# On Coset Weighted Potential Game

## TL;DR

This paper introduces coset weighted potential games, a generalized class of potential games, and provides algebraic and geometric methods for their verification, analysis, and decomposition.

## Contribution

It defines coset weighted potential games, develops an algebraic verification method, and proposes a new orthogonal decomposition based on coset weights.

## Key findings

- A new class of potential games called coset weighted potential games is defined.
- An algebraic method using semi-tensor product of matrices is developed for verification.
- A novel orthogonal decomposition of finite games based on coset weights is proposed.

## Abstract

In this paper we first define a new kind of potential games, called coset weighted potential game, which is a generalized form of weighted potential game. Using semi-tensor product of matrices, an algebraic method is provided to verify whether a finite game is a coset weighted potential game, and a simple formula is obtained to calculate the corresponding potential function. Then some properties of coset weighted potential games are revealed. Finally, by resorting to the vector space structure of finite games, a new orthogonal decomposition based on coset weights is proposed, the corresponding geometric and algebraic expressions of all the subspaces are given by providing their bases.