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

3. Preferences Assumptions: Useful observations
monotonicity of optimal actions
separating type

4. Equilibrium
Equilibrium satisfies 1. 2. 3.

5. 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
So far we have shown that there exists such that cardinality of set of induced actions in any PBN is no grater than

6. Step 2 Wlog Claim: For any PBN with there exists unique cutoff vector.
such that each type induces action with probability one.

8. Step 3
Claim: For any PBN set of induced actions satisfies:

9. Sufficiency
Let be a cutoff vector such that each cutoff type , is indifferent between neighboring actions
P: There exists a partition equilibrium with the cutoff thresholds

10. Characterization result
Set of all PNB equilibria is fully characterized by the set of monotonic solutions to the equation for any
Observations:
Equality is called an arbitrage condition
Defines a second order difference equation with boundary conditions
DE has some solution for any But does solution define cutoffs?

11. Quadratic Example Preferences Remarks
Arbitrage conditions is a second order linear difference equation
All equilibria can be derived in closed form

12. Partition equilibrium (with N=1,2,…,? cutoffs)
Optimal R choice, conditional on

13. Partition equilibrium (with N=1,2,…,? cutoffs)
Arbitrage condition
Monotonicity

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