Harsanyi Type Spaces and Final Coalgebras Constructed from Satisfied Theories

This paper connects coalgebra with a long discussion in the foundations of game theory on the modeling of type spaces. We argue that type spaces are coalgebras, that universal type spaces are final coalgebras, and that the modal logics already proposed in the economic theory literature are closely related to those in recent work in coalgebraic modal logic. In the other direction, the categories of interest in this work are usually measurable spaces or compact (Hausdorff) topological spaces. A coalgebraic version of the construction of the universal type space due to Heifetz and Samet [4] is generalized for some functors in those categories. Since the concrete categories of interest have not been explored so deeply in the coalgebra literature, we have some new results. We show that every functor on the category of measurable spaces built from constant functors, products, coproducts, and the probability measure space functor has a final coalgebra. Moreover, we construct this final coalgebra from the relevant version of coalgebraic modal logic. Specifically, we consider the set of theories of points in all coalgebras and endow this set with a measurable and coalgebra structure.