On the Cooperative Behavior in Multistage Multicriteria Game with Chance Moves

  • Denis Kuzyutin
  • Ekaterina GromovaEmail author
  • Nadezhda Smirnova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12095)


We consider a class of multistage multicriteria games in extensive form with chance moves where the players cooperate to maximize their expected joint vector payoff. Assuming that the players have agreed to accept the minimal sum of relative deviations rule in order to choose a unique Pareto optimal payoffs vector, we prove the time consistency of the optimal cooperative strategy profile and corresponding optimal bundle of the cooperative trajectories. Then, if the players adopt a vector analogue of the Shapley value as the solution concept, they need to design an appropriate imputation distribution procedure to ensure the sustainability of the achieved cooperative agreement. We provide a generalization of the incremental payment schedule that is applicable for the games with chance moves and satisfies such advantageous properties as the efficiency, strict balance condition and the time consistency property in the whole game. We illustrate our approach with an example of the extensive-form game tree with chance moves.


Multicriteria game Multistage game Cooperative behavior Time consistency Shapley value Chance moves 


  1. 1.
    Climaco, J., Romero, C., Ruiz, F.: Preface to the special issue on multiple criteria decision making: current challenges and future trends. Int. Trans. Oper. Res. 25, 759–761 (2018). Scholar
  2. 2.
    Crettez, B., Hayek, N.: A dynamic multi-objective duopoly game with pollution and depollution (2020, submitted to Dynamic Games and Applications)Google Scholar
  3. 3.
    Finus, M.: Game Theory and International Environmental Cooperation. Edward Elgar, Cheltenham (2001)CrossRefGoogle Scholar
  4. 4.
    Gromova, E.V., Petrosyan, L.A.: On an approach to constructing a characteristic function in cooperative differential games. Autom. Remote Control 78(9), 1680–1692 (2017). Scholar
  5. 5.
    Gromova, E.V., Plekhanova, T.M.: On the regularization of a cooperative solution in a multistage game with random time horizon. Discrete Appl. Math. 255, 40–55 (2019). Scholar
  6. 6.
    Haurie, A.: A note on nonzero-sum diferential games with bargaining solution. J. Optim. Theory Appl. 18, 31–39 (1976)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Haurie, A., Krawczyk, J.B., Zaccour, G.: Games and Dynamic Games. Scientific World, Singapore (2012)CrossRefGoogle Scholar
  8. 8.
    Hayek, N.: Infinite-horizon multiobjective optimal control problems for bounded processes. Discrete Continuous Dyn. Syst. Ser. S 11(6), 1121–1141 (2018). Scholar
  9. 9.
    Kuhn, H.: Extensive games and the problem of information. Ann. Math. Stud. 28, 193–216 (1953)MathSciNetGoogle Scholar
  10. 10.
    Kuzyutin, D.: On the problem of the stability of solutions in extensive games. Vestnik St. Petersburg Univ. Math. 4(22), 18–23 (1995). (in Russian)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kuzyutin, D., Gromova, E., Pankratova, Y.: Sustainable cooperation in multicriteria multistage games. Oper. Res. Lett. 46(6), 557–562 (2018). Scholar
  12. 12.
    Kuzyutin, D., Nikitina, M., Razgulyaeva, L.: On the A-equilibria properties in multicriteria extensive games. Appl. Math. Sci. 9(92), 4565–4573 (2015)Google Scholar
  13. 13.
    Kuzyutin, D., Nikitina, M.: Time consistent cooperative solutions for multistage games with vector payoffs. Oper. Res. Lett. 45(3), 269–274 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kuzyutin, D., Nikitina, M.: An irrational behavior proof condition for multistage multicriteria games. In: Consrtuctive Nonsmooth Analysis and Related Topics (dedic. to the memory of V.F. Demyanov), CNSA 2017, Proceedings, pp. 178–181. IEEE (2017)Google Scholar
  15. 15.
    Kuzyutin, D., Pankratova, Y., Svetlov, R.: A-subgame concept and the solutions properties for multistage games with vector payoffs. In: Petrosyan, L.A., Mazalov, V.V., Zenkevich, N.A. (eds.) Frontiers of Dynamic Games. SDGTFA, pp. 85–102. Springer, Cham (2019). Scholar
  16. 16.
    Kuzyutin, D., Smirnova, N., Gromova, E.: Long-term implementation of the cooperative solution in multistage multicriteria game. Oper. Res. Persp. 6, 100107 (2019). Scholar
  17. 17.
    Madani, K., Lund, J.R.: A Monte-Carlo game theoretic approach for multi-criteria decision making under uncertainty. Adv. Water Resour. 34, 607–616 (2011)CrossRefGoogle Scholar
  18. 18.
    Mendoza, G.A., Martins, H.: Multi-criteria decision analysis in natural resource management: a critical review of methods and new modelling paradigms. Forest Ecol. Manage. 230, 1–22 (2006)CrossRefGoogle Scholar
  19. 19.
    Moulin, H.: Axioms of Cooperative Decision Making. Cambridge University Press, Cambridge (1988)CrossRefGoogle Scholar
  20. 20.
    Myerson, R.: Game Theory. Analysis of Conflict. Harvard University Press, Cambridge (1997)zbMATHGoogle Scholar
  21. 21.
    Pankratova, Y., Tarashnina, S., Kuzyutin, D.: Nash equilibria in a group pursuit game. Appl. Math. Sci. 10(17), 809–821 (2016)Google Scholar
  22. 22.
    Parilina, E., Zaccour, G.: Node-consistent core for games played over event trees. Automatica 55, 304–311 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Parilina, E., Zaccour, G.: Node-consistent Shapley value for games played over event trees with random terminal time. J. Opt. Theory Appl. 175(1), 236–254 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Petrosyan, L.: Stable solutions of differential games with many participants. Vestn. Leningrad Univ. 19, 46–52 (1977). (in Russian)Google Scholar
  25. 25.
    Petrosyan, L., Danilov, N.: Stability of the solutions in nonantagonistic differential games with transferable payoffs. Vestn. Leningrad Univ. 1, 52–59 (1979). (in Russian)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Petrosyan, L.A., Kuzyutin, D.V.: On the stability of E-equilibrium in the class of mixed strategies. Vestnik St. Petersburg Univ. Math. 3(15), 54–58 (1995). (in Russian) MathSciNetGoogle Scholar
  27. 27.
    Petrosyan, L., Kuzyutin, D.: Games in Extensive Form: Optimality and Stability. Saint Petersburg University Press, Saint Petersburg (2000). (in Russian)zbMATHGoogle Scholar
  28. 28.
    Petrosyan, L., Zaccour, G.: Time-consistent Shapley value allocation of pollution cost reduction. J. Econ. Dyn. Control 27(3), 381–398 (2003)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Pieri, G., Pusillo, L.: Interval values for multicriteria cooperative games. AUCO Czech Econ. Rev. 4, 144–155 (2010)Google Scholar
  30. 30.
    Pieri, G., Pusillo, L.: Multicriteria partial cooperative games. Appl. Math. 6, 2125–2131 (2015)CrossRefGoogle Scholar
  31. 31.
    Podinovskii, V., Nogin, V.: Pareto-Optimal Solutions of Multicriteria Problems. Nauka, Moscow (1982). (in Russian)Google Scholar
  32. 32.
    Puerto, J., Perea, F.: On minimax and Pareto optimal security payoffs in multicriteria games. J. Math. Anal. Appl. 457(2), 1634–1648 (2018). Scholar
  33. 33.
    Reddy, P., Shevkoplyas, E., Zaccour, G.: Time-consistent Shapley value for games played over event trees. Automatica 49(6), 1521–1527 (2013)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Rettieva, A.N.: Cooperation in dynamic multicriteria games with random horizons. J. Glob. Optim. 76(3), 455–470 (2018). Scholar
  35. 35.
    Shapley, L.: A value for n-person games. In: Kuhn, H., Tucker, A.W. (eds.) Contributions to the Theory of Games, II, pp. 307–317. Princeton University Press, Princeton (1953)Google Scholar
  36. 36.
    Shapley, L.: Equilibrium points in games with vector payoffs. Naval Res. Logistics Q. 6, 57–61 (1959)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Voorneveld, M., Vermeulen, D., Borm, P.: Axiomatizations of Pareto equilibria in multicriteria games. Games Econ. Behav. 28, 146–154 (1999)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Saint Petersburg State UniversitySt. PetersburgRussia
  2. 2.National Research University Higher School of Economics (HSE)St. PetersburgRussia

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