Insights into Persuasion and Equilibrium in Multidimensional Cheap Talk
Explore the dynamics of multidimensional cheap talk, focusing on sender-receiver interactions, influential equilibrium, welfare rankings, and fragility to asymmetries. Lessons touch on bubbling equilibrium, influential equilibrium issues, welfare rankings preferences, and the impact of asymmetric preferences on equilibrium stability.
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Multidimensional Cheap Talk L4 Persuasion by Cheap Talk (AER 2010) Comparative cheap talk (JET 2007) Chakraborty an Harbaugh
Multidimensional Cheap Talk Two agents: Sender (S) and Receiver (R) Timing and actions: for each k=1, .K - Sender observes state , sends message - Receiver observes message , choses action k=1, K potential issues - R preferences - S preferences Examples: professor with K students, biased media outlet
Type independent preferences, 1 issue (dimension) Type independent monotonic preferences - Large biases in CS model - Type independent utilities One dimension: only non influential equilibrium Argument: Example: media outlet (S) and voter (R) - Type measure of honesty - Action voting effort Communication impossible to sustain
Bubbling equilibrium issues (uniform distribution) S preferences Next: 3 key lessons
Influential equilibrium (lesson 1) issues (uniform distribution), receiver S preferences (change to symmetric) Messages could be interpreted as ``rankings (Comparative cheap talk) With strict preference
Welfare Rankings (lesson 2) R (always) prefers informative equilibrium to bubbling (Blackwell) S preferences (quasiconvex, quasiconcave, linear) How comes that S might strictly prefer bubbling equilibrium? Examples of quasiconvex preferences
Fragility to asymmetries (lesson 3) issues (uniform distribution) Asymmetric preferences (change to symmetric) What if Influential equilibrium disappears
P1: Comparative Cheap Talk (JET 2007) K symmetric issues: - Separable S utilities, symmetric across issues - Symmetric prior distribution - Weak supermodularity (e.g., type independent) Results (K alternatives) - Complete or partial rankings (``top 3 ) supported in equilibriu - Almost fully revealing equilibrium with Asymmetric issues: - Example: type independent utilities (weak supermodularity) - Strict supermodularity (strict incentives in symmetric settings) - Influential equilibrium exist with sufficiently small perturbations Levy and Razin (ECMA 2007) non-existence of R equilibrium with large asymmetries
Persuasion by Cheap Talk (AER 2010) Assumption: Type independent, possibly non-additive utility of S Arbitrary asymmetries with respect to - utilities - distributions Main Results: - informative equilibrium exists with ``sophisticated messages - Full revelation along K-1 dimensions)
Asymmetry in utilities Spinning argument
Problem issues (uniform distribution), receiver Asymmetric S preferences Exists partition for which expected values fall on the same indifference curve
General ``spinning argument Sphere Function is odd if P: Continuous and odd function has an origin. (Borsuk-Ulam)
General ``spinning argument compact and convex, absolutely continuous, full support R preferences S preferences, type independent, continuous Observation: function is continuous and odd
General argument compact and convex, absolutely continuous, full support R preferences S preferences, type independent, continuous s.t. Borsuk-Ulam imply that for any there exists P: There exists an influential equilibrium Constructive argument How large is the set of PBN
Finer partition (lesson 4) Linear utility function For N=1,2,.. one can construct 2^N element partition, Probability mass of each element goes to zero Sender reveals all the information in K-1 dimensions
Nonlinear preferences: problem and solution Argument extends for strictly quasivonvex preferences
Substantive insight Partly revealing (influential) equilibrium - Exists! - R prefers revealing equilibrium to bubbling - S prefers revealing equilibrium if preferences strictly quasiconvex
Quasiconvex preferences: Desirability of quasiconvex preferences: - Partly revealing equilibria improve S (ex ante) welfare - For such preferences infinite partitions exist Former property important given easy commitment to ``not to talk
Benefits from randomness of ? Which economic settings give rise to quasiconvex preferences Let , When variation in is good? 1. Separable convex utility per each issue (advertising) Four settings: 2. Settings in which determines the outcome - unit demand (recommendation game)
Application : Recommendation game 2 objects, quality observed by a seller R: buyer, unit demand, outside option S: salesperson maximizes probability of selling Interpretation: - Professor with Ph.D. two students on the market, one position - Dealer charging a commission fee - Lobbyist advising a senator on several bill proposals
