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The Communication Complexity of Graphical Games on Grid Graphs

  • Jen-Hou Chou
  • Chi-Jen LuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11316)

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.

Keywords

Nash equilibrium Communication complexity 

References

  1. 1.
    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)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    Cover, T., Thomas, J.: Elements of Information Theory. Wiley, Hoboken (1991)CrossRefGoogle Scholar
  4. 4.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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_9CrossRefGoogle Scholar
  6. 6.
    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)Google Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  9. 9.
    Gottlob, G., Greco, G., Scarcello, F.: Pure Nash equilibria: hard and easy games. J. Artif. Intell. Res. 24, 357–406 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hart, S., Mansour, Y.: How long to equilibrium? the communication complexity of uncoupled equilibrium procedures. Games Econ. Behav. 69(1), 107–126 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    Papadimitriou, C.H., Roughgarden, T.: Computing correlated equilibria in multiplayer games. J. ACM 55(3), 14 (2008)CrossRefGoogle Scholar
  14. 14.
    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)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institute of Information Science, Academia SinicaTaipeiTaiwan

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