Strategic Information Transmission Basic Cheap Talk (2) 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: Today: Prior distribution of types (uniform)

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 Corresponding vector of actions is monotonic. Cutoff types are indifferent between neighbouring actions

Quadratic Model Consider problem Foc Action: unbiased estimator conditional on signal

Quadratic Model Define Observe that in partition equilibrium Then

Necessary and sufficient condition for PBN Necessary condition Sufficiency (constructive argument)

Characterization result Let Set of all PNB equilibria is fully characterized by the set of monotonic solutions to the equation for any Observations: Condition is called an arbitrage condition It defines second order difference equation with 2 boundary conditions DE has unique solution for any . Not all solution define cutoffs.

Partition equilibrium (with N=1,2,…,? cutoffs) Arbitrage condition Receiver response Combining these two gives

Quadratic model Arbitrage condition defines a linear difference equation With initial conditions solution is given by Parameter is going to be determined from terminal condition For some and hence solution will not be monotonic

Maximal number of cutoffs ` Let Monotonic solution if and only if For fixed terminal condition

Maximal number of cutoffs

Maximal N?

Equilibrium, closed form

Characterization of PBN (Quadratic model) P: Set of all PBN equilibria in a quadratic model is fully characterized by family of Where Example

Multiplicity of equilibria Off equilibrium beliefs not uniquely defined (for messages not used) Indeterminacy of a language (permutations of messages) Multiplicity of equilibrium outcomes Equilibrium selection Neologism free equilibria Evolutionary arguments (Robson Kim and Sobel) Behavioral types (Kartik, Che) Efficient equilibria (Pareto superior)

Ex post welfare With b= 0.05 three partitions N=1,2,3 None of the equilibria dominates the other in Pareto sense

Ex post welfare With b= 0.05 three partitions N=1,2,3 None of the equilibria dominates the other in Pareto sense

Ex ante welfare Ex ante welfare of R and S is perfectly aligned Ex ante welfare fully determined by residual variance Residual variance decreasing in

State independent preferences State independent preferences of S Persuasion games, recommendation games, monetary games Examples Large biases State independent utilities Bubbling is the only equilibrium

Takeaway message Technical insight: sorting of types and partition equilibrium Substantive message Bubbling equilibrium always exists Fully revealing equilibrium possible only if preferences perfectly aligned Partly revealing equilibrium exists only if R-S preferences partly aligned Ex ante welfare increases with equilibrium informativeness No communication with state independent preferences General message: communication hard to sustain in equilibrium

Calculation 1 Optimal R choice, conditional on

Variance

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

Monotonicity of welfare in N Claim: ex ante welfare of R and S is increasing in (up to )