Web Appendix for Critical Types

In this web appendix we demonstrate the construction of a regular type. The idea is to construct hierarchies out of “stacking” finite truncations of other hierarchies on top of each other. We illustrate with a simple example with 2 players. Let p̄ = (p1, p2, . . .) be an infinite sequence of numbers pn ∈ [0, 1] . Consider a hierarchy u p̄ according to which player i believes that the state of nature is equal to ω = +1 with probability p1 and that player −i believes that the state is equal to ω = +1 with probability p2 and that −i believes that i believes that the state is equal to ω = +1 with probability p3 and so on. Now, take two sequences p̄ and p̄′. For each k, we can construct a hierarchy in which the first k-order beliefs are given by u p̄, and after that, the beliefs are described by u p̄′ : formally, let u p̄ ∧k u p̄′ = u(p1,p2,...,pk,p′ 1,p 2,...) We say that the hierarchy u p̄ ∧k u p̄′ is constructed by “stacking” u p̄′ on the k-th level of u p̄. We can extend the “stacking” operation to general hierarchies u and u′. Additionally, we can consider sequences of hierarchies u1,u2,. . . and levels k1,k2,. . . and construct a new hierarchy by first “stacking” u2 on the k1-level of u1, then add u3 at the (k1 + k2)-level of the previously constructed hierarchy, and so on. We use this operation to construct a hierarchy without non-trivial common beliefs. To see how it works, start with hierarchy u(1,1,...). This hierarchy exhibits a common belief in a non-trivial event, namely common knowledge that state is equal to ω = +1. We can eliminate such common belief by “stacking” hierarchy u(0,0,...) on the top of u(1,1,...). So constructed hierarchy û = u(1,1,...,1,0,0,...) exhibits common belief in the event that “both players either know (i.e., belief with full probability) that the state is equal to ω + 1 or they know that the state is equal to ω = −1.” The latter common belief can be eliminated by “stacking” the hierarchy u( 2 , 1 2 ,...) on the top of ∗Department of Economics, Northwestern University, [email protected]. †Department of Economics, University of Texas at Austin, [email protected]