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Checking Cheap Talk

Ian Ball, Xin Gao

Abstract

We consider a sender-receiver game in which the receiver's action is binary and the sender's preferences are state-independent. The state is multidimensional. The receiver can select one dimension of the state to check (i.e., observe) before choosing his action. We identify a class of influential equilibria in which the sender's message reveals which components of the state are highest, and the receiver selects one of these components to check. The sender can benefit from communication if and only if she prefers one of these equilibria to the no-communication outcome. Similar equilibria exist when the receiver can check multiple dimensions.

Checking Cheap Talk

Abstract

We consider a sender-receiver game in which the receiver's action is binary and the sender's preferences are state-independent. The state is multidimensional. The receiver can select one dimension of the state to check (i.e., observe) before choosing his action. We identify a class of influential equilibria in which the sender's message reveals which components of the state are highest, and the receiver selects one of these components to check. The sender can benefit from communication if and only if she prefers one of these equilibria to the no-communication outcome. Similar equilibria exist when the receiver can check multiple dimensions.
Paper Structure (26 sections, 10 theorems, 66 equations, 2 figures)

This paper contains 26 sections, 10 theorems, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Constructing the top equilibria
  4. Benchmark: No verification
  5. Top equilibria
  6. Equilibrium welfare
  7. Equilibrium bounds
  8. Benefiting from communication
  9. Optimality of top equilibria
  10. Checking multiple attributes
  11. Conclusion
  12. Proofs
  13. Favorability
  14. Proof of Theorem \ref{['res:top_single']}
  15. Proof of Lemma \ref{['res:buying_prob']}
  16. ...and 11 more sections

Key Result

Theorem 1

For $k =1, \ldots, N-1$, the top-$k$ equilibrium exists if and only if Moreover, these price thresholds are ordered as follows:

Figures (2)

  • Figure 1: Top equilibrium conditions and associated buying probabilities
  • Figure 2: The set $\mathcal{X}$ and its upper envelope

Theorems & Definitions (11)

  • Theorem 1: Top-$k$ equilibrium condition
  • Lemma 1: Equilibrium buying probabilities
  • Theorem 2: Benefiting from communication
  • Theorem 3: Optimality of top equilibria
  • Theorem 4: Optimal price and equilibrium
  • Theorem 5: Top-$k$ equilibrium condition with multiple attributes checked
  • Remark : Adaptive checking
  • Lemma 2: Price thresholds
  • Lemma 3: Upper envelope
  • Lemma 4: Favorability
  • ...and 1 more