Basic Cheap Talk L 2 Strategic Information Transmission

Basic Cheap Talk L 2 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 - Receiver observes message • Preferences: • Prior distribution of types (uniform) • Cheap talk (why? ) , sends message , choses action

Preferences • Assumptions: • Useful facts: 1) Optimal action function is well defined and strictly monotone. 2) Suppose (Topkis, Theorem 3. 10. 1) . Then

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

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

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 • Type partitions type space induces action if if D: PBN is a partition equilibrium if there exists a cutoff vector type in induces unique action with probability one. such that each

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 grater than. such that in any PBN cardinality of set Proof: Fix equilibrium and set is no

Step 1 (cd)

Step 1 (cd)

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

Sufficiency • Let be a cutoff vector such that each cutsoff type between neighboring actions is indifferent 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