Comparative Cheap Talk Comparative cheap talk (JET 2007) Persuasion by Cheap Talk (AER 2010) 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 Each issue k=1,…K is symmetric in terms of R and S (additive) preferences type distribution Important classes of preferences:

Previous lecture We assumed K=2, We have focused on ``ranking’’ equilibria, R-messages Type independent preferences of S Revealing R-messages is (weakly) incentive compatible Ranking equilibrium is partly informative, influential R-equilibrium dominates B-equilibirum if quasiconvex Type dependent (supermodual) preferences of S and R Revealing R-messages is (strictly) incentive compatible R-equilibrium may dominate B-equilibirum even with quasiconcave

Today Many symmetric issues K>2, Limit of complete ranking equilibrium as K Partial rankings (sender welfare non-monotonic) Asymmetric issues (biases) Small asymmetries (strict supermodularity) Non-existence of R equilibrium with large asymmetries Existence of incentive compatible type partitions (not R partitions) Next lecture: 4 applications

K issues K! number of distinct rankings (permutations) for K=5 this number is 120 for K=10 this number is 3628800 Type independent utility of S indifference curve is a K-1 dimensional manifold For each individual issue ranking more informative for higher K

Limit K be a c.d.f over types For each issue type. i.i.d P: Ex ante (per issue) welfare of S (R) asymptotically converges to fully informative one as

Heuristic argument

Quadratic CS model Preferences (Best) partition equilibrium versus (best) relative cheap talk equilbirium Ex ante welfare of S (welfare =R –b^2)

Partial Rankings Coarse equivalent classes (top ten students in a class) Complete ranking and bubbling are two extremes Partial rankings induce (first order) ranked posteriors, each for one class Sender’s welfare need not be monotone in completeness of a ranking

Example: Recommendation game Players: S: professor with 3 students each with quality , Message space =rankings R: future employer, binary action hire, not hire) Preferences Equilibria with (complete or partial) rankings exist

Example: Recommendation game Bubbling equilibrium Complete ranking equilibrium Partial rankings (my top 2 students are …)

Asymmetric issues Asymmetries - types are drawn from different distributions - sender may have stronger preferences for some issues - receiver may have biases too Asymmetries are common

Problem issues (uniform distribution), receiver Asymmetric S preferences Incentives to misreport 2 alternative solutions

Solution 1 Strict supermodularity of preferences Strict incentives to reveal rankings in symmetric setting R equilibrium exist with sufficiently small perturbations Levy and Razin (ECMA 2007) – non-existence of R equilibrium with large assymetries

Solution 2 (idea) Persuasion by Cheap Talk (AER 2010)

Problem issues (uniform distribution), receiver Asymmetric S preferences Exists partition for which expected values fall on the same indifference curve 2 alternative solutions

General argument compact and convex, absolutely continuous, full support R preferences S preferences, type independent, continuous

Arbitrary fine partitions (linear S utility) Argument extends for strictly quasivonvex preferences