GAMES AND ECONOMIC BEHAVIOR 8, 56--90 (1995) Learning To Be Imperfect: The Ultimatum Game* JOHN GALE Department of Economics, University of Wisconsin, I180 Observatory Drive, Madison, Wisconsin 53706 KENNETH G. BINMORE Department of Economics, University College London, Gower Street, London WCIE 6BT England AND LARRY SAMUELSON Department of Economics, University of Wisconsin, 1180 Observatory Drive, Madison, Wisconsin 53706 Received August 22, 1994 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. © 1995 Academic Press, Inc. * Financial support from National Science Foundation Grant SES-9122176 and the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303 at the University of Bonn, is gratefully acknowledged. We thank Drew Fudenberg and Joseph Harrington for helpful discussions, and thank two referees and an associate editor for helpful comments. We are grateful to the Department of Economics at the University of Bonn and the Institute for Advanced Studies at the Hebrew University of Jerusalem, where part of this work was done, for their hospitality. The authors first encountered the possibility of using the replicator dynamics to obtain unusual results in the Ultimatum Game in a manuscript of James Andreoni and John Miller. This paper was prepared for the Nobel Symposium at Bj6rkborn, Sweden, 18-20 June 1993. 56 0899-8256/95 $6.00 Copyright © 1995 by Academic Press, Inc. All fights of reproduction in any form reserved.LEARNING TO BE IMPERFECT 57 1. THE ULTIMATUM GAME Consider two players with a dollar to divide. The rules of the Ultimatum Game specify that player I begins by making an offer of x ~ [0, 1] to player II, who then accepts or refuses. If player II accepts, player I gets I - x and player II gets x. If player II refuses, both get nothing. Traditional game theory predicts that the play of this game will result in the unique subgame-perfect equilibrium in which player II plans to accept whatever she is offered and player I offers player II nothing. J In the first of many experiments on this and related games by numerous authors, Giath et al. (1982) found that the modal offer was ½ and that player I had roughly half a chance of being rejected if he offered about ] of the sum of money available. Binmore et al. (1989) reported qualitatively simi- lar results in their replication of the Ultimatum Game experiment. There have been many related studies in the interim, surveyed by Bolton and Zwick (1993), Giath and Tietz (1990), Roth (1994), and Thaler (1988). Critics of traditional game theory have quoted these results (along with the early results on the finitely repeated Prisoners' Dilemma and games involving the private provision of public goods) as demonstrating that the optimizing paradigm on which game theory is based is fundamentally mistaken. Instead, so the story goes, people simply honor whatever social norm is appropriate to the situation. Frank (1988) is particularly eloquent on this subject. In bargaining games, for example, it is popular to assert that people "just play fair." Many game theorists have responded by dismissing laboratory results as irrelevant to actual behavior. We agree that the results of poorly designed experiments are irrelevant. Binmore (1992, p. 51) stresses that an experi- mentalist or game theorist should be cautious about making predictions unless the following criteria are satisfied: 2 • The game is reasonably simple; • The incentives are adequate; • Sufficient opportunity for trial-and-error learning is provided. I If offers must be made in whole numbers of cents, other subgame-perfect equilibria also exist, but player II never gets more than one cent in any of these. 2 As experimental techniques in economics have become increasingly sophisticated, the importance of these factors has come to the fore. As advocated by Smith (1991), it is now commonplace to offer experimental subjects large incentives instead of the negligible amounts considered appropriate by many psychologists. At the same time, the introduction of com- puter technology has made it possible to use interactive demonstrations to teach subjects the rules of the game quickly and efficiently and to give the subjects the experience of large numbers of repetitions of the game. A survey to Ledyard (1992) of recent experiments concerning the private provision of public goods is revealing. In experiment after experiment, subjects are reported to approach the game-theoretic equilibrium as the incentives increase and the subjects' experience with the game becomes extensive.58 GALE, BINMORE, AND SAMUELSON On the other hand, game theorists cannot ignore experiments that persis- tently refute their predictions when all three criteria are satisfied. In the case of the Ultimatum Game, the relevant experiments have been repli- cated too often for doubts about the data to persist. A theory predicting that real people will use the subgame-perfect equilibrium in the Ultimatum Game is therefore open to question. At first glance, the case for subgame-perfection in the Ultimatum Game seems ironclad. This is a two-player game of perfect information in which each player moves only once. 3 Player I need only believe that player II will not play a weakly dominated strategy to arrive at the subgame-perfect offer. But the deletion of weakly dominated strategies is an eductive principle (cf. Binmore (1987, 1988)), whereas we believe that the principles to which one must appeal when predicting actual behavior, in the labora- tory or elsewhere, are almost always evolutive in character. That is to say, the outcomes we observe are not the product of careful reasoning but of trial-and-error learning. This paper demonstrates that interactive learning processes readily lead to outcomes in the
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