Complexity and Mixed Strategy Equilibria∗

We propose a theory of mixed strategies in zero-sum two-person games. Given a finite zero-sum two-person game g, we extend it to collective games g∞ and g∞,S, which are infinite repetitions of the game g. Players in the collective games are restricted to use computable strategies only, but each has a complex sequence that can be used in the computation. We adopt kolmogorov complexity to define complex sequences, which are also called random sequences in the literature. The two random sequences are assumed to be independent, and so they can be used to generate complex strategies in g∞ and g∞,S with all possible rational relative frequencies that are unpredictable to their opponents. These complex strategies are analogous to mixed strategies in g. In g∞, however, there are strategies that do not correspond to any mixed strategies. In g∞,S, players are allowed to use only those strategies analogous to pure or mixed strategies in g. We show that the collective games g∞ and g∞,S are solvable, and they both have the same value as that of g. Moreover, we are able to show that any equilibrium strategy in g∞,S has the relative frequency as the probability value of an equilibrium mixed strategy in g.