Abstract
We consider the problem of deciding the existence of pure Nash equilibrium and the problem of finding mixed Nash equilibrium in graphical games defined on the two dimensional \(d \times m\) grid graph. Unlike previous works focusing on the computational complexity of centralized algorithms, we study the communication complexity of distributed protocols for these problems, in the setting that each player initially knows only his private input of constant length describing his utility function and each player can only communicate directly with his neighbors. For the pure Nash equilibrium problem, we show that in any protocol, the players in some game must communicate a total of at least \(\varOmega (dm^2)\) bits when \(d \ge \log m\) and at least \(\varOmega (d 2^d m)\) bits when \(d < \log m\). For the mixed Nash equilibrium problem, we show that in any protocol, the players in some game must communicate at least \(\varOmega (d^2 m^2)\) bits in total, and moreover, every player must communicate at least \(\varOmega (dm)\) bits. We also provide protocols with matching or almost matching upper bounds.
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Notes
- 1.
Our algorithms actually work for any set U of constant size. We make this restriction on U to make our lower bound results stronger—the problems remain hard even when specialized to such a set U. In fact, our lower bound in Sect. 3 even holds when U is restricted to \(\{0, 1\}\).
References
Bar-Yossef, Z., Jayram, T.S., Kumar, R., Sivakumar, D.: An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci. 68(4), 702–732 (2004)
Babichenko, Y., Rubinstein, A.: Communication complexity of approximate Nash equilibria. In: Proceedings of 49th Annual ACM SIGACT Symposium on Theory of Computing, pp. 878–889 (2017)
Cover, T., Thomas, J.: Elements of Information Theory. Wiley, Hoboken (1991)
Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009)
Daskalakis, K., Papadimitriou, C.H.: The complexity of games on highly regular graphs. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 71–82. Springer, Heidelberg (2005). https://doi.org/10.1007/11561071_9
Daskalakis, C., Papadimitriou, C.H.: Computing pure Nash equilibria in graphical games via Markov random fields. In: Proceedings of ACM 7th Conference on Electronic Commerce, pp. 91–99 (2006)
Elkind, E., Goldberg, L.A., Goldberg, P.W.: Nash equilibria in graphical games on trees revisited. In: Proceedings of 7th ACM Conference on Electronic Commerce, pp. 100–109 (2006)
Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1996)
Gottlob, G., Greco, G., Scarcello, F.: Pure Nash equilibria: hard and easy games. J. Artif. Intell. Res. 24, 357–406 (2005)
Hart, S., Mansour, Y.: How long to equilibrium? the communication complexity of uncoupled equilibrium procedures. Games Econ. Behav. 69(1), 107–126 (2010)
Kakade, S., Kearns, M., Langford, J., Ortiz, L.: Correlated equilibria in graphical games. In: Proceeding of 4th ACM Conference on Electronic Commerce, pp. 42–47 (2003)
Kearns, M., Littman, M., Singh, S.: Graphical models for game theory. In: Proceedings of 17th Conference in Uncertainty in Artificial Intelligence, pp. 253–260 (2001)
Papadimitriou, C.H., Roughgarden, T.: Computing correlated equilibria in multiplayer games. J. ACM 55(3), 14 (2008)
Schoenebeck, G., Vadhan, S.: The computational complexity of Nash equilibria in concisely represented games. In: Proceedings of 7th ACM Conference on Electronic Commerce, pp. 270–279 (2006)
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Chou, JH., Lu, CJ. (2018). The Communication Complexity of Graphical Games on Grid Graphs. In: Christodoulou, G., Harks, T. (eds) Web and Internet Economics. WINE 2018. Lecture Notes in Computer Science(), vol 11316. Springer, Cham. https://doi.org/10.1007/978-3-030-04612-5_8
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