If a class of games is known to have a Nash equilibrium with probability values that are either zero or Ω(1) -and thus with support of bounded size -then obviously this equilibrium can be found exhaustively in polynomial time. Somewhat surprisingly, we show that there is a PTAS for the class of games whose equilibria are guaranteed to have ) -values, and therefore large -Ω(n) -supports. We also point out that there is a PTAS for games with sparse payoff matrices, a family for which the exact problem is known to be PPAD-complete [6] . Both algorithms are of a special kind that we call oblivious: The algorithm just samples a fixed distribution on pairs of mixed strategies, and the game is only used to determine whether the sampled strategies comprise an -Nash equilibrium; the answer is "yes" with inverse polynomial probability (in the second case, the algorithm is actually deterministic). These results bring about the question: Is there an oblivious PTAS for finding a Nash equilibrium in general games? We answer this question in the negative; our lower bound comes close to the quasi-polynomial upper bound of [18] . Another recent PTAS for anonymous games [13, 14, 7] is also oblivious in a weaker sense appropriate for this class of games (it samples from a fixed distribution on unordered collections of mixed strategies), but its running time is exponential in 1 . We prove that any oblivious PTAS for anonymous games with two strategies and three player types must have , rendering the algorithm in [7] (which works with any bounded number of player types) essentially optimal within oblivious algorithms. In contrast, we devise a poly(n) · (1/ ) O(log 2 (1/ )) non-oblivious PTAS for anonymous games with two strategies and any bounded number of player types. The key idea of our algorithm is to search not over unordered * This work was done while the author was a postdoctoral researcher at Microsoft Research, New England. † Supported by NSF grant CCF -0635319, a gift from Yahoo! Research, and a MICRO grant. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. sets of mixed strategies, but over a carefully crafted set of collections of the first O(log 1 ) moments of the distribution of the number of players playing strategy 1 at equilibrium. The algorithm works because of a probabilistic result of more general interest that we prove: the total variation distance between two sums of independent indicator random variables decreases exponentially with the number of moments of the two sums that are equal, independent of the number of indicators.
Furthermore, REF proved a lower bound on the running time needed by any oblivious algorithm, which lets the latter algorithm be essentially optimal.
On oblivious PTAS's for nash equilibrium
STOC '09
Mathematics Computer Science