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

Leonardo Pedroso, Andrea Agazzi, W. P. M. H. Heemels, Mauro Salazar

TL;DR

This work studies dynamic mean-field games with continuous-time, finite-state players whose rewards depend on population state-action distributions. It introduces a Mixed Stationary Nash Equilibrium (MSNE) to provide an evolutionary interpretation absent in traditional BSNE concepts, and develops a mean-field evolutionary dynamics that connects MSNE to rest points under broad revision protocols. The paper proves MSNE existence, analyzes its relation to finite-population equilibria, and demonstrates the framework through a Medium Access Game, illustrating distinct BSNE and MSNE predictions. The results offer qualitative insights and strong approximation guarantees for large populations, laying groundwork for Part II’s stability analysis and broader applicability to congestible resource settings.

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.

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

TL;DR

This work studies dynamic mean-field games with continuous-time, finite-state players whose rewards depend on population state-action distributions. It introduces a Mixed Stationary Nash Equilibrium (MSNE) to provide an evolutionary interpretation absent in traditional BSNE concepts, and develops a mean-field evolutionary dynamics that connects MSNE to rest points under broad revision protocols. The paper proves MSNE existence, analyzes its relation to finite-population equilibria, and demonstrates the framework through a Medium Access Game, illustrating distinct BSNE and MSNE predictions. The results offer qualitative insights and strong approximation guarantees for large populations, laying groundwork for Part II’s stability analysis and broader applicability to congestible resource settings.

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.

Paper Structure

This paper contains 30 sections, 14 theorems, 31 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. State-of-the-art
  3. State-of-the-art
  4. Statement of Contributions
  5. Notation
  6. Model
  7. Finite-population Model
  8. Policies
  9. Mean Field Model
  10. Assumptions
  11. Equilibria
  12. Definition of BSNE and MSNE
  13. Existence
  14. Uniqueness under Congestion Game Payoff Structure
  15. Approximation
  16. ...and 15 more sections

Key Result

Lemma 1

For any class $c\in [C]$, a solution to eq:ODE_mu_u_S with initial condition $\mu^c(0) \in X^c$ exists in $t\in [0,\infty)$, is unique, and is Lipschitz continuous w.r.t. $\mu^c(0)$. Furthermore, if $\lim_{N\to \infty}\hat{\mu}^c(0)= \mu^c(0)$ almost surely, then $\lim_{N\to \infty}\hat{\mu}^c(t)= \

Figures (3)

  • Figure 1: Example of MSNE $\mu$ that is not a rest point.
  • Figure 3: Graphical interpretation of BSNE: For many values of $h$, evolution with $q \in [0,1]$ of the payoff of deviating to policy $u_q$ while the rest of the population uses $u_h$.
  • Figure 4: Graphical interpretation of MSNE: Evolution with $x\in [0,1]$ of the payoff of playing policies $u_1$ and $u_0$ while the proportion of the remainder of the population playing $u_1$ is $x$ and playing $u_0$ is $1-x$.

Theorems & Definitions (41)

  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1: BSNE
  • Definition 2: MSNE
  • Definition 3
  • Definition 4
  • Lemma 3
  • ...and 31 more