Evolutionary Dynamics in Continuous-time Finite-state Mean Field Games - Part I: Equilibria

2025-11-03

AI Summary

This paper studies large-scale dynamic games in continuous time, where players possess finite state and action spaces, their states evolve stochastically based on actions, and payoffs depend on both individual strategies and the population’s joint state–action distribution (e.g., traffic congestion). Addressing the limitation of classical evolutionary game theory—which neglects individual state dynamics—we first formulate a mean-field approximation model incorporating state evolution and introduce the concept of mixed stationary Nash equilibrium (MSNE) with an evolutionary interpretation. Theoretically, we establish strong convergence of finite-population games to the mean-field limit, prove equivalence between MSNEs and equilibria of the mean-field dynamics, and derive sufficient conditions for evolutionary stability. Methodologically, we integrate continuous-time Markov decision processes, distribution-dependent reward modeling, and evolutionary dynamics analysis—thereby filling a critical gap in the theory of dynamic evolutionary equilibria.

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
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

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. While prior work in evolutionary game theory has primarily focused on static games without individual player state dynamics, we present the first comprehensive evolutionary analysis of such dynamic games. We propose an evolutionary model together with a mean field approximation of the finite-population game and establish strong approximation guarantees. We show that standard solution concepts for dynamic games lack an evolutionary interpretation, and we propose a new concept - the Mixed Stationary Nash Equilibrium (MSNE) - which admits one. We analyze the relationship between MSNE and the rest points of the mean field evolutionary model and study the evolutionary stability of MSNE.

Problem

• Modeling evolutionary dynamics in continuous-time finite-state mean field games
• Analyzing player interactions with population-dependent rewards and state transitions
• Establishing new equilibrium concepts with evolutionary stability guarantees

Innovation

• Mean field approximation for finite-population dynamic games
• Mixed Stationary Nash Equilibrium as new solution concept
• Evolutionary stability analysis of MSNE equilibria