On the complexity of succinct zero-sum games

We study the complexity of solving succinct zero-sum games, i.e., the games whose payoff matrix M is given implicitly by a Boolean circuit C such that M(i, j) = C(i, j). We complement the known EXP-hardness of computing the exact value of a succinct zero-sum game by several results on approximating the value. (1) We prove that approximating the value of a succinct zero-sum game to within an additive factor is complete for the class promise-S₂^p, the "promise" version of S₂^p. To the best of our knowledge, it is the first natural problem shown complete for this class. (2) We describe a ZPP^NP algorithm for constructing approximately optimal strategies, and hence for approximating the value, of a given succinct zero-sum game. As a corollary, we obtain, in a uniform fashion, several complexity-theoretic results, e.g., a ZPP^NP algorithm for learning circuits for SAT (Bshouty et al., 1996) and a recent result by Cai (2001) that S₂^p ⊆ ZPP^NP. (3) We observe that approximating the value of a succinct zero-sum game to within a multiplicative factor is in PSPACE, and that it cannot be in promise-S₂^p unless the polynomial-time hierarchy collapses. Thus, under a reasonable complexity-theoretic assumption, multiplicative factor approximation of succinct zero-sum games is strictly harder than additive factor approximation.