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

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)

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

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

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

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

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

Obedience constraints obedience condition Identical Slope

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

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

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

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

Optimal decision rule Omniscient S Implementation

INTEGRATION OVER SIGNALS Unconditional probabilities

BCE Comparative statics

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’’

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)

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 relax obedience constraint State G State B Optimal rule Public signals

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

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