Abstract
We consider a two-player nonzero-sum differential game in the case where players use nonanticipative strategies. We define the Nash equilibrium in this case and obtain a characterization of Nash equilibrium strategies. We show that a Nash equilibrium solution can be approximately realized by control-with-guide strategies.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Warga, J., Optimal Control of Differential and Functional Equations, New York: Academic, 1972. Translated under the title Optimal’noe upravlenie differentsial’nymi i funktsional’nymi uravneniyami, Moscow: Nauka, 1977.
Kleimenov, A.F., Neantagonisticheskie pozitsionnye differentsial’nye igry (Non-Zero-Sum Positional Differential Games), Yekaterinburg: Nauka, 1993.
Kononenko, A.F., On Equilibrium Positional Strategies in Nonantagonistic Differential Games, Dokl. Akad. Nauk SSSR, 1976, vol. 231, no. 2, pp. 285–288.
Krasovskii, N.N. and Subbotin, A.I., Pozitsionnye differentsial’nye igry (Positional Differential Games), Moscow: Nauka, 1974. Translated into English under the title Game Theoretical Control Problems, New York: Springer-Verlag, 1988.
Krasovskii, N.N., A Differential Game of Approach and Evasion. II, Izv. Akad. Nauk SSSR, Tekh. Kibernet., 1973, no. 3, pp. 22–42.
Krasovskii, N.N., Upravlenie dinamicheskoi sistemoi. Zadacha o minimume garantirovannogo resul’tata (Control of a Dynamical System. Minimum Guaranteed Result), Moscow: Nauka, 1985.
Kryazhimskii, A.V. and Chentsov, A.G., O strukture igrovogo upravleniya v zadachakh sblizheniya i ukloneniya (On the Game Control Structure in Approach-Evasion Problems). Available from VINITI, 1979, Sverdlovsk, no. 1729-80.
Subbotin, A.I. and Chentsov, A.G., Optimizatsiya garantii v zadachakh upravleniya (Optimization of the Guarantee in Control Problems), Moscow: Nauka, 1981.
Chistyakov, S.V., On Noncooperative Differential Games, Dokl. Akad. Nauk SSSR, 1981, vol. 259, no. 5, pp. 1052–1055.
Cardaliaguet, P., Buckdahn, R., and Rainer, C., Nash Equilibrium Payoffs for Nonzero-Sum Stochastic Differential Games, SIAM J. Control Optimiz., 2004, vol. 43, no. 2, pp. 624–642.
Elliot, R.J. and Kalton, N., The Existence of Value for Differential Games, in Memoir Am. Math. Soc., Providence: AMS, 1972, vol. 126.
Evans, L.C. and Sougandinis, P.E., Differential Games and Representation Formulas for Solutions of Hamilton-Jacobi-Bellman Equations, Indiana Univ. Math. J., 1984, vol. 33, pp. 773–797.
Roxin, E., Axiomatic Approach in Differential Games, J. Optimiz. Theory Appl., 1969, vol. 3, no. 3, pp. 153–163.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © Yu.V. Averboukh, 2012, published in Matematicheskaya Teoriya Igr i Prilozheniya, 2012, No. 3, pp. 3–20.
Rights and permissions
About this article
Cite this article
Averboukh, Y.V. Nash equilibrium in differential games and the quasi-strategy formalism. Autom Remote Control 75, 1491–1502 (2014). https://doi.org/10.1134/S000511791408013X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000511791408013X