Econometrica: Sep, 1993, Volume 61, Issue 5

Rational Learning Leads to Nash Equilibrium

https://doi.org/0012-9682(199309)61:5<1019:RLLTNE>2.0.CO;2-Z
p. 1019-1045

Ehud Kalai, Ehud Lehrer

Each of $n$ players, in an infinitely repeated game, starts with subjective beliefs about his opponents' strategies. If the individual beliefs are compatible with the true strategies chosen, then Bayesian updating will lead in the long run to accurate prediction of the future play of the game. It follows that individual players, who know their own payoff matrices and choose strategies to maximize their expected utility, must eventually play according to a Nash equilibrium of the repeated game. An immediate corollary is that, when playing a Harsanyi-Nash equilibrium of a repeated game of incomplete information about opponents' payoff matrices, players will eventually play a Nash equilibrium of the real game, as if they had complete information.