Bayesian Persuasion Online

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 two important observations about this game. The first observation is that, as long as Receiver knows which π Sender chose, it is not important to assume that she directly observes the realization s from π. Specifically, if we instead assume that Sender observes s privately and then sends a verifiable message about s, the same information is revealed in equilibrium and we obtain the same outcomes. The second observation is that Sender's gain from persuasion is weakly greater in the game that we study than in any other communication game, including Spence (1973), Crawford and Sobel (1982), or Milgrom (1981). In this section of the Online Appendix we provide a formal statement and proof of these two claims. 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 the next section. 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 3) and are used solely for notational convenience. A persuasion mechanism defines a game. The timing is …