Social optimum of finite mean field games: existence and uniqueness of equilibrium solutions in the finite horizon and stationary solutions in the infinite horizon

TL;DR

This paper studies the social optimality in finite state mean field games, establishing conditions for the existence and uniqueness of equilibrium solutions in both finite and infinite horizon settings.

Contribution

It provides new sufficient conditions for the existence and uniqueness of social optimal strategies and equilibrium solutions in finite and infinite horizon mean field games.

Findings

Conditions for existence and uniqueness of equilibrium solutions.

Existence of stable solutions in infinite horizon cases.

Examples satisfying the theoretical conditions.

Abstract

In this paper, we consider the social optimal problem of discrete time finite state space mean field games (referred to as finite mean field games [1]). Unlike the individual optimization of their own cost function in competitive models, in the problem we consider, individuals aim to optimize the social cost by finding a fixed point of the state distribution to achieve equilibrium in the mean field game. We provide a sufficient condition for the existence and uniqueness of the individual optimal strategies used to minimize the social cost. According to the definition of social optimum and the derived properties of social optimal cost, the existence and uniqueness conditions of equilibrium solutions under initial-terminal value constraints in the finite horizon and the existence and uniqueness conditions of stable solutions in the infinite horizon are given. Finally, two examples that satisfy the conditions for the above solutions are provided.