Guaranteeing Control Strategies and Best Response Dynamics in Game Investments Models

The paper analyzes the behaviour of equilibrium trajectories in dynamic bimatrix games that have applications for modelling investment processes. The authors consider a system of differential equations that describes the evolutionary dynamics of the behavior of two players on an infinite time horizon. First, based on the idea of guaranteeing strategies in the sense of N.N. Krasovskii, we study the constructions of dynamic Nash equilibrium. Such strategies take into account the long-term interests of players. At the second step, the dynamics of the players’ best replies is considered, taking into account short-term interests. In this case, the equilibrium trajectory converges to the point of static Nash equilibrium. As an example, we consider a model of a bimatrix game on the financial market, in which players are coalitions of traders (bulls and bears) with two investment strategies (investing in either stocks or bonds). At the third step, the so-called “mixed” dynamics is investigated, in which the first player uses guaranteeing strategies, and the second player is guided by the strategy of best reply dynamics. For all three cases, equilibrium trajectories are constructed and the values of payoff functionals of the players at the equilibrium points are compared. The findings show that the value of payoffs on the trajectory of dynamic Nash equilibrium is better than the properties of the trajectory of best reply dynamics. It is also demonstrated that the quality indices at the finite point of the movement for equilibrium trajectories of “mixed” dynamics dominate over the first two cases.