# Linear Representation of Symmetric Games

## TL;DR

This paper introduces a linear algebraic framework using semi-tensor products to analyze symmetric games, characterizing their symmetries and potential game relationships through linear representations of the symmetric group.

## Contribution

It develops a novel linear representation approach for symmetric games, providing algebraic conditions for various symmetries and linking symmetric Boolean games to potential games.

## Key findings

- Algebraic conditions for symmetries derived from linear representations.
- Symmetric Boolean games are shown to be potential games.
- Some Boolean games outside symmetry classes can also be potential games.

## Abstract

Using semi-tensor product of matrices, the structures of several kinds of symmetric games are investigated via the linear representation of symmetric group in the structure vector of games as its representation space. First of all, the symmetry, described as the action of symmetric group on payoff functions, is converted into the product of permutation matrices with structure vectors of payoff functions. Using the linear representation of the symmetric group in structure vectors, the algebraic conditions for the ordinary, weighted, renaming and name-irrelevant symmetries are obtained respectively as the invariance under the corresponding linear representations. Secondly, using the linear representations the relationship between symmetric games and potential games is investigated. This part is mainly focused on Boolean games. An alternative proof is given to show that ordinary, renaming and weighted symmetric Boolean games are also potential ones under our framework. The corresponding potential functions are also obtained. Finally, an example is given to show that some other Boolean games could also be potential games.