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

TL;DR

This paper introduces a new evolutionary framework for continuous-time finite-state mean field games, analyzing equilibria, stability, and proposing the novel Mixed Stationary Nash Equilibrium concept.

Contribution

It presents the first evolutionary analysis of dynamic finite-state mean field games, including a new equilibrium concept and approximation guarantees.

Findings

Established strong approximation between finite-population and mean field models.

Proposed the Mixed Stationary Nash Equilibrium (MSNE) as an evolutionary solution concept.

Analyzed the stability and relationship of MSNE with mean field rest points.

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.