Abstract
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.
This research was supported by the DFG Research Training Group 1298 (AlgoSyn).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
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)
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)
Canny, J.: Some algebraic and geometric computations in PSPACE. In: Proc. STOC 1988, pp. 460–469. ACM Press, New York (1988)
Chatterjee, K., Henzinger, T.A., Jurdziński, M.: Games with secure equilibria. Theoretical Computer Science 365(1-2), 67–82 (2006)
Chatterjee, K., Jurdziński, M., Henzinger, T.A.: Quantitative stochastic parity games. In: Proc. SODA 2004, pp. 121–130. ACM Press, New York (2004)
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)
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)
Condon, A.: The complexity of stochastic games. Information and Computation 96(2), 203–224 (1992)
Conitzer, V., Sandholm, T.: Complexity results about Nash equilibria. In: Proc. IJCAI 2003, pp. 765–771. Morgan Kaufmann, San Francisco (2003)
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)
de Alfaro, L., Henzinger, T.A.: Concurrent omega-regular games. In: Proc. LICS 2000, pp. 141–154. IEEE Computer Society Press, Los Alamitos (2000)
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)
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)
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)
Filar, J., Vrieze, K.: Competitive Markov decision processes. Springer, Heidelberg (1997)
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)
Nash Jr., J.F.: Equilibrium points in N-person games. Proc. National Academy of Sciences of the USA 36, 48–49 (1950)
Neyman, A., Sorin, S. (eds.): Stochastic Games and Applications. NATO Science Series C, vol. 570. Springer, Heidelberg (1999)
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)
Selten, R.: Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit. Zeitschrift für die gesamte Staatswissenschaft 121, 301–324, 667–689 (1965)
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)
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)
Ummels, M., Wojtczak, D.: The complexity of Nash equilibria in simple stochastic multiplayer games. Technical report, University of Edinburgh (2009)
Zielonka, W.: Perfect-information stochastic parity games. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 499–513. Springer, Heidelberg (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ummels, M., Wojtczak, D. (2009). The Complexity of Nash Equilibria in Simple Stochastic Multiplayer Games. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds) Automata, Languages and Programming. ICALP 2009. Lecture Notes in Computer Science, vol 5556. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02930-1_25
Download citation
DOI: https://doi.org/10.1007/978-3-642-02930-1_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02929-5
Online ISBN: 978-3-642-02930-1
eBook Packages: Computer ScienceComputer Science (R0)