Information Design: A unified Perspective L9 Bergmann and Morris 2017

Schedule of presentations December 12: Srinivas Kartik and Weelden 2017 December 7: Yuteng, Che and Kartik 2007 Chen: Glazer Rubinstein 2004 Shuhei: Ely 2016

Plan Today: General Information Design Problem Revelation principle and BCE Two step procedure KG example reconsidered Next lecture: prior information, multiple receivers By this we illustrate the key substantive findings in the information design

Basic Game Sender faces many Receives who ``play a game ’’ among each other A game: I players (receivers) Finite action space State space: , prior Preferences: ``Prior’’ information structure Finite set of signals , Signal distribution We call it a basic game (of incomplete information),

Designer’s instruments Designer observes (3 variants) Payoff state and types for all Payoff state only, can elicit types Payoff state only, cannot elicit types Designer provides `supplemental’’ information to players Sends message to each player (here called signal) Choice: Communication rule C Remark: Without knowledge the designer essentially is a mediator from the literature on correlated equilibrium, Forges (1993, 2006)

Designer’s preferences over C is an augmented incomplete information game Strategy of each player Profile is a BNE if … Each BNE induces some decision rule Equilibrium correspondence

Designer’s preferences over C Ex post utility. implies ex ante preferences over decision rules Complication: Equilibrium correspondence is not a function does not define preferences over message strategies We need ``selection’’ criterion Two alternative approaches

Designer’s preferences over C Designer choses as well as - Objective of a designer - Most papers (all discussed in this lecture) For any choice C nature selects adverse equilibrium - Robust (adversarial) information design - Carroll (2016) , Goldstein and Huang 2016, Inostroza and Pavan 2017

Correlated Equilibrium Aumann 1974 Consider complete information (coordination) game Nash equilibria Decision rules induced by (mixed Nash) equilibria Restriction Generalization: Correlated equilibrium Set of CE is a polytope

Bayes Correlated Equilibrium (BCE) D: Decision rule is a BCE in the basic game if for any Obedience conditions Let be the set of all BCE in game G Revelation principle (Bergmann Morris 2016) T: A decision rule is BCE in a basic game if and only if it is a BNE in the augmented game, i.e., Proof

Implications Max max problem equivalent to choosing preferred BCE in Optimal communication rule can be found in two-step procedure Characterize the set of all BCE (obedience conditions) Find BCE that maximizes S preferences on this set Find the corresponding communication rule Benefits: Linear programing: finite set of extreme points Optimal message strategy is well defined Comparative statics of BCE in abstraction of R preferences Derivation of equilibrium without concavification Max min problem, set of feasible decision rules is smaller than

Plan We apply these observations to characterize equilibrium in several examples Today: One R with no prior information (KG example) Next lecture: modifications of this example Effects of private information Effects of multiple receivers

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

Decision rule Decision rule In the binary model a decision rule is summarized by Geometric representation: Interpretation: Stochastic recommendation from a designer Which of the decision rules can be implemented with some ?

Step 1: Set of BCE Given , ex ante distribution over states and actions Recommendation ``invest’’ is followed if Recommendation not invest is followed if The latter condition is redundant

BCE Set Polytope How to implement extreme points of. ?

Step 2: Optimal information design S maximizes the expected probability of investment Optimal choice Indifference curves

Step 2: Optimal information design S maximizes the expected probability of investment Message strategy Lessons (as in KM): Obfuscation of information ``not invest’’ is ex post optimal given bad message ``not invest’’ and ``invest’’ are equally attractive given good message

Next lecture Next lecture: One player with prior information (comparative statics) Two players, no prior information (public versus private signals) Two players, prior information (generalized comparative statics) Left out: Design with private information Elicitation of information No elicitation