The Complexity of Nash Equilibria in Limit-Average Games

  • Michael Ummels
  • Dominik Wojtczak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6901)

Abstract

We study the computational complexity of Nash equilibria in concurrent games with limit-average objectives. In particular, we prove that the existence of a Nash equilibrium in randomised strategies is undecidable, while the existence of a Nash equilibrium in pure strategies is decidable, even if we put a constraint on the payoff of the equilibrium. Our undecidability result holds even for a restricted class of concurrent games, where nonzero rewards occur only on terminal states. Moreover, we show that the constrained existence problem is undecidable not only for concurrent games but for turn-based games with the same restriction on rewards. Finally, we prove that the constrained existence problem for Nash equilibria in (pure or randomised) stationary strategies is decidable and analyse its complexity.

Keywords

Nash Equilibrium Pure Strategy Terminal State Stationary Strategy Conjunctive Normal Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Allender, E., Bürgisser, P., Kjeldgaard-Pedersen, J., Miltersen, P.B.: On the complexity of numerical analysis. SIAM Journal on Computing 38(5), 1987–2006 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alur, R., Degorre, A., Maler, O., Weiss, G.: On omega-languages defined by mean-payoff conditions. In: de Alfaro, L. (ed.) FOSSACS 2009. LNCS, vol. 5504, pp. 333–347. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Bouyer, P., Brenguier, R., Markey, N.: Nash equilibria for reachability objectives in multi-player timed games. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 192–206. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Canny, J.: Some algebraic and geometric computations in PSPACE. In: STOC 1988, pp. 460–469. ACM Press, New York (1988)Google Scholar
  5. 5.
    Chatterjee, K., Doyen, L., Henzinger, T.A., Raskin, J.-F.: Generalized mean-payoff and energy games. In: FSTTCS 2010. LIPICS, vol. 8. Schloss Dagstuhl (2010)Google Scholar
  6. 6.
    Chen, X., Deng, X., Teng, S.-H.: Settling the complexity of computing two-player Nash equilibria. Journal of the ACM 56(3) (2009)Google Scholar
  7. 7.
    Conitzer, V., Sandholm, T.: Complexity results about Nash equilibria. In: IJCAI 2003, pp. 765–771. Morgan Kaufmann, San Francisco (2003)Google Scholar
  8. 8.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM Journal on Computing 39(1), 195–259 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    de Alfaro, L., Henzinger, T.A., Kupferman, O.: Concurrent reachability games. Theoretical Computer Science 386(3), 188–217 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ehrenfeucht, A., Mycielski, J.: Positional strategies for mean payoff games. International Journal of Game Theory 8, 109–113 (1979)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Etessami, K., Yannakakis, M.: On the complexity of Nash equilibria and other fixed points. SIAM Journal on Computing 39(6), 2531–2597 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Everett, H.: Recursive games. In: Dresher, M., Tucker, A.W., Wolfe, P. (eds.) Contributions to the Theory of Games III. Annals of Mathematical Studies, vol. 39, pp. 47–78. Princeton University Press, Princeton (1957)Google Scholar
  13. 13.
    Fink, A.M.: Equilibrium in a stochastic n-person game. Journal of Science in Hiroshima University 28(1), 89–93 (1964)MathSciNetMATHGoogle Scholar
  14. 14.
    Fisman, D., Kupferman, O., Lustig, Y.: Rational synthesis. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 190–204. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Garey, M.R., Graham, R.L., Johnson, D.S.: Some NP-complete geometric problems. In: STOC 1976, pp. 10–22. ACM Press, New York (1976)Google Scholar
  16. 16.
    Gillette, D.: Stochastic games with zero stop probabilities. In: Dresher, M., Tucker, A.W., Wolfe, P. (eds.) Contributions to the Theory of Games III. Annals of Mathematical Studies, vol. 39, pp. 179–187. Princeton University Press, Princeton (1957)Google Scholar
  17. 17.
    Henzinger, T.A.: Games in system design and verification. In: TARK 2005, pp. 1–4. National University of Singapore (2005)Google Scholar
  18. 18.
    Karp, R.M.: A characterization of the minimum cycle mean in a digraph. Discrete Mathematics 23(3), 309–311 (1978)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mertens, J.-F., Neyman, A.: Stochastic games. International Journal of Game Theory 10(2), 53–66 (1981)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Nash Jr., J.F.: Equilibrium points in N-person games. Proceedings of the National Academy of Sciences of the USA 36, 48–49 (1950)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Neyman, A., Sorin, S. (eds.): Stochastic Games and Applications. NATO Science Series C, vol. 570. Springer, Heidelberg (2003)MATHGoogle Scholar
  22. 22.
    Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley and Sons, Chichester (1994)CrossRefMATHGoogle Scholar
  23. 23.
    Shapley, L.S.: Stochastic games. Proceedings of the National Academy of Sciences of the USA 39, 1095–1100 (1953)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Thuijsman, F., Raghavan, T.E.S.: Perfect-information stochastic games and related classes. International Journal of Game Theory 26, 403–408 (1997)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Ummels, M.: The complexity of nash equilibria in infinite multiplayer games. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 20–34. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  26. 26.
    Ummels, M., Wojtczak, D.: The complexity of nash equilibria in simple stochastic multiplayer games. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 297–308. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  27. 27.
    Ummels, M., Wojtczak, D.: The complexity of Nash equilibria in limit-average games. Tech. Rep. LSV-11-15, ENS Cachan (2011)Google Scholar
  28. 28.
    Velner, Y., Rabinovich, A.: Church synthesis problem for noisy input. In: Hofmann, M. (ed.) FOSSACS 2011. LNCS, vol. 6604, pp. 275–289. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  29. 29.
    Vielle, N.: Two-player stochastic games I: A reduction. Israel Journal of Mathematics 119(1), 55–91 (2000)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Vielle, N.: Two-player stochastic games II: The case of recursive games. Israel Journal of Mathematics 119(1), 93–126 (2000b)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Washburn, A.R.: Deterministic graphical games. Journal of Mathematical Analysis and Applications 153, 84–96 (1990)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Zwick, U., Paterson, M.: The complexity of mean payoff games on graphs. Theoretical Computer Science 158(1-2), 343–359 (1996)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Michael Ummels
    • 1
  • Dominik Wojtczak
    • 2
    • 3
  1. 1.LSV, CNRS & ENS CachanFrance
  2. 2.University of LiverpoolUK
  3. 3.Computing LaboratoryOxford UniversityUK

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