Complexity and Optimality of the Best Response Algorithm in Random Potential Games

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9928)

Abstract

In this paper we compute the worst-case and average execution time of the Best Response Algorithm (BRA) to compute a pure Nash equilibrium in finite potential games. Our approach is based on a Markov chain model of BRA and a coupling technique that transform the average execution time of this discrete algorithm into the solution of an ordinary differential equation. In a potential game with N players and A strategies per player, we show that the worst case complexity of BRA (number of moves) is exactly \(N A^{N-1}\), while its average complexity over random potential games is equal to \(e^\gamma N + O(N)\), where \(\gamma \) is the Euler constant. We also show that the effective number of states visited by BRA is equal to \(\log N + c + O(1/N)\) (with \( c \leqslant e^\gamma \)), on average. Finally, we show that BRA computes a pure Nash Equilibrium faster (in the strong stochastic order sense) than any local search algorithm over random potential games.

Notes

Acknowledgement

This work was partially supported by LabEx Persyval-Lab.

References

  1. 1.
    Daskalakis, P.G.C., Papadimitriou, C.: The complexity of computing a nash equilibrium. SIAM J. Comput. 39(3), 195–259 (2009)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Rosenthal, R.W.: A class of games possessing pure-strategy nash equilibria. Int. J. Game Theory 2(1), 65–67 (1973)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Beckman, M., McGuire, C.B., Winsten, C.B.: Studies in the Economics of Transportation. Yale University Press, New Haven (1956)Google Scholar
  4. 4.
    Orda, A., Rom, R., Shimkin, N.: Competitive routing in multi user communication networks. IEEE/ACM Trans. Netw. 1(5), 510–521 (1993)CrossRefGoogle Scholar
  5. 5.
    Gallager, R.G.: A minimum delay routing algorithm using distributed computation. IEEE Trans. Commun. 25(1), 73–85 (1977)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Wardrop, J.: Some theoretical aspects of road traffic research, part ii. In: Proceedings of the Institute of Civil Engineers, vol. 1, pp. 325–378 (1954)Google Scholar
  7. 7.
    Roughgarden, T.: Selfish Routing and the Price of Anarchy. MIT Press, Cambridge (2005)MATHGoogle Scholar
  8. 8.
    Fabrikant, A., Papadimitriou, C., Talwar, K.: The complexity of pure nash equilibria. In: Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, ser. (STOC 2004), pp. 604–612. ACM (2004)Google Scholar
  9. 9.
    Ackermann, H., Röglin, H., Vöcking, B.: On the impact of combinatorial structure on congestion games. In: FOCS 2006, pp. 613–622 (2006)Google Scholar
  10. 10.
    Nash, J.: Equilibrium points in n-person game. Proc. Nat. Acad. Sci. 38, 48–49 (1950)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Monderer, D., Shapley, L.: Potential games. Games Econ. Behav. 14(1), 124–143 (1996). ElsevierMathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Babichenko, Y., Tamuz, O.: Graphical potential games. arXiv.org, Tech. Rep.: 1405.1481v2Google Scholar
  13. 13.
    Even-Dar, E., Kesselman, A., Mansour, Y.: Convergence time to nash equilibria. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719. Springer, Heidelberg (2003)Google Scholar
  14. 14.
    Feldman, M., Tamir, T.: Convergence of best-response dynamics in games with conflicting congestion effects. In: Goldberg, P.W. (ed.) WINE 2012. LNCS, vol. 7695, pp. 496–503. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Imre, B., Vempala, S., Vetta, A.: Nash equilibria in random games. Random Structures Algorithms 31(4), 391–405 (2007)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Rinott, Y., Scarsini, M.: On the number of pure strategy nash equilibria in random games. Games Econ. Behav. 33(2), 274–293 (2000)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Durand, S., Gaujal, B.: Complexity, optimality of the best response algorithm in random potential games. Inria, Research Report RR-8925 (2016). https://hal.inria.fr/hal-01330805
  18. 18.
    Voorneveld, M.: Best-response potential games. Econ. Lett. 66(3), 289–295 (2000)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Series in Probability and Statistics. Wiley, Chichester (2002)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Univ. Grenoble AlpesGrenobleFrance
  2. 2.InriaGrenobleFrance

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