On Characterizations of Potential and Ordinal Potential Games

TL;DR

This paper extends the theory of potential games to multi-dimensional action spaces with differentiable costs, providing new conditions for potential and ordinal potential games, and demonstrating their application in network congestion scenarios.

Contribution

It introduces necessary and sufficient conditions for potential and ordinal potential games in multi-dimensional settings, expanding classical results and offering a systematic construction method.

Findings

Extended classical potential game results to multi-dimensional actions.

Provided a systematic way to construct potential functions.

Applied conditions to network congestion games.

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.