Abstract
We deal with multistage multicriteria games in extensive form and employ so-called “A-subgame” concept to examine dynamical properties of some non-cooperative and cooperative solutions. It is proved that if we take into account only the active players at each A-subgame the set of all strong Pareto equilibria is time consistent but does not satisfy dynamical compatibility.
We construct an optimal cooperative trajectory and vector-valued characteristic function using the refined leximin algorithm. To ensure the sustainability of a cooperative agreement we design the A-incremental imputation distribution procedure for the Shapley value which provides a better incentive for cooperation than classical incremental allocation procedure. This specific payment schedule corresponds to the A-subgame concept satisfies time consistency and efficiency condition and implies non-zero current payment to the active player immediately after her move.
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References
Bergstresser, K., Yu, P.: Domination structures and multicriteria problems in N-person games. Theor. Decis. 8, 5–48 (1977)
Borm, P., Tijs, S., Van Den Aarssen, J.: Pareto equilibrium in multiobjective games. Methods Oper. Res. 60, 303–312 (1990)
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). https://doi.org/10.1111/itor.12515
Finus, M.: Game Theory and International Environmental Cooperation. Edward Elgar, Cheltenham (2001)
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)
Gromova, E.V., Plekhanova, T.M.: On the regularization of a cooperative solution in a multistage game with random time horizon. Discret. Appl. Math. (2018). https://doi.org/10.1016/j.dam.2018.08.008
Haddad, M., Sanders, D.: Selection of discrete multiple criteria decision making methods in the presence of risk and uncertainty. Oper. Res. Perspect. 5, 357–370 (2018). https://doi.org/10.1016/j.orp.2018.10.003
Haurie, A.: A note on nonzero-sum differential games with bargaining solution. J. Optim. Theory Appl. 18, 31–39 (1976)
Jahn, J.: Vector Optimization: Theory, Applications, and Extensions, 2nd edn. Springer, Berlin (2011)
Kuhn, H.: Extensive games and the problem of information. Ann. Math. Stud. 28, 193–216 (1953)
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)
Kuzyutin, D.: Properties of solutions of positional games with incomplete information. J. Comput. Syst. Sci. Int. 34(1), 157–162 (1996)
Kuzyutin, D.: Reexamination of the strong time consistency concept in extensive games. In: Proceedings of the 7th International Symposium on Dynamic Games and Application, Kanagava, pp. 564–572 (1996)
Kuzyutin, D., Nikitina, M.: Time consistent cooperative solutions for multistage games with vector payoffs. Oper. Res. Lett. 45(3), 269–274 (2017)
Kuzyutin, D., Nikitina, M.: An irrational behavior proof condition for multistage multicriteria games. In: Proceedings of Constructive nonsmooth analysis and related topics (dedic. to the memory of V.F. Demyanov), CNSA 2017, pp. 178–181. IEEE, Piscataway (2017)
Kuzyutin, D., Romanenko, I.: On properties of equilibrium solutions for n-person games in extensive form. Vestnik St. Petersburg Univ. Math. 3(15), 17–27 (1998) (in Russian)
Kuzyutin, D., Nikitina, M., Pankratova, Y.: O Svoystvakh Ravnovesiy v Mnogokriterialnikh Pozicionnikh Igrakh N Lic. Matematicheskaya Teoriya Igr i eyo Prilojeniya 6(1), 19–40 (2014) (in Russian)
Kuzyutin, D., Nikitina, M., Razgulyaeva, L.: On the A-equilibria properties in multicriteria extensive games. Appl. Math. Sci. 9(92), 4565–4573 (2015)
Kuzyutin, D., Gromova, E., Pankratova, Ya.: Sustainable cooperation in multicriteria multistage games. Oper. Res. Lett. 46(6), 557–562 (2018). https://doi.org/10.1016/j.orl.2018.09.004
Kuzyutin, D., Smirnova, N., Gromova, E.: Long-term implementation of the cooperative solution in multistage multicriteria game. Oper. Res. Perspect. 6, 100107 (2019). https://doi.org/10.1016/j.orp.2019.100107
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)
Mendoza, G.A., Martins, H.: Multi-criteria decision analysis in natural resource management: A critical review of methods and new modelling paradigms. For. Ecol. Manag. 230, 1–22 (2006)
Pankratova, Y., Tarashnina, S., Kuzyutin, D.: Nash equilibria in a group pursuit game. Appl. Math. Sci. 10(17), 809–821 (2016)
Petrosyan, L.: Stable solutions of differential games with many participants. Vestn. Leningr. Univ. 19, 46–52 (1977) (in Russian)
Petrosyan, L., Danilov, N.: Stability of the solutions in nonantagonistic differential games with transferable payoffs. Vestn. Leningr. Univer. 1, 52–59 (1979) (in Russian)
Petrosyan, L.A., Kuzyutin, D.V.: On the stability of E-equilibrium in the class of mixed strategies. Vestn. St.Petersburg Univ. Math. 3(15), 54–58 (1995) (in Russian)
Petrosyan, L., Kuzyutin, D.: Games in Extensive Form: Optimality and Stability. Saint Petersburg University Press, Saint Petersburg (2000) (in Russian)
Petrosyan, L., Zaccour, G.: Time-consistent Shapley value allocation of pollution cost reduction. J. Econ. Dyn. Control. 27(3), 381–398 (2003)
Pieri, G., Pusillo, L.: Interval values for multicriteria cooperative games. AUCO Czech. Econ. Rev. 4, 144–155 (2010)
Pieri, G., Pusillo, L.: Multicriteria partial cooperative games. Appl. Math. 6, 2125–2131 (2015)
Podinovskii, V., Nogin, V.: Pareto-optimal solutions of multicriteria problems. Nauka (1982) (in Russian)
Puerto, J., Perea, F.: On minimax and Pareto optimal security payoffs in multicriteria games. J. Math. Anal. Appl. 457(2), 1634–1648 (2018). https://doi.org/10.1016/j.jmaa.2017.01.002
Reddy, P., Shevkoplyas, E., Zaccour, G.: Time-consistent Shapley value for games played over event trees. Automatica 49(6), 1521–1527 (2013)
Rettieva, A.: Cooperation in dynamic multicriteria games with random horizons. J. Glob. Optim. (2018). https://doi.org/10.1007/s10898-018-0658-6.
Shapley, L.: A value for n-person games. In: Contributions to the Theory of Games, II, Kuhn, H., Tucker, A.W. (eds.), pp. 307–317. Princeton University Press, Princeton (1953)
Shapley, L.: Equilibrium points in games with vector payoffs. Nav. Res. Logist. Q. 6, 57–61 (1959)
Voorneveld, M., Vermeulen, D., Borm, P.: Axiomatizations of Pareto equilibria in multicriteria games. Games Econom. Behav. 28, 146–154 (1999)
Voorneveld, M., Grahn, S., Dufwenberg, M.: Ideal equilibria in noncooperative multicriteria games. Math. Methods Oper. Res. 52, 65–77 (2000)
Wang, Y.: Existence of a Pareto equilibrium. J. Optim. Theory Appl. 79, 373–384 (1993)
Yeung, D.: An irrational-behavior-proof condition in cooperative differential games. Int. Game Theory Rev. 8(4), 739–744 (2006)
Acknowledgements
The research of the first and the second author was funded by RFBR under the research project 18-00-00727 (18-00-00725). The research of the third author was funded by RFBR under the research project 18-00-00727 (18-00-00628).
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Kuzyutin, D., Pankratova, Y., Svetlov, R. (2019). A-Subgame Concept and the Solutions Properties for Multistage Games with Vector Payoffs. In: Petrosyan, L., Mazalov, V., Zenkevich, N. (eds) Frontiers of Dynamic Games. Static & Dynamic Game Theory: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23699-1_6
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