1. Information Design: A unified PerspectiveL9
Bergmann and Morris 2017

2. Schedule of presentations
December 12: Srinivas Kartik and Weelden 2017
December 7: Yuteng, Che and Kartik 2007
Chen: Glazer Rubinstein 2004
Shuhei: Ely 2016

3. Plan Today: General Information Design Problem
Revelation principle and BCE
Two step procedure
KG example reconsidered
Next lecture: prior information, multiple receivers
By this we illustrate the key substantive findings in the information design

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

5. Designer’s instruments
Designer observes (3 variants)
Payoff state and types for all
Payoff state only, can elicit types
Payoff state only, cannot elicit types
Designer provides `supplemental’’ information to players
Sends message to each player (here called signal)
Choice: Communication rule C
Remark: Without knowledge the designer essentially is a mediator from the literature on correlated equilibrium, Forges (1993, 2006)

6. Designer’s preferences over C
is an augmented incomplete information game
Strategy of each player
Profile is a BNE if …
Each BNE induces some decision rule
Equilibrium correspondence

7. Designer’s preferences over C
Ex post utility implies ex ante preferences over decision rules
Complication: Equilibrium correspondence is not a function
does not define preferences over message strategies
We need ``selection’’ criterion
Two alternative approaches

8. Designer’s preferences over C
Designer choses as well as
- Objective of a designer
- Most papers (all discussed in this lecture)
For any choice C nature selects adverse equilibrium
- Robust (adversarial) information design
- Carroll (2016) , Goldstein and Huang 2016, Inostroza and Pavan 2017

9. Correlated Equilibrium Aumann 1974
Consider complete information (coordination) game
Nash equilibria
Decision rules induced by (mixed Nash) equilibria
Restriction
Generalization: Correlated equilibrium
Set of CE is a polytope

10. Bayes Correlated Equilibrium (BCE)
D: Decision rule is a BCE in the basic game if for any
Obedience conditions
Let be the set of all BCE in game G
Revelation principle (Bergmann Morris 2016)
T: A decision rule is BCE in a basic game if and only if it is a BNE in the augmented game, i.e.,
Proof

11. Implications Max max problem equivalent to choosing preferred BCE in
Optimal communication rule can be found in two-step procedure
Characterize the set of all BCE (obedience conditions)
Find BCE that maximizes S preferences on this set
Find the corresponding communication rule
Benefits:
Linear programing: finite set of extreme points
Optimal message strategy is well defined
Comparative statics of BCE in abstraction of R preferences
Derivation of equilibrium without concavification
Max min problem, set of feasible decision rules is smaller than

12. Plan
We apply these observations to characterize equilibrium in several examples
Today: One R with no prior information (KG example)
Next lecture: modifications of this example
Effects of private information
Effects of multiple receivers

13. KG reconsidered Binary state space , equally likely states
One player (Receiver) interpreted as firm
Binary action space
Payoffs (assume )
No ``prior’’ information about a state
Designer S observes , commits to message structure
Objective: maximizes sum of probabilities of investment:
This is a KG example (modulo changes in labeling) when

14. Decision rule Decision rule
In the binary model a decision rule is summarized by
Geometric representation:
Interpretation: Stochastic recommendation from a designer
Which of the decision rules can be implemented with some ?

15. Step 1: Set of BCE
Given , ex ante distribution over states and actions
Recommendation ``invest’’ is followed if
Recommendation not invest is followed if
The latter condition is redundant

16. BCE Set
Polytope
How to implement extreme points of ?

17. Step 2: Optimal information design
S maximizes the expected probability of investment
Optimal choice
Indifference curves

18. Step 2: Optimal information design
S maximizes the expected probability of investment
Message strategy
Lessons (as in KM):
Obfuscation of information
``not invest’’ is ex post optimal given bad message
``not invest’’ and ``invest’’ are equally attractive given good message

19. Next lecture Next lecture:
One player with prior information (comparative statics)
Two players, no prior information (public versus private signals)
Two players, prior information (generalized comparative statics)
Left out: Design with private information
Elicitation of information
No elicitation