Finite State Mean Field Games

Abstract

Mean field game is a powerful framework for studying the strategic interactions within a large population of rational agents. Although existing research has predominantly relied on diffusion models to depict agents’ states, numerous applications, such as epidemic control and botnet defense, can best be modeled by systems of particles in discrete state space. This thesis tackles finite state mean field games. In the first part of the thesis, we develop a probabilistic approach for finite state mean field games. Based on the weak formulation of optimal control, the approach accommodates the interactions through the players’ strategies and flexible information structures.

The second part of the thesis is devoted to finite state mean field games involving a player possessing dominating influence. Two different mechanisms are explored. We first study a form of Stackelberg games, in which the dominating player, referred to as principal, moves first and chooses its strategy which impacts the dynamics and ob- jective functions of every remaining player, referred to as agent. Having observed the principal’s strategy, the agents reach a Nash equilibrium. We seek optimal strategies of the principal, whose objective function depends on the statistical distribution of the agents’ states in equilibrium. Using the weak formulation of finite state mean field games developed previously in the thesis, we transform the principal’s optimization problem into a McKean-Vlasov control problem, and provide a semi-explicit solution under the assumptions of linear transition rate, quadratic cost and risk-neutral utility.

In the second model, we assume that all players move simultaneously and we study Nash equilibria formed jointly by major and minor players. We introduce finite player games and derive mean field game formulation in the limit of infinitely many minor players. In this limit, we characterize the best responses of major and minor players via viscosity solutions of HJB equations, and we prove existence of Nash equilibria under reasonable assumptions. We also derive approximate Nash equilibria for the finite player game from the solution of the mean field game.