Algorithms for Playing Games with Limited Randomness

摘要

We study multiplayer games in which the participants have access to only limited randomness. This constrains both the algorithms used to compute equilibria (they should use little or no randomness) as well as the mixed strategies that the participants are capable of playing (these should be sparse). We frame algorithmic questions that naturally arise in this setting, and resolve several of them. We give deterministic algorithms that can be used to find sparse e-equilibria in zero-sum and non-zero-sum games, and a randomness-efficient method for playing repeated zero-sum games. These results apply ideas from derandomization (expander walks, and δ-independent sample spaces) to the algorithms of Lipton, Markakis, and Mehta [LMM03], and the online algorithm of Freund and Schapire [FS99]. Subsequently, we consider a large class of games in which sparse equilibria are known to exist (and are therefore amenable to randomness-limited players), namely games of small rank. We give the first "fixed-parameter" algorithms for obtaining approximate equilibria in these games. For rank-k games, we give a deterministic algorithm to find (k + 1)-sparse e-equilibria in time polynomial in the input size n and some function f(k, 1/∈). In a similar setting Kannan and Theobald [KT07] gave an algorithm whose run-time is n~(O(k)). Our algorithm works for a slightly different notion of a game's "rank," but is fixed parameter tractable in the above sense, and extends easily to the multi-player case.