A Note On Orthogonal Decomposition of Finite Games

TL;DR

This paper investigates how different inner products affect the orthogonal decomposition of finite games, revealing conditions under which a unified decomposition approach is possible.

Contribution

It identifies the compatible condition necessary for a common decomposition induced by standard and weighted inner products in finite games.

Findings

Only compatible inner products induce a common decomposition.

Analyzed potential, zero-sum, and symmetry-based decompositions.

Established conditions for unified orthogonal decomposition.

Abstract

Various decomposition of finite games have been proposed. The inner product of vectors plays a key role in the decomposition of finite games. This paper considers the effect of different inner products on the orthogonal decomposition of finite games. We find that only when the compatible condition is satisfied, a common decomposition can be induced by the standard inner product and the weighted inner product. To explain the result, we studied the existing decompositions, including potential based decomposition, zero-sum based decomposition, and symmetry based decomposition.