Basic Cheap Talk (Welfare) Strategic Information Transmission Crawford an Sobel (1982)

Cheap talk game Two agents: Sender (S) Receiver (R) Timing and actions: Sender observes state , sends message Receiver observes message , choses action Preferences: Quadratic model Uniform prior distribution of types

Partition equilibrium Cutoff vector satisfies Best reply of R to type distribution Arbitrage condition for n=1,….,N-1 Set of PNB equilibria is characterized by the set of monotonic solutions to (the implicit) difference equation, for for N=1,2, …..

Quadratic model For quadratic preferences R response to uniform distribution Arbitrage condition defines a linear difference equation With initial conditions solution is given by Parameter is going to be determined from terminal condition

Maximal number of cutoffs ` consider Monotonic solution if and only if For any. terminal condition

Maximal number of cutoffs

Maximal N?

Equilibrium, closed form

Multiplicity of equilibria Off equilibrium beliefs not uniquely defined (for messages not used) Indeterminacy of a language (permutations of messages) Multiplicity of equilibrium outcomes

Equilibrium Selection (Welfare Criterion) Large literature on equilibrium selection Proposed criteria (the most efficient equilibrium): Evolutionary arguments (Robson Kim and Sobel) Behavioral types (Kartik) Prominent (focal) equilibria (Shelling): Efficient equilibria (Pareto superior) Which equilibrium is welfare superior?

Ex post welfare With b= 0.05 three partitions N=1,2,3 None of the equilibria dominates the other in Pareto sense

Ex ante welfare Action = unbiased conditional estimate of type Ex ante welfare of R and S is perfectly aligned Ex ante welfare fully determined by residual variance

Variance

Monotonicity of welfare in N Claim: ex ante welfare of R and S is increasing in (up to )

Monotonicity in N? Yes

Beyond Crawford-Sobel What if More generally, suppose that Product quality, talent, payment, etc. Observation: Only one action induced in eqilibrium

Takeaway message Technical insight: sorting of types and partition equilibrium Substantive message Bubbling equilibrium always exists Fully revealing equilibrium possible only if preferences perfectly aligned Partly revealing equilibrium exists only if R-S preferences partly aligned Ex ante welfare increases with equilibrium informativeness No communication with independent preferences General message: communication hard to sustain in equilibrium