Evolutionary Dynamics in Continuous-time Finite-state Mean Field Games -- Part II: Stability

TL;DR

This paper analyzes the stability of mixed stationary Nash equilibria in continuous-time finite-state mean field games, providing conditions for their robustness and long-term persistence in large population dynamic settings.

Contribution

It introduces stability conditions for MSNE in continuous-time mean field games, extending the understanding of equilibrium robustness in population dynamics.

Findings

Derived conditions for local stability of MSNE

Established criteria for global stability under evolutionary dynamics

Provided insights into long-term viability of equilibria in large populations

Abstract

We study a dynamic game with a large population of players who choose actions from a finite set in continuous time. Each player has a state in a finite state space that evolves stochastically with their actions. A player's reward depends not only on their own state and action but also on the distribution of states and actions across the population, capturing effects such as congestion in traffic networks. In Part I, we introduced an evolutionary model and a new solution concept - the mixed stationary Nash Equilibrium (MSNE) - which coincides with the rest points of the mean field evolutionary model under meaningful families of revision protocols. In this second part, we investigate the evolutionary stability of MSNE. We derive conditions on both the structure of the MSNE and the game's payoff map that ensure local and global stability under evolutionary dynamics. These results characterize when MSNE can robustly emerge and persist against strategic deviations, thereby providing insight into its long-term viability in large population dynamic games.