# Decomposition of games: some strategic considerations

## TL;DR

This paper explores how to decompose strategically equivalent games into components, emphasizing the importance of defining specific classes of decompositions to maintain consistency under game transformations.

## Contribution

It introduces the concept of strategic equivalence in game decomposition and proposes classes of decompositions that preserve properties under transformations.

## Key findings

- Decomposition must account for strategic equivalence.
- Classes of decompositions ensure commutativity with game transformations.
- Highlights the need for tailored decomposition methods for equivalent games.

## Abstract

Candogan et al. (2011) provide an orthogonal direct-sum decomposition of finite games into potential, harmonic and nonstrategic components. In this paper we study the issue of decomposing games that are strategically equivalent from a game-theoretical point of view, for instance games obtained via transformations such as duplications of strategies or positive affine mappings of of payoffs. We show the need to define classes of decompositions to achieve commutativity of game transformations and decompositions.