Cooperative differential games with additive payoff functions and finite duration are considered. Players can choose cooperation partners and have the ability to reduce—but not fully terminate—the level of cooperation with opponents. A novel approach to constructing the characteristic function for such games is proposed. The properties of optimality principles, including the core and the Shapley value, derived using this characteristic function, are investigated. The theoretical results are illustrated through a numerical example.
Keywords: differential game, additive payoffs, cooperative game, characteristic function, core, Shapley value
Received March 10, 2026
Revised March 25, 2026
Accepted March 30, 2026
Anna Viktorovna Tur, Cand. Sci. (Phys.-Math.), St.Petersburg State University, St.Petersburg, 198504 Russia, e-mail: a.tur@spbu.ru
Leon Aganesovich Petrosyan, Dr. Phys.-Math. Sci., Prof., St.Petersburg State University, St.Petersburg, 198504 Russia, e-mail: l.petrosyan@spbu.ru
REFERENCES
1. Isaacs R. Differential games. NY, John Wiley and Sons, 1965, 384 p. ISBN: 978-0471428602 . Translated to Russian under the title Differentsial’nye igry, Moscow, Mir Publ., 1967, 480 p.
2. Krasovskii N.N., Subbotin A.I. Game-theoretical control problems. NY, Springer, 1988, 517 p. ISBN: 978-1-4612-8318-8 . Original Russian text was published in Pozitsionnye differentsial’nye igry, Moscow, Nauka Publ., 1974, 456 p.
3. Petrosyan L., Tur A. Differential games with additive payoffs. Inter. Game Theory Review, 2024, vol. 26, no. 2, art. no. 2440010. https://doi.org/10.1142/s0219198924400103
4. von Neumann J., Morgenstern O. Theory of games and economic behavior. Princeton, Princeton Univer. Press, 1953, 641 p. Translated to Russian under the title Teoriya igr i ekonomicheskoye povedeniye, Moscow, Nauka Publ., 1970, 707 p.
5. Gromova E., Marova E., Gromov D. A substitute for the classical Neumann — Morgenstern characteristic function in cooperative differential games. J. Dynam. Games, 2020, vol. 7, no. 2, pp. 105–122. https://doi.org/10.3934/jdg.2020007
6. Reddy P.V., Zaccour G. A friendly computable characteristic function. Math. Soc. Sci., 2016, vol. 82, pp. 18–25. https://doi.org/10.1016/j.mathsocsci.2016.03.008
7. Petrosyan L., Yeung D. Construction of dynamically stable solutions in differential network games. In: Tarasyev A., Maksimov V., Filippova T. (eds.) Proc. Inter. Conf. “Stability, control, differential games” (SCDG2019). Cham, Springer, 2020, pp. 79–92. https://doi.org/10.1007/978-3-030-42831-0_5
8. Vinogradov O.L. Matematicheskiy analiz [Mathematical analysis]. St. Petersburg, BHV-Petersburg Publ., 2017, 752 p. ISBN: 978-5-9775-3815-2 .
9. Shapley L.S. Cores of convex games. Int. J. Game Theory, 1971, vol. 1, no. 1, pp. 11–26. https://doi.org/10.1007/BF01753431
10. Petrosyan L.A. Strongly dynamically stable differential optimality principles. Vestn. Leningr. Univer. Ser. 1: Matem., mekh., astron., 1993, no. 4, pp. 35–40.
11. Petrosyan L.A., Danilov N.N. Stability of solutions to non-antagonistic differential games with transferable payoffs. Vestn. Leningr. Univer. Ser. 1: Matem., mekh., astron., 1979, no. 1, pp. 52–59.
12. Shapley L.S. A value for n-person games. In: Kuhn H., Tucker A. (eds.) Contributions to the theory of games, vol. II, Princeton, Princeton Univ. Press, 1953, pp. 307–317. https://doi.org/10.1515/9781400881970-018
13. Mazalov V.V., Trukhina L.I. Generating functions and the Myerson vector in communication networks. Discrete Math. Appl., 2014, vol. 24, no. 5, pp. 295–303. https://doi.org/10.1515/dma-2014-0026
14. Dockner E., Jorgensen S., Long N.V., Sorger G. Differential games in economics and management science. Cambridge, Cambridge Univer. Press, 2000, 375 p. https://doi.org/10.1017/CBO9780511805127
15. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. The mathematical theory of optimal processes, NY, London, Sydney, John Wiley & Sons, 1962, 360 p. ISBN-13: 978-0470693810 . Original Russian text published in Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mishchenko E. F. Matematicheskaya teoriya optimal’nykh protsessov, Moscow, Fizmatlit Publ., 1961, 394 p.
Cite this article as: A.V. Tur, L.A. Petrosyan. Cooperation in differential games with additive payoffs. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 3, pp. 267–280.
