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

• Published in: Lobachevskii journal of mathematics Vol. 46; no. 1; pp. 341 - 350
• Main Author: Trusov, N. V.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.01.2025; Springer Nature B.V
• Subjects: Algebra; Analysis; Bellman theory; Boundary conditions; Boundary value problems; Density distribution; Fokker-Planck equation; Game theory; Geometry; Group dynamics; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Partial differential equations; Probability Theory and Stochastic Processes; numerical solution; mean field games; optimal control; 2010 Mathematics Subject Classification: 49K20, 93A30, 35Q93; turnpike effect
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080224608002

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.