but considerations which nonetheless color strategy. The problem of Germany's failure to ally with Russia on terms equivalent with its alliance with Austria-Hungary, however, remains. Although our analysis in Chapter 6 of geography's role applies only to three-country systems, geography seems relevant nonetheless. Given a Russo-German alliance in the absence of an Austro-German alliance, we must look at the countermoves available to Austria. Militarily, its most viable counter is an Austro-French coalition, which at some point would almost certainly involve the British as well. Germany would be faced with a multifront war threat - a war in which the proximity of Austria and France to the primary fields of battle would discount Austro-French resources to a lesser degree than Russian resources (see Heller 1926). In addition, we can view an Austro-German alliance as fulfilling the conditions for system stability that Remark 6.4 describes. With Russia an ineffectual military force against France (i.e., dRF