The Stability of Stationary Equilibria of Mean Field Games With Finite State and Action Space

The aim of this paper is to study some new stability results on the essential component and generic uniqueness of stationary equilibria for mean field games with finite state and action space (briefly, FSASMFG). A group of FSASMFGs comprises a dense residual subset such that every point of stationary mean field equilibrium (briefly, SMFE) is essential. However, it is not true that any mapping has at least one essential fixed point; for example, the identity mapping of the interval [0, 1] onto itself has no essential fixed point. Thus, we further investigate that every FSASMFG has at least one essential component of its SMFE by proving the connectivity of minimal essential sets of the SMFE. Furthermore, we verify that, in the sense of Baire’s category, most FSASMFGs have a unique deterministic stationary equilibrium by means of nonlinear analysis and set‐valued analysis.