Game Theorists Research Articles

Equal Pay for All Prisoners

- l By prisoners we mean, of course, players of the well-known Prisoner's Dilemma game (to be described presently). We shall show that there exist simple strategies for the infinitely iterated Prisoner's Dilemma that act as equalizers in the sense that all co-players receive the same payoff, no matter what their strategies are like. The Prisoner's Dilemma game, a favorite with game theorists, social scientists, philosophers, and evolutionary biologists, displays the vulnerability of cooperation in a minimalistic model (see [1] to [5]). The two players engaged in this game can choose whether to cooperate or to defect. If both defect, they gain 1 point each; if both cooperate, they gain 3 points; but if one player defects and the other does not, then the defector receives 5 points and the other player only 0. The right move is obviously to defect, no matter what the other player does. As a result, both players earn 1 point instead of 3. But if the same two players repeat the game very frequently, there exists no strategy that is best against all comers. The diversity of strategies is staggering. If we simulate on a computer populations of strategies evolving under a mutationselection regime (with mutation introducing new strategies and selection weening out those with lowest payoff), we observe a rich variety of evolutionary histories frequently leading to cooperative regimes dominated by strategies like Pavlov (cooperate whenever the opponent's move, in the previous round, matched yours) or Generous Tit For Tat (always reciprocate your opponent's cooperative move, but reciprocate only two-thirds of the defections). Remarkably, all strategies of the iterated Prisoner's Dilemma, which can be very complex and make up a huge set, obtain the same payoff against some rather simple equalizer strategies. More generally, let us consider a two-player game where both players have the same two strategies and the same payoff matrix. We denote the first strategy (row 1) by C (for 'cooperate') and the second (row 2) by D (for 'defect') and write the payoff matrix as

A system dynamics model for a mixed-strategy game between police and driver

Game theorists have recommended many reasonable strategies in policy problems, using, in general, the concept of equilibrium strategy for analyzing the dynamic consequences of available policy options. One of the best known recommendations is that of George Tsebelis, which contends that increasing penalties is not a viable policy tool for decreasing the lawviolation tendencies of drivers. This is because the interactions between police and driver can best be represented as a mixed strategy in which the players choose their alternative actions based upon a probability, but the probability of driver’s law violation cannot be decreased by increasing the penalty. Our system dynamics model for a mixed-strategy game shows that it takes a very long time for a game-theoretic equilibrium to appear. Therefore, game players cannot, and should not, depend on the equilibrium for choosing their actions. Furthermore, our mixed-game model shows that an increase in penalty can induce compliance from the people, contradictory to the game-theoretic solution, but consistent with real-world behaviors. © 1997 by John Wiley & Sons, Ltd. Syst. Dyn. Rev. 13, 33‐52,

On Game Theory and the Law

Michael Alexeev* Douglas Baird, Robert Gertner, & Randall Picker, Game Theory and the Law. Cambridge, MA: Harvard University Press, 1994. xii + 330 pages. $45.00. On 18 January 1996 the Indiana University Law and Economics Lunch Bunch1 met to discuss the book Game Theory and the Law by Douglas Baird, Robert Gertner, and Randall Picker. Professor Dau-Schmidt wanted to review the book and, being rather a strong believer in social cooperation, thought a group effort would be a good idea. The rest of us, being eager to express our opinions, agreed. Dau-Schmidt: I want to read a section from the preface of the book that sets forth the purposes the authors had in mind, and then ask your responses. This book rests on the premise that game theory can offer insights to those who want to understand how laws affect the way people behave.... First we wanted to introduce the formal tools of modern game theory to a wide audience.... Second, and as important, we wanted to show how modern game theory allows us to sharpen our intuitions and provides us with new ways of looking at familiar problems. In short, we have tried to write a book that offers those interested in law a new way of thinking about legal rules, and a book that shows those interested in game theory a and largely unexplored in which its tools have many applications. (P. xi). The book goes on to describe a variety of game-theoretic tools, including the normal form of a game, the extensive form of a game, modeling imperfect information, signaling, screening, reputation, repeated games, collective action problems, noncooperative bargaining, and bargaining and information problems; and then it applies these game-theoretic tools to a variety of legal problems, among them tort theory, contract law, antitrust law, bankruptcy law, employment law, and labor law. Have the authors succeeded in their purposes? Mike? Alexeev: To some extent they did, and to some extent they did not. My impression was that the book was written mostly for lawyers. I think that the authors succeeded in demonstrating the usefulness of game theory to people who do not have any previous exposure to game theory. The authors present a number of simple, but useful, applications to legal doctrine including tort law, antitrust, and labor and employment law. But the book does not really show a fertile and largely unexplored domain of applications to game theorists and economists. Most of the book's material comes from economics articles, and even the trial cases are mostly the cases that have already been discussed in the economics literature, so the book is much less useful for the economists or game theorists in particular. But I enjoyed reading it, and I think it will be useful for lawyers. Dau-Schmidt: Eric? Rasmusen: I agree that it may be most useful for people in law who would like to see applications of game theory to particular areas of law. A tort expert might want to read the chapter on torts for its descriptions and numerical examples demonstrating how particular game theory models can be applied to tort law. The book is weaker as a general reference for somebody who's in a hurry to find something. Although the index and glossary2 are good, the structure isn't convenient for reference. This isn't a mathematical book, but it is a dense one which requires careful reading.3 Stake: The authors are quite successful in showing how game theory can provide marvelous counterintuitive insights. But I'm not confident that my law students could slog through the material without some help. I found the book challenging to read, for two opposing reasons. There are portions that are overexplained, even for lawyers. On the other hand, there were places where the explanation seemed thin. For example, it was not immediately obvious to me where the authors got the two sides of the inequality on page 169. It turns out that the same formula was used for both sides, but one side had already been simplified. …

Knowledge, Belief and Counterfactual Reasoning in Games

Deliberation about what to do in any context requires reasoning about what will or would happen in various alternative situations, including situations that the agent knows will never in fact be realized. In contexts that involve two or more agents who have to take account of each others' deliberation, the counterfactual reasoning may become quite complex. When I deliberate, I have to consider not only what the causal effects would be of alternative choices that I might make, but also what other agents might believe about the potential effects of my choices, and how their alternative possible actions might affect my beliefs. Counterfactual possibilities are implicit in the models that game theorists and decision theorists have developed – in the alternative branches in the trees that model extensive form games and the different cells of the matrices of strategic form representations – but much of the reasoning about those possibilities remains in the informal commentary on and motivation for the models developed. Puzzlement is sometimes expressed by game theorists about the relevance of what happens in a game ‘off the equilibrium path’: of what would happen if what is (according to the theory) both true and known by the players to be true were instead false.

Factional Competition for the Party Endorsement: The Case of Japan's Liberal Democratic Party

This Note explores the candidate-endorsement process in the Liberal Democratic Party (LDP) of Japan during its period of hegemony (1955–90). Even in parties without an enduring factional structure such as the LDP, nominations are often troublesome – witness, for example, the reselection controversy in Britain's Labour party at the end of the 1970s or the perennially damaging fights in American primary elections. Moreover, it is easy to understand why nomination politics is so consistently problematic: the gist of the problem is simply that different groups within a party may differ as to who should receive the party endorsement in a given district (or, in list systems, who should get the safe spots on the list). Group A naturally wants its candidate(s) endorsed (there may be more than one in multi-member districts), but so do groups B, C and D. The resulting interaction between groups can be what a game theorist would call a co-ordination, or Battle of the Sexes, game.

Learning to be imperfect: The ultimatum game

This paper studies interactive learning processes that are subject to constant perturbations or “noise.” We argue that payoffs in the Ultimatum Game are such that responders are more apt to be “noisy” than are proposers and show that as a result the learning process readily leads to outcomes that are Nash equilibria but not subgame-perfect. We conclude that game theorists should not restrict attention to the subgame-perfect equilibrium when predicting laboratory behavior in the Ultimatum Game. Journal of Economic Literature Classification Numbers: C70, C72.

Game-theoretic axioms for local rationality and bounded knowledge

We present an axiomatic approach for a class of finite, extensive form games of perfect information that makes use of notions like “rationality at a node” and “knowledge at a node.” We distinguish between the game theorist's and the players' own “theory of the game.” The latter is a theory that is sufficient for each player to infer a certain sequence of moves, whereas the former is intended as a justification of such a sequence of moves. While in general the game theorist's theory of the game is not and need not be axiomatized, the players' theory must be an axiomatic one, since we model players as analogous to automatic theorem provers that play the game by inferring (or computing) a sequence of moves. We provide the players with an axiomatic theory sufficient to infer a solution for the game (in our case, the backwards induction equilibrium), and prove its consistency. We then inquire what happens when the theory of the game is augmented with information that a move outside the inferred solution has occurred. We show that a theory that is sufficient for the players to infer a solution and still remains consistent in the face of deviations must be modular. By this we mean that players have distributed knowledge of it. Finally, we show that whenever the theory of the game is group-knowledge (or common knowledge) among the players (i.e., it is the same at each node), a deviation from the solution gives rise to inconsistencies and therefore forces a revision of the theory at later nodes. On the contrary, whenever a theory of the game is modular, a deviation from equilibrium play does not induce a revision of the theory.

Rational choice and political economy

The purpose of rational choice is to provide a grand theoretical framework for designing human institutions. Once theoretical work had shown how markets optimally aggregated preferences, attempts were made to extend the theory from markets to politics. Attempts by Downs and Olson to describe elections and collective action produced relatively poor predictions, but impelled game theorists to generalize preference‐based theories to include belief formation. A consequence of this change is that the theory is no longer purely axiomatic, but draws on insights about human behavior from other disciplines and empirical analysis of the role institutions play in determining beliefs.

Tit-for-Tat among the Iroquois: A Game Theoretic Perspective on Inter-tribal Political Organization

The Iroquois have intrigued scholars since anthropology's Nineteenth Century beginnings, in part, because their system of the League epitomizes intertribal politics unsullied by the presence of the State. We suggest that the emergence of an inter-group form of political management lacking the usual hierarchical institutions of stratified and statist societies can arise in an environment of tribal warfare and mistrust, if there are rather minimal conditions governing social interaction. These conditions have been investigated by game theorists analyzing the Tit-for-Tat solution to the "Prisoner's Dilemma," by biologists studying the evolution of cooperation, and by social scientists analyzing competition and cooperation in human societies. The Prisoner's Dilemma model includes a number of predictions that can be tested in the Iroquois by means of archaeological and ethnohistorical data and most of these predictions are confirmed. We conclude that the formalisms of game theory as well as the individualistic orientation of evolutionary biology provide new insights into the study of social organization and political anthropology.

Democratization and economic liberalization: Which is the chicken and which is the egg?

The literature on the connections between democratization and economic liberalization is briefly reviewed. There are two main approaches, associated with statistical generalization and with historical sociology. Both can be instructive, and each should be studied by partisans of the other. But they are inadequate on their own. This article suggests that the problem may be broken up by considering aspects of it under the links, respectively, from order to growth; democracy to liberty; and liberty to growth. These are briefly reviewed, drawing attention to the contributions of classical political theorists as well as of modern political scientists, economists, game theorists, and evolutionary biologists.

Computability, complexity and economics

Herbert Simon advocates that economists should study procedureal rationality instead of substantive rationality. One approach for studying procedural rationality is to consider algorithmic representations of procedures, which can then be studied using the concepts of computability and complexity. For some time, game theorists have considered the issue of computability and have employed automata to study bounded rationality. Outside game theory very little research has been performed. Very simple examples of the traditional economic optimization models can require transfinite computations. The impact of procedural rationality on economics depends on the computational resources available to economic agents.

On Models for Product Markets

The first lecture I ever gave was, inevitably, on the theory of the firm. The immediate consequence was rather unexpected: my dean called me into his office and explained to me in so many words that I should refrain from teaching such material, since it was reserved for my colleagues from the business school. My answer was immediate: But then, Sir, I am also a business economist. Since that day I have always, perhaps naively, tried to think as a business economist would. By definition, it seems, business economics is about the study of what business men actually do. ( I avoid the word 'managers' on purpose). By definition, microeconomics is also about what business men actually do. Unfortunately, the second statement is not a correct description of the current state of affairs. My point is that it should be, in the sense that microeconomists should rationalise observed behaviour in particular markets in the same way as botanists should explain why certain flowers are blue and other flowers are yellow. I can see one area of research where business economists and microeconomists can meet: the construction of game theoretic models of particular product markets. That we should all be or become applied game theorists cannot come as a surprise. But to use these skills for the analysis of a particular product may seem a waste of time. Indeed, during a recent rather thorough search for empirical studies of particular product markets, I found only a few-very few-published in refereed journals. The current flow of publications on duopoly models, comparing one-shot and repeated game equilibria, or Cournot and Bertrand strategies, or pure and mixed strategies, and so on, are of general interest. But isn't the market for such papers getting saturated? Shouldn't we try to find out under which institutional circumstances a particular type of equilibrium is relevant? Is generality really more important than relevance? Is a science on a sound footing when a paper is rejected because its assumptions correspond to existing contractual and institutional arrangements and are, therefore, 'too specific to warrant publication'? T o continue in a more positive mode, I do expect market models, inspired by a particular case study, to provide insights that are of general interest. (This is not to say-as the pre-war experience has shown-that an accumulation of booklength industry studies would lead to theoretical generalisations!). In fact, such models have already inspired interesting theoretical developments.

Game Theory and the History of Ideas about Rationality: An Introductory Survey

Although it may seem from its formalism that game theory must have sprung from the mind of John von Neumann as a corollary of his work on computers or theoretical physics, it should come as no real surprise to philosophers that game theory is the articulation of a historically developing philosophical conception of rationality in thought and action. The history of ideas about rationality is deeply contradictory at many turns. While there are theories of rationality that claim it is fundamentally social and aims at understanding and molding all facets of human psychological life, game theory takes rationality to be essentially located in individuals and to concern only the means to achieve predetermined ends. Thus, there are some thinkers who have made important contributions to this history who do not appear in the story of game theory at all, among them, Plato, Kant, and Hegel. There is, however, a clear trail to follow linking theories of instrumental rationality from Aristotle to the nineteenth-century marginalist economists and ultimately to von Neumann and Morgenstern and contemporary game theorists, that historically grounds game theory as a model of rational interaction.

A "Reputation" Refinement without Equilibrium

THE ECONOMIC LITERATURE concerning agents' reputations has grown steadily since the seminal work of Kreps, Milgrom, Roberts, and Wilson.2 Early work focused on how incomplete information leads to equilibria that are vastly different (but more intuitive) than those possible in the complete information game. Recently, however, game theorists have been studying how incomplete information might refine the set of equilibria.3 One important class of games is that in which a single long-run agent plays a simultaneous move (stage) game with a sequence of opponents, each of whom plays only once, yet observes all previous play. Fudenberg and Levine (1989) study the reputation of the long-run player in this type of game. They argue that the 'most reasonable' equilibrium is the one which the long-run player most prefers. Their intuition is sustained when one perturbs the game with the Stackelberg strategy. Fudenberg and Levine show that in the perturbed game the equilibrium payoffs of the long-run player are bounded below by a number that converges to the Stackelberg payoff.

Knowledge and Equilibrium in Games

This paper describes an approach to noncooperative game theory that aims to capture considerations that exercise the minds of real-world strategists. The most commonly used tool of noncooperative game theory is the Nash equilibrium. This raises the question: Are there assumptions on what the players in a game think—including what they think other players think, and so on—that lead to consideration of Nash equilibrium? The paper provides answers to this, and related, questions. The approach of this paper involves analyzing the decision problem facing each player in a strategic (“interactive”) situation. In addition to grounding game theory in considerations that are of the essence in actual strategic situations, the approach has a number of other objectives: 1) to make game theory more immediately accessible to people who are trained in decision theory but who are not “game theorists” and 2) to make game theory easier to teach to students. Finally, the approach suggests new directions for research into the nature of strategic situations.

Doxastic paradoxes without self-reference

Certain doxastic paradoxes (paradoxes analogous to the Paradox of the Liar but involving ideal belief instead of truth ) demonstrate that some formal paradoxes cannot be avoided simply by limiting the expressiveness of one's formal language in order to exclude the very possibility of self-referential thoughts and beliefs. These non-self-referential paradoxes, moreover, should be of special interest to such investigators of rationality as game theorist, economists and cognitive psychologists, since they occur more frequently in the world beyond the Godel-theorist's laboratory.

A General Theory of Equilibrium Selection in Games.

The authors, two of the most prominent game theorists of this generation, have devoted a number of years to the development of the theory presented here, and to its economic applications. They propose rational criteria for selecting one particular uniformly perfect equilibrium point as the solution of any noncooperative game. And, because any cooperative game can be remodelled as a noncooperative bargaining game, their theory defines a one-point solution for any cooperative game as well. By providing solutions - based on the same principles of rational behavior - for all classes of games, both cooperative and noncooperative, both those with complete and with incomplete information, Harsanyi and Selten's approach achieves a remarkable degree of theoretical unification for game theory as a whole and provides a deeper insight into the nature of game-theoretic rationality. The book applies this theory to a number of specific game classes, such as unanimity games; bargaining with transaction costs; trade involving one seller and several buyers; two-person bargaining with incomplete information on one side, and on both sides. The last chapter discusses the relationship of the authors' theory to other recently proposed solution concepts, particularly the Kohberg-Mertens stability theory.

Compromise

Human conflict and its resolution is obviously a subject of great practical importance. Equally obviously, it is a vast subject, ranging from total war at one end of the spectrum to negotiated settlement at its other end. The literature on the subject is correspondingly vast and, in recent times, technical, thanks to the valuable contributions made to it by game theorists, economists, and writers on industrial and international relations. In this essay, however, I shall discuss only one familiar form of conflict-resolution. There is room for such a discussion, because philosophers have lately neglected compromise, despite the interest shown in it by the aforementioned experts, and despite the classic treatments of it by Halifax, Burke and Morley. Truly, ‘…compromise is not so widely discussed by philosophers as one might expect’, and ‘…the idea of compromise has been largely neglected by Anglo-American jurisprudence’.

Games Economists Play: A Noncooperative View

* For some thirty years following World War II, most bright young economists did not go into industrial organization. It was not hard to see why this was so. The quick, big payoffs in economics tend to go to theorists, and industrial organization was a subject in which theory was not only unsatisfactory, but moribund. The promise of the Chamberlin-Robinson revolution of the early 1930s had not been fulfilled, and, while institutional and verbaltheoretic writings on the subject were sometimes illuminating, there was no hard analytic theory formalizing the structure-conduct-performance paradigm. Oligopoly, in particular, showed no signs of analytic tractability. Today, the atmosphere is altogether different. The profession is full of young theorists whose principal occupation is industrial organization. Indeed, industrial organization has become the hot topic for microtheorists. Journals, meetings, and seminars abound in research on oligopoly, and, judging by the market, this is a booming industry. Of course, the principal reason for the change is the rise of game theory and its applications in these areas. Von Neumann and Morgenstern's great work was published in 1944 and hailed immediately by theorists (von Neumann and Morgenstern, 1944). Jacob Marschak, indeed, stated in reviewing it (Marschak, 1946, p. 1 15 ): Ten more such books and the progress of economics is assured. Nevertheless, it was not until the last decade that game theory came to the ascendant as the premier fashionable tool of microtheorists. That ascendancy appears fairly complete. Bright young theorists today tend to think of every problem in game-theoretic terms, including problems that are easier to deal with in other forms. Every department feels it needs at least one game theorist or at least one theorist who thinks in game-theoretic terms. Oligopoly theory in particular is totally dominated by the game-theoretic approach. The field appears to be in an exciting stage of ferment. To understand why I believe that such excitement is not warranted, it is necessary to step back, briefly review the history of oligopoly theory before the game-theory revolution, and consider what it is that one wants theory to deliver.

A Game-Theoretical Approach to the Results of Parliamentary Elections in Belgium between the Wars

This paper deals with the distribution of power among Belgian political parties during the interwar period. In the 1930s Belgium, like most European countries, was confronted with the electoral success of extreme right- and left-wing parties that wanted to change the existing political system into an authoritarian one. Usually, historians draw attention to the rapidly growing share of seats in Parliament held by extreme parties as a sign of their increasing influence on Belgian politics. Among game theorists, however, it is widely accepted that the proportion of seats is a poor proxy for power relations (Schotter, 1979). It is indeed possible that a political party acquiring a higher proportion of seats in Parliament loses its capacity to influence the outcome of a vote, and vice versa.