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Information Design: A unified Perspective
L19 Bergmann and Morris 2017
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Schedule of presentations
April 18 : Yue Li April 20: Quiran Shao April 25: Alaxander Clark April 27: Gabriel Martinez May 2: Ziwei Wang May 4: Yixi Yang
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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
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Basic Game Sender faces many Receives who ``play a game ’’ among each other A game: I players (receivers) Finite action space Type space: , prior Preferences: ``Prior’’ information structure Finite set of signals , Signal distribution We call it a basic game (of incomplete information),
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Designer’s instruments
Designer observes (Literature assumes 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 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)
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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
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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 We need some selection criterion Two alternative approaches
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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 - Latter: Robust information design - Carroll (2016) , Goldstein and Huang 2016, Inostroza and Pavan 2017
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Bayes Correlated Equilibrium (BCE)
D: Decision rule is a BCE in the basic game if for any Let be the set of all BSE in game G Revelation principle (Bergmann Morris 2016) T1: A decision rule is BCE in a basic game if and only if it is a BNE in the augmented game, i.e., for some Proof
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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
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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
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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)
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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 ?
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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
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BCE Set Polytope How to implement extreme points of ?
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Step 2: Optimal message strategy
S maximizes the expected probability of investment Optimal choice 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
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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
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