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 error is complete for the class promise-S-2(p), the "promise" version of S-2(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., JCSS, 1996) and a recent result by Cai (JCSS, 2007) that S-2(p) subset of 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-S-2(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-error approximation.