1. Information Design: A unified Perspective Prior information
L10
Bergmann and Morris 2017

2. Information design
Sender faces many Receives who ``play a game ’’ among each other
A basic game
I players (receivers)
Finite action space
State space: , prior
Preferences:
``Prior’’ information structure
Finite set of signals ,
Signal distribution
S: supplements prior information with messages (communication rule)

3. Information design Ex post preferences of S
In Bayesian game a BNE is sufficiently summarized by decision rule
Let
Information design problem
The problem seems extremely hard

4. Bayes Correlated equilibrium (BCE)
Decision rule Is BCE if
Let be a collection of BCE decision rules in game
Revelation principle
Implication: problem equivalent to
Two steps procedure (linear programming)
- find set
- find best on

5. Prior beliefs Binary state space , prior
One receiver ( interpreted AS firm)
Binary action space
R payoff
default action
Designer S observes , commits to message structure
S maximizes probabilities of investment

6. Decision rule SPACE Decision rule 2 dimensional manifold
S preferences over
MRS

7. Asymmetric prior Prior distribution:
Given , ex ante distribution over states and actions
BCE is given by two linear obedience conditions

8. Obedience constraints
obedience condition
Identical Slope

9. Set of BCE equilibria For For Comparative static with respect to
extreme prior beliefs
Uninformative beliefs

10. Optimal decision rule For For Optimal choices
Extreme points of a polytope
Implementation?

11. Player with prior information
R receives signal (message, type) according to distribution
Signals split prior into 2 “interim-posteriors”
Experiment is more informative than if

12. Problem of Omniscient S
Omniscient designer observes (and conditions on) signal
Two independent problems with different ``interim posteriors’’

13. Optimal decision rule Omniscient S
Implementation

14. INTEGRATION OVER SIGNALS
Unconditional probabilities

15. BCE Comparative statics

16. General lessons Example
More informative initial signal makes obedience constraints tighter
BCE set shrinking with higher q
Single agent information structure is an experiment (Blackwell sense)
Partial (more informative) orders on set of signals
Blackwell ``sufficiency’’ (statistical) order
Blackwell ``more valuable’’ order
Bergmann and Morris ``more incentive constrained’’ order
Equivalence of the thee orders (Bergmann Morris 2013)
Bergmann Morris 2016 generalizes this to games with many players
Define ``sufficiency’’ (statistical) order on information structures
Show equivalence with ``more constrained order’’

17. Next lecture Strategic complementarities among many players Set of BCE
Optimal choice
Instrumental preferences over correlations
Private vs public signals
Elicitation of private information (non-omniscient designer)

18. Two Firms (Many Players)
Objective: sum of investment probabilities for both firms
Designer has no intrinsic preferences for correlation
If no strategic interactions then optimization firm by firm
Firm 1 payoff with strategic complementarities
Strategic complements (substitutes) if ( )

19. Decision rule Decision rule (6 numbers +2 )
Wlog symmetric decision rules (4 numbers, 2 for each state)
is the probability that firm invests regardless of the other firm
Restriction

20. Obedience (BCE) constraints
Obedience of ``invest’’ recommendation
With obedience condition for “do not invest” is redundant

21. BCE set Set of all BCE symmetric equilibria (4 dimensional manifold)
Given by the following inequalities:
Its projections to space is given by
The BCE set is monotonic in degree of complementarity
Optimal points?

22. Optimal decision rule (for small )
Observation: Correlations relax obedience constraint
State G
State B
Optimal rule
Public signals

23. Optimal decision rule (for small )
Observation: Correlations tightens obedience constraint
State G
Assume
State B
Optimal rule
Private signal

24. General lessons
No intrinsic preference over correlation (sum of probabilities)
Correlation: instrument to relax obedience constraint
Strategic complements (substitutes) positive (negative) correlation
Public vs private signals
Papers that use this this mechanism
One sided complementarity Madhavet Perego Taneva 2016
Two sided complementarity Bergmann and Morris 2016
Strategic substitutes (Cournot) Bergmann and Morris 2013
Intrinsic motives (objective: at least one firm invests)
Ely 2017 (private signals)
Bergmann Heumann and Morris 2016
Arieli and Babicenko 2016