Coherent beliefs are not always types

For two interacting agents, we construct a space of nature states S and a coherent Ž . hierarchy of beliefs s-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 s-additive coherent extension over S and the hierarchies of the other agent. Thus, this hierarchy of beliefs w 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 x 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. q 1999 Elsevier Science S.A. All rights reserved.