On Skew-Symmetric Games

TL;DR

This paper introduces skew-symmetric games as a natural subspace of finite games, explores their properties, and presents a symmetry-based decomposition of finite games into orthogonal subspaces.

Contribution

It defines skew-symmetric games within the vector space of finite games and proposes a decomposition into symmetric, skew-symmetric, and asymmetric subspaces.

Findings

Skew-symmetric games form an orthogonal complement of symmetric games for two players.

A linear representation for verifying skew-symmetry in finite games is provided.

Finite games can be decomposed into three orthogonal subspaces: symmetric, skew-symmetric, and asymmetric.

Abstract

By resorting to the vector space structure of finite games, skew-symmetric games (SSGs) are proposed and investigated as a natural subspace of finite games. First of all, for two player games, it is shown that the skew-symmetric games form an orthogonal complement of the symmetric games. Then for a general SSG its linear representation is given, which can be used to verify whether a finite game is skew-symmetric. Furthermore, some properties of SSGs are also obtained in the light of its vector subspace structure. Finally, a symmetry-based decomposition of finite games is proposed, which consists of three mutually orthogonal subspaces: symmetric subspace, skew-symmetric subspace and asymmetric subspace. An illustrative example is presented to demonstrate this decomposition.