The class of states of the world as an -groupoid: identifying states of the world and hierarchies of belief

Abstract

A natural problem in the methodology of Economics is how to model interactions in which the decisions made by agents depend on the beliefs held about both objective aspects of the relevant situation and the beliefs ascribed to other agents. There are two ways of representing such contexts. One is by means of a hierarchy of beliefs, the other as a state of the world. These two can be seen, from a Systems-theoretical perspective, as representing streams generated by a process and its fixed points, respectively. These two representations are in fact equivalent. The finer question we address here is whether they are identical. Resorting to a type-theoretical syntax, we obtain a unified interpretation in which both entities give rise to ∞-groupoids. Voevodsky´s univalent axiom then ensures their identity in small types universes.