1. Information Design: A unified Perspective
L19
Bergmann and Morris 2017

2. Schedule of presentations
April 18 : Yue Li
April 20: Quiran Shao
April 25: Alaxander Clark
April 27: Gabriel Martinez
May 2: Ziwei Wang
May 4: Yixi Yang

3. Plan Today: General Information Design Framework
Revelation principle and BCE
Two step procedure
KG example reconsidered
Next lecture: we modify the KG example
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
Type 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 (Literature assumes 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 messages to each player (here called signal)
Communication rule C
Remark: Without knowledge the designer essentially becomes 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
We need some 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 review)
For any designer choice C nature selects adverse equilibrium
- Latter: Robust information design
- Carroll (2016) , Goldstein and Huang 2016, Inostroza and Pavan 2017

9. Bayes Correlated Equilibrium (BCE)
D: Decision rule is a BCE in the basic game if for any
Let be the set of all BSE in game G
Revelation principle (Bergmann Morris 2016)
T1: A decision rule is BCE in a basic game if and only if it is a BNE in the augmented game, i.e., for some
Proof

10. 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
Find BCE that maximizes S preferences on this set
Find the corresponding communication rule
Benefits:
Problem has a structure of linear programming
Optimal message strategy is well defined
Comparative statics of BCE
Derivation of equilibrium without concavification
Max min problem, set of feasible decision rules is smaller than

11. Plan
We apply these results to characterize equilibrium in a sequence of examples
Today: One with no prior information (KG example)
Next lecture modifications of this example

12. 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)

13. 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 ?

14. 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

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

16. Step 2: Optimal message strategy
S maximizes the expected probability of investment
Optimal choice
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

17. 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)
Design with private information