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Towards An Analytical Framework for Dynamic Potential Games

Xin Guo, Yufei Zhang

TL;DR

An analytical framework for dynamic potential games is built and it is proved that a game is a dynamic potential game if and only if each player's value function can be decomposed as a potential function and a residual term which is solely dependent on other players' policies.

Abstract

Potential game is an emerging notion and framework for studying N-player games, especially with heterogeneous players. In this paper, we build an analytical framework for dynamic potential games. We prove that a game is a dynamic potential game if and only if each player's value function can be decomposed as a potential function and a residual term which is solely dependent on other players' policies. This decomposition is consistent with the result in the static setting and enables us to identify and analyze an important and new class of dynamic potential games called the distributed game. Moreover, we prove that a game is a dynamic potential game if the value function has a symmetric Jacobian. This generalizes the differential characterization for static potential games by replacing the classical derivative with a new notation of functional derivative with respect to Markov policies. For a general class of continuous-time stochastic games, we explicitly characterize their potential functions. In particular, we show that the potential function of linear-quadratic games can be studied through a system of linear ODEs. Furthermore, under a rank condition on control coefficients, we prove a linear-quadratic game is a Markov potential game if and only if all players have identical cost functions.

Towards An Analytical Framework for Dynamic Potential Games

TL;DR

An analytical framework for dynamic potential games is built and it is proved that a game is a dynamic potential game if and only if each player's value function can be decomposed as a potential function and a residual term which is solely dependent on other players' policies.

Abstract

Potential game is an emerging notion and framework for studying N-player games, especially with heterogeneous players. In this paper, we build an analytical framework for dynamic potential games. We prove that a game is a dynamic potential game if and only if each player's value function can be decomposed as a potential function and a residual term which is solely dependent on other players' policies. This decomposition is consistent with the result in the static setting and enables us to identify and analyze an important and new class of dynamic potential games called the distributed game. Moreover, we prove that a game is a dynamic potential game if the value function has a symmetric Jacobian. This generalizes the differential characterization for static potential games by replacing the classical derivative with a new notation of functional derivative with respect to Markov policies. For a general class of continuous-time stochastic games, we explicitly characterize their potential functions. In particular, we show that the potential function of linear-quadratic games can be studied through a system of linear ODEs. Furthermore, under a rank condition on control coefficients, we prove a linear-quadratic game is a Markov potential game if and only if all players have identical cost functions.
Paper Structure (23 sections, 15 theorems, 110 equations)

This paper contains 23 sections, 15 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Our work.
  3. Our approach.
  4. Notation.
  5. (Dynamic) Potential Game and Its Nash Equilibrium
  6. Distributed Game as Markov Potential Game
  7. Characterization of Differentiable Potential Game
  8. Probabilistic Characterization of Continuous-time Potential Game
  9. PDE Characterization of Continuous-time Potential Games
  10. Continuous-time Linear Quadratic Potential Game
  11. Probabilistic characterization in LQ games.
  12. PDE characterization in LQ games.
  13. Proofs of Main Results
  14. Proofs of Theorems \ref{['thm:separation_value']}, \ref{['thm:distributed_game']} and \ref{['thm:distributed_game_differentiable']}
  15. Proof of Theorem \ref{['thm:symmetry_value_sufficient']}
  16. ...and 8 more sections

Key Result

Proposition 2.1

If there exists $s_0\in \mathcal{S}$ such that $\mathcal{G}$ with initial state $s_0$ is a CLPG with potential function $\Phi^{s_0}$, then any $\phi^*\in \mathop{\mathrm{arg\,min}}\limits_{\phi\in \pi^{(N)} }\Phi^{s_0}(\phi)$ is a closed-loop Nash equilibrium of $\mathcal{G}$ with initial state $s_

Theorems & Definitions (46)

  • Definition 2.1: Nash equilibrium
  • Definition 2.2: Dynamic potential game
  • Proposition 2.1
  • Example 2.1: Team Markov Game
  • Theorem 2.2
  • Remark 2.1: Dynamic vs static potential games
  • Definition 3.1: Distributed game $\mathcal{G}_{\rm dist}$
  • Theorem 3.1
  • Remark 3.1
  • Example 3.1: Crowd motion
  • ...and 36 more