Equilibrium and Guaranteeing Solutions in Evolutionary Nonzero Sum Games

Advanced methods of theory of optimal guaranteeing control and techniques of generalized (viscosity, minimax) solutions of Hamilton-Jacobi equations are applied to nonzero game interaction between two large groups (coalitions) of agents (participants) arising in economic and biological evolutionary models. Random contacts of agents from different groups happen according to a control dynamical process which 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. Payoffs of coalitions are determined by the functionals of different types on infinite horizon. The notion of a dynamical Nash equilibrium is introduced in the class of control feedbacks. A solution feedbacks maximizing with the guarantee the own payoffs (guaranteeing feedback) is proposed. Guaranteeing feedbacks are constructed in the framework of the the theory of generalized solutions of Hamilton-Jacobi equations. The analytical formulas are obtained for corresponding value functions. The equilibrium trajectory is generated by guaranteeing feedbacks and its properties are investigated. The considered approach provides new qualitative results for the equilibrium trajectory in evolutionary models. The first striking result consists in the fact that the designed equilibrium trajectory provides better (in some bimatrix games strictly better) index values for both coalitions than trajectories which converge to static Nash equilibria (as, for example, trajectories of classical models with the replicator dynamics). The second principle result implies both evolutionary properties of the equilibrium trajectory: evolution takes place in the characteristic domains of Hamilton-Jacobi equations and revolution at switching curves of guaranteeing feedbacks. The third specific feature of the proposed solution is "positive" nature of guaranteeing feedbacks which maximize the own payoff unlike the "negative" nature of punishing feedbacks which minimize the opponent payoff and lead to static Nash equilibrium. The fourth concept takes into account the foreseeing principle in constructing feedbacks due to the multiterminal character of payoffs in which future |states are also evaluated. The fifth idea deals with the venturous factor of the equilibrium trajectory and prescribes the risk barrier surrounding it. These results indicate promising applications of theory of guaranteeing control for constructing solutions in evolutionary models

Demand Functions in Dynamic Games

The paper is devoted to construction of solutions in dynamic bimatrix games. In the model, the payoffs are presented by discounted integrals on the infinite time horizon. The dynamics of the game is subject to the system of the A.N. Kolmogorov type differential equations. The problem of construction of equilibrium trajectories is analyzed in the framework of the minimax approach proposed by N.N. Krasovskii and A.I. Subbotin in the differential games theory. The concept of dynamic Nash equilibrium developed by A.F. Kleimenov is applied to design the structure of the game solution. For obtaining constructive control strategies of players, the maximum principle of L.S. Pontryagin is used in conjunction with the generalized method of characteristics for Hamilton-Jacobi equations. The impact of the discount index is indicated for equilibrium strategies of the game and demand functions in the dynamic bimatrix game are constructed. © 2018The paper is supported by Russin Foundation for Basic Reseaarch (Project No. 18-01-0264a)

A Differential Model for a 2x2-Evolutionary Game Dynamics

A dynamical model for an evolutionary nonantagonistic (nonzero sum) game between two populations is considered. A scheme of a dynamical Nash equilibrium in the class of feedback (discontinuous) controls is proposed. The construction is based on solutions of auxiliary antagonistic (zero-sum) differential games. A method for approximating the corresponding value functions is developed. The method uses approximation schemes for constructing generalized (minimax, viscosity) solutions of first order partial differential equations of Hamilton-Jacobi type. A numerical realization of a grid procedure is described. Questions of convergence of approximate solutions to the generalized one (the value function) are discussed, and estimates of convergence are pointed out. The method provides equilibrium feedbacks in parallel with the value functions. Implementation of grid approximations for feedback control is justified. Coordination of long- and short-term interests of populations and individuals is indicated. A possible relation of the proposed game model to the classical replicator dynamics is outlined