1. Strategic Information Transmission
Basic Cheap Talk L2
Strategic Information Transmission
Crawford an Sobel (1982)

2. Road map Today We introduce a basic cheap talk game
Fully characterize the set of PNB in terms of cutoffs
Remarks:
We will use alternative notation relative to the paper
Use some more ``modern’’ arguments
Next class:
Derive equilibria in closed form for quadratic preferences
Compare them in terms of ex ante welfare (both S and R)
Discuss some selection criteria

3. Cheap talk game Two agents: Sender (S) Receiver (R)
Timing and actions:
Sender observes state , sends message
Receiver observes message , choses action
Preferences:
Prior distribution of types (uniform)
Cheap talk (why?)

4. Preferences Assumptions: 1) Strict concavity (in )
2) Single peak (for any )
3) Supermodularity
Well-defined optimal choice function
Example

5. Properties Optimal action. function strictly increasing.
This generalizes as follows. Suppose
(Topkis, Theorem )

6. Properties (fix Let ``difference’’ function be
1) Function is continuous and strictly increasing in
2) Zero of the function partitions type space
3) Optimal action of a threshold type satisfies

7. PBN Equilibrium Sender Receiver beliefs strategy Equilibrium satisfies1. 2. 3.

8. Simplifying observation
R objective function is strictly concave – randomizing suboptimal
Equilibrium satisfies 1. 2. 3.

9. Two (straightforward) observations
Bubbling equilibrium exists for any preferences
Assume no preference bias, Fully revealing equilibrium exists.
How about equilibrium with senders preference ares bias?
In what follows we assume
Useful fact:

10. Partition equilibrium (Definition)
Cutoff vector partitions type space if
Type induces action if
D: PBN is a partition equilibrium if there exists a cutoff vector such that all types in induce same action with probability one.

11. Partition equilibrium (Necessity)
P: There exists such that any PBN equilibrium takes a form of a partition equilibrium with cutoffs. Moreover
Corresponding vector of actions is monotonic.
Cutoff types are indifferent between neighbouring actions
Significance of this result:
Any equilibrium at most partly revealing
Characterization of equilibrium tgrough indifferent cutoff types

12. Claim 1 Set of actions induced in equilibrium:
Claim 1: There exists such that in any PBN cardinality of set is no grater than
Proof: Fix PBN and corresponding set

13. CN

14. CN

15. Claim 2 Fix finite . Wlog let
Claim: There exists unique cutoff vector such that
induces action with probability one
Cutoff is indifferent between

17. Sufficiency For a cutoff vector define
Suppose cutoff types are indifferent about actions
P: There exists a partition equilibrium with the cutoff thresholds

18. Quadratic model
T: Set of all PNB equilibria is fully characterized by the set of solutions to the difference equation
Observations:
Second order non-linear difference equation
If it has a solution with N cutoffs, then it also has a solution with N-1
Some equilibria are better in than others in terms of welfare
Within a quadratic setting equilibria can be derived in closed form