Abstract
We present a new, distributed method to compute approximate Nash equilibria in bimatrix games. In contrast to previous approaches that analyze the two payoff matrices at the same time (for example, by solving a single LP that combines the two players’ payoffs), our algorithm first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then computes an approximate Nash equilibrium using only limited communication between the players. Our method gives improved bounds on the complexity of computing approximate Nash equilibria in a number of different settings. Firstly, it gives a polynomial-time algorithm for computing approximate well supported Nash equilibria (WSNE) that always finds a 0.6528-WSNE, beating the previous best guarantee of 0.6608. Secondly, since our algorithm solves the two LPs separately, it can be applied to give an improved bound in the limited communication setting, giving a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and finds a 0.6528-WSNE, which beats the previous best known guarantee of 0.732. It can also be applied to the case of approximate Nash equilibria, where we obtain a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and always finds a 0.382-approximate Nash equilibrium, which improves the previous best guarantee of 0.438. Finally, the method can also be applied in the query complexity setting to give an algorithm that makes \(O(n \log n)\) payoff queries and always finds a 0.6528-WSNE, which improves the previous best known guarantee of 2/3.
The first, third and fifth author are partially supported by Research partially supported by the Centre for Discrete Mathematics and its Applications (DIMAP) and by EPSRC award EP/D063191/1. The second, fourth and sixth author are supported by EPSRC grant EP/L011018/1. The second author is also supported by ISF grant #2021296. The full version of this paper, with complete proofs, is available at http://arxiv.org/abs/1512.03315.
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Notes
- 1.
The statements of our results can easily be extended to the case where all payoffs can be represented using b bits by including a factor b in all our communication complexity bounds.
References
Bosse, H., Byrka, J., Markakis, E.: New algorithms for approximate Nash equilibria in bimatrix games. Theoret. Comput. Sci. 411(1), 164–173 (2010)
Chen, X., Deng, X., Teng, S.-H.: Settling the complexity of computing two-player Nash equilibria. J. ACM 56(3), 14:1–14:57 (2009)
Conitzer, V., Sandholm, T.: Communication complexity as a lower bound for learning in games. In: Proceedings of ICML, pp. 185–192 (2004)
Czumaj, A., Fasoulakis, M., Jurdzinski, M.: Approximate well-supported Nash equilibria in symmetric bimatrix games. In: Proceedings of SAGT, pp. 244–254 (2014)
Daskalakis, C., Mehta, A., Papadimitriou, C.H.: Progress in approximate Nash equilibria. In: Proceedings of EC, pp. 355–358 (2007)
Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009)
Daskalakis, C., Mehta, A., Papadimitriou, C.H.: A note on approximate Nash equilibria. Theoret. Comput. Sci. 410(17), 1581–1588 (2009)
Fearnley, J., Savani, R.: Finding approximate Nash equilibria of bimatrix games via payoff queries. In: Proceedings of EC, pp. 657–674 (2014)
Fearnley, J., Goldberg, P.W., Savani, R., Sørensen, T.B.: Approximate well-supported Nash equilibria below two-thirds. In: Serna, M. (ed.) SAGT 2012. LNCS, vol. 7615, pp. 108–119. Springer, Heidelberg (2012). doi:10.1007/978-3-642-33996-7_10
Fearnley, J., Gairing, M., Goldberg, P.W., Savani, R.: Learning equilibria of games via payoff queries. In: Proceedings of EC, pp. 397–414 (2013)
Fearnley, J., Igwe, T.P., Savani, R.: An empirical study of finding approximate equilibria in bimatrix games. In: Proceedings of SEA, pp. 339–351 (2015)
Goldberg, P.W., Pastink, A.: On the communication complexity of approximate Nash equilibria. Games Econ. Behav. 85, 19–31 (2014)
Goldberg, P.W., Roth, A.: Bounds for the query complexity of approximate equilibria. In: Proceedings of EC, pp. 639–656 (2014)
Hart, S., Mansour, Y.: How long to equilibrium? The communication complexity of uncoupled equilibrium procedures. Games Econ. Behav. 69(1), 107–126 (2010)
Kontogiannis, S.C., Spirakis, P.G.: Well supported approximate equilibria in bimatrix games. Algorithmica 57(4), 653–667 (2010)
Lipton, R.J., Markakis, E., Mehta, A.: Playing large games using simple strategies. In: Proceedings of EC, pp. 36–41 (2003)
Nash, J.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)
Rubinstein, A.: Settling the complexity of computing approximate two-player Nash equilibria. In: Proceedings of FOCS, pp. 258–265 (2016)
Tsaknakis, H., Spirakis, P.G.: An optimization approach for approximate Nash equilibria. Internet Math. 5(4), 365–382 (2008)
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Czumaj, A., Deligkas, A., Fasoulakis, M., Fearnley, J., Jurdziński, M., Savani, R. (2016). Distributed Methods for Computing Approximate Equilibria. In: Cai, Y., Vetta, A. (eds) Web and Internet Economics. WINE 2016. Lecture Notes in Computer Science(), vol 10123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54110-4_2
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