Chapter 7 - Coordination of Long-Term and Short-Term Payoffs in Dynamic Bimatrix Games

Abstract

This chapter discusses the coordination of long-term and short-term payoffs in dynamic bi-matrix games. Dynamic optimality principles are applied to construct equilibrium solutions in evolutionary bi-matrix games. The dynamics of the model can be interpreted as Kolmogorov's differential equations in which coefficients describing flows are not fixed a priori and can be chosen on the feedback principle. Long-term payoffs of coalitions are defined on the infinite horizon by the integral functionals. Generalizing the considered three-step optimal control problem one can formulate the result that arises from the optimization nature of the problem and provide better index values than values of trajectories tending to static Nash equilibria. A peculiarity of the model is the coordination of velocities of solutions constructed for long-term payoff integrals with directions of gradients of short-term payoffs. The notion of a dynamic Nash equilibrium is introduced in the class of control feedbacks. The chapter proposes a solution based on the notion of guaranteeing feedbacks.