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 for quadratic preferences 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: 1) Strict concavity (in ) 2) Single peak (for any ) 3) Supermodularity Well-defined optimal choice function Example

Properties Optimal action. function strictly increasing. This generalizes as follows. Suppose (Topkis, Theorem 3.10.1)

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

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:

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.

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

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

CN

CN

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

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

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