Table Of Content

Game Theory: A critical text Routledge 2004 This book started life as the second edition of an earlier one, also written in collaboration with Shaun Hargreaves-Heap (of the University of East Anglia), published back in 1995. The latter was entitled Game Theory: A critical introduction. Around 2000 we started working on a second edition, following the unexpected success of the 1995 effort. However, as the writing continued, we realised that we were, in effect, writing a totally new book; one that shared very few passages with the 1995 edition (and was much longer and more comprehensive to boot). To make this clear, we effected a subtle but important change in the subtitle: The 2004 edition is referred to as a Text, in contrast to the 1995 one which was a 'mere' Introduction. Below you can read the Preface of the 2004 text, followed by the table of contents. Individual chapters are downloadable by pressing on the relevant chapter title below (the versions available are preproduction drafts - not the final copy as it appears in the book) GAME THEORY: A Critical Text PREFACE As ever there are people and cats to thank. Some have left us, others have joined in, all are crucial players in our minds. There is also on this occasion electronic mail. Separated by oceans, endless landmasses, and the odd desert, we could not have written this book otherwise. Its genesis goes back to a time in the eighties when we were colleagues at the University of East Anglia, Norwich, where game theory had become the object of interdisciplinary scrutiny. Both of us had been toying with game theory in an idiosyncratic way (see SHH’s 1989 and YV’s 1991 books) and it was a matter of time before we did it in an organised manner. The excuse for the book developed out of some joint work we did in Sydney in 1990, where YV had sought refuge (from Maggie’s England) and SHH was visiting (on a year’s ‘release’). During this gestation period a number of colleagues played an important role in helping us see the trees from the forest and guided us around many pitfalls. The late Martin Hollis was one of them and we miss him dearly. The first draft of this book took shape in various cafeterias in Florence during YV’s visit to Europe in 1992 and matured on beaches and in restaurants during SHH’s second visit to Sydney in 1993. For the next two years, the email wires between Sydney and Norwich, or wherever they are, were rarely anything other than warm to hot. When the first edition of this book came out in 1995, we blamed the Internet for all errors. On viewing the galley proofs of the first edition, the uncertainty of our feelings about the whole enterprise was such that we almost fell out. The actual reception was far more gratifying than at least one of us had imagined. Many social theorists, from distant lands and near, wrote in appreciation. It is they, and Rob Langham (our indefatigable Routledge editor), who must be blamed entirely for this new effort. Had their reaction not been as heart warming, we doubt we would have found the energy to go back to the drawing board. For this is precisely what we did: While maintaining the original book’s style and philosophy, we started from scratch, only occasionally plundering the first edition. Help came our way from two quarters: The Australian Research Council which funded the experimental work referred to in Chapters 6&7, and EUSSIRF which funded YV’s research at the LSE Library in 2002. The current book’s creation coincided with YV’s latest migration, this time to his native Greece and to hitherto unknown administrative chores. It also coincided with SHH going ‘native’ at East Anglia; namely, becoming that University’s Pro-Vice-Chancellor. Evidently, unlike the first edition, this one did not come of age on golden Antipodean beaches or in Florentine cafeterias. We hope we succeeded in concealing this sad reality in the pages that follow. It is natural to reflect on whether the writing of a book exemplifies its theme. Has the production of these two books been a game? In a sense it has. The opportunities for conflict abounded within a two-person interaction which would have not generated this book unless strategic compromise was reached and co-operation prevailed. In another sense, however, this was definitely no mere game. The point about games is that objectives and rules are known in advance. The writing of a book (let alone two in succession, and on the same subject) is a different type of game, one that game theory does not consider. It not only involves moving within the rules, but also requires the ongoing creation of the rules. And if this were not enough, it involves the ever-shifting profile of objectives, beliefs and concerns of each author as the writing proceeds. Our one important thought in this book is that game theory will remain deficient until it develops an interest in games like the one we experienced while writing this book over the last ten years or so. Is it any wonder that this is A Critical Text? Lastly, there are the people and the cats: Empirico, Joε, Lindsay, Margarita, Pandora, Thibeau and Tolstoy – thank you. Yanis Varoufakis and Shaun Hargreaves-Heap, July 2003 Chapter 1: OVERVIEW 1.1 INTRODUCTION 1.1.1 Why study game theory? 1.1.2 What is game theory? 1.1.3 Why this book? 1.1.4 Why a second edition? 1.2 THE ASSUMPTIONS OF GAME THEORY 1.2.1 Individual action is instrumentally rational 1.2.2 Common knowledge of rationality 1.2.3 Common priors 1.2.4 Action within the rules of the game 1.3 LIBERAL INDIVIDUALISM, THE STATE AND GAME THEORY 1.3.1 Methodological individualism 1.3.2 Game theory’s contribution to liberal individualism 1.4 A GUIDE TO THE REST OF THE BOOK 1.4.1 Three classic games: Hawk-dove, Co-ordination and the Prisoner’s Dilemma 1.4.2 Chapter-by-chapter guide 1.5 CONCLUSION Chapter 2: THE ELEMENTS OF GAME THEORY 2.1 INTRODUCTION 2.2 THE REPRESENTATION OF STRATEGIES, GAMES AND INFORMATION SETS 2.2.1 Pure and mixed strategies 2.2.2 The Normal Form, the Extensive Form and the Information Set 2.3 DOMINANCE REASONING 2.3.1 Strict and weak dominance 2.3.2 Degrees of common knowledge of instrumental rationality 2.4 RATIONALISABLE BELIEFS AND ACTIONS 2.4.1 The successive elimination of strategically inferior moves 2.4.2 Rationalisable strategies and their connection with Nash’s equilibrium 2.5 NASH EQUILIBRIUM 2.5.1 John Nash’s beautiful idea 2.5.2 Consistently aligned beliefs, the hidden Principle of Rational Determinacy and the Harsanyi-Aumann Doctrine 2.5.3 Some Objections to Nash: Part I 2.6 NASH EQUILIBRIUM IN MIXED STRATEGIES 2.6.1 The scope and derivation of Nash equilibria in mixed strategies 2.6.2 The reliance of NEMS on CAB and the Harsanyi doctrine 2.6.3 Aumann’s defence of CAB and NEMS 2.7 CONCLUSION Chapter 3: BATTLING INDETERMINACY - Refinements of Nash’s equilibrium in static and dynamic games 3.1 Introduction 3.2 THE STABILITY OF NASH EQUILIBRIA 3.2.1 Trembling hand perfect Nash equilibria 3.2.2 Harsanyi’s Bayesian Nash equilibria and his defence of NEMS 3.3 DYNAMIC GAMES 3.3.1 Extensive form and backward induction 3.3.2 Subgame perfection, Nash and CKR 3.3.3 Sequential equilibria 3.3.4 Bayesian learning, sequential equilibrium and the importance of reputation 3.3.5 Signalling equilibria 3.4 FURTHER REFINEMENTS 3.4.1 Proper equilibria 3.4.2 Forward induction 3.4 SOME LOGICAL OBJECTIONS TO NASH, PART III 3.4.1 A critique of subgame perfection 3.4.2 A negative rejoinder (based on the Harsanyi-Aumann doctrine) 3.4.3 A positive rejoinder (based on sequential equilibrium) 3.3.4 Conclusion: Out-of-equilibrium beliefs, patterned trembles and consistency 3.5 CONCLUSION 3.5.1 The status of Nash and Nash refinements 3.5.2 In defence of Nash 3.5.3 Why has game theory been attracted ‘so uncritically’ to Nash? Chapter 4: BARGAINING GAMES - Rational agreements, bargaining power and the Social Contract • 4.1 INTRODUCTION • 4.2 CREDIBLE AND INCREDIBLE TALK IN SIMPLE BARGAINING GAMES • 4.3 JOHN NASH’S GENERIC BARGAINING PROBLEM AND HIS SOLUTION 4.3.1 The bargaining problem 4.3.2 Nash’s solution – An example 4.3.3 Nash’s solution as an equilibrium of fear 4.3.4 Nash’s axiomatic account 4.3.5 Do the axioms apply? 4.3.6 Nash’s solution – a summary • 4.4 ARIEL RUBINSTEIN AND THE BARGAINING PROCESS: The return of Nash backward induction 4.4.1 Rubinstein’s solution to the bargaining problem 4.4.2 A proof of Rubinstein’s theorem 4.4.3 The (trembling hand) defence of Rubinstein’s solution 4.4.4 A final word on Nash, trembling hands and Rubinstein’s bargaining solution • 4.5 JUSTICE IN POLITICAL AND MORAL PHILOSOPHY 4.5.1 The negative result and the opening to Rawls and Nozick 4.5.2 Procedures and outcomes (or ‘means’ and ends) and axiomatic bargaining theory • 4.6 CONCLUSION Chapter 5: THE PRISONERS’ DILEMMA - The riddle of co-operation and its implications for collective agency 5.1 INTRODUCTION: THE STATE AND THE GAME THAT POPULARISED GAME THEORY 5.2 EXAMPLES OF HIDDEN PRIOSNERS’ DILEMMAS AND FREE RIDERS IN SOCIAL LIFE 5.3 SOME EVIDENCE ON HOW PEOPLE PLAY THE GAME 5.4 EXPLAINING COOPEATION 5.4.1 Kant and morality: Is it rational to defect? 5.4.2 Altruism 5.4.3 Inequality aversion 5.4.4 Choosing a co-operative disposition instrumentally 5.5 CONDITIONAL CO-OPERATION IN REPEATED PRISONERS’ DILEMMAS 5.5.1 Tit-for-tat in Axelrod’s tournaments 5.5.2 Tit-for-tat as a Nash equilibrium strategy when the horizon is unknown 5.5.3 Spontaneous public good provision 5.5.4 The Folk Theorem, Indeterminacy and the State 5.5.5 Does a finite horizon wreck co-operation? The theory and the evidence 5.6 CONCLUSION: COOPERATION AND THE STATE IN LIBERAL THEORY 5.6.1 Rational cooperation? 5.6.2 Liberalism and the prisoners’ dilemma 5.6.3 The limits of the prisoners’ dilemma Chapter 6: EVOLUTIONARY GAMES - Evolution, Games and Social Theory 6.1 GAME THEORY’S ENCOUNTER WITH EVOLUTIONARY BIOLOGY 6.1.1 The origins of Evolutionary Game Theory 6.1.2 Evolutionary stability and equilibrium: An introduction 6.1.3 Spontaneous order versus political rationalism 6.2 SYMMETRICAL EVOLUTION IN HOMOGENEOUS POPULATIONS 6.2.1 Static games 6.2.1 Dynamic games 6.3 EVOLUTION IN HETEROGENEOUS POPULATIONS 6.3.1 Asymmetrical (or two-dimensional) evolution and the demise of Nash equilibria in mixed strategies (NEMS) 6.3.2 Does Evolutionary Game Theory apply to humans as well as it does to birds, ants, etc.? An experiment with 2-dimensional evolution in the Hawk-Dove game 6.3.3 Multi-dimensional evolution and the conflict of conventions 6.3.4 The origin of conventions and the challenge to methodological individualism 6.3.5 The politics of mutations: Conventions, inequality and revolt 6.3.6 Discriminatory Conventions: A brief synopsis 6.4 SOCIAL EVOLUTION: POWER, MORALITY AND HISTORY 6.4.1 Social versus natural selection 6.4.2 Conventions as covert social power 6.4.3 The evolution of predictions into moral beliefs: Hume on morality 6.4.4 Gender, class and functionalism 6.4.5 The evolution of predictions into ideology: Marx against morality 6.5 CONCLUSION Chapter 7: PSYCHOLOGICAL GAMES - Demolishing the divide between motives and beliefs 7.1 INTRODUCTION 7.2 DIFFERENT KINDS OF ‘OTHER REGARDING’ MOTIVES 7.2.1 The ‘other’-regarding motives of Homo Economicus 7.2.2 Beliefs as predictions and as motives 7.3 THE MOTIVATING POWER OF NORMATIVE BELIEFS 7.3.1 Fairness equilibria 7.3.2 Computing Fairness equilibria 7.3.3 Assessing Rabin 7.3.4 An alternative formulation linking entitlements to intentions 7.3.5 Team thinking 7.4 PSYCHOLOGY AND EVOLUTION 7.4.1 On the origins of normative beliefs: an adaptation to experience 7.4.2 On the origins of normative beliefs: The resentment-aversion versus the subversion- proclivity hypotheses 7.5 CONCLUSION: SHARED PRAXIS, SHARED MEANINGS POSTSCRIPT SOLUTIONS TO PROBLEMS NOTES AUTHORS’ INDEX SUBJECT INDEX REFERENCES CHAPTER 1 OVERVIEW 1.1 INTRODUCTION 1.1.1 Why study game theory? 1.1.2 What is game theory? 1.1.3 Why this book? 1.1.4 Why this edition? 1.2 THE ASSUMPTIONS OF GAME THEORY 1.2.1 Individual action is instrumentally rational 1.2.2 Common knowledge of rationality 1.2.3 Common priors 1.2.4 Action within the rules of the game 1.3 LIBERAL INDIVIDUALISM, THE STATE AND GAME THEORY 1.3.1 Methodological individualism 1.3.2 Game theory’s contribution to liberal individualism 1.4 A GUIDE TO THE REST OF THE BOOK 1.4.1 Three classic games: Hawk-Dove, Co-ordination and the Prisoner’s Dilemma 1.4.2 Chapter-by-chapter guide 1.5 CONCLUSION 1 1 OVERVIEW 1.1 INTRODUCTION 1.1.1 Why study game theory? This book’s first edition began with the observation that game theory was everywhere; that after thrilling a whole generation of post-1970 economists, it was spreading like a bushfire through the social sciences. In addition, game theorists had begun to advance some pretty ambitious claims regarding the potential of the theory to become for the social sciences what mathematics is to the natural sciences: a unifying force able to bring together politics, economics, sociology, anthropology etc. under one roof and turn them into sub-disciplines of some broader ‘science of society’. As a glimpse of game theory’s increasing confidence, we cited two prominent game theorists’ explanation of the attraction: Game Theory may be viewed as a sort of umbrella or ‘unified field’ theory for the rational side of social science . . . [it] does not use different, ad hoc constructs… it develops methodologies that apply in principle to all interactive situations. (Aumann and Hart, 1992) To overcome the reader’s suspicion that such exuberance was confined to game theory’s practitioners, we also cited Jon Elster, a well-known social theorist with very diverse interests, whose views on the usefulness of game theory did not differ significantly from that of the practitioners: [I]f one accepts that interaction is the essence of social life, then… game theory provides solid microfoundations for the study of social structure and social change. (Elster, 1982) Our point was that, if seemingly disinterested social theorists held game theory in such esteem, those studying social processes and institutions can ill-afford to pass game theory by. Our book intended to subject its grand claims to critical scrutiny while, at the same time, presenting a concise, simplified yet analytically advanced account of game theory’s techniques. We concluded our inquiry by arguing that the study of game theory is extremely useful for social scientists, albeit not for the reasons put forward by the game theorists. In the first few years after our first edition saw the light of day, the enthusiasm continued to grow unabated. In his impressive 1999 survey, Roger Myerson compared the discovery of game theory’s main concept (the Nash Equilibrium) with that of the DNA double helix and claimed that it has transformed economics to such a remarkable degree that the latter can now pose credibly as the foundational ‘science of society’. It is often said that the seed of Decline begins to take root at the height of an Empire’s power and optimism. As game theory was conquering in the 1990s diverse fields from economics and anthropology to philosophy and biology, doubt emerged concerning its real value for social theorists. Interestingly, this apostasy came not from some radical group opposed to game theory for self-interested reasons but, rather, from practising game theorists (see Mailath, 1998, and Samuelson, 2002). Our book’s point in 1995 was that game theory is best studied critically (hence our subtitle). Insights of great substance are to be had from understanding two things at once: 2 (A) Why game theory inspired such enthusiasm among intelligent social theorists spanning many disciplines, and (B) Why the jury is still out regarding all of the theory’s foundational notions. At the time of our first edition’s publication, our commitment to seeking enlightenment about social processes through immanent criticism of game theory’s concepts was treated by some as eccentric (even heretical). Naturally we feel vindicated by the game theorists’ recent espousal of the method of immanent criticism. However, what is of greater importance is that, precisely because the game theorists themselves are becoming increasingly interested in the weaknesses of their foundations, social theorists stand to gain substantially from a critical engagement with the debates within game theory. To put it bluntly, understanding why game theory does not, in the end, constitute the science of society (even though it comes close) is terribly important in understanding the nature and complexity of social processes. This is, in our view, the primary reason why social theorists should be studying game theory. 1.1.2 What is Game Theory? Game theory really begins with the publication of The Theory of Games and Economic Behaviour by John von Neumann and Oskar Morgenstern (first published in 1944 with second and third editions in 1947 and 1953). They defined a game as any interaction between agents that is governed by a set of rules specifying the possible moves for each participant and a set of outcomes for each possible combination of moves. One is hard put to find an example of social phenomenon that cannot be so described. Thus a theory of games promises to apply to almost any social interaction where individuals have some understanding of how the outcome for one is affected not only by his or her own actions but also by the actions of others. This is quite extraordinary. From crossing the road in traffic, to decisions to disarm, raise prices, give to charity, join a union, produce a commodity, have children, and so on, the claim was made that we shall now be able to draw on a single mode of analysis: the theory of games. 1.1.3 Why this book? Our motivation for writing this book originally was an interesting contradiction. On the one hand, we doubted that the claim in Section 1.1.2 was warranted. This explains the book’s subtitle. On the other hand, however, we enjoyed game theory and had spent many hours pondering its various twists and turns. Indeed it had helped us on many issues. However, we believed that this is predominantly how game theory makes a contribution: It is useful mainly because it helps clarify some fundamental issues and debates in social science, for instance those within and around the political theory of liberal individualism. In this sense, we believed the contribution of game theory to be largely paedagogical. Such contributions are not to be sneezed at. We also felt that game theory’s further substantial contribution was a negative one. The contribution comes through demonstrating the limits of a particular form of individualism in social science: one based exclusively on the model of persons as preference-satisfiers. This model is often regarded as the direct heir of David Hume’s (the 18th century philosopher) conceptualisation of human reasoning and motivation. It is principally associated with what is known today as Rational Choice Theory, or with the (neoclassical) Economic Approach to social life (see Downs, 1957, and Becker, 1976). Our first edition’s main conclusion (which was developed through the book) was that game theory exposes the limits of these models of human agency. In other words, game theory does not actually deliver Jon Elster’s ‘solid microfoundations’ for all social science; and this tells us something about the inadequacy of its chosen ‘microfoundations’. 3