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Dynamic Population Games: A Tractable Intersection of Mean-Field Games and Population Games

Ezzat Elokda, Saverio Bolognani, Andrea Censi, Florian Dörfler, Emilio Frazzoli

TL;DR

Dynamic Population Games (DPGs) address the computational bottleneck of stationary mean-field formulations by proving that SNE in a discrete-time, finite-state-and-action setting can be exactly reduced to NE in a static population game. For any DPG ${\mathcal G}$, there exists an equivalent $F^{\mathcal G}$ whose NE $s^*=(d^*,\pi^*)$ coincides with the SNE of ${\mathcal G}$, enabling existence guarantees under mild continuity and opening the door to evolutionary-dynamics based computation and stability analysis. This reduction unlocks classical population-game tools, providing a practical pathway for computing and analyzing equilibria in large-scale, strategic dynamic systems, with applications ranging from fair resource allocation to epidemic modelling and control. The paper also releases open-source software to compute SNEs, highlighting the approach’s potential for real-world design and policy evaluation.

Abstract

In many real-world large-scale decision problems, self-interested agents have individual dynamics and optimize their own long-term payoffs. Important examples include the competitive access to shared resources (e.g., roads, energy, or bandwidth) but also non-engineering domains like epidemic propagation and control. These problems are natural to model as mean-field games. Existing mathematical formulations of mean field games have had limited applicability in practice, since they require solving non-standard initial-terminal-value problems that are tractable only in limited special cases. In this letter, we propose a novel formulation, along with computational tools, for a practically relevant class of Dynamic Population Games (DPGs), which correspond to discrete-time, finite-state-and-action, stationary mean-field games. Our main contribution is a mathematical reduction of Stationary Nash Equilibria (SNE) in DPGs to standard Nash Equilibria (NE) in static population games. This reduction is leveraged to guarantee the existence of a SNE, develop an evolutionary dynamics-based SNE computation algorithm, and derive simple conditions that guarantee stability and uniqueness of the SNE. We provide two examples of applications: fair resource allocation with heterogeneous agents and control of epidemic propagation. Open source software for SNE computation: https://gitlab.ethz.ch/elokdae/dynamic-population-games

Dynamic Population Games: A Tractable Intersection of Mean-Field Games and Population Games

TL;DR

Dynamic Population Games (DPGs) address the computational bottleneck of stationary mean-field formulations by proving that SNE in a discrete-time, finite-state-and-action setting can be exactly reduced to NE in a static population game. For any DPG , there exists an equivalent whose NE coincides with the SNE of , enabling existence guarantees under mild continuity and opening the door to evolutionary-dynamics based computation and stability analysis. This reduction unlocks classical population-game tools, providing a practical pathway for computing and analyzing equilibria in large-scale, strategic dynamic systems, with applications ranging from fair resource allocation to epidemic modelling and control. The paper also releases open-source software to compute SNEs, highlighting the approach’s potential for real-world design and policy evaluation.

Abstract

In many real-world large-scale decision problems, self-interested agents have individual dynamics and optimize their own long-term payoffs. Important examples include the competitive access to shared resources (e.g., roads, energy, or bandwidth) but also non-engineering domains like epidemic propagation and control. These problems are natural to model as mean-field games. Existing mathematical formulations of mean field games have had limited applicability in practice, since they require solving non-standard initial-terminal-value problems that are tractable only in limited special cases. In this letter, we propose a novel formulation, along with computational tools, for a practically relevant class of Dynamic Population Games (DPGs), which correspond to discrete-time, finite-state-and-action, stationary mean-field games. Our main contribution is a mathematical reduction of Stationary Nash Equilibria (SNE) in DPGs to standard Nash Equilibria (NE) in static population games. This reduction is leveraged to guarantee the existence of a SNE, develop an evolutionary dynamics-based SNE computation algorithm, and derive simple conditions that guarantee stability and uniqueness of the SNE. We provide two examples of applications: fair resource allocation with heterogeneous agents and control of epidemic propagation. Open source software for SNE computation: https://gitlab.ethz.ch/elokdae/dynamic-population-games

Paper Structure

This paper contains 14 sections, 4 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Dynamic Population Games (DPGs)
  4. DPG Setting
  5. Solution Concept: SNE
  6. Main Result: Reduction to Population Game
  7. Consequences of the Reduction
  8. Existence of
  9. Evolutionary Dynamics for Computation
  10. Stability and Uniqueness of
  11. Applications
  12. Fair Resource Allocation
  13. Epidemic Modelling and Control
  14. Conclusion

Key Result

Theorem 1

For every ${\mathcal{G}}$, there exists a static population game $F^{\mathcal{G}}$ whose Nash Equilibria (NE) coincide with the Stationary Nash Equilibria (SNE) of ${\mathcal{G}}$, i.e., it holds that $(d^*,\pi^*)$ is a of ${\mathcal{G}}$ if and only if $s^* = (d^*,\pi^*)$ is a of $F^{\mathcal{G}}$.

Figures (2)

  • Figure 1: in karma with heterogeneous discount factors.
  • Figure 2: Effect of vaccine cost and availability on epidemic spread.

Theorems & Definitions (8)

  • Definition 1: SNE
  • Theorem 1: Reduction of
  • proof
  • Proposition 1: Existence of
  • proof
  • Proposition 2
  • Definition 2: Stable
  • Proposition 3