Abstract
We present a novel polynomial time approximation scheme for two-strategy anonymous games, in which the players’ utility functions, although potentially different, do not differentiate among the identities of the other players. Our algorithm computes an ε-approximate Nash equilibrium of an n-player 2-strategy anonymous game in time \(\text{poly}(n) \cdot (1/\epsilon)^{O(1/\epsilon^2)}\), which significantly improves upon the running time \(n^{O(1/\epsilon^2)}\) required by the algorithm of Daskalakis & Papadimitriou, 2007. The improved running time is based on a new structural understanding of approximate Nash equilibria: We show that, for any ε, there exists an ε-approximate Nash equilibrium in which either only O(1/ε 3) players randomize, or all players who randomize use the same mixed strategy. To show this result we employ tools from the literature on Stein’s Method.
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References
Barbour, A.D., Holst, L., Janson, S.: Poisson Approximation. Oxford University Press, New York (1992)
Barbour, A.D., Lindvall, T.: Translated Poisson Approximation for Markov Chains. Journal of Theoretical Probability 19(3) (July 2006)
Blonski, M.: Anonymous Games with Binary Actions. Games and Economic Behavior 28(2), 171–180 (1999)
Blonski, M.: The women of Cairo: Equilibria in large anonymous games. Journal of Mathematical Economics 41(3), 253–264 (2005)
Bosse, H., Byrka, J., Markakis, E.: New Algorithms for Approximate Nash Equilibria in Bimatrix Games. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 17–29. Springer, Heidelberg (2007)
Chen, X., Deng, X.: 3-NASH is PPAD-Complete. Electronic Colloquium in Computational Complexity, TR05-134 (2005)
Chen, X., Deng, X.: Settling the Complexity of Two-Player Nash Equilibrium. In: FOCS (2006)
Chien, S., Sinclair, A.: Convergence to Approximate Nash Equilibria in Congestion Games. In: SODA (2007)
Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)
Daskalakis, C.: An Efficient PTAS for Two-Strategy Anonymous Games. ArXiv Report (2008)
Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The Complexity of Computing a Nash Equilibrium. In: STOC (2006)
Daskalakis, C., Mehta, A., Papadimitriou, C.H.: A Note on Approximate Nash Equilibria. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 297–306. Springer, Heidelberg (2006)
Daskalakis, C., Mehta, A., Papadimitriou, C.H.: Progress in Approximate Nash Equilibria. In: EC (2007)
Daskalakis, C., Papadimitriou, C.H.: Three-Player Games Are Hard. Electronic Colloquium in Computational Complexity, TR05-139 (2005)
Daskalakis, C., Papadimitriou, C.H.: Computing Equilibria in Anonymous Games. In: FOCS (2007)
Daskalakis, C., Papadimitriou, C.H.: Discretized Multinomial Distributions and Nash Equilibria in Anonymous Games. In: FOCS (2008)
Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The Complexity of Pure Nash Equilibria. In: STOC (2004)
Feder, T., Nazerzadeh, H., Saberi, A.: Approximating Nash Equilibria Using Small-Support Strategies. In: EC (2007)
Goldberg, P.W., Papadimitriou, C.H.: Reducibility Among Equilibrium Problems. In: STOC (2006)
Kalai, E.: Partially-Specified Large Games. In: Deng, X., Ye, Y. (eds.) WINE 2005. LNCS, vol. 3828, pp. 3–13. Springer, Heidelberg (2005)
Kannan, R., Theobald, T.: Games of Fixed Rank: A Hierarchy of Bimatrix Games. In: SODA (2007)
Kontogiannis, S.C., Panagopoulou, P.N., Spirakis, P.G.: Polynomial Algorithms for Approximating Nash Equilibria of Bimatrix Games. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 286–296. Springer, Heidelberg (2006)
Lipton, R., Markakis, E., Mehta, A.: Playing Large Games Using Simple Strategies. In: EC (2003)
Milchtaich, I.: Congestion Games with Player-Specific Pay off Functions. Games and Economic Behavior 13, 111–124
von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)
Papadimitriou, C.H., Roughgarden, T.: Computing Equilibria in Multi-Player Games. In: SODA (2005)
Röllin, A.: Translated Poisson Approximation Using Exchangeable Pair Couplings. ArXiv Report (2006)
Tsaknakis, H., Spirakis, P.G.: An Optimization Approach for Approximate Nash Equilibria. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 42–56. Springer, Heidelberg (2007)
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Daskalakis, C. (2008). An Efficient PTAS for Two-Strategy Anonymous Games. In: Papadimitriou, C., Zhang, S. (eds) Internet and Network Economics. WINE 2008. Lecture Notes in Computer Science, vol 5385. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92185-1_26
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DOI: https://doi.org/10.1007/978-3-540-92185-1_26
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