We show that the problem of finding an -approximate Nash equilibrium in an anonymous game with seven pure strategies is complete in PPAD, when the approximation parameter is exponentially small in the number of players.
In @cite_17 , Daskalakis and Papadimitriou presented a PTAS for finding an @math -approximate Nash equilibrium in an anonymous game with two pure strategies, with running time @math , where @math denotes the number of bits required to describe the payoffs. The running time was subsequently improved in @cite_15 to @math . The first PTAS in @cite_17 is based on the existence of an @math -approximate Nash equilibrium consisting of integer multiples of @math , while the second PTAS in @cite_15 is based on the existence of an @math -approximate Nash equilibrium satisfying the following special property: either at most @math players play mixed strategies, or all players who mix play the same mixed strategy. Later @cite_41 extended the result of @cite_17 , giving the only known PTAS for anonymous games with any bounded number of pure strategies with time @math for some function @math of @math , number of pure strategies, and @math .
@cite_41 @cite_15 @cite_17
2950776615 1855561663 2040155008 2740520762
We present efficient approximation algorithms for finding Nash equilibria in anonymous games, that is, games in which the players utilities, though different, do not differentiate between other players. Our results pertain to such games with many players but few strategies. We show that any such game has an approximate pure Nash equilibrium, computable in polynomial time, with approximation O(s^2 L), where s is the number of strategies and L is the Lipschitz constant of the utilities. Finally, we show that there is a PTAS for finding an epsilon We present a novel polynomial time approximation scheme fortwo-strategy anonymous games, in which the players’ utilityfunctions, although potentially different, do not differentiateamong the identities of the other players. Our algorithm computesan e-approximate Nash equilibrium of an n-player 2-strategyanonymous game in time @math , which significantly improves uponthe running time @math required by the algorithm ofDaskalakis & Papadimitriou, 2007. The improved running time isbased on a new structural understanding of approximate Nashequilibria: We show that, for any e, there exists ane-approximate Nash equilibrium in which either onlyO(1 e 3) players randomize, or all players who randomizeuse the same mixed strategy. To show this result we employ toolsfrom the literature on Stein’s Method. Motivated by multi-budgeted optimization and other applications, we consider the problem of randomly rounding a fractional solution x in the (non-bipartite graph) matching and matroid intersection polytopes. We show that for any fixed Δ > 0, a given point x can be rounded to a random solution R such that E[1R] = (1 − Δ)x and any linear function of x satisfies dimension-free Chernoff-Hoeffding concentration bounds (the bounds depend on Δ and the expectation μ). We build on and adapt the swap rounding scheme in our recent work [9] to achieve this result. Our main contribution is a non-trivial martingale based analysis framework to prove the desired concentration bounds. In this paper we describe two applications. We give a randomized PTAS for matroid intersection and matchings with any fixed number of budget constraints. We also give a deterministic PTAS for the case of matchings. The concentration bounds also yield related results when the number of budget constraints is not fixed. As a second application we obtain an algorithm to compute in polynomial time an ∈-approximate Pareto-optimal set for the multi-objective variants of these problems, when the number of objectives is a fixed constant. We rely on a result of Papadimitriou and Yannakakis [26]. Abstract We study the computation of Nash equilibria of anonymous games, via algorithms that use adaptive queries to a game's payoff function. We show that exact equilibria cannot be found via query-efficient algorithms, and exhibit a two-strategy, 3-player anonymous game whose exact equilibria require irrational numbers. We obtain positive results for known sub-classes of anonymous games. Our main result is a new randomized query-efficient algorithm for approximate equilibria of two-strategy anonymous games that improves on the running time of previous algorithms. It is the first to obtain an inverse polynomial approximation in poly-time, and yields an efficient polynomial-time approximation scheme.