A categorical representation of games

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 ${\mathbf{G}\mathbf{a}\mathbf{m}}_I$ is built from this representation, and a more general category $\mathbf{G}\mathbf{a}\mathbf{m}$ 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 $\mathbf{G}\mathbf{a}\mathbf{m}$, $\mathbf{N}\mathbf{E}$ and ${\mathbf{G}\mathbf{a}\mathbf{m}}^{NE}$ in which the morphisms are equilibria-preserving. We illustrate the expressivity and usefulness of this framework with some examples.