Bayesian Persuasion Online Appendix

1 Persuasion mechanisms In this paper we study a particular game where Sender chooses a signal π whose realization is observed by Receiver who then takes her action. In Subsection 2.3 we made an observation that Sender's gain from persuasion is weakly greater in this game than in any other communication game. In this section of the Online Appendix we provide a formal statement and proof of this claim. To do so, we introduce the notion of a persuasion mechanism. As before, Receiver has a continuous utility function u (a, ω) that depends on her action a ∈ A and the state of the world ω ∈ Ω. Sender has a continuous utility function v (a, ω) that depends on Receiver's action and the state of the world. Sender and Receiver share a prior µ 0 ∈ int (∆ (Ω)). Let a * (µ) denote the set of actions that maximize Receiver's expected utility given her belief is µ. We assume that there are at least two actions in A and that for any action a there exists a µ s.t. a * (µ) = {a}. The action space A is compact and the state space Ω is finite. We will relax the latter assumption in Section 3. A persuasion mechanism (π, c) is a combination of a signal and a message technology. Sender's private signal π consists of a finite realization space S and a family of distributions {π (·|ω)} ω∈Ω over S. A message technology c consists of a finite message space M and a family of functions c (·|s) : M → R + ; c (m|s) denotes the cost to Sender of sending message m after receiving signal realization s. 1 The assumptions that S and M are finite are without loss of generality (cf. Proposition 4) and are used solely for notational convenience. A persuasion mechanism defines a game. The timing is as follows. First, nature selects ω from Ω according to µ 0. Neither Sender nor Receiver observe nature's move. Then, Sender privately observes a realization s ∈ S from π (·|ω) and chooses a message m ∈ M. Finally, Receiver observes m and chooses an action a ∈ A. Sender's payoff is v (a, ω)−c (m|s) and Receiver's payoff is u (a, ω). We represent the Sender's and Receiver's (possibly stochastic) strategies by σ and ρ, respectively. We use µ (ω|m) to denote …