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On Equilibria in Quantitative Games with Reachability/Safety Objectives

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Abstract

In this paper, we study turn-based multiplayer quantitative non zero-sum games played on finite graphs with reachability objectives. In this framework each player aims at reaching his own goal as soon as possible. We focus on existence results for two solution concepts: Nash equilibrium and secure equilibrium. We prove the existence of finite-memory Nash (resp. secure) equilibria in n-player (resp. 2-player) games. For the case of Nash equilibria, we extend our result in two directions. First, we show that finite-memory Nash equilibria still exist when the model is enriched by allowing n-tuples of positive costs on edges (one cost by player). Secondly, we prove the existence of Nash equilibria in quantitative games with both reachability and safety objectives.

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Notes

  1. The general case of reachability/safety objectives is handled in Sect. 3.3.1.

  2. Note that the length is not defined as the number of vertices.

  3. Our definition naturally extends the notion of secure equilibrium proposed in [6] to the quantitative reachability framework. A longer discussion comparing the two notions can be found in Sect. 2.4.

  4. We will keep this convention through the article.

  5. Note that player 1 has no choice in vertices C and D, that is, σ 1(hv) is necessarily equal to A for v∈{C,D}.

  6. For qualitative games, we use the notion of payoff rather than the notion of cost since Win (resp. Lose) can be seen as a payoff of 1 (resp. 0) and the aim of the players is to maximize their payoffs.

  7. Indeed when j>k, i.e. when player j has not reached his goal set, the coalition punishes him in the exact same way as Lemma 16 by preventing him from visiting his goal set.

  8. Remark that ≾1 (or ≾2) is a kind of lexicographic order on \((\mathbb{N}\cup\{+\infty\}) \times (\mathbb{N} \cup\{+\infty\})\).

  9. We are conscious that it is counterintuitive to use the particular value −1, but it is helpful in the proofs.

  10. Notice that in the second case, when ρ does not visit \({\sf{Goal}}_{1}\) in \({\sf{Trunc}}_{d}(\mathcal{T})\), player 1 may reach his goal set in \(\mathcal{T}\) when deviating in this way, and this would be profitable for him in this game.

References

  1. Alur, R., Kanade, A., Weiss, G.: Ranking automata and games for prioritized requirements. In: Computer Aided Verification, 20th International Conference, CAV 2008. Lecture Notes in Computer Science, vol. 5123, pp. 240–253. Springer, Berlin (2008)

    Google Scholar 

  2. Bloem, R., Chatterjee, K., Henzinger, T., Jobstmann, B.: Better quality in synthesis through quantitative objectives. In: CAV: Computer-Aided Verification. Lecture Notes in Computer Science, vol. 5643, pp. 140–156. Springer, Berlin (2009)

    Chapter  Google Scholar 

  3. Boros, E., Gurvich, V.: Why chess and back gammon can be solved in pure positional uniformly optimal strategies. Rutcor Research Report 21-2009, Rutgers University, (2009)

  4. Brihaye, T., Bruyère, V., De Pril, J.: Equilibria in quantitative reachability games. In: Ablayev, F.M., Mayr, E.W. (eds.) CSR. Lecture Notes in Computer Science, vol. 6072, pp. 72–83. Springer, Berlin (2010)

    Google Scholar 

  5. Brihaye, T., Bruyère, V., De Pril, J.: On subgame perfection in quantitative reachability games. Log. Methods Comput. Sci. 9(1) (2012)

  6. Chatterjee, K., Henzinger, T., Jurdziński, M.: Games with secure equilibria. Theor. Comput. Sci. 365(1–2), 67–82 (2006)

    Article  MATH  Google Scholar 

  7. Chatterjee, K., Henzinger, T.A.: Finitary winning in omega-regular games. In: TACAS. Lecture Notes in Computer Science, vol. 3920, pp. 257–271. Springer, Berlin (2006)

    Google Scholar 

  8. Chatterjee, K., Majumdar, R., Jurdzinski, M.: On Nash equilibria in stochastic games. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL. Lecture Notes in Computer Science, vol. 3210, pp. 26–40. Springer, Berlin (2004)

    Google Scholar 

  9. Clarke, E., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (2000)

    Google Scholar 

  10. Grädel, E., Thomas, W., Wilke, T.: Automata, Logics, and Infinite Games. Lecture Notes in Computer Science., vol. 2500. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  11. Grädel, E., Ummels, M.: Solution concepts and algorithms for infinite multiplayer games. In: Apt, K., van Rooij, R. (eds.) New Perspectives on Games and Interaction. Texts in Logic and Games, vol. 4, pp. 151–178. Amsterdam University Press, Amsterdam (2008)

    Google Scholar 

  12. Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley Series in Computer Science. Addison-Wesley, Reading (1979)

    MATH  Google Scholar 

  13. Horn, F., Thomas, W., Wallmeier, N.: Optimal strategy synthesis in request-response games. In: Automated Technology for Verification and Analysis, 6th International Symposium, ATVA 2008. Lecture Notes in Computer Science, vol. 5311, pp. 361–373. Springer, Berlin (2008)

    Google Scholar 

  14. Kuhn, H.: Extensive games and the problem of information. In: Classics in Game Theory, pp. 46–68 (1953)

    Google Scholar 

  15. Martin, D.: Borel determinacy. Ann. Math. 102(2), 363–371 (1975)

    Article  MATH  Google Scholar 

  16. Nash, J.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. USA 36(1), 48–49 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  17. Osborne, M., Rubinstein, A.: A Course in Game Theory. MIT Press, Cambridge (1994)

    MATH  Google Scholar 

  18. Thomas, W.: On the synthesis of strategies in infinite games. In: STACS 95, Munich, 1995. Lecture Notes in Computer Science, vol. 900, pp. 1–13. Springer, Berlin (1995)

    Chapter  Google Scholar 

  19. Zimmermann, M.: Time-optimal winning strategies for poset games. In: CIAA. Lecture Notes in Computer Science, vol. 5642, pp. 217–226. Springer, Berlin (2009)

    Google Scholar 

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Acknowledgements

This work has been partly supported by the ESF project GASICS and a grant from the National Bank of Belgium. The third author is supported by a grant from L’Oreal-UNESCO/F.R.S.-FNRS. The authors are grateful to Jean-François Raskin and Hugo Gimbert for useful discussions.

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  1. University of Mons—UMONS, Place du Parc 20, 7000, Mons, Belgium

    Thomas Brihaye, Véronique Bruyère & Julie De Pril

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  1. Thomas Brihaye
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  2. Véronique Bruyère
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  3. Julie De Pril
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Correspondence to Thomas Brihaye.

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Brihaye, T., Bruyère, V. & De Pril, J. On Equilibria in Quantitative Games with Reachability/Safety Objectives. Theory Comput Syst 54, 150–189 (2014). https://doi.org/10.1007/s00224-013-9495-7

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