1. Information Design: A unified Perspective cn
L20
Bergmann and Morris 2017

2. Today Alternative priors (comparative statics)
Effect of players private information on
set of BCE equilibria
Optimal choice
Strategic complementarities among many players
Set of BCE
Instrumental preferences over correlations
Private vs public signals

3. In the previous lecture
Basic game of incomplete information
Communication rule C
Decision rule
Let be a collection of BNE decision rules in game
Let be a collection of BCE decision rules in game
Characterization:
Implication: design problem equivalent to
Two step procedure (linear programming)

4. KG reconsidered Binary state space , equally likely states
One player interpreted (firm)
Binary action space
Payoffs (assume x= 0.55)
No ``prior’’ information about a state
Designer S observes , commits to message structure
Objective: max sum of probabilities of investment:
This is KG example (modulo changes in labeling)

5. Decision rule Decision rule (2 dimensional manifold)
BCE decision rules
Optimal decision rule

6. Asymmetric prior Prior distribution:
Given , ex ante distribution over states and actions
BCE is given by two linear obedience conditions
Which of the two ``obedience’’ conditions is binding?

7. Obedience constraints
Condition 1
Condition 2
Threshold

8. Set of BCE equilibria For For
Full information transmission and no information
With highly informative prior
With lower bound on

9. Players ``prior’’ information
Firm receives signal (type) correct with probability
Omniscient designer observes (and hence conditions on)
Two independent problems with different ``posterior-priors’’

10. Decision rule Supose Signal Signal
Integrating out types gives the set of all

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

12. Questions BCE set for different precisions of prior signal
This comparative statics extends to an example with many players

13. 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’’

14. 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 ( )

15. 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

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

17. 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?

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

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

20. 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