On the Complexity of Succinct Zero-Sum Games

• Authors: Lance Fortnow;Russell Impagliazzo;Valentine Kabanets;Christopher Umans
• Venue: CCC '05 Proceedings of the 20th Annual IEEE Conference on Computational Complexity
• Year: 2005

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-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 [7] and a recent result by Cai [9] that S_2^P 驴 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-factor approximation.