Comparative Analysis of Game Equilibrium Trajectories and Replicator Dynamics in Models of Investments

The paper provides an analysis for the behavior of equilibrium trajectories of two various types in dynamic bimatrix games, which are adequate models of evolutionary processes occurring in economic and biological systems. The model, which describes the players’ interaction, is given by differential equations with control parameters. The game dynamics has the property of strong invariance with respect to the space of the game given by unit square. Different types of game equilibria are studied for the considered dynamic model. The first type of equilibrium is obtained within the theory of differential games and based on the concept of guaranteed strategies proposed by N.N. Krasovskii. In this case, equilibrium trajectories are generated by guaranteed feedbacks and form the basic construction for the dynamic Nash equilibrium. In the second variant, equilibrium trajectories are determined on the basis of ideas of replicator dynamics from the theory of evolutionary games – a classic tool for describing competition in economic models and processes of population interaction in biology. In both cases, the designs of equilibrium trajectories are presented, for which a comparative analysis of the payoff indices is performed. A study was conducted for the trends of the trajectories of “mixed” game dynamics, within which the first player is guided by the guaranteed principles when making decisions and the second player is guided by the properties of trajectories of the replicator dynamics. The results of the comparative analysis demonstrate that the quality indices of the dynamic Nash equilibrium with guaranteed strategies dominate the characteristics of the replicator dynamics. A variant of the investment model on the financial markets of stocks and bonds is considered, which demonstrates the results of constructing equilibrium trajectories in dynamic bimatrix games.