Information Design A unified Perspective L 9 Bergmann

Information Design: A unified Perspective L 9 Bergmann and Morris 2018

Schedule of presentations Schedule: 12/06 Yuanzhe Liu Natalia Serna Zhenrui Liu 12/11 Yiyou Zhang Wenbo Min Jonathan Becker

Plan • Today: - General Information Design Problem - BCE and Revelation principle - 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: - Preferences: , prior • ``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 - Payoff state only, can elicit types - Payoff state only, cannot elicit types for all. (omniscient) • 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 - Each BNE induces some decision rule - Equilibrium correspondence is a BNE if …

Designer’s preferences over C • Ex post utility. implies ex ante preferences over decision rules • Complication: Equilibrium correspondence • need not be 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 course) • For any choice C nature selects adverse equilibrium - Objective of a designer - Robust (adversarial) information design - Carroll (2016) , Goldstein and Huang 2016, Inostroza and Pavan 2017

Decision rules in Nash • Consider complete information (coordination) game • Nash equilibria • Decision rules induced by (mixed Nash) equilibria • Restriction

Correlated Equilibrium Aumann 1974 • Consider complete information (coordination) game • Common randomization mechanism (weather) • 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 commits to message structure, observes • 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 • One of the 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): 1) Obfuscation of information 2) ``not invest’’ is ex post optimal given bad message 3) ``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