Prior information Bergmann and Morris 2017 L10 Information design Sender faces many Receives who play a game among each other A basic game I players receivers Finite action space ID: 713134
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Slide1
Information Design: A unified Perspective Prior information
Bergmann and Morris 2017
L10Slide2
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 structureFinite set of signals , Signal distribution S: supplements prior information with messages (communication rule)Slide3
Information design
Ex post preferences of S
In Bayesian game a BNE is sufficiently summarized by decision ruleLetInformation design problemThe problem seems extremely hardSlide4
Bayes Correlated equilibrium (BCE)
Decision rule. Is BCE if
Let be a collection of BCE decision rules in game Revelation principleImplication: problem equivalent toTwo steps procedure (linear programming) - find set - find best on Slide5
Prior beliefs
Binary state space , prior
One receiver ( interpreted AS firm) Binary action space R payoff default actionDesigner S observes , commits to message structureS maximizes probabilities of investmentSlide6
Decision rule SPACE
Decision rule
2 dimensional manifoldS preferences over MRSSlide7
Asymmetric prior
Prior distribution:
Given , ex ante distribution over states and actionsBCE is given by two linear obedience conditions Slide8
Obedience constraints
obedience condition
obedience conditionIdentical SlopeSlide9
Set of BCE equilibria
For
For Comparative static with respect to extreme prior beliefsUninformative beliefs Slide10
Optimal decision rule
For
For Optimal choicesExtreme points of a polytopeImplementation?Slide11
Player with prior information
Prior
R receives signal (message, type) according to distributionSignals split prior into 2 “interim-posteriors”Experiment is more informative than. if Slide12
Problem of Omniscient S
Omniscient designer observes (and conditions on) signal
Two independent problems with different ``interim posteriors’’Slide13
Optimal decision rule Omniscient S
Optimal decision rule
ImplementationSlide14
INTEGRATION OVER SIGNALS
Unconditional probabilitiesSlide15
BCE Comparative staticsSlide16
General lessons
Example
More informative initial signal makes obedience constraints tighterBCE set shrinking with higher qSingle 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’’ orderEquivalence of the thee orders (Bergmann Morris 2013)Bergmann Morris 2016 generalizes this to games with many playersDefine ``sufficiency’’ (statistical) order on information structuresShow equivalence with ``more constrained order’’ Slide17
Next lecture
Strategic complementarities among many players
Set of BCEOptimal choiceInstrumental preferences over correlations Private vs public signalsElicitation of private information (non-omniscient designer)Slide18
Two Firms (Many Players)
Objective: sum of investment probabilities for both firms
Designer has no intrinsic preferences for correlationIf no strategic interactions then optimization firm by firmFirm 1 payoff with strategic complementaritiesStrategic complements (substitutes) if ( )Slide19
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 RestrictionSlide20
Obedience (BCE) constraints
Obedience of ``invest’’ recommendation
With obedience condition for “do not invest” is redundantSlide21
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 complementarityOptimal points?Slide22
Optimal decision rule (for small )
Observation: Correlations relax obedience constraint
State GState BOptimal rule Public signalsSlide23
Optimal decision rule (for small )
Observation: Correlations. tightens obedience constraint
State GAssume State BOptimal rule Private signalSlide24
General lessons
No intrinsic preference over correlation (sum of probabilities)
Correlation: instrument to relax obedience constraint Strategic complements (substitutes) positive (negative) correlationPublic vs private signalsPapers that use this this mechanismOne sided complementarity Madhavet Perego Taneva 2016Two sided complementarity Bergmann and Morris 2016Strategic substitutes (Cournot) Bergmann and Morris 2013Intrinsic motives (objective: at least one firm invests)Ely 2017 (private signals) Bergmann Heumann and Morris 2016Arieli and Babicenko 2016