On the Existence of Weak Subgame Perfect Equilibria

  • Véronique Bruyère
  • Stéphane Le Roux
  • Arno Pauly
  • Jean-François Raskin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10203)


We study multi-player turn-based games played on a directed graph, where the number of players and vertices can be infinite. An outcome is assigned to every play of the game. Each player has a preference relation on the set of outcomes which allows him to compare plays. We focus on the recently introduced notion of weak subgame perfect equilibrium (weak SPE), a variant of the classical notion of SPE, where players who deviate can only use strategies deviating from their initial strategy in a finite number of histories. We give general conditions on the structure of the game graph and the preference relations of the players that guarantee the existence of a weak SPE, which moreover is finite-memory.


  1. 1.
    Berwanger, D.: Admissibility in infinite games. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Brenguier, R., Clemente, L., Hunter, P., Pérez, G.A., Randour, M., Raskin, J.-F., Sankur, O., Sassolas, M.: Non-zero sum games for reactive synthesis. In: Dediu, A.-H., Janoušek, J., Martín-Vide, C., Truthe, B. (eds.) LATA 2016. LNCS, vol. 9618, pp. 3–23. Springer, Cham (2016). doi: 10.1007/978-3-319-30000-9_1 CrossRefGoogle Scholar
  3. 3.
    Brenguier, R., Raskin, J.-F., Sankur, O.: Assume-admissible synthesis. In: CONCUR, LIPIcs, vol. 42, pp. 100–113. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)Google Scholar
  4. 4.
    Brenguier, R., Raskin, J.-F., Sassolas, M.: The complexity of admissibility in omega-regular games. In: CSL-LICS, pp. 23:1–23:10. ACM (2014)Google Scholar
  5. 5.
    Brihaye, T., Bruyère, V., Meunier, N., Raskin, J.-F.: Weak subgame perfect equilibria and their application to quantitative reachability. In: CSL, LIPIcs, vol. 41, pp. 504–518. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)Google Scholar
  6. 6.
    Bruyère, V., Meunier, N., Raskin, J.-F.: Secure equilibria in weighted games. In: CSL-LICS, pp. 26:1–26:26. ACM (2014)Google Scholar
  7. 7.
    Chatterjee, K., Henzinger, T.A., Jurdzinski, M.: Games with secure equilibria. Theor. Comput. Sci. 365, 67–82 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chatterjee, K., Doyen, L., Filiot, E., Raskin, J.-F.: Doomsday equilibria for omega-regular games. In: McMillan, K.L., Rival, X. (eds.) VMCAI 2014. LNCS, vol. 8318, pp. 78–97. Springer, Heidelberg (2014). doi: 10.1007/978-3-642-54013-4_5 CrossRefGoogle Scholar
  9. 9.
    Chatterjee, K., Doyen, L., Henzinger, T.A.: Quantitative languages. ACM Trans. Comput. Log. 11, 23:1–23:38 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chatterjee, K., Henzinger, T.A.: Assume-guarantee synthesis. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 261–275. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-71209-1_21 CrossRefGoogle Scholar
  11. 11.
    Pril, J., Flesch, J., Kuipers, J., Schoenmakers, G., Vrieze, K.: Existence of secure equilibrium in multi-player games with perfect information. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014. LNCS, vol. 8635, pp. 213–225. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-44465-8_19 Google Scholar
  12. 12.
    Flesch, J., Kuipers, J., Mashiah-Yaakovi, A., Schoenmakers, G., Solan, E., Vrieze, K.: Perfect-information games with lower-semicontinuous payoffs. Math. Oper. Res. 35, 742–755 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Flesch, J., Predtetchinski, A.: A characterization of subgame perfect equilibrium plays in borel games of perfect information. Math. Oper. Res. (2017, to appear)Google Scholar
  14. 14.
    Fudenberg, D., Levine, D.: Subgame-perfect equilibria of finite- and infinite-horizon games. J. Econ. Theor. 31, 251–268 (1983)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Grädel, E., Ummels, M.: Solution concepts and algorithms for infinite multiplayer games. In: New Perspectives on Games and Interaction, vol. 4, pp. 151–178. University Press, Amsterdam (2008)Google Scholar
  16. 16.
    Kuhn, H.W.: Extensive games and the problem of information, pp. 46–68. Classics in Game Theory (1953)Google Scholar
  17. 17.
    Kupferman, O., Perelli, G., Vardi, M.Y.: Synthesis with rational environments. Ann. Math. Artif. Intell. 78(1), 3–20 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Le Roux, S.: Infinite subgame perfect equilibrium in the Hausdorff difference. In: Hajiaghayi, M.T., Mousavi, M.R. (eds.) TTCS 2015. LNCS, vol. 9541, pp. 147–163. Springer, Cham (2016)CrossRefGoogle Scholar
  19. 19.
    Le Roux, S., Pauly, A.: Infinite sequential games with real-valued payoffs. In: CSL-LICS, pp. 62:1–62:10. ACM (2014)Google Scholar
  20. 20.
    Nash, J.F.: Equilibrium points in \(n\)-person games. In: PNAS, vol. 36, pp. 48–49. National Academy of Sciences (1950)Google Scholar
  21. 21.
    Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: POPL, pp. 179–190. ACM Press (1989)Google Scholar
  22. 22.
    Purves, R.A., Sudderth, W.D.: Perfect information games with upper semicontinuous payoffs. Math. Oper. Res. 36(3), 468–473 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rubinstein, A.: Comments on the interpretation of game theory. Econometrica 59, 909–924 (1991)CrossRefGoogle Scholar
  24. 24.
    Selten, R.: Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit. Zeitschrift für die gesamte Staatswissenschaft 121, 301–324, 667–689 (1965)Google Scholar
  25. 25.
    Shen, X.S., Yu, H., Buford, J., Akon, M.: Handbook of Peer-to-Peer Networking. Springer, Heidelberg (2010)CrossRefMATHGoogle Scholar
  26. 26.
    Solan, E., Vieille, N.: Deterministic multi-player Dynkin games. J. Math. Econ. 39, 911–929 (2003)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    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). doi: 10.1007/11944836_21 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Véronique Bruyère
    • 1
  • Stéphane Le Roux
    • 2
  • Arno Pauly
    • 2
  • Jean-François Raskin
    • 2
  1. 1.Département d’informatiqueUniversité de Mons (UMONS)MonsBelgium
  2. 2.Département d’informatiqueUniversité Libre de Bruxelles (ULB)BrusselsBelgium

Personalised recommendations