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

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 Tree, whose objects are " functioned trees", which are specially designed to be incorporated into preforms. I show that Tree is isomorphic to the full subcategory of Grph whose objects are converging arborescences. Then the paper defines the category 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 NCP isomorphisms, define a forgetful functor from NCP to Tree, and show that Tree is equivalent to the full subcategory of 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 NCP morphisms and isomorphisms.