Analysis of Equilibrium Trajectories in Dynamic Bimatrix Games

Abstract

The paper is devoted to the analysis of behavior of equilibrium trajectories for game dynamic systems arising in solution of bimatrix games. At the first stage, the approach is considered based on the ideas of guaranteed strategies in the sense of N.N. Krasovskii. In the framework of guaranteed solutions, we propose algorithms for constructing the value functions, positional strategies and equilibrium trajectories using the definition of the dynamic Nash equilibrium. At the second stage, we analyze equilibrium trajectories of the replicator dynamics relating to the theory of evolutionary games. At the third stage, we examine the dynamic system generated by the strategies of best replies similar to the Cournot model. The comparison is carried out for the objective indices of the equilibrium trajectories of all three dynamic systems. It is shown that the characteristics of the trajectories of the dynamic Nash equilibrium are better than the properties of the trajectories of the replicator dynamics or the best reply dynamics. In addition, the numerical experiments are implemented for the so-called mixed dynamics in which the first player uses the guaranteed strategy and the strategy of the second player is formed either by the replicator formulas or by the best reply dynamics. The simulation results for the mixed dynamics demonstrate that the values of players’ payoff functionals in the final points of trajectories are better in comparsion with the indices for trajectories of the replicator dynamics and the best reply dynamics and even better than at the final point of the dynamic Nash equilibrium.