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Bayesian Persuasion with Selective Disclosure

Yifan Dai, Drew Fudenberg, Harry Pei

TL;DR

This paper analyzes Bayesian persuasion where a sender publicly commits to an initial experiment but privately conducts follow-up experiments and selectively discloses results, with the sender privately knowing their capacity $t$. It shows that the sender’s ability to attain the commitment payoff, denoted $oldsymbol{ar V}(oldsymbol{ ho}_0)$, depends on the receiver’s uncertainty about the sender’s capacity and on whether the optimal initial experiment induces non-credible beliefs; if non-credible beliefs exist and capacity is uncertain, equilibria yield payoffs strictly below the commitment level, while in monotone environments with credible beliefs the commitment payoff can be attained, and with a dominating type the sender can approximate it. The paper develops a robust analysis under Perfect Extended Bayesian Equilibrium (PEBE), provides an illustrative example, and extends results to settings where the sender learns their type before the initial experiment or faces a constant marginal cost for additional experiments. Overall, it clarifies the fragility of commitment credibility in information-design problems with endogenous follow-up investigation and selective disclosure, highlighting the conditions under which commitment-like persuasion remains sustainable in practice.

Abstract

A sender first publicly commits to an experiment and then can privately run additional experiments and selectively disclose their outcomes to a receiver. The sender has private information about the maximal number of additional experiments they can perform (i.e., their type). We show that the sender cannot attain their commitment payoff in any equilibrium if (i) the receiver is sufficiently uncertain about their type and (ii) the sender could benefit from selective disclosure after conducting their full-commitment optimal experiment. Otherwise, there can be equilibria where the sender obtains their commitment payoff.

Bayesian Persuasion with Selective Disclosure

TL;DR

This paper analyzes Bayesian persuasion where a sender publicly commits to an initial experiment but privately conducts follow-up experiments and selectively discloses results, with the sender privately knowing their capacity . It shows that the sender’s ability to attain the commitment payoff, denoted , depends on the receiver’s uncertainty about the sender’s capacity and on whether the optimal initial experiment induces non-credible beliefs; if non-credible beliefs exist and capacity is uncertain, equilibria yield payoffs strictly below the commitment level, while in monotone environments with credible beliefs the commitment payoff can be attained, and with a dominating type the sender can approximate it. The paper develops a robust analysis under Perfect Extended Bayesian Equilibrium (PEBE), provides an illustrative example, and extends results to settings where the sender learns their type before the initial experiment or faces a constant marginal cost for additional experiments. Overall, it clarifies the fragility of commitment credibility in information-design problems with endogenous follow-up investigation and selective disclosure, highlighting the conditions under which commitment-like persuasion remains sustainable in practice.

Abstract

A sender first publicly commits to an experiment and then can privately run additional experiments and selectively disclose their outcomes to a receiver. The sender has private information about the maximal number of additional experiments they can perform (i.e., their type). We show that the sender cannot attain their commitment payoff in any equilibrium if (i) the receiver is sufficiently uncertain about their type and (ii) the sender could benefit from selective disclosure after conducting their full-commitment optimal experiment. Otherwise, there can be equilibria where the sender obtains their commitment payoff.
Paper Structure (22 sections, 32 equations, 3 figures)

This paper contains 22 sections, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Baseline Model
  3. Analysis
  4. Assumptions & Preliminaries
  5. Main Results
  6. Example:
  7. Discussion
  8. PEBE vs wPBE:
  9. The Sender's Worst Equilibrium Payoff:
  10. The Sender Observes Their Type Before Conducting the Initial Experiment:
  11. Constant Marginal Cost of Additional Experiments:
  12. Conclusion
  13. PEBE in Games with Random Successors
  14. Implications of Assumption \ref{['Ass1']}
  15. Proof of Proposition \ref{['Prop1']}
  16. ...and 7 more sections

Figures (3)

  • Figure 1: The sender's utility under the receiver's best reply as a function of the receiver's belief (red) and its concave closure (blue), with non-credible beliefs in yellow and credible beliefs in green.
  • Figure 2: The sender's indirect utility as a function of the receiver's belief $\overline{u}(\pi)$ (red) and its concave closure (blue) in the example.
  • Figure 3: The sender's payoff function $\overline{u}(\pi)$ (red) and its concave closure (blue). The green arrows represent type $1$ sender's profitable deviation after observing the outcome of the initial experiment that leads to naive belief $\frac{1}{3}$.

Theorems & Definitions (6)

  • proof : Proof of Proposition 2:
  • proof
  • proof
  • proof : Proof of Lemma \ref{['lem: approx opt dist lower bound']}
  • proof
  • proof