Solutions of cooperative differential games with partner sets

Leon A. Petrosyan   ; Yaroslavna B. Pankratova

Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 15 (2023) no. 2, pp. 105-121

Résumé

Network differential games with partner sets are considered. The payoff of each player depends on his actions and the actions of the players from his partner set. The article proposes a cooperative version of the game and introduces a special type of characteristic function that takes into account players in partner sets and proves its superadditivity. As a solution, we propose a $C$-core which non-emptiness is proved, as well as a Shapley value and a $\tau$-value.

Détail

Keywords: Shapley value, differential network game, partner sets.

Bibliographie

[1] Tur A. V., Petrosyan L. A., “Kooperativnye printsipy optimalnosti v differentsialnykh igrakh na setyakh”, MTIP, 12:4 (2020), 93–111 ; Tur A. V., Petrosyan L. A., “Cooperative Optimality Principles in Differential Games on Networks”, Autom. Remote Control, 82:6 (2021), 1095–1106 | Zbl | DOI | MR | Zbl

[2] Bulgakova M., Petrosyan L., “About one multistage non-antagonistic network game”, Vestnik Sankt-Peterburgskogo Universiteta, Prikladnaya Matematika, Informatika, Protsessy Upravleniya, 5:4 (2019), 603–615 | DOI | MR

[3] Gao H. and Pankratova Y., “Cooperation in dynamic network games”, Contributions to Game Theory and Management, 10 (2017), 42–67 | MR

[4] Gromova E., “The Shapley value as a sustainable cooperative solution in differential games of three players”, Recent Advances in Game Theory and Applications. Static Dynamic Game Theory: Foundations Applications, eds. L.A. Petrosyan and V.V. Mazalov (eds.), Birkhäuser, Cham, 2016, 67–89 | DOI | MR | Zbl

[5] Isaacs R., Differential games, Wiley, NY, 1965, 384 pp. | Zbl

[6] Meza M.A.G. and Lopez-Barrientos J. D., “A differential game of a duopoly with network externalities”, Recent Advances in Game Theory and Applications. Static Dynamic Game Theory: Foundations Applications, eds. L.A. Petrosyan and V.V. Mazalov, Birkhäuser, Cham, 2016, 49–66 | DOI | MR | Zbl

[7] Krasovsky N.N, Subbotin A.I., Positional differential games, Nauka, M., 1974, 456 pp. (in Russian) | MR

[8] Pai H.M., “A differential game formulation of a controlled network”, Queueing Systems: Theory and Applications Archive, 64:4 (2010), 325–358 | DOI | MR | Zbl

[9] Petrosyan L.A., “Cooperative differential games on networks”, Trudy Inst. Mat. i Mekh. UrO RAN, 16, no. 5, 2010, 143–150 (in Russian)

[10] Petrosyan L., Zaccour G., “Time-consistent Shapley value allocation of pollution cost reduction”, J. Economic Dynamics and Control, 27 (2003), 381–398 | DOI | MR | Zbl

[11] Petrosyan L. A. and Yeung D. W. K., “Shapley value for differential network games: Theory and application”, J. Dynamics and Games, 8:2 (2020), 151–166 | DOI | MR

[12] Petrosyan L., Yeung D.W.K., Pankratova Y., “Cooperative Differen-tial Games with Partner Sets on Networks”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 27, no. 3, 2021, 286–295 | DOI | MR

[13] Petrosyan L., Pankratova Y., “Two Level Cooperation in Dynamic Network Games with Partner Sets”, International Conference on Mathematical Optimization Theory and Operations Research, Springer, Cham, 2022 | DOI | MR

[14] Shapley L.S., “A value for N-person games”, Contributions to the Theory of Games, eds. H. Kuhn and A. Tucker, Princeton Univ. Press, Princeton, 1953, 307–317 | MR

[15] Tijs S.H., “An axiomatization of the $\tau $-value”, Math. Social Sciences, 13 (1987), 177–181 | DOI | MR | Zbl

[16] Wie B.W., “A differential game model of Nash equilibrium on a congested traffic network”, Networks, 23 (1993), 557–565 | DOI | MR | Zbl

[17] Yeung D.W.K. and Petrosyan L.A., “Subgame consistent cooperative solution in stochastic differential games”, J. Optim. Theory Appl., 120:3 (2004), 651–666 | DOI | MR | Zbl

[18] Yeung D.W.K. and Petrosyan L.A., Subgame Consistent Cooperation: A Comprehensive Treatise, Springer Singapore, Singapore, 2016 | MR | Zbl

[19] Zhan G.H., Jiang L.V., Huang S., Wang J., and Zhang Y., “Attack-defense differential game model for network defense strategy selection”, IEEE Access, 7 (2018), 2169–3536 | DOI