The Complexity of Nash Equilibria in Simple Stochastic Multiplayer Games

  • Michael Ummels
  • Dominik Wojtczak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5556)


We analyse the computational complexity of finding Nash equilibria in simple stochastic multiplayer games. We show that restricting the search space to equilibria whose payoffs fall into a certain interval may lead to undecidability. In particular, we prove that the following problem is undecidable: Given a game \(\mathcal G\), does there exist a pure-strategy Nash equilibrium of \(\mathcal G\) where player 0 wins with probability 1. Moreover, this problem remains undecidable if it is restricted to strategies with (unbounded) finite memory. However, if mixed strategies are allowed, decidability remains an open problem. One way to obtain a provably decidable variant of the problem is to restrict the strategies to be positional or stationary. For the complexity of these two problems, we obtain a common lower bound of NP and upper bounds of NP and PSpace respectively.


Nash Equilibrium IEEE Computer Society Pure Strategy Markov Decision Process Stochastic Game 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allender, E., Bürgisser, P., Kjeldgaard-Pedersen, J., Miltersen, P.B.: On the complexity of numerical analysis. In: Proc. CCC 2006, pp. 331–339. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  2. 2.
    Brázdil, T., Brožek, V., Forejt, V., Kučera, A.: Stochastic games with branching-time winning objectives. In: Proc. LICS 2006, pp. 349–358. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  3. 3.
    Canny, J.: Some algebraic and geometric computations in PSPACE. In: Proc. STOC 1988, pp. 460–469. ACM Press, New York (1988)Google Scholar
  4. 4.
    Chatterjee, K., Henzinger, T.A., Jurdziński, M.: Games with secure equilibria. Theoretical Computer Science 365(1-2), 67–82 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chatterjee, K., Jurdziński, M., Henzinger, T.A.: Quantitative stochastic parity games. In: Proc. SODA 2004, pp. 121–130. ACM Press, New York (2004)Google Scholar
  6. 6.
    Chatterjee, K., Majumdar, R., Jurdziński, M.: On Nash equilibria in stochastic games. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 26–40. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Chen, X., Deng, X.: Settling the complexity of two-player Nash equilibrium. In: Proc. FOCS 2006, pp. 261–272. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  8. 8.
    Condon, A.: The complexity of stochastic games. Information and Computation 96(2), 203–224 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Conitzer, V., Sandholm, T.: Complexity results about Nash equilibria. In: Proc. IJCAI 2003, pp. 765–771. Morgan Kaufmann, San Francisco (2003)Google Scholar
  10. 10.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. In: Proc. STOC 2006, pp. 71–78. ACM Press, New York (2006)Google Scholar
  11. 11.
    de Alfaro, L., Henzinger, T.A.: Concurrent omega-regular games. In: Proc. LICS 2000, pp. 141–154. IEEE Computer Society Press, Los Alamitos (2000)Google Scholar
  12. 12.
    de Alfaro, L., Henzinger, T.A., Kupferman, O.: Concurrent reachability games. In: Proc. FOCS 1998, pp. 564–575. IEEE Computer Society Press, Los Alamitos (1998)Google Scholar
  13. 13.
    Etessami, K., Kwiatkowska, M.Z., Vardi, M.Y., Yannakakis, M.: Multi-objective model checking of Markov decision processes. Logical Methods in Computer Science 4(4) (2008)Google Scholar
  14. 14.
    Etessami, K., Yannakakis, M.: On the complexity of Nash equilibria and other fixed points. In: Proc. FOCS 2007, pp. 113–123. IEEE Computer Society Press, Los Alamitos (2007)Google Scholar
  15. 15.
    Filar, J., Vrieze, K.: Competitive Markov decision processes. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  16. 16.
    Garey, M.R., Graham, R.L., Johnson, D.S.: Some NP-complete geometric problems. In: Proc. STOC 1976, pp. 10–22. ACM Press, New York (1976)Google Scholar
  17. 17.
    Nash Jr., J.F.: Equilibrium points in N-person games. Proc. National Academy of Sciences of the USA 36, 48–49 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Neyman, A., Sorin, S. (eds.): Stochastic Games and Applications. NATO Science Series C, vol. 570. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  19. 19.
    Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals. Journal of Symbolic Computation 13(3), 255–352 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Selten, R.: Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit. Zeitschrift für die gesamte Staatswissenschaft 121, 301–324, 667–689 (1965)Google Scholar
  21. 21.
    Ummels, M.: Rational behaviour and strategy construction in infinite multiplayer games. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 212–223. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  22. 22.
    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
  23. 23.
    Ummels, M., Wojtczak, D.: The complexity of Nash equilibria in simple stochastic multiplayer games. Technical report, University of Edinburgh (2009)Google Scholar
  24. 24.
    Zielonka, W.: Perfect-information stochastic parity games. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 499–513. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Michael Ummels
    • 1
  • Dominik Wojtczak
    • 2
    • 3
  1. 1.RWTH Aachen UniversityGermany
  2. 2.CWIAmsterdamThe Netherlands
  3. 3.University of EdinburghUK

Personalised recommendations