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 Contributions to Game Theory and Management, 2020, Volume 13, Pages 218–243 (Mi cgtm367)

Mechanism for shifting Nash equilibrium trajectories to cooperative Pareto solutions in dynamic bimatrix games

Nikolay A. Krasovskiia, Alexander M. Tarasyevba

a Krasovskii Institute of Mathematics and Mechanics UrB RAS, S. Kovalevskoy Str., 16, Yekaterinburg, 620990, Russia
b Ural Federal University named after the first President of Russia B.N. Yeltsin, Mira Str., 19, Yekaterinburg, 620002, Russia

Abstract: In the paper, constructions of the generalized method of characteristics are applied for calculating the generalized minimax (viscosity) solutions of Hamilton-Jacobi equations in dynamic bimatrix games. The structure of the game presumes interactions of two players in the framework of the evolutionary game model. Stochastic contacts between players occur according to the dynamic process, which can be interpreted as a system of Kolmogorov's differential equations with controls instead of probability parameters. It is assumed that control parameters are not fixed and can be constructed by the feedback principle. Two types of payoff functions are considered: short-term payoffs are determined in the current moments of time, and long-term payoffs are determined as limit functionals on the infinite time horizon. The notion of dynamic Nash equilibrium in the class of controlled feedbacks is considered for the long-term payoffs. In the framework of constructions of dynamic equilibrium, the solutions are designed on the basis of maximization of guaranteed payoffs. Such guaranteeing strategies are built in the framework of the theory of minimax (viscosity) solutions of Hamilton-Jacobi equations. The analytical formulas are obtained for the value functions in the cases of different orientations for the “zigzags” (broken lines) of acceptable situations in the static game. The equilibrium trajectories generated by the minimax solutions shift the system in the direction of cooperative Pareto points. The proposed approach provides new qualitative properties of the equilibrium trajectories in the dynamic bimatrix games which guarantee better results of payoffs for both players than static Nash equilibria. As an example, interactions of two firms on the market of innovative electronic devices are examined within the proposed approach for treating dynamic bimatrix games.

Keywords: optimal control, dynamic bimatrix games, value functions, minimax solutions of Hamilton-Jacobi equations, dynamic Nash equilibrium trajectories, shift to Pareto maximum.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00221_a The first author, Nikolay A. Krasovskii, is supported by the Russian Foundation for Basic Research (Project No. 18-01-00221a).

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Citation: Nikolay A. Krasovskii, Alexander M. Tarasyev, “Mechanism for shifting Nash equilibrium trajectories to cooperative Pareto solutions in dynamic bimatrix games”, Contributions to Game Theory and Management, 13 (2020), 218–243

Citation in format AMSBIB
\Bibitem{KraTar20} \by Nikolay~A.~Krasovskii, Alexander~M.~Tarasyev \paper Mechanism for shifting Nash equilibrium trajectories to cooperative Pareto solutions in dynamic bimatrix games \jour Contributions to Game Theory and Management \yr 2020 \vol 13 \pages 218--243 \mathnet{http://mi.mathnet.ru/cgtm367}