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

2025-11-05

AI Summary

This paper investigates the evolutionary stability of mixed stationary Nash equilibria (MSNE) in continuous-time finite-state mean-field games. In large populations, individual strategy updates and state evolution are intrinsically coupled, complicating stability analysis. Method: We propose a stochastic evolutionary dynamics framework wherein individual behavior follows a revision protocol driven by feedback from the population distribution, integrated with continuous-time mean-field approximation and finite-state Markov transitions. Contribution/Results: We establish necessary and sufficient conditions for both local and global stability of MSNE—first revealing how payoff mapping structure and equilibrium mixing degree fundamentally determine stability. Furthermore, we prove robustness of MSNE under strategic perturbations and its long-term persistence. These results provide a rigorous theoretical foundation for the existence and selection of robust equilibria in mean-field games.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Machine Learning: Evolutionary Learning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Analyzing evolutionary stability of mixed stationary Nash Equilibria in dynamic games
• Establishing conditions for local and global stability under evolutionary dynamics
• Determining when equilibria persist against strategic deviations in large populations

Innovation

• Continuous-time finite-state evolutionary dynamics model
• Mixed stationary Nash Equilibrium as solution concept
• Stability conditions for MSNE under strategic deviations