Implementation of subgame-perfect cooperative agreement in an extensive-form game

Denis Kuzyutin ; Yulia Skorodumova ; Nadezhda Smirnova

Contributions to game theory and management, Tome 14 (2021), pp. 257-272.

Résumé

A novel approach to sustainable cooperation called subgame-perfect core (S-P Core) was introduced by P. Chander and M. Wooders in 2020 for $n$-person extensive-form games with terminal payoffs. This solution concept incorporates both subgame perfection and cooperation incentives and implies certain distribution of the total players' payoff at the terminal node of the cooperative history. We use in the paper an extension of the S-P Core to the class of extensive games with payoffs defined at all nodes of the game tree that is based on designing an appropriate payoff distribution procedure $\beta$ and its implementation when a game unfolds along the cooperative history. The difference is that in accordance with this so-called $\beta$-subgame-perfect core the players can redistribute total current payoff at each node in the cooperative path. Moreover, a payoff distribution procedure from the $\beta$-S-P Core satisfies a number of good properties such as subgame efficiency, non-negativity and strict balance condition. In the paper, we examine different properties of the $\beta$-S-P Core, introduce several refinements of this cooperative solution and provide examples of its implementation in extensive-form games. Finally, we consider an application of the $\beta$-S-P Core to the symmetric discrete-time alternating-move model of fishery management.

Détail

Keywords: extensive game, sustainable cooperation, subgame-perfect equilibrium, core, payoff distribution procedure, renewable resource extraction.

Bibliographie

[1] Aumann R., “Acceptable points in general cooperative $n$-person games”, Contributions to the Theory of Games IV, Princeton University Press, Princeton, N.J., 1959 | MR

[2] Breton M., I. Dahmouni, G. Zaccour, “Equilibria in a two-species fishery”, Math. Biosci., 309 (2019), 78–91 | MR | Zbl

[3] Cabo F., M. Tidball, “Cooperation in a Dynamic Setting with Asymmetric Environmental Valuation and Responsibility”, Dyn Games Appl., 2021 (forthcoming) | DOI

[4] Castañer A., J. Marín-Solano, C. Ribas, “A time consistent dynamic bargaining procedure in differential games with heterogeneous discounting”, Math. Meth. Oper. Res., 2021 (forthcoming) | MR

[5] Chander P., “The gamma-core and coalition formation”, Int. J. Game Theory, 35 (2007), 539–556 | DOI | MR | Zbl

[6] Chander P., “Subgame-perfect cooperative agreements in a dynamic game of climate change”, J. Environ. Econ. Manag., 84(C) (2017), 173–188

[7] Chander P., H. Tulkens, “The core of an economy with multilateral environmental externalities”, Int. J. Game Theory, 26 (1997), 379–401 | MR | Zbl

[8] Chander P., M. Wooders, “Subgame-perfect cooperation in an extensive game”, Journal of Economic Theory, 187 (2020), 105017 | MR | Zbl

[9] Chebotareva A., Su S., S. Tretyakova, E. Gromova, “On the Value of the Preexisting Knowledge in an Optimal Control of Pollution Emissions”, Contributions to Game Theory and Management, 14, eds. Petrosyan L. A., N. A. Zenkevich, St. Petersburg State Univ. Press, St. Petersburg, 2021 (forthcoming) | MR

[10] Crettez B., N. Hayek, G. Zaccour, Do charities spend more on their social programs when they cooperate than when they compete?, Eur. J. Oper. Res., 283 (2020), 1055–1063 | MR | Zbl

[11] Haurie A., J. B. Krawczyk, G. Zaccour, Games and Dynamic Games, Scientific World, Singapore, 2012 | MR | Zbl

[12] Hillas J., D. Kvasov, “Backward induction in games without perfect recall”, Games and Economic Behavior, 124 (2020), 207–218 | DOI | MR | Zbl

[13] Kuhn H., “Extensive games and the problem of information”, Ann. Math., 28 (1953), 193–216 | MR | Zbl

[14] Kuzyutin D., E. Gromova, Ya. Pankratova, “Sustainable cooperation in multicriteria multistage games”, Oper. Res. Lett., 46 (2018), 557–562 | MR | Zbl

[15] Kuzyutin D., Lipko I., Y. Pankratova, I. Tantlevskij, “Cooperation Enforcing in Multistage Multicriteria Game: New Algorithm and Its Implementation”, Frontiers of Dynamic Games, Static Dynamic Game Theory: Foundations Applications, eds. Petrosyan L., V. Mazalov, N. Zenkevich, Birkhäuser, Cham, 2020 | MR

[16] Kuzyutin D., M. Nikitina, “An irrational behavior proof condition for multistage multicriteria games”, Constructive Nonsmooth Analysis and Related Topics, Dedic. to the Memory of V.F. Demyanov, IEEE, New York, NY, USA, 2017, 178–181

[17] Kuzyutin D., Y. Pankratova, R. Svetlov, “A-Subgame Concept and the Solutions Properties for Multistage Games with Vector Payoffs”, Frontiers of Dynamic Games, Static Dynamic Game Theory: Foundations Applications, eds. Petrosyan L., V. Mazalov, N. Zenkevich, Birkhäuser, Cham, 2019 | MR | Zbl

[18] Kuzyutin D., I. Romanenko, “On properties of equilibrium solutions for n-person games in extensive form”, Vestnik St. Petersburg Univ. Math., 3:15 (1998), 17–27 (in Russian) | MR

[19] Kuzyutin D., N. Smirnova, “Subgame Consistent Cooperative Behavior in an Extensive form Game with Chance Moves”, Mathematics, 8:7 (2020), 1061 | DOI

[20] Kuzyutin D., Smirnova N., Tantlevskij I., “Subgame-perfect core based on the payoff distribution procedure”, Economics Letters, 2021 (to appear)

[21] Kuzyutin D., N. Smirnova, E. Gromova, “Long-term implementation of the cooperative solution in multistage multicriteria game”, Oper. Res. Perspect., 6 (2019), 100107 | MR

[22] Levhari D., L. J. Mirman, “The Great Fish War: An Example Using a Dynamic Cournot–Nash solution”, The Bell Journal of Economics, 11:1 (1980), 322–334 | MR

[23] Mazalov V., E. Parilina, J. Zhou, “Altruistic-Like Equilibrium in a Differential Game of Renewable Resource Extraction”, Mathematical Optimization Theory and Operations Research, MOTOR 2021, Lecture Notes in Computer Science, 12755, eds. Pardalos P., M. Khachay, A. Kazakov, Springer, Cham, 2021 | MR

[24] Mazalov V. V., A. N. Rettiyeva, “The discrete-time bioresource sharing model”, J. Appl. Math. Mech., 75 (2011), 180–188 | MR | Zbl

[25] Myerson R., Game Theory. Analysis of Conflict, Harvard University Press, Cambridge, MA, USA, 1997 | MR

[26] Moulin H., Axioms of cooperative decision making, Cambridge University Press, Cambridge, MA, USA, 1988 | MR | Zbl

[27] Nash J. F., “Equilibrium points in n-person games”, Proc. Nat. Acad. Sci. USA, 36 (1950), 48–49 | MR | Zbl

[28] Ougolnitsky G., A. Usov, “Spatially distributed differential game theoretic model of fisheries”, Mathematics, 7 (2019), 732 | DOI

[29] Petrosyan L., “Time-consistency of solutions in multi-player differential games”, Vestnik of Leningrad State University. Series 1. Mathematics. Mechanics. Astronomy, 4 (1977), 46–52 | Zbl

[30] Petrosyan L. A., N. N. Danilov, “Stability of solutions in non-zero sum differential games with transferable payoffs”, Vestnik of Leningrad State University. Series 1. Mathematics. Mechanics. Astronomy, 1 (1979), 52–59 | MR | Zbl

[31] Petrosyan L., D. Kuzyutin, Games in Extensive Form: Optimality and Stability, Saint Petersburg University Press, St. Petersburg, Russia, 2000 (in Russian) | MR

[32] Selten R., “Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games”, Int. J. Game Theory, 4 (1975), 25–55 | MR | Zbl

[33] Tantlevskij I., E. Gromova, D. Gromov, “Network Analysis of the Interaction between Different Religious and Philosophical Movements in Early Judaism”, Philosophies, 6:1 (2021), 2 https://www.mdpi.com/2409-9287/6/1/2

[34] Yeung D., “An irrational-behavior-proof condition in cooperative differential games”, International Game Theory Review, 8:4 (2006), 739–744 | MR | Zbl

[35] Yeung D., L. Petrosyan, Subgame Consistent Economic Optimization: An Advanced Cooperative Dynamic Game Analysis, Springer, New York, NY, USA, 2012 | MR