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Persuading while Learning

Itai Arieli, Yakov Babichenko, Dimitry Shaiderman, Xianwen Shi

TL;DR

The paper develops a dynamic information-design model where a private, learning sender updates beliefs via a martingale $\mathbf{X}$ on $[0,1]$ and commits to revelation policies to persuade a sequence of receivers. It proves that, under Blackwell order-preserving martingales, optimal strategies are interval policies, which reduces the problem to selecting stopping masses in each period. Applying this to random-walk martingales on grids, it shows that for sufficiently impatient senders the greedy policy is optimal and can be implemented with near-complete transparency before adoption; it also characterizes the threshold $1/D$ that governs these results and analyzes several grid-based examples. The work contributes to dynamic persuasion literature by providing a tractable, measure-theoretic framework and principled conditions under which disclosure is either fully transparent or optimally selective. Overall, it offers precise guidance on information disclosure policies when the sender learns privately over time and faces sequential, binary adoption decisions.

Abstract

We propose a dynamic product adoption persuasion model involving an impatient partially informed sender who gradually learns the state. In this model, the sender gathers information over time, and hence her posteriors' sequence forms a discrete-time martingale. The sender commits to a dynamic revelation policy to persuade the agent to adopt a product. We demonstrate that under the assumption that the sender's martingale possesses Blackwell-preserving kernels, the family of optimal strategies for the sender takes an interval form; namely, in every period the set of martingale realizations in which adoption occurs is an interval. Utilizing this, we prove that if the sender is sufficiently impatient, then under a random walk martingale, the optimal policy is fully transparent up to the moment of adoption; namely, the sender reveals the entire information she privately holds in every period.

Persuading while Learning

TL;DR

The paper develops a dynamic information-design model where a private, learning sender updates beliefs via a martingale on and commits to revelation policies to persuade a sequence of receivers. It proves that, under Blackwell order-preserving martingales, optimal strategies are interval policies, which reduces the problem to selecting stopping masses in each period. Applying this to random-walk martingales on grids, it shows that for sufficiently impatient senders the greedy policy is optimal and can be implemented with near-complete transparency before adoption; it also characterizes the threshold that governs these results and analyzes several grid-based examples. The work contributes to dynamic persuasion literature by providing a tractable, measure-theoretic framework and principled conditions under which disclosure is either fully transparent or optimally selective. Overall, it offers precise guidance on information disclosure policies when the sender learns privately over time and faces sequential, binary adoption decisions.

Abstract

We propose a dynamic product adoption persuasion model involving an impatient partially informed sender who gradually learns the state. In this model, the sender gathers information over time, and hence her posteriors' sequence forms a discrete-time martingale. The sender commits to a dynamic revelation policy to persuade the agent to adopt a product. We demonstrate that under the assumption that the sender's martingale possesses Blackwell-preserving kernels, the family of optimal strategies for the sender takes an interval form; namely, in every period the set of martingale realizations in which adoption occurs is an interval. Utilizing this, we prove that if the sender is sufficiently impatient, then under a random walk martingale, the optimal policy is fully transparent up to the moment of adoption; namely, the sender reveals the entire information she privately holds in every period.
Paper Structure (31 sections, 25 theorems, 64 equations)

This paper contains 31 sections, 25 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Related Literature
  4. Model and Preliminaries
  5. Persuasion as a stopping problem
  6. Measure Theoretic Formulation
  7. Blackwell Order Preserving Kernels
  8. Examples of Blackwell preserving martingales
  9. Conditionally Independent Signals
  10. Random Walk on a Grid
  11. Classes of policies
  12. The greedy policy
  13. Interval policies
  14. Properties of interval policies
  15. Existence and uniqueness
  16. ...and 16 more sections

Key Result

Lemma 1

An equivalent reformulation of the sender optimization problem given in Equation eq:sender-u is the following: maximize $\sum_{t=1}^T |\nu_t|w_t$ subject to the following recursively defined constraints: $X_1\sim\mu_1$, $\nu_t\leq\mu_t$ for every $t=1,\ldots ,T$, $\overline{\nu_t}\geq l$ for every $

Theorems & Definitions (49)

  • Example 1
  • Lemma 1
  • Definition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 2
  • Definition 3
  • Lemma 5
  • Lemma 6
  • ...and 39 more