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

2. 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

3. 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

4. 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

5. 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)

6. 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?

7. 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

8. 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

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

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

11. BCE
BCE set for and

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

13. 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

14. 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

15. 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

16. 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

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

18. What are we loosing relative to elicitation

19. 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