Symmetric Decomposition of Asymmetric Games

TL;DR

This paper presents a novel theoretical framework that decomposes asymmetric two-population games into symmetric single-population games, simplifying analysis of Nash equilibria and evolutionary dynamics.

Contribution

It introduces a symmetric decomposition method for asymmetric games, revealing formal relationships between asymmetric and symmetric game equilibria, aiding analysis.

Findings

Decomposition reduces complexity of asymmetric game analysis.

Nash equilibria of asymmetric games relate to those of symmetric counterparts.

Formal relationships facilitate understanding of evolutionary dynamics.

Abstract

We introduce new theoretical insights into two-population asymmetric games allowing for an elegant symmetric decomposition into two single population symmetric games. Specifically, we show how an asymmetric bimatrix game (A,B) can be decomposed into its symmetric counterparts by envisioning and investigating the payoff tables (A and B) that constitute the asymmetric game, as two independent, single population, symmetric games. We reveal several surprising formal relationships between an asymmetric two-population game and its symmetric single population counterparts, which facilitate a convenient analysis of the original asymmetric game due to the dimensionality reduction of the decomposition. The main finding reveals that if (x,y) is a Nash equilibrium of an asymmetric game (A,B), this implies that y is a Nash equilibrium of the symmetric counterpart game determined by payoff table A, and x is a Nash equilibrium of the symmetric counterpart game determined by payoff table B. Also the reverse holds and combinations of Nash equilibria of the counterpart games form Nash equilibria of the asymmetric game. We illustrate how these formal relationships aid in identifying and analysing the Nash structure of asymmetric games, by examining the evolutionary dynamics of the simpler counterpart games in several canonical examples.