Games with Small Forgetfulness

While it is known how players may learn to play in a game they know, the issue of how their model of the game evolves over time is largely unexplored. This paper introduces small forgetfulness and shows that it may destabilize standard full-memory solutions. Players are repeatedly matched to play a game. After any match, they forget with in ̄nitesimal probability the feasibility of any opponents' unobserved action, and they are reminded of all actions that they observe. During each period, they play an equilibrium consistent with their perception of the game. We show that the unique backward induction path drifts into a non-Nash, self-con ̄rming equilibrium, in a class of extensive-form games that are fully characterized. Such a long-run prediction is always Pareto-undominated, and may Pareto dominate the original backward induction path. In one-shot simultaneous-move games, forgetfulness yields a re ̄nement stronger than trembling hand perfection. Our results imply that there are games that players may never fully learn. JEL CLASSIFICATION: C71, C73, D83 ¤I am indebted to Je®rey Ely, Drew Fudenberg, Ehud Kalai, George Mailath, Steven Matthews, Larry Samuelson, and especially Eddie Dekel, and Juuso Valimaki, for guiding me in developing the ideas contained in this paper, and in organizing the exposition. I would also like to thank all the participants of a Math Center seminar at Northwestern University, of the 1999 North American Summer Meeting of the Econometric Society at the University of Wisconsin at Madison, of the 1999 Summer in Tel Aviv, and of the 1999 Stony Brook International Conference on Game Theory. The usual caveat applies. Department of Economics, 2005 Sheridan Rd. Evanston, Il, 60208-2009. E-mail: [email protected] Webpage: http://pubweb.nwu.edu/~fsq395/