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

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

TL;DR

The paper addresses stability of mean-field equilibria in continuous-time finite-state evolutionary games by introducing an explicit master-equation dynamic and analyzing the evolutionary stability of mixed stationary Nash equilibria (MSNE). It proves non-MSNE rest points are Lyapunov unstable under evolutionary dynamics, establishes local stability for strict MSNE under several revision protocols, and develops a two-time-scale framework showing convergence to MSNE sets in potential and stable games. A regular ESS framework with a practical eigenvalue test is introduced to certify local stability of MSNE subsets via a mapping between policy distributions and steady-state payoffs. The medium access (MAC) example demonstrates the theory's applicability, showing MSNE existence and global/local stability properties and illustrating fast-state dynamics coupled with slower revision dynamics. Overall, the work provides robust, design-oriented conditions under which a desired population state can robustly emerge and persist in large-population dynamic games, with implications for congestion control, networked systems, and strategic diffusion models.

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.

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

TL;DR

The paper addresses stability of mean-field equilibria in continuous-time finite-state evolutionary games by introducing an explicit master-equation dynamic and analyzing the evolutionary stability of mixed stationary Nash equilibria (MSNE). It proves non-MSNE rest points are Lyapunov unstable under evolutionary dynamics, establishes local stability for strict MSNE under several revision protocols, and develops a two-time-scale framework showing convergence to MSNE sets in potential and stable games. A regular ESS framework with a practical eigenvalue test is introduced to certify local stability of MSNE subsets via a mapping between policy distributions and steady-state payoffs. The medium access (MAC) example demonstrates the theory's applicability, showing MSNE existence and global/local stability properties and illustrating fast-state dynamics coupled with slower revision dynamics. Overall, the work provides robust, design-oriented conditions under which a desired population state can robustly emerge and persist in large-population dynamic games, with implications for congestion control, networked systems, and strategic diffusion models.

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.

Paper Structure

This paper contains 21 sections, 23 theorems, 59 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-MSNE rest points of imitative via comparison revision protocols
  4. Strict MSNE
  5. Two Time-scale Stability of non-strict MSNE
  6. Global Stability of MSNE
  7. Local Stability of $\mathcal{M}$
  8. Medium Access Game: Stability
  9. Model
  10. Stability of MSNE
  11. Conclusion
  12. Proofs of Section \ref{['sec:local_ev_stability']}
  13. Proof of Theorem \ref{['th:imitative_rest_unstable']}
  14. Proof of Theorem \ref{['th:stability_strict']}
  15. Proof of Lemma \ref{['lem:ODE_x_z']}
  16. ...and 6 more sections

Key Result

Theorem 1

Consider an imitative via comparison, excess payoff, or pairwise comparison revision protocol $\rho^c$ for each class. Let $\mu^\star$ be a rest point of the evolutionary dynamics eq:ODE_mu_ev. If $\mu^\star$ is not a MSNE, then $\mu^\star$ is not Lyapunov stable under eq:ODE_mu_ev and no solution t

Figures (3)

  • Figure 1: Trajectories of mass on two policies about a regular ESS $\mathcal{M}_\mathcal{F}$.
  • Figure 2: Trajectories of mass on two policies about a non-ESS $\mathcal{M}_\mathcal{F}$.
  • Figure 3: Trajectories of a finite-population in the illustrative MAC game.

Theorems & Definitions (53)

  • Theorem 1
  • proof
  • Corollary 1
  • Definition 1
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 3
  • ...and 43 more