Multidimensional Cheap Talk 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 Borsuk-Ulam imply that for any there exists s.t. 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’’ Argument extends for strictly quasivonvex preferences

Benefits from randomness of ? Which economic settings give rise to quasiconvex preferences Let , When variation in is good? Four settings: 1. Separable convex utility per each issue (advertising) 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