Solution of Evolutionary Games via Hamilton-Jacobi-Bellman Equations

The paper is focused on construction of solution for bimatrix evolutionary games basing on methods of the theory of optimal control and generalized solutions of Hamilton-Jacobi-Bellman equations. It is assumed that the evolutionary dynamics describes interactions of agents in large population groups in biological and social models or interactions of investors on financial markets. Interactions of agents are subject to the dynamic process which provides the possibility to control flows between different types of behavior or investments. Parameters of the dynamics are not fixed a priori and can be treated as controls constructed either as time programs or feedbacks. Payoff functionals in the evolutionary game of two coalitions are determined by the limit of average matrix gains on infinite horizon. The notion of a dynamical Nash equilibrium is introduced in the class of control feedbacks within Krasovskii’s theory of differential games. Elements of a dynamical Nash equilibrium are based on guaranteed feedbacks constructed within the framework of the theory of generalized solutions of Hamilton-Jacobi-Bellman equations. The value functions for the series of differential games are constructed analytically and their stability properties are verified using the technique of conjugate derivatives. The equilibrium trajectories are generated on the basis of positive feedbacks originated by value functions. It is shown that the proposed approach provides new qualitative results for the equilibrium trajectories in evolutionary games and ensures better results for payoff functionals than replicator dynamics in evolutionary games or Nash values in static bimatrix games. The efficiency of the proposed approach is demonstrated by applications to construction of equilibrium dynamics for agents’ interactions on financial markets.