Information Design: A unified Perspective cn
L20
Bergmann and Morris 2017
Today
• Alternative priors (comparative statics)
• Effect of players private information on
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set of BCE equilibria
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Optimal choice
• Strategic complementarities among many players
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Set of BCE
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Optimal choice
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Instrumental preferences over correlations
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Private vs public signals
In the previous lecture
• Basic game of incomplete information
• Communication rule C
• Decision rule
-
Let
be a collection of BNE decision rules in game
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Let
be a collection of BCE decision rules in game
• Characterization:
• Implication: design problem
• Two step procedure (linear programming)
equivalent to
KG reconsidered
• Binary state space
, equally likely states
• One player interpreted (firm)
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Binary action space
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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)
Decision rule
• Decision rule (2 dimensional manifold)
• BCE decision rules
• Optimal decision rule
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?
Obedience constraints
• Condition 1
• Condition 2
• Threshold
Set of BCE equilibria
• For
For
• Full information transmission and no information
• With highly informative prior
• With
lower bound on
Players ``prior’’ information
• Firm receives signal (type) correct with probability
• Omniscient designer observes (and hence conditions on)
• Two independent problems with different ``posterior-priors’’
• Supose
• Signal
Decision rule
Signal
• Integrating out types gives the set of all
Obedience constraints
• Suppose
• Signal
Signal
• Integrating out types gives the set of all
Questions
• BCE set for different precisions of prior signal
• This comparative statics extends to an example with many players
General lessons
• Example
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More informative initial signal makes obedience constraints tighter
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BCE set shrinking with higher q
• Single agent information structure is an experiment (Blackwell sense)
• Partial (more informative) orders on set of signals
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Blackwell ``sufficiency’’ (statistical) order
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Blackwell ``more valuable’’ order
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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
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Define ``sufficiency’’ (statistical) order on information structures
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Show equivalence with ``more constrained order’’
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
(
)
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
Obedience (BCE) constraints
• Obedience of ``invest’’ recommendation
• With
obedience condition for “do not invest” is redundant
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?
Optimal decision rule (for small
• Observation: Correlations
• State G
• State B
• Optimal rule
• Public signals
)
relax obedience constraint
Optimal decision rule (for small
• Observation: Correlations.
• State G
• Assume
• State B
• Optimal rule
• Private signal
)
tightens obedience constraint
General lessons
• No intrinsic preference over correlation (sum of probabilities)
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Correlation: instrument to relax obedience constraint
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Strategic complements (substitutes)
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Public vs private signals
positive (negative) correlation
• Papers that use this this mechanism
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One sided complementarity Madhavet Perego Taneva 2016
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Two sided complementarity Bergmann and Morris 2016
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Strategic substitutes (Cournot) Bergmann and Morris 2013
• Intrinsic motives (objective: at least one firm invests)
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Ely 2017 (private signals)
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Bergmann Heumann and Morris 2016
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Arieli and Babicenko 2016