A categorical representation of games

TL;DR

This paper introduces a categorical framework for representing strategic games using multi-graph structures, generalizing Nash equilibria and defining subcategories that preserve equilibria, with applications demonstrated through examples.

Contribution

It develops a novel categorical approach to model and analyze games, extending the concept of Nash equilibrium within this framework.

Findings

The framework is complete and cocomplete.

Nash equilibrium is generalized in the categorical setting.

Examples demonstrate the framework's expressivity and utility.

Abstract

Strategic games admit a multi-graph representation, in which two kinds of relations, accessibility, and preferences, are used to describe how the players compare the possible outcomes. A category of games with a fixed set of players $\mathbf{Gam}_I$ is built from this representation, and a more general category $\mathbf{Gam}$ is defined with games having different sets of players, both being complete and cocomplete. The notion of Nash equilibrium can be generalized in this context. We then introduce two subcategories of $\mathbf{Gam}$, $\mathbf{NE}$ and $\mathbf{Gam}^{NE}$ in which the morphisms are equilibria-preserving. We illustrate the expressivity and usefulness of this framework with some examples.