Information Design A unified Perspective L 19 Bergmann
Information Design: A unified Perspective L 19 Bergmann and Morris 2017
Schedule of presentations Schedule: April 18 : Yue Li April 20: Quiran Shao April 25: Alaxander Clark April 27: Gabriel Martinez May 2: Ziwei Wang May 4: Yixi Yang
Plan • Today: - General Information Design Framework - Revelation principle and BCE - Two step procedure - KG example reconsidered • Next lecture: we modify the KG example • 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 - Type 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 (Literature assumes 3 variants) - Payoff state and types - Payoff state only, can elicit types - Payoff state only, cannot elicit types for all • Designer provides `supplemental’’ information to players • Sends messages to each player (here called signal) • Communication rule C • Remark: Without knowledge the designer essentially becomes 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 • does not define preferences over • We need some selection criterion • Two alternative approaches is not a function
Designer’s preferences over C • Designer choses as well as - Objective of a designer - Most papers (all discussed in this review) • For any designer choice C nature selects adverse equilibrium - Objective of a designer - Latter: Robust information design - Carroll (2016) , Goldstein and Huang 2016, Inostroza and Pavan 2017
Bayes Correlated Equilibrium (BCE) D: Decision rule • Let is a BCE in the basic game if for any be the set of all BSE in game G • Revelation principle (Bergmann Morris 2016) T 1: A decision rule is BCE in a basic game augmented game, i. e. , • Proof for some if and only if it is a BNE in the
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 - Find BCE that maximizes S preferences on this set - Find the corresponding communication rule • Benefits: - Problem has a structure of linear programming - Optimal message strategy is well defined - Comparative statics of BCE - Derivation of equilibrium without concavification • Max min problem, set of feasible decision rules is smaller than
Plan • We apply these results to characterize equilibrium in a sequence of examples • Today: One with no prior information (KG example) • Next lecture modifications of this example
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)
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 message strategy • S maximizes the expected probability of investment • Optimal choice • 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) - Design with private information
- Slides: 17