The Group Behavioral Characteristics in a Mean Field Game Model with a Turnpike Effect

We describe the agents’ group behavior by the concept of mean field games with a turnpike effect. The problem is formalized by a system of PDEs: a Kolmogorov–Fokker–Planck equation that describes the evolution of the agents’ density distribution, and a Hamilton–Jacobi–Bellman equation that describes the optimal strategy of the agents. The boundary conditions pose at the initial moment of time for a Kolmogorov–Fokker–Planck equation and at the final moment of time for a Hamilton–Jacobi–Bellman equation. The considered system of PDEs is coupled due to the imitation behavior of the agents. To solve the boundary value problem of PDEs, we use a reduction to an extremal problem and present its numerical solution. In numerical results we examine the optimal strategies of the agents considering different behavioral characteristics.