Dynamic Non-Bayesian Persuasion

1. Problem Statement

The paper investigates whether a sender in a persuasion game can achieve a higher expected payoff by using a sequence of experiments (dynamic persuasion) rather than a single experiment (static persuasion) when the receiver is non-Bayesian.

• Context: Standard Bayesian persuasion (Kamenica & Gentzkow, 2011) assumes the receiver updates beliefs via Bayes' rule. In that framework, dynamic and static persuasion yield the same value because the order of information revelation does not matter (timing invariance).
• The Gap: Empirical evidence suggests receivers often update non-Bayesianly (e.g., underreaction, base-rate neglect). The paper asks: Does the timing of information delivery matter when the receiver systematically distorts beliefs?
• Goal: To characterize the specific class of non-Bayesian updating rules under which dynamic persuasion offers no advantage over static persuasion (i.e., when static persuasion is "without loss").

2. Methodology and Framework

A. The Model

• Environment: A finite state space $\Theta$, a prior $p$, and a receiver who chooses an action $a$ to maximize utility. The sender designs information structures to influence the receiver's action.
• Non-Bayesian Updating (Systematic Distortions): The paper adopts the framework of De Clippel and Zhang (2022). The receiver's posterior belief $\mu(\sigma, p)(\cdot|s)$ is formed by first computing the Bayesian posterior $q = \mu^B(\sigma, p)(\cdot|s)$ and then applying a distortion rule $D_p$ dependent on the prior:
  $$\mu(\sigma, p)(\cdot|s) = D_p(q)$$
  Examples include Grether's $\alpha$-$\beta$ rule and linear underreaction.
• Dynamic Setup: The sender commits to a two-step strategy:
  1. Choose a first-stage experiment $\sigma_1$.
  2. Upon observing signal $s_1$, the receiver updates to $D_p(q)$ where $q$ is the Bayesian posterior.
  3. The sender then chooses a second-stage experiment $\sigma_2^{s_1}$ contingent on $s_1$.
  4. The receiver updates again from the distorted prior $D_p(q)$ using the second signal.

B. Analytical Approach

The author utilizes a concavification approach adapted for non-Bayesian receivers.

1. Interim Problem: For a given first-stage posterior $q$, the sender solves a static persuasion problem where the receiver's "prior" is the distorted belief $D_p(q)$. This is transformed into a standard Bayesian persuasion problem with a modified distortion rule $D^{II}_q(\cdot; p)$.
2. Dynamic Value: The sender's total dynamic value is obtained by concavifying the value of the interim problem over the first-stage signal distribution.
3. Key Condition: The core of the analysis compares the set of feasible joint distributions of sender/receiver beliefs in static ($T^1$) versus dynamic ($T^2$) settings.

3. Key Contributions and Results

A. The Concept of Divisibility

The paper introduces Divisibility as the necessary and sufficient condition for the sender to be indifferent between static and dynamic persuasion.

• Definition: A distortion rule $D$ is divisible if the distortion applied in the first step coincides with the effective distortion in the second step from the sender's perspective. Formally, for any full-support beliefs $p, q, r$:
  $$D_p(r) = D^{II}_q(r; p)$$
  where $D^{II}$ represents the composite distortion in the dynamic setting.
• Intuition: Under divisible rules, the "timing" of information does not alter the set of achievable posterior beliefs. Sequential updating is equivalent to simultaneous updating.

B. Main Characterization Theorem (Theorem 1)

Under mild regularity conditions (continuity, non-decreasing support, injectivity) and an axiom of Prior Independence under Certainty (fully revealing signals yield the same posterior regardless of the prior), the following are equivalent:

1. The distortion rule $D$ is divisible.
2. The set of feasible ex-ante payoffs for the sender is identical under static and dynamic persuasion ($V^1(p, D) = V^2(p, D)$).
3. The sender is indifferent between static and dynamic persuasion in any environment.

Implication: If a rule is divisible (e.g., Bayesian updating, Geometric distortion), the analyst can safely restrict attention to the tractable static framework without loss of generality.

C. Characterization of Divisible Rules

• Geometric Distortion: The paper proves that Geometric Distortion (a special case of Grether's $\alpha$-$\beta$ rule where $\alpha=1$) is divisible. This includes Bayesian updating ($\beta=1$) and geometric under/over-reaction ($\beta \neq 1$).
• Functional Form: Divisible rules admit a specific functional form characterized by a homeomorphism $F$:
  $$D_p(q) = F^{-1} \left( \frac{F(p)}{p} q \cdot \frac{1}{\langle \frac{F(p)}{p}, q \rangle} \right)$$
  This generalizes the result of Cripps (2018) to the persuasion context.

D. Extension: Weak Divisibility and Linear Underreaction

The main theorem relies on "Prior Independence under Certainty," which Linear Underreaction (where the posterior is a convex combination of the Bayesian posterior and the prior) fails to satisfy.

• Alternative Axiom: The paper introduces Invariance to No Information (if the signal is uninformative, the belief remains the prior).
• Weak Divisibility: Under this axiom, the paper defines Weak Divisibility, which allows the first and second-step distortions to differ by an affine transformation (represented by a column-stochastic matrix).
• Result (Theorem 3): Weak divisibility characterizes indifference between static and dynamic persuasion for environments where the sender has state-independent preferences.
• Corollary: Linear underreaction is not weakly divisible. Therefore, in environments with linear underreaction, dynamic persuasion can strictly outperform static persuasion.

E. Motivating Example (Proposition 1)

In a "Judge-Prosecutor" environment with Grether's $\alpha$-$\beta$ rule:

• If $\alpha < 1$ (Base-rate neglect): The sender strictly prefers dynamic persuasion. By withholding information in the first step, the sender allows the receiver's distorted belief to drift toward the uniform prior (which is closer to the sender's preferred threshold), making the second-step persuasion more effective.
• If $\alpha \geq 1$: The sender is indifferent. The rule is divisible in this case.

4. Significance and Implications

1. Theoretical Boundary: The paper establishes the precise frontier where the standard static Bayesian persuasion framework remains valid even with non-Bayesian receivers. It clarifies that "timing invariance" is not a universal property of non-Bayesian models but depends specifically on the structure of the distortion.
2. Practical Guidance: For analysts modeling strategic information design (e.g., in marketing, political campaigning, or judicial settings), the paper provides a checklist:
   • If the receiver's bias is geometric (distorting likelihoods but not base rates in a specific way), static models suffice.
   • If the bias involves linear underreaction or base-rate neglect ($\alpha < 1$), dynamic strategies (sequencing information) can yield strictly higher payoffs, and static models will underestimate the sender's value.
3. Methodological Contribution: The paper extends the "concavification" technique to multi-stage non-Bayesian problems, demonstrating how to transform complex dynamic non-Bayesian problems into tractable static problems with modified utility functions, provided the divisibility condition holds.

Summary of Key Findings

Updating Rule Type	Divisible?	Static vs. Dynamic Value
Bayesian	Yes	Indifferent
Geometric Distortion ($\alpha=1, \beta \neq 1$)	Yes	Indifferent
Base-Rate Neglect ($\alpha < 1$)	No	Dynamic > Static
Linear Underreaction	No (Not Weakly Divisible)	Dynamic > Static (in some env.)

The paper concludes that while dynamic persuasion is often more powerful, the static framework is robust and sufficient if and only if the receiver's updating rule satisfies the specific structural property of divisibility.