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

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 in the quadratic model Compare them in terms of ex ante welfare (both S and R) Discuss some selection criteria

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?)

Preferences Assumptions: Useful facts: 1) Optimal action function is well defined and strictly monotone. 2) Suppose . Then (Topkis, Theorem 3.10.1)

Preferences 3) Let be such that. . Then 4) Increasing differences. Let . Then

PBN Equilibrium Sender Receiver beliefs strategy Equilibrium satisfies 1. 2. 3.

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

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 5:

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 each type in induces unique action with probability one.

Partition equilibrium (Necessity) P: There exists such that any PBN equilibrium takes a form of a partition equilibrium with . cutoffs. Significance of this result: Any equilibrium at most partly revealing Any equilibrium defines a finite cutoff vector such that cutoff types are indifferent between neighbouring actions.

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

Step 1 (cd)

Step 1 (cd)

Step 2 Claim: There exists unique cutoff vector such that for each type induces action with probability one.

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

Main theorem 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