Hamilton–Jacobi equations in evolutionary games

Advanced methods of the theory of optimal control and generalized minimax solutions of Hamilton–Jacobi equations are applied to a nonzero sum game between two large groups of agents in the framework of economic and biological evolutionary models. Random contacts of agents from different groups happen according to a control dynamic process which can be interpreted as Kolmogorov's differential equations. Coefficients of equations are not fixed a priori and can be chosen as control parameters on the feedback principle. Payoffs of coalitions are determined by the limit functionals on infinite horizon. The notion of a dynamical Nash equilibrium is considered in the class of control feedbacks. A solution is proposed basing on feedbacks maximizing with the guarantee the own payoffs. Guaranteed feedbacks are constructed in the framework of the theory of generalized solutions of Hamilton–Jacobi equations. The analytical formulas are obtained for corresponding value functions. The equilibrium trajectory is generated and its properties are investigated. The considered approach provides new qualitative results for the equilibrium trajectory in evolutionary games.