ABSTRACT
Dynamical game interactions are relevant to both differential and evolutionary game-theoretical models. This chapter introduces a notion of a dynamical Nash equilibrium in a class of feedback controls. Feedbacks driving the coalitions to the classical “punishment” solutions in static bimatrix games give a natural and elementary example of a dynamical Nash equilibrium. The chapter proposes another solution which provides a better (at least not worse) long-term result. Our approach originates from theory of positional differential games and rests on the idea of optimal guaranteeing feedbacks in the associated zero sum games. The chapter defines relevant zero sum games and studies them within the framework of the theory of viscosity (minimax) solutions of Hamilton-Jacobi equations. It shows that the equilibrium trajectories generated under the optimal feedbacks stay, in the long run, within a domain in which the current payoffs to each coalition are better (no worse) than the payoff at a static Nash equilibrium point.
