Information Design: Unobserved types L21 Bergmann and Morris 2017 Kolotilin, Li Mylovanow Zapelchelynuk 2015

Question So far: designer observes (and hence condition on) private information What if designer does not have access to such information? Today: design without private players information Two scenarios Designer can elicit private information Designer cannot elicit information

Basic Game Sender faces many Receives who ``play a game ’’ among each other A game: I players (receivers) Finite action space Type space: , prior Preferences: ``Prior’’ information structure Finite set of signals , Signal distribution We call it a basic game (of incomplete information), Communication rule

Obedience BNE in game (G,C) induces (expected) decision rule, Let be set of all BN decision rules in (G,C) Characterization: iff it satisfies obedience condition for G: Recommended action needs to be interim optimal Optimization domain for a designer is a polytope

Elicitation (Private Persuasion) What if designer observes state but not types Suppose designer can privately ask players about their types Communication rule and hence BNE decision rules are contingent on Incentives to tell the truth What is the adequate incentive compatibility condition? General mechanism design (Myerson 1991) Bayesian collective choice problems ( traditional MD) Bayesian games with communication (here)

Standard truth telling Decision rule satisfies truth telling condition if In ``basic’’ mechanism design, actions are controlled by the designer Suppose decision rule satisfies obedience and truthtelling Is attainable in BNE with communication?

Incentive compatibility Player simultaneously makes two choices Obedience + truthtelling: necessary but not sufficient for attainability We need immunity to double deviations Incentive compatibility condition (Myerson 91, Section 6.3) Let be a deviation function

Incentive compatibility (Myerson 91) D: Decision rule is incentive compatible iff it is immune to double deviations i.e., In binary state and action model obedience and truthtelling is sufficient for IC We show such example next

Example (Elicitation) Firm example Omniscient designer observes and hence conditions on Two independent problems with different ``posterior-priors’’

Obedience constraints Suppose Signal Signal Integrating out types gives the set of all

BCE BCE set for and

Additional truthtelling constraints Truthtelling constraint for signal g Truthtelling constraint for signal b

Obedience and truthtelling Implications for the feasible set Set of attainable BCE is strictly smaller than for omniscient designer More generally such set is weakly smaller and typically strictly smaller

No Elicitation (Public Persuasion) What if designer cannot privately ask players about their types? One option: recommendation Dramatically reduces set of attainable Can we do better than that? Let and Strategy recommendation This way recommendations can condition on agents types

Which are we loosing relative to elicitation? Two further restrictions on BCE set from no elicitation - public feasibility (mechanical constraint) -alternative (stronger) obedience constraint In the example none of the two restricts the set of attainable BCE No elucidation condition is vacuous for binary model

Public feasibility Heuristic argument Elicitation: arbitrary s.t. incentive compatibility constraints Public feasibility: payer’s strategy cannot vary in others type Let . Rule: both firms invest iff firm’s one signal is Is incentive compatible Is not publicly feasible In one player game public feasibility is vacuous

Obedience Tighter obedience constraint The player receives recommendation for any type Example Observation: public feasibility is vacuous

What are we loosing relative to elicitation

Two General results Set of attainable BCE with elicitation weakly larger than with no elicitation Binary model the two methods are equivalent For each departure example with strict inclusion