Geodesic
Nash equilibrium in differential games and the construction of the programmed iteration method
Sbornik. Mathematics, Tome 202 (2011) no. 5, pp. 621-647

Voir la notice de l'article provenant de la source Math-Net.Ru

This work is devoted to the study of nonzero-sum differential games. The set of payoffs in a situation of Nash equilibrium is examined. It is shown that the set of payoffs in a situation of Nash equilibrium coincides with the set of values of consistent functions which are fixed points of the program absorption operator. A condition for functions to be consistent is given in terms of the weak invariance of the graph of the functions under a certain differential inclusion. Bibliography: 18 titles.
Keywords: differential games, Nash equilibrium, programmed iteration method. 
Yu. V. Averboukh. Nash equilibrium in differential games and the construction of the programmed iteration method. Sbornik. Mathematics, Tome 202 (2011) no. 5, pp. 621-647. http://geodesic.mathdoc.fr/item/SM_2011_202_5_a0/
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[1] N. N. Krasovskii, A. I. Subbotin, Game-theoretical control problems, Springer Ser. Soviet Math., Springer-Verlag, New York, 1988 | MR | MR | Zbl | Zbl

[2] N. N. Krasovskii, Upravlenie dinamicheskoi sistemoi. Zadacha o minimume garantirovannogo rezultata, Nauka, M., 1985 | MR | Zbl

[3] A. G. Chentsov, “On the structure of a game problem of convergence”, Soviet Math. Dokl., 16:5 (1976), 1404–1408 | MR | Zbl

[4] A. G. Čencov, “On a game problem of converging at a given instant of time”, Math. USSR-Sb., 28:3 (1976), 353–376 | DOI | MR | Zbl

[5] A. I. Subbotin, A. G. Chentsov, Optimizatsiya garantii v zadachakh upravleniya, Nauka, M., 1981 | MR | Zbl

[6] A. F. Kleimenov, Neantagonisticheskie pozitsionnye differentsialnye igry, Nauka, Ekaterinburg, 1993 | MR | Zbl

[7] J. F. Nash, jr., “Equilibrium points in $n$-person games”, Proc. Nat. Acad. Sci. U.S.A., 36:1 (1950), 48–49 | DOI | MR | Zbl

[8] J. Nash, “Non-cooperative games”, Ann. of Math. (2), 54:2 (1951), 286–295 | DOI | MR | Zbl

[9] T. Basar, G. J. Olsder, Dynamic noncooperative game theory, Academic Press, London–New York, 1995 | MR | Zbl

[10] A. F. Kononenko, “On equilibrium positional strategies in nonantagonistic differential games”, Soviet Math. Dokl., 17:6 (1976), 1557–1560 | MR | Zbl

[11] A. F. Kononenko, Yu. E. Chistyakov, “On equilibrium positional strategies in $n$-person differential games”, Soviet Math. Dokl., 37:2 (1988), 514–517 | MR | Zbl

[12] L. A. Petrosyan, N. N. Danilov, Kooperativnye differentsialnye igry i ikh prilozheniya, Izd-vo Tomsk. un-ta, Tomsk, 1985 | MR | Zbl

[13] Yu. E. Chistyakov, “On coalition-free differential games”, Soviet Math. Dokl., 24:1 (1981), 166–169 | MR | Zbl

[14] R. V. Gamkrelidze, Principles of optimal control theory, Plenum Press, New York–London, 1978 | MR | MR | Zbl

[15] J. Warga, Optimal control of differential and functional equations, Academic Press, New York–London, 1972 | MR | MR | Zbl

[16] A. I. Subbotin, Obobschennye resheniya differentsialnykh uravnenii 1-go poryadka. Perspektivy dinamicheskoi optimizatsii, RKhD, Izhevsk, 2003

[17] H. G. Guseinov, A. I. Subbotin, V. N. Ushakov, “Derivatives for multivalued mappings with applications to game-theoretical problems of control”, Problems Control Inform. Theory/Problemy Upravlen. Teor. Inform., 14:3 (1985), 155–167 | MR | Zbl

[18] N. N. Krasovskiy, “A differential game of approach and evasion. I”, Engrg. Cybernetics, 11:2 (1973), 189–203 | MR | Zbl