Geodesic
Asymptotic behavior of solutions in dynamical bimatrix games with discounted indices
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 27 (2017) no. 2, pp. 193-209 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper is devoted to the analysis of dynamical bimatrix games with integral indices discounted on an infinite time interval. The system dynamics is described by differential equations in which players' behavior changes according to incoming control signals. For this game, a problem of construction of equilibrium trajectories is considered in the framework of minimax approach proposed by N. N. Krasovskii and A. I. Subbotin in the differential games theory. The game solution is based on the structure of dynamical Nash equilibrium developed in papers by A. F. Kleimenov. The maximum principle of L. S. Pontryagin in combination with the method of characteristics for Hamilton–Jacobi equations are applied for the synthesis of optimal control strategies. These methods provide analytical formulas for switching curves of optimal control strategies. The sensitivity analysis for equilibrium solutions is implemented with respect to the discount parameter in the integral payoff functional. It is shown that equilibrium trajectories in the problem with the discounted payoff functional asymptotically converge to the solution of a dynamical bimatrix game with average integral payoff functionals examined in papers by V. I. Arnold. Obtained results are applied to a dynamical model of investments on financial markets.
Keywords: dynamical games, Pontryagin maximum principle, Hamilton–Jacobi equations, equilibrium trajectories. 
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N. A. Krasovskii; A. M. Tarasyev. Asymptotic behavior of solutions in dynamical bimatrix games with discounted indices. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 27 (2017) no. 2, pp. 193-209. http://geodesic.mathdoc.fr/item/VUU_2017_27_2_a3/

[1] Arnold V. I., “Optimization in mean and phase transitions in controlled dynamical systems”, Funct. Anal. Appl., 36:2 (2002), 83–92 | DOI | DOI | MR | Zbl

[2] Vorob'yev N. N., Game theory for economists and system scientists, Nauka, M., 1985, 271 pp.

[3] Kleimenov A. F., Nonantagonistic positional differential games, Nauka, Yekaterinburg, 1993, 184 pp.

[4] Kolmogorov A. N., “On analytical methods in probability theory”, Uspekhi Matematicheskikh Nauk, 5 (1938), 5–41 (in Russian)

[5] Krasovskiy N. A., Kryazhimskiy A. V., Tarasyev A. M., “Hamilton–Jacobi equations in evolutionary games”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 20, no. 3, 2014, 114–131 (in Russian) | Zbl

[6] Krasovskii N. A., Taras'ev A. M., “Search for maximum points of a vector criterion based on decomposition properties”, Proceedings of the Steklov Institute of Mathematics, 269, suppl. 1 (2010), S174–S190 | DOI

[7] Krasovskii N. A., Taras'yev A. M., Equilibrium solutions in dynamical games, Ural State Agrarian University, Yekaterinburg, 2015, 128 pp.

[8] Krasovskii N. A., Tarasyev A. M., “Equilibrium trajectories in dynamical bimatrix games with average integral payoff functionals”, Mat. Teor. Igr Prilozh., 8:2 (2016), 58–90 (in Russian) | Zbl

[9] Krasovskii N. N., Subbotin A. I., Game-theoretical control problems, Springer, New York, 1988, 517 pp. | MR | Zbl

[10] Kryazhimskii A. V., Osipov Yu. S., “On differential-evolutionary games”, Proceedings of the Steklov Institute of Mathematics, 211 (1995), 234–261 | MR | Zbl

[11] Kurzhanskii A. B., Control and observation under uncertainty, Nauka, M., 1977, 392 pp.

[12] Petrosjan L. A., Zenkevich N. A., “Conditions for sustainable cooperation”, Autom. Remote Control, 76:10 (2015), 1894–1904 | DOI | MR

[13] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mishchenko E. F., The mathematical theory of optimal processes, John Wiley Sons, New York, 1962, viii+360 pp. | MR | Zbl

[14] Forex Market

[15] Subbotin A. I., Minimax inequalities and Hamilton–Jacobi equations, Nauka, M., 1991, 214 pp.

[16] Subbotina N. N., “The method of Cauchy characteristics and generalized solutions of the Hamilton–Jacobi–Bellman equation”, Sov. Math., Dokl., 44:2 (1992), 501–506 | MR | Zbl

[17] Ushakov V. N., Uspenskii A. A., Lebedev P. D., “Geometry of singular curves for one class of velocity”, Vestn. St-Peterbg. Univ., Ser. 10, Prikl. Mat. Inf. Prots. Upr., 2013, no. 3, 157–167 (in Russian)

[18] Basar T., Olsder G. J., Dynamic noncooperative game theory, Academic Press, London, 1982, 519 pp. | MR | Zbl

[19] CNN Money, http://money.cnn.com/

[20] Friedman D., “Evolutionary games in economics”, Econometrica, 59:3 (1991), 637–666 https://pdfs.semanticscholar.org/acd0/73087a5ac678e44e61ced6c015cdf98397f9.pdf | DOI | MR | Zbl

[21] Intriligator M., Mathematical optimization and economic theory, Prentice–Hall, New York, 1971, 508 pp. | MR

[22] Krasovskii A. N., Krasovskii N. N., Control under lack of information, Birkhäuser, Boston, 1995, 322 pp. | DOI | MR

[23] Krasovskii N. A., Tarasyev A. M., “Decomposition algorithm of searching equilibria in a dynamic game”, Autom. Remote Control, 76:10 (2015), 1865–1893 | DOI | MR

[24] Tarasyev A. M., A Differential model for a $2 \times 2$-evolutionary game dynamics, IIASA Working Paper, No WP-94-063, IIASA, Laxenburg, Austria, 1994 http://pure.iiasa.ac.at/4148/1/WP-94-063.pdf