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## Archive of Issues

Russia Yekaterinburg
Year
2017
Volume
27
Issue
3
Pages
299-308
 Section Mathematics Title Randomized Nash equilibrium for differential games Author(-s) Averboukh Yu.V.a Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa Abstract The paper is concerned with the randomized Nash equilibrium for a nonzero-sum deterministic differential game of two players. We assume that each player is informed about the control of the partner realized up to the current moment. Therefore, the game is formalized in the class of randomized non-anticipative strategies. The main result of the paper is the characterization of a set of Nash values considered as pairs of expected players' outcomes. The characterization involves the value functions of the auxiliary zero-sum games. As a corollary we get that the set of Nash values in the case when the players use randomized strategies is a convex hull of the set of Nash values in the class of deterministic strategies. Additionally, we present an example showing that the randomized strategies can enhance the outcome of the players. Keywords differential games, Nash equilibrium, randomized strategies UDC 517.977.8 MSC 91A23, 91A10, 91A05 DOI 10.20537/vm170301 Received 2 August 2017 Language English Citation Averboukh Yu.V. Randomized Nash equilibrium for differential games, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 3, pp. 299-308. References Averboukh Yu. Nash equilibrium for differential games and nonanticipative strategies, IFAC Proceedings Volumes, 2011, vol. 44, issue 1, pp. 9340-9342. DOI: 10.3182/20110828-6-IT-1002.00967 Averboukh Yu. Nash equilibrium in differential games and the quasi-strategy formalism, Automation and Remote Control, 2014, vol. 75, issue 8, pp. 1491-1502. DOI: 10.1134/S000511791408013X Bressan A., Shen W. Semi-cooperative strategies for differential games, Internat. J. Game Theory, 2004, vol. 32, issue 4, pp. 561-593. DOI: 10.1007/s001820400180 Bressan A., Shen W. Small BV solutions of hyperbolic noncooperative differential games, SIAM J. Control Optim., 2004, vol. 43, issue 1, pp. 194-215. DOI: 10.1137/S0363012903425581 Buckdahn R., Cardaliaguet P., Rainer C. Nash equilibrium payoffs for nonzero-sum stochastic differential games, SIAM J. Control Optim., 2004, vol. 43, issue 2, pp. 624-642. DOI: 10.1137/S0363012902411556 Case J.H. Towards a theory of many players differential games, SIAM Journal on Control, 1969, vol. 7, issue 2, pp. 179-197. DOI: 10.1137/0307013 Chistyakov S.V. On noncooperative differential games, Dokl. Akad. Nauk SSSR, 1981, vol. 259, pp. 1052-1055 (in Russian). Elliot R.J., Kalton N. The existence of value in differential games, in Memoir Am. Math. Soc., Providence: AMS, 1972, vol. 126. Friedman A. Differential games, New York: Wiley, 1971. Kleimenov A.F. Neantagonisticheskie pozitsionnye differentsial'nye igry (Non-antagonistic positional differential games), Ekaterinburg: Nauka, 1993. Kononenko A.F. On equilibrium positional strategies in nonantagonistic differential games, Dokl. Akad. Nauk SSSR, 1976, vol. 231, pp. 285-288 (in Russian). Kononenko A.F., Chistyakov Yu.E. On equilibrium positional strategies in $n$-person differential games, Soviet Math. Dokl., 1988, vol. 37, issue 2, pp. 514-517. Nash J. Equilibrium points in $n$-person games, Proceedings of the National Academy of Sciences of the United States of America, 1950, vol. 36, no. 1, pp. 48-49. DOI: 10.1073/pnas.36.1.48 Roxin E. Axiomatic approach in differential games, J. Optimiz. Theory Appl., 1969, vol. 3, no. 3, pp. 153-163. DOI: 10.1007/BF00929440 Subbotin A.I., Chentsov A.G. Optimizatsiya garantii v zadachakh upravleniya (Optimization of the guarantee in control problems), Moscow: Nauka, 1981. Tolwinski B., Haurie A., Leitman G. Cooperate equilibria in differential games, J. Math. Anal. Appl., 1986, vol. 119, pp. 182-202. DOI: 10.1016/0022-247X(86)90152-6 Warga J. Optimal control of differential and functional equations, New York: Academic Press, 1972. Full text