Supplement to “Market Signaling with Grades”

C Supplementary Appendix: More than Two Types Let us expand the type space to be {1, ..., N}, N > 2. The receivers’ prior, μ0, is a probability distribution with full support over the type space. When describing beliefs (be they prior, interim or final) we use superscripts to denote the probability assigned to the various types (e.g., μ0 is the probability the prior assigns to type t). We now assume that after observing x and g, the receivers each choose some action, and that all payoff-relevant information (from the sender’s perspective) from a profile of receivers’ actions can be summarized by a scalar a ∈ R+. For example, in equity issuance, a represents the market clearing price of the sale—the values of other bids have no direct bearing on the sender’s payoff. We therefore, continue to represent the sender’s utility, Ut, as a function of two arguments, with the first argument remaining x and the second argument now a instead of μf . Ut remains differentiable, with Ut,2 > 0. Let a(μf , x) denote the unique value of a that results when each receiver is playing a best response in the continuation game that follows the sender’s choice of x and the common final belief μf . For all x and t, a ∗ is differentiable in x and μt. The sender’s utility functions continues to satisfy A.1–A.2.1 To simplify analysis, we replace A.3 and A.4 with the following A.3′ For any t and fixed μf , Ut(x, a (μf , x)) is strictly decreasing in x. A.4′ For any x, a and t′ 6= t, Ut,2(x, a) = Ut′,2(x, a). A.3′ says that the signaling action is wasteful. This implies that, in equilibrium, the sender never directly gains from signaling and that xt = 0 for all t. A.4 ′ implies that types do not differ in their risk preference regarding lotteries over values of a.2 Finally, in line with our applications, we assume that