Construction of Equilibrium Trajectories in Dynamical Bimatrix Coordination Games

The paper deals with the problem of construction of equilibrium trajectories in dynamical bimatrix coordination games. Players manage the investment motion, which is inertial with respect to control signals. Control strategies are constructed via the feedback principle from the differential games theory, developed in N.N. Krasovskii’s scientific school. The universal formulation of the game problem is investigated, in which the dynamics of the controlled system operates on the infinite time horizon, and players’ functionals are determined as the limit of the expected values of bimatrix games payoffs. The solution for the problem of construction of equilibrium trajectories is implemented in the framework of the differential games theory and the theory of generalized minimax solutions of partial differential equations of the Hamilton-Jacobi type. The technique of conjugate derivatives is used to verify the stability property for the proposed solutions. An algorithm for construction of equilibrium trajectories is designed based on control strategies, generated by switching lines of the value function. The asymptotic behavior of equilibrium trajectories is investigated, which reveals important trends for the investments dynamics analysis. The proposed approach generates new qualitative properties of equilibrium trajectories and provides better results than values of competitive static Nash equilibria. The obtained result implements the idea for shifting the dynamical system from unfavorable competitive Nash equilibria to cooperative Pareto maximum points with better payoffs. As an application, a model of investments is examined for East Siberian gas pipeline projects, in which equilibrium trajectories are constructed and investment efficiency is analyzed.