1. Strategic Information Transmission
Basic Cheap Talk (2) L3
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:
Today:
Prior distribution of types (uniform)

3. Partition equilibrium (Definition)
D: Cutoff vector satisfies
D: PBN is a partition equilibrium if there exists a cutoff vector such that each type in induces unique action with probability one.
P: There exists such that any PBN equilibrium takes a form of a partition equilibrium with
Corresponding vector of actions is monotonic.
Cutoff types are indifferent between neighbouring actions

4. Quadratic Model Consider problem Foc
Action: unbiased estimator conditional on signal

5. Quadratic Model
Define
Observe that in partition equilibrium
Then

6. Necessary and sufficient condition for PBN
Necessary condition
Sufficiency (constructive argument)

7. Characterization result
Let
Set of all PNB equilibria is fully characterized by the set of monotonic solutions to the equation for any
Observations:
Condition is called an arbitrage condition
It defines second order difference equation with 2 boundary conditions
DE has unique solution for any Not all solution define cutoffs.

8. Partition equilibrium (with N=1,2,…,? cutoffs)
Arbitrage condition
Receiver response
Combining these two gives

9. Quadratic model
Arbitrage condition defines a linear difference equation
With initial conditions solution is given by
Parameter is going to be determined from terminal condition
For some and hence solution will not be monotonic

10. Maximal number of cutoffs
` Let
Monotonic solution if and only if
For fixed terminal condition

11. Maximal number of cutoffs

12. Maximal N?

13. Equilibrium, closed form

14. Characterization of PBN (Quadratic model)
P: Set of all PBN equilibria in a quadratic model is fully characterized by family of
Where
Example

15. 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
Neologism free equilibria
Evolutionary arguments (Robson Kim and Sobel)
Behavioral types (Kartik, Che)
Efficient equilibria (Pareto superior)

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

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

18. Ex ante welfare Ex ante welfare of R and S is perfectly aligned
Ex ante welfare fully determined by residual variance
Residual variance decreasing in

19. State independent preferences
State independent preferences of S
Persuasion games, recommendation games, monetary games
Examples
Large biases
State independent utilities
Bubbling is the only equilibrium

20. 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 state independent preferences
General message: communication hard to sustain in equilibrium

21. Calculation 1
Optimal R choice, conditional on

22. Variance

23. Solution to SODE
Homogenous equation
Bias term (non-homogenous part)

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