Rationalizability

Abstract: If it is common knowledge that the players in a game are Bayesian utility maximizers who treat uncertainty about other players' actions like any other uncertainty, then the outcome is necessarily a correlated equilibrium. Random strategies appear as an expression of each player's uncertainty about what the others will do, not as the result of willful randomization. Use is made of the common prior assumption, according to which differences in probability assessments by different individuals are due to the different information that they have (where "information" may be interpreted broadly, to include experience, upbringing, and genetic makeup). Copyright 1987 by The Econometric Society.

Abstract: Consider the following game: a large number of players have to state simultaneously a number in the closed interval [0, 100]. The winner is the person whose chosen number is closest to the mean of all chosen numbers multiplied by a parameter p, where p is a predetermined positive parameter of the game; p is common knowledge. The payoff to the winner is a fixed amount, which is independent of the stated number and p. If there is a tie, the prize is divided equally among the winners. The other players whose chosen numbers are further away receive nothing.' The game is played for four rounds by the same group of players. After each round, all chosen numbers, the mean, p times the mean, the winning numbers, and the payoffs are presented to the subjects. For 0 c p < 1, there exists only one Nash equilibrium: all players announce zero. Also for the repeated supergame, all Nash equilibria induce the same announcements and payoffs as in the one-shot game. Thus, game theory predicts an unambiguous outcome. The structure of the game is favorable for investigating whether and how a player's mental process incorporates the behavior of the other players in conscious reasoning. An explanation proposed, for out-of-equilibrium behavior involves subjects engaging in a finite depth of reasoning on players' beliefs about one another. In the simplest case, a player selects a strategy at random without forming beliefs or picks a number that is salient to him (zero-order belief). A somewhat more sophisticated player forms first-order beliefs on the behavior of the other players. He thinks that others select a number at random, and he chooses his best response to this belief. Or he forms second-order beliefs on the first-order beliefs of the others and maybe nth order beliefs about the (n I )th order beliefs of the others, but only up to a finite n, called the ndepth of reasoning. The idea that players employ finite depths of reasoning has been studied by various theorists (see e.g., Kenneth Binmore, 1987, 1988; Reinhard Selten, 1991; Robert Aumann, 1992; Michael Bacharach, 1992; Cristina Bicchieri, 1993; Dale 0. Stahl, 1993). There is also the famous discussion of newspaper competitions by John M. Keynes (1936 p. 156) who describes the mental process of competitors confronted with picking the face that is closest to the mean preference of all competitors.2 Keynes's game, which he considered a Gedankenexperiment, has p = 1. However, with p = 1, one cannot distinguish between different steps of reasoning by actual subjects in an experiment. There are some experimental studies in which reasoning processes have been analyzed in ways similar to the analysis in this paper. Judith Mehta et al. (1994), who studied behavior in two-person coordination games, suggest that players coordinate by either applying depth of reasoning of order I or by picking a focal point (Thomas C. Schelling, 1964), which they call "Schelling salience." Stahl and Paul W. Wilson (1994) analyzed behavior in symmetric 3 x 3 games and concluded that subjects were using depths of reasoning of orders 1 or 2 or a Nash-equilibrium strategy. * Department of Economics, Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, Spain. Financial support from Deutsche Forschungsgemeinschaft (DFG) through Sonderforschungsbereich 303 and a postdoctoral fellowship from the University of Pittsburgh are gratefully acknowledged. I thank Reinhard Selten, Dieter Balkenborg, Ken Binmore, John Duffy, Michael Mitzkewitz, Alvin Roth, Karim Sadrieh, Chris Starmer, and two anonymous referees for helpful discussions and comments. I learned about the guessing game in a game-theory class given by Roger Guesnerie, who used the game as a demonstration experiment. 'The game is mentioned, for example, by Herve Moulin (1986), as an example to explain rationalizability, and by Mario H. Simonsen (1988). 2 This metaphor is frequently mentioned in the macroeconomic literature (see e.g., Roman Frydman, 1982).

Abstract: Game-theoretic reasoning pervades economic theory and is used widely in other social and behavioural sciences. An Introduction to Game Theory International Edition, by Martin J. Osborne, presents the main principles of game theory and shows how they can be used to understand economics, social, political, and biological phenomena. The book introduces in an accessible manner the main ideas behind the theory rather than their mathematical expression. All concepts are defined precisely, and logical reasoning is used throughout. The book requires an understanding of basic mathematics but assumes no specific knowledge of economics, political science, or other social or behavioural sciences. Coverage includes the fundamental concepts of strategic games, extensive games with perfect information, and coalitional games; the more advanced subjects of Bayesian games and extensive games with imperfect information; and the topics of repeated games, bargaining theory, evolutionary equilibrium, rationalizability, and maxminimization. The book offers a wide variety of illustrations from the social and behavioural sciences. Each topic features examples that highlight theoretical points and illustrations that demonstrate how the theory may be used.

Abstract: This paper explores the fundamental problem of what can be inferred about the outcome of a noncooperative game, from the rationality of the players and from the information they possess. The answer is summarized in a solution concept called rationalizability. Strategy profiles that are rationalizable are not always Nash equilibria; conversely, the information in an extensive form game often allows certain "unreasonable" Nash equilibria to be excluded from the set of rationalizable profiles. A stronger form of rationalizability is appropriate if players are known to be not merely "rational" but also "cautious." "WHAT CONSTITUTES RATIONAL BEHAVIOR in a noncooperative strategic situation?" This paper explores the issue in the context of a wide class of finite noncooperative games in extensive form. The traditional answer relies heavily upon the idea of Nash equilibrium (Nash [17]). The position developed here, however, is that as a criterion for judging a profile of strategies to be "reasonable" choices for players in a game, the Nash equilibrium property is neither necessary nor sufficient. Some Nash equilibria are intuitively unreasonable, and not all reasonable strategy profiles are Nash equilibria. The fact that a Nash equilibrium can be intuitively unattractive is well-known: the equilibrium may be "imperfect." Introduced into the literature by Selten [20], the idea of imperfect equilibria has prompted game theorists to search for a narrower definition of equilibrium. While this research, some of which will be discussed here, has been extremely instructive, it remains inconclusive. Theorists often agree about what should happen in particular games, but to capture this intuition in a general solution concept has proved to be very difficult. If this paper is successful it should make some progress in that direction. The other side of the coin has received less scrutiny. Can all non-Nash profiles really be excluded on logical grounds? I believe not. The standard justifications for considering only Nash profiles are circular in nature, or make gratuitous assumptions about players' decision criteria or beliefs. The following discussion of these points is extremely brief, due to space constraints; more detailed arguments may be found in Pearce [18].