General Procedures Leading To Correlated Equilibria

Hart and Mas-Colell (2000) show that if all players play “regret matching” strategies, i.e., they play with probabilities proportional to the regrets, then the empirical distribution of play converges to the set of correlated equilibria, and the regrets of every player converge to zero. Here we show that if only one player, say player i, plays with these probabilities, while the other players are “not too sophisticated,” then the result that player i's regrets converge to zero continues to hold. The condition of “not too sophisticated” essentially says that the effect of one change of action of player i on the future actions of the other players decreases to zero as the horizon goes to infinity. Furthermore, we generalize all these results to a whole class of “regret-based” strategies introduced in Hart and Mas-Colell (2001). In particular, these simplify the “conditional smooth fictitious play” of Fudenberg and Levine (1999).