Coherent beliefs are not always types

Abstract For two interacting agents, we construct a space of nature states S and a coherent hierarchy of beliefs ( σ -additive probability measures) of one agent about S , about S and the beliefs of the other agent about S , and so on—a hierarchy that has no σ -additive coherent extension over S and the hierarchies of the other agent. Thus, this hierarchy of beliefs cannot be the description of the beliefs of some type in some Harsanyi [Harsanyi, J.C., 1967–1968. Games with incomplete information played by Bayesian players, parts I, II, and III. Man. Sc. 14, 159–182, 320–334, 486–502] type space. Therefore, the space C of coherent hierarchies over S properly contains the universal space T * of `all possible types' over S . We show how to extract T * out of C in a transfinite process.