Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation

• Published in: Stochastic processes and their applications Vol. 129; no. 11; pp. 4510 - 4555
• Main Authors: Cecchin, Alekos, Pelino, Guglielmo
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.11.2019
• Subjects: [formula omitted]-person games; Central Limit Theorem; Finite state space; Jump Markov processes; Large Deviation Principle; Master Equation; Mean field games; Nash equilibrium; Propagation of chaos; 60F10; Nash equilibrium; Master Equation; N-person games; 91A10; 93E20; Finite state space; Jump Markov processes; Propagation of chaos; Large Deviation Principle; 60J27; Central Limit Theorem; 60K35; Mean field games; 60F05
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2018.12.002

We show the convergence of finite state symmetric N-player differential games, where players control their transition rates from state to state, to a limiting dynamics given by a finite state Mean Field Game system made of two coupled forward–backward ODEs. We exploit the so-called Master Equation, which in this finite-dimensional framework is a first order PDE in the simplex of probability measures, obtaining the convergence of the feedback Nash equilibria, the value functions and the optimal trajectories. The convergence argument requires only the regularity of a solution to the Master Equation. Moreover, we employ the convergence results to prove a Central Limit Theorem and a Large Deviation Principle for the evolution of the N-player empirical measures. The well-posedness and regularity of solution to the Master Equation are also studied, under monotonicity assumptions.