Unraveling in Guessing Games: An Experimental Study
通过一个多人猜数游戏实验，研究参与者如何通过有限深度的推理（如零阶、一阶信念）选择数字，并观察其行为与纳什均衡（全报零）的偏离。
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).