# Rank Reduction in Bimatrix Games

## TL;DR

This paper introduces a method to reduce the rank of bimatrix games without altering their equilibria, using matrix pencil theory and Wedderburn's formula, with implications for game complexity and computation.

## Contribution

It presents a novel rank reduction technique for bimatrix games that preserves equilibria, leveraging matrix pencil theory and providing constructive proofs for generic cases.

## Key findings

- Rank can be reduced by 1 in generic square games.
- Rank can be reduced by 2 in generic rectangular games.
- The method preserves the game's equilibrium.

## Abstract

The rank of a bimatrix game is defined as the rank of the sum of the payoff matrices of the two players. The rank of a game is known to impact both the most suitable computation methods for determining a solution and the expressive power of the game. Under certain conditions on the payoff matrices, we devise a method that reduces the rank of the game without changing the equilibrium of the game. We leverage matrix pencil theory and the Wedderburn rank reduction formula to arrive at our results. We also present a constructive proof of the fact that in a generic square game, the rank of the game can be reduced by 1, and in generic rectangular game, the rank of the game can be reduced by 2 under certain assumptions.