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

2. 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

3. 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, …..

4. 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

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

6. Maximal number of cutoffs

7. Maximal N?

8. Equilibrium, closed form

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

10. 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?

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

12. 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

13. Variance

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

15. Monotonicity in N? Yes

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

17. 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