Concurrent Games with Ordered Objectives

  • Patricia Bouyer
  • Romain Brenguier
  • Nicolas Markey
  • Michael Ummels
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7213)


We consider concurrent games played on graphs, in which each player has several qualitative (e.g. reachability or Büchi) objectives, and a preorder on these objectives (for instance the counting order, where the aim is to maximise the number of objectives that are fulfilled).

We study two fundamental problems in that setting: (1) the value problem, which aims at deciding the existence of a strategy that ensures a given payoff; (2) the Nash equilibrium problem, where we want to decide the existence of a Nash equilibrium (possibly with a condition on the payoffs). We characterise the exact complexities of these problems for several relevant preorders, and several kinds of objectives.


Nash Equilibrium Winning Strategy Nash Equilibrium Problem Boolean Circuit Winning Region 
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.


  1. 1.
    Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. J. ACM 49(5), 672–713 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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
  3. 3.
    Bouyer, P., Brenguier, R., Markey, N., Ummels, M.: Concurrent games with ordered objectives. Research Report LSV-11-22, LSV, ENS Cachan, France (2011)Google Scholar
  4. 4.
    Bouyer, P., Brenguier, R., Markey, N., Ummels, M.: Nash equilibria in concurrent games with Büchi objectives. In: FSTTCS 2011. LIPIcs, vol. 13, pp. 375–386. Leibniz-Zentrum für Informatik, LZI (2011)Google Scholar
  5. 5.
    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
  6. 6.
    Henzinger, T.A.: Games in system design and verification. In: TARK 2005, pp. 1–4 (2005)Google Scholar
  7. 7.
    Hunter, P., Dawar, A.: Complexity Bounds for Regular Games. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 495–506. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Jurdzinski, M.: Deciding the winner in parity games is in UP ∩ co-UP. Inf. Process. Lett. 68(3), 119–124 (1998)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Laroussinie, F., Markey, N., Oreiby, G.: On the expressiveness and complexity of ATL. Logicical Methods in Computer Science 4(2) (2008)Google Scholar
  10. 10.
    McNaughton, R.: Infinite games played on finite graphs. Annals of Pure and Applied Logic 65(2), 149–184 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Nash Jr., J.F.: Equilibrium points in n-person games. Proc. National Academy of Sciences of the USA 36(1), 48–49 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Neumann, J., Szepietowski, A., Walukiewicz, I.: Complexity of weak acceptance conditions in tree automata. Inf. Process. Lett. 84(4), 181–187 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Papadimitriou, C.H.: Complexity Theory. Addison-Wesley (1994)Google Scholar
  14. 14.
    Paul, S., Simon, S.: Nash equilibrium in generalised Muller games. In: FSTTCS 2009. LIPIcs, vol. 4, pp. 335–346. LZI (2010)Google Scholar
  15. 15.
    Thomas, W.: Infinite Games and Verification (Extended abstract of a tutorial). In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 58–64. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  16. 16.
    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
  17. 17.
    Ummels, M., Wojtczak, D.: The Complexity of Nash Equilibria in Limit-Average Games. In: Katoen, J.-P., König, B. (eds.) CONCUR 2011 – Concurrency Theory. LNCS, vol. 6901, pp. 482–496. Springer, Heidelberg (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Patricia Bouyer
    • 1
  • Romain Brenguier
    • 1
  • Nicolas Markey
    • 1
  • Michael Ummels
    • 1
  1. 1.LSV, CNRS & ENS CachanFrance

Personalised recommendations