The Category of Node-and-Choice Preforms for Extensive-Form Games

Abstract

It would be useful to have a category of extensive-form games whose isomorphisms specify equivalences between games. Since working with entire games is too large a project for a single paper, I begin here with preforms, where a “preform” is a rooted tree together with choices and information sets. In particular, this paper first defines the category $\mathbf {Tree}$, whose objects are “functioned trees”, which are specially designed to be incorporated into preforms. I show that $\mathbf {Tree}$ is isomorphic to the full subcategory of $\mathbf {Grph}$ whose objects are converging arborescences. Then the paper defines the category $\mathbf {NCP}$, whose objects are “node-and-choice preforms”, each of which consists of a node set, a choice set, and an operator mapping node-choice pairs to nodes. I characterize the $\mathbf {NCP}$ isomorphisms, define a forgetful functor from $\mathbf {NCP}$ to $\mathbf {Tree}$, and show that $\mathbf {Tree}$ is equivalent to the full subcategory of $\mathbf {NCP}$ whose objects are perfect-information preforms. The paper also shows that many game-theoretic entities can be derived from preforms, and that these entities are well-behaved with respect to $\mathbf {NCP}$ morphisms and isomorphisms.