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On Characterizations of Potential and Ordinal Potential Games

Sina Arefizadeh, Angelia Nedich, Gautam Dasarathy

TL;DR

This work develops necessary and sufficient, cost-difference-based conditions for identifying potential games in multidimensional action spaces $K_i\subset\mathbb{R}^{n_i}$, avoiding differentiation/integration in general. It introduces a constructive framework: a joint-deviation increment relation $\phi(z+y)-\phi(z)=\sum_i[f_i(z_1+y_1,\ldots,z_i+y_i,z_{i+1},\ldots,z_N)\!-\!f_i(z_1+y_1,\ldots,z_{i-1}+y_{i-1},z_i,\ldots,z_N)]$ characterizes potential games and under symmetry yields explicit $\phi$ via $\phi(z)=C-h_P(z,-z)$. The paper then specializes to aggregative games with two-player subgame analysis (Theorem 7) and a constructive global potential (Theorem 8), and extends the framework to ordinal and generalized ordinal potentials under convexity and Lipschitz-gradient conditions, with examples including a three-player Cournot game and a network congestion game. Overall, the results facilitate verification and construction of potential-based dynamics in multi-agent systems and broaden the applicability of potential-game theory to richer action spaces and non-smooth costs.

Abstract

This paper investigates some necessary and sufficient conditions for a game to be a potential game. At first, we extend the classical results of Slade and Monderer and Shapley from games with one-dimensional action spaces to games with multi-dimensional action spaces, which require differentiable cost functions. Then, we provide a necessary and sufficient conditions for a game to have a potential function by investigating the structure of a potential function in terms of the players' cost differences, as opposed to differentials. This condition provides a systematic way for construction of a potential function, which is applied to network congestion games, as an example. Finally, we provide some sufficient conditions for a game to be ordinal potential and generalized ordinal potential.

On Characterizations of Potential and Ordinal Potential Games

TL;DR

This work develops necessary and sufficient, cost-difference-based conditions for identifying potential games in multidimensional action spaces , avoiding differentiation/integration in general. It introduces a constructive framework: a joint-deviation increment relation characterizes potential games and under symmetry yields explicit via . The paper then specializes to aggregative games with two-player subgame analysis (Theorem 7) and a constructive global potential (Theorem 8), and extends the framework to ordinal and generalized ordinal potentials under convexity and Lipschitz-gradient conditions, with examples including a three-player Cournot game and a network congestion game. Overall, the results facilitate verification and construction of potential-based dynamics in multi-agent systems and broaden the applicability of potential-game theory to richer action spaces and non-smooth costs.

Abstract

This paper investigates some necessary and sufficient conditions for a game to be a potential game. At first, we extend the classical results of Slade and Monderer and Shapley from games with one-dimensional action spaces to games with multi-dimensional action spaces, which require differentiable cost functions. Then, we provide a necessary and sufficient conditions for a game to have a potential function by investigating the structure of a potential function in terms of the players' cost differences, as opposed to differentials. This condition provides a systematic way for construction of a potential function, which is applied to network congestion games, as an example. Finally, we provide some sufficient conditions for a game to be ordinal potential and generalized ordinal potential.
Paper Structure (9 sections, 13 theorems, 98 equations)

This paper contains 9 sections, 13 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Notation and Terminology
  3. Preliminaries
  4. Characterization of Potential Games
  5. Necessary Condition
  6. Necessary and Sufficient Conditions
  7. Examples
  8. Some Characterization of Ordinal and Generalized Ordinal Potential Games
  9. Conclusions

Key Result

Theorem 1

Let $u_i(\cdot)$ be a continuously differentiable function on a convex set $E\subset \mathbb{R}^n$, for all $i=1,2,\ldots,n$, for some $n\ge1$. Then, the one-form $\omega_\mathbf{a}(d\mathbf{a})=\sum_{i=1}^{n} u_i(\mathbf{a})da_i$, with $\mathbf{a}\in E$, is exact on the set $E$ if and only if we ha

Theorems & Definitions (35)

  • Definition 1: Potential Game
  • Definition 2: Ordinal Potential Game
  • Definition 3: Generalized Ordinal Potential Game
  • Definition 4: Aggregative Game C4
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • ...and 25 more