Strategic Information Transmission Basic Cheap Talk L3 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: Prior distribution of types (uniform)

Preferences Assumptions: Useful observations monotonicity of optimal actions separating type

Equilibrium Equilibrium satisfies 1. 2. 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 So far we have shown that there exists such that cardinality of set of induced actions in any PBN is no grater than

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

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

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 .

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?

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

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

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

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