Information Design Unobserved types L 21 Bergmann and

Information Design: Unobserved types L 21 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: - Preferences: , prior • ``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 truthtelling Decision rule satisfies truthtelling 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 • 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