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An $α$-potential game framework for $N$-player dynamic games

Xin Guo, Xinyu Li, Yufei Zhang

TL;DR

An $α$-potential framework extends static potential games to continuous-time stochastic differential games, capturing heterogeneity and interaction strength via $α$. It reduces the search for $α$-Nash equilibria to minimizing an $α$-potential $Φ$, constructed via linear derivatives and bounded by the asymmetry of second-order derivatives. The authors quantify $α$ for general stochastic differential games, embed the optimization into a conditional McKean–Vlasov control problem with a verification theorem, and illustrate the approach with a linear-quadratic example where an explicit $α$-Nash equilibrium can be computed through finite-dimensional ODEs. The framework accommodates distributed and mean-field interactions, enabling analysis beyond homogeneous mean-field models and offering practical algorithms for large $N$.

Abstract

This paper proposes and studies a general form of dynamic $N$-player non-cooperative games called $α$-potential games, where the change of a player's value function upon her unilateral deviation from her strategy is equal to the change of an $α$-potential function up to an error $α$. Analogous to the static potential game (which corresponds to $α=0$), the $α$-potential game framework is shown to reduce the challenging task of finding $α$-Nash equilibria for a dynamic game to minimizing the $α$-potential function. Moreover, an analytical characterization of $α$-potential functions is established, with $α$ represented in terms of the magnitude of the asymmetry of value functions' second-order derivatives. For stochastic differential games in which the state dynamic is a controlled diffusion, $α$ is characterized in terms of the number of players, the choice of admissible strategies, and the intensity of interactions and the level of heterogeneity among players. Two classes of stochastic differential games, namely distributed games and games with mean field interactions, are analyzed to highlight the dependence of $α$ on general game characteristics that are beyond the mean-field paradigm, which focuses on the limit of $N$ with homogeneous players. To analyze the $α$-NE, the associated optimization problem is embedded into a conditional McKean-Vlasov control problem. A verification theorem is established to construct $α$-NE based on solutions to an infinite-dimensional Hamilton-Jacobi-Bellman equation, which is reduced to a system of ordinary differential equations for linear-quadratic games.

An $α$-potential game framework for $N$-player dynamic games

TL;DR

An -potential framework extends static potential games to continuous-time stochastic differential games, capturing heterogeneity and interaction strength via . It reduces the search for -Nash equilibria to minimizing an -potential , constructed via linear derivatives and bounded by the asymmetry of second-order derivatives. The authors quantify for general stochastic differential games, embed the optimization into a conditional McKean–Vlasov control problem with a verification theorem, and illustrate the approach with a linear-quadratic example where an explicit -Nash equilibrium can be computed through finite-dimensional ODEs. The framework accommodates distributed and mean-field interactions, enabling analysis beyond homogeneous mean-field models and offering practical algorithms for large .

Abstract

This paper proposes and studies a general form of dynamic -player non-cooperative games called -potential games, where the change of a player's value function upon her unilateral deviation from her strategy is equal to the change of an -potential function up to an error . Analogous to the static potential game (which corresponds to ), the -potential game framework is shown to reduce the challenging task of finding -Nash equilibria for a dynamic game to minimizing the -potential function. Moreover, an analytical characterization of -potential functions is established, with represented in terms of the magnitude of the asymmetry of value functions' second-order derivatives. For stochastic differential games in which the state dynamic is a controlled diffusion, is characterized in terms of the number of players, the choice of admissible strategies, and the intensity of interactions and the level of heterogeneity among players. Two classes of stochastic differential games, namely distributed games and games with mean field interactions, are analyzed to highlight the dependence of on general game characteristics that are beyond the mean-field paradigm, which focuses on the limit of with homogeneous players. To analyze the -NE, the associated optimization problem is embedded into a conditional McKean-Vlasov control problem. A verification theorem is established to construct -NE based on solutions to an infinite-dimensional Hamilton-Jacobi-Bellman equation, which is reduced to a system of ordinary differential equations for linear-quadratic games.
Paper Structure (20 sections, 17 theorems, 123 equations)

This paper contains 20 sections, 17 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Outline of main results
  4. Analytical framework for general $\alpha$-potential games
  5. $\alpha$-Potential games and approximate Nash equilibria
  6. Characterization of $\alpha$-potential games via linear derivatives
  7. Stochastic differential game and its $\alpha$-potential function
  8. Quantifying $\alpha$ for stochastic differential game
  9. Conditional McKean-Vlasov control problem for $\alpha$-NE
  10. Conditional MKV control problem
  11. HJB equation for the $\alpha$-potential function
  12. A toy example: linear-quadratic $\alpha$-potential games
  13. Quantifying $\alpha$ for $\mathcal{G}_{\textrm{LQ}}$
  14. Constructing $\alpha$-NE for $\mathcal{G}_{\textrm{LQ}}$
  15. Proofs of main results
  16. ...and 5 more sections

Key Result

Proposition 2.1

Let $\mathcal{G}$ be an $\alpha$-potential game for some $\alpha$ and $\Phi$ be an $\alpha$-potential function. For each $\varepsilon\ge 0$, if there exists $\overline {\textbf{a}} \in \mathcal{A}^{(N)}$ such that $\Phi(\overline {\textbf{a}} )\le \inf_{{\textbf{a}} \in \mathcal{A}^{(N)} }\Phi({\te

Theorems & Definitions (36)

  • Definition 2.1: $\alpha$-potential game
  • Definition 2.2
  • Proposition 2.1
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Proposition 2.2
  • Theorem 3.1
  • Proposition 3.1
  • Remark 4.1
  • ...and 26 more