Bayesian Persuasion cn L18 Kamienica and Genzkow (AER 2011)

Basic Bayesian Persuasion Two agents: Sender (S) and Receiver (R) Type space Action space . Message space Preferences S sends message, , R responds with an action S ex ante commits to Solution concept Important tool

Concavification Aumann and Mashler (1995) set of all Bayes plausible distributions of posteriors Proof (straightforward) Implication: Concavification of value function

Example 2: KG example Story: prosecutor S and judge R Binary model Preferences For beliefs R optimal choice Expected S utility given common beliefs (no persuasion)

Obfuscation of information KG example: function is neither concave nor concave

Lessons from the example Partial transmission may be optimal (obfuscation of information) Information transmission iff default action (beliefs ) reveal low type, (worst) action is optimal ex post (beliefs ) does not reveal high type, No action is optimal ex post Actions are indifferent KG generalize these observations

Transmission of information (necessity) Transmission of information=benefits from persuasion Fix prior and let D: S would share beliefs with R if Example: P: S has incentives to transmit information only if exist that S would share Remark: verification of the condition requires graph of

Heuristic proof

Transmission of information (sufficiency) D: R preferences are discrete at if Remarks: preferences are discrete, generically and continuous no is optimal for posterior in the neighborhood of P: Suppose at R preferences are discrete and there is that S would share. Then S benefits from transmission of information

Heuristic proof

Ex post optimality of worst action In KG example for message (posterior ) r optimal ex post D: is worst action if D: Action is ex post optimal for if Suppose one of optimal posteriors leads to worst action, P: Action is ex post optimal at Proof Does worst action always exist? Indifference condition generalizes under stronger conditions

Expected state model hard to work with when Suppose Expected S utility Exists Let be a concave closure of Can be used to make predictions

Expected state model For any prior is not a value function of the persuasion problem but its upper bound P: S benefits from transmission of information iff Not helpful in finding optimal message strategy

Summary Value function in the persuasion problem is concave A geometric tool to find optimal message strategy in 2 state example Some results characterizing persuasion Not clear if they this help to find optimal mechanism in a general model Limitations: settings with more than two states? Settings with two or more agents? Solutions: Settings with two or more agents 2-stage approach