Correlated Equilibrium in Two-Person Zero-Sum Games

Abstract

but any convex combination of pairs of optimal strategies such that p(2, 2) = 0 satisfies p(1, 1) > 2 (with the obvious notation p(i, j) for the induced probability of row i and column j). However, the following is easily checked. Let I and J be the sets of pure strategies of player 1 and player 2 respectively in a zero-sum game G with value v. Then p = [p(i, i)I(, J) IXJ is a correlated equilibrium distribution for G if and only if for every E J such that p(jo) > 0, the conditional probability of player 2 over player l's actions given jo' [p (iIjo)]I , is an optimal strategy for player 1, yielding exactly v against jo and similarly for [p(jlio)]jEj, io E , p(io) > 0. Hence as conjectured by R. Aumann, if a pure strategy pair occurs with positive probability in a correlated equilibrium, then it occurs with positive probability in a pair of optimal strategies. Also, if one of the players has a unique optimal strategy, then every correlated equilibrium distribution concentrates on a pair of optimal strategies.