Geodesic
Markov approximations of nonzero-sum differential games
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 30 (2020) no. 1, pp. 3-17 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper is concerned with approximate solutions of nonzero-sum differential games. An approximate Nash equilibrium can be designed by a given solution of an auxiliary continuous-time dynamic game. We consider the case when dynamics is determined by a Markov chain. For this game the value function is determined by an ordinary differential inclusion. Thus, we obtain a construction of approximate equilibria with the players' outcome close to the solution of the differential inclusion. Additionally, we propose a way of designing a continuous-time Markov game approximating the original dynamics.
Keywords: nonzero-sum differential games, approximate Nash equilibria, Markov games, differential inclusion. 
@article{VUU_2020_30_1_a0,
     author = {Yu. V. Averboukh},
     title = {Markov approximations of nonzero-sum differential games},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {3--17},
     year = {2020},
     volume = {30},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VUU_2020_30_1_a0/}
}
TY  - JOUR
AU  - Yu. V. Averboukh
TI  - Markov approximations of nonzero-sum differential games
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2020
SP  - 3
EP  - 17
VL  - 30
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VUU_2020_30_1_a0/
LA  - en
ID  - VUU_2020_30_1_a0
ER  - 
%0 Journal Article
%A Yu. V. Averboukh
%T Markov approximations of nonzero-sum differential games
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2020
%P 3-17
%V 30
%N 1
%U http://geodesic.mathdoc.fr/item/VUU_2020_30_1_a0/
%G en
%F VUU_2020_30_1_a0
Yu. V. Averboukh. Markov approximations of nonzero-sum differential games. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 30 (2020) no. 1, pp. 3-17. http://geodesic.mathdoc.fr/item/VUU_2020_30_1_a0/

[1] Averboukh Y., “Universal Nash equilibrium strategies for differential games”, Journal of Dynamical and Control Systems, 21:3 (2015), 329–350 | DOI | MR | Zbl

[2] Averboukh Y., “Approximate public-signal correlated equilibria for nonzero-sum differential games”, SIAM Journal on Control and Optimization, 57:1 (2019), 743–772 | DOI | MR | Zbl

[3] Başar T., Olsder G., Dynamic noncooperative game theory, SIAM, Philadelphia, 1998 | MR

[4] Bressan A., Shen W., “Semi-cooperative strategies for differential games”, International Journal of Game Theory, 32:4 (2004), 561–593 | DOI | MR | Zbl

[5] Bressan A., Shen W., “Small BV solutions of hyperbolic noncooperative differential games”, SIAM Journal on Control and Optimization, 43:1 (2004), 194–215 | DOI | MR | Zbl

[6] Cardaliaguet P., Plaskacz S., “Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game”, International Journal of Game Theory, 32:1 (2003), 33–71 | DOI | MR | Zbl

[7] Chistyakov S. V., “On coalition-free differential games”, Soviet Mathematics. Doklady, 24 (1981), 166–169 | MR | Zbl

[8] Fleming W. H., Soner H. M., Controlled Markov processes and viscosity solutions, Springer, New York, 2006 | DOI | MR | Zbl

[9] Friedman A., Differential games, Dover Publications, New York, 2013 | MR

[10] Gihman I. I., Skorohod A. V., Controlled stochastic processes, Springer, New York, 1979 | DOI | MR

[11] Hamadène S., Mannucci P., “Regularity of Nash payoffs of Markovian nonzero-sum stochastic differential games”, Stochastics, 91:5 (2019), 695–715 | DOI | MR

[12] Kleimenov A., Non-antagonistic positional differential games, Nauka, Yekaterinburg, 1993

[13] Kolokoltsov V. N., Markov processes, semigroups and generators, De Gruyter, Berlin, 2011 | MR | Zbl

[14] Kononenko A. F., “On equilibrium positional strategies in nonantagonistic differential games”, Soviet Mathematics. Doklady, 17 (1976), 1557–1560 | MR | Zbl

[15] Krasovskii N. N., Subbotin A. I., Game-theoretical control problems, Springer, New York, 1988 | MR | Zbl

[16] Levy Y., “Continuous-time stochastic games of fixed duration”, Dynamic Games and Applications, 3:2 (2013), 279–312 | DOI | MR | Zbl

[17] Mannucci P., “Nonzero-sum stochastic differential games with discontinuous feedback”, SIAM Journal on Control and Optimization, 43:4, 1222–1233 | DOI | MR | Zbl

[18] Mannucci P., “Nash points for nonzero-sum stochastic differential games with separate Hamiltonians”, Dynamic Games and Applications, 4:3 (2014), 329–344 | DOI | MR | Zbl

[19] Meyer P. A., Probability and potentials, Blaisdell Publishing Company, Waltham, Mass., 1966 | MR | Zbl

[20] Tolwinski B., Haurie A., Leitman G., “Cooperative equilibria in differential games”, Journal of Mathematical Analysis and Applications, 119:1–2 (1986), 182–202 | DOI | MR | Zbl