The Minmax Value in Three-Person Repeated Games with Stationary Bounded Recall

Abstract: A k-recall strategy in a repeated game is a strategy that relies only on the last k periods of history. For a game in strategic form G, G[k_1,…,k_n] denotes the repeated version of G in which each Player i is resticted to (mixtures of) k_i-recall strategies. Peretz (2013) showed that for any ϵ > 0 and every C > 0 there is a stage game G in which the minmax value of Player 3 is 1 and maxmin value of Player 3 is 0, such that the minmax value of Player 3 in G[k,k,Ck] is less than ϵ (for any k large enough). We present a converse result: For any game G and ϵ > 0, there is C > 0 such that the minmax value of Player 3 in G[k,k,Ck] is at least his minmax value in G minus ϵ (for any k large enough).