📈 economics

Sequential Non-Bayesian Persuasion

This paper demonstrates that when a receiver's belief-updating rule systematically deviates from Bayesian rationality, a sender can strategically benefit from providing information sequentially, a result that contrasts with the classic finding that sequential persuasion offers no advantage in standard Bayesian settings.

Original authors: Yaron Azrieli, Rachana Das

Published 2026-08-06

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Yaron Azrieli, Rachana Das

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to convince a friend to take a specific action, like buying a ticket to a concert or voting for a candidate. In the world of economics, this is called "persuasion." For decades, the standard rulebook for this game assumed that people are perfect calculators. If you showed them new facts, they would instantly and perfectly update their beliefs using a strict mathematical formula called "Bayes' rule." Under this old rulebook, there was a surprising twist: if you wanted to convince someone, it didn't matter if you gave them all the information at once or dripped it out slowly over time. A smart, perfect calculator would end up with the exact same opinion either way, so a "one-shot" reveal was just as good as a long, drawn-out campaign.

But real people aren't perfect calculators. We have biases. Sometimes we ignore new evidence because it feels too shocking (we are "conservative"), and sometimes we get stuck on our first impression. This paper explores what happens when the person you are trying to convince has these human quirks. The researchers ask a simple but powerful question: If your friend updates their beliefs in a messy, biased way, does it help to tell them things slowly, one piece at a time, rather than all at once? The answer turns out to be a resounding "yes," but only under specific conditions. The paper proves that by carefully timing your information, you can actually guide a biased person to make a decision they would never have made if you had just dumped the facts on them in a single burst.

The Game of "Convince Me"

Let's set the stage with a simple game. There are two players: the Sender (you, the persuader) and the Receiver (your friend, the decision-maker). The world has different possible "states" (like "the concert is amazing" or "the concert is terrible"), but your friend doesn't know which one is true. They start with a guess, called a "prior." You know the truth, and you want them to take a specific action, like buying a ticket.

In the classic version of this game, the Receiver is a "Bayesian" thinker. This means they are like a flawless robot. If you show them a piece of evidence, they instantly mix it with their old guess to form a new, perfect opinion. The big discovery in previous research was that for these perfect robots, timing doesn't matter. Whether you show them one big report or a thousand tiny clues, the final distribution of their opinions is the same. You can't trick a perfect calculator by stretching out the conversation.

However, this paper assumes the Receiver is biased. They don't update perfectly. Instead, they have a "bias function" that distorts the truth. A common example used in the paper is "Conservative Bayesianism." Imagine a person who is very stubborn. When they see new evidence, they only update their belief a little bit, pulling it slightly toward the new truth but mostly sticking to their old guess. They never fully rule out a possibility, no matter how much evidence you pile up; they just get a little more convinced.

The Magic of the Slow Burn

The authors of this paper discovered that when the Receiver is this kind of stubborn, biased person, sequential persuasion (telling them things over time) becomes a superpower.

Here is the magic trick:

The One-Shot Limit: If you try to convince the stubborn friend with just one big piece of information, their bias acts like a rubber band. They pull the new evidence back toward their old belief. If the evidence needs to be extreme to change their mind, the rubber band snaps them back before they ever take the action you want.
The Sequential Drift: But if you give them the information in steps, you can use their own stubbornness against them. You give them a small hint. They update a little. Then, you give them another hint based on their new (slightly updated) belief. Because they are updating from a slightly different starting point each time, their belief can "drift" further and further away from their original guess.

Think of it like pushing a heavy boulder up a hill. If you try to push it in one giant shove, it might not move because of the friction (the bias). But if you push it a little, let it settle, and then push it again from its new position, you can eventually get it to the top. The paper shows that by repeating the process, the Receiver's belief can drift so far that they finally choose the action the Sender wants, an action they would have refused if the information had been presented all at once.

What the Paper Actually Proves

The authors didn't just guess this; they built a mathematical proof to show exactly when this trick works. They looked at several classic scenarios found in economics textbooks:

The Two-Choice Scenario: Imagine a judge deciding whether a defendant is guilty or innocent. The prosecutor (Sender) wants a conviction. If the judge is a "Conservative Bayesian" (stubborn), the prosecutor can't just show one piece of evidence to get a conviction if the starting belief is too low. However, the paper proves that if the judge's bias is "weakly Bayes plausible" (a technical way of saying their stubbornness isn't too weird or extreme), the prosecutor can use a sequence of signals to eventually push the judge's belief over the edge and get that conviction.
The Linear Scenario: Imagine a manager (Sender) trying to get an employee (Receiver) to set a price. If the employee is biased, the manager can't just give one report. But by revealing information step-by-step, the manager can guide the employee to a price that yields a higher profit than any single report could achieve.
The "Crawford-Sobel" Scenario: This is a famous model where the Sender and Receiver have slightly different goals (like a politician and a voter). The paper proves that even here, if the Receiver is biased in a "balanced" way (meaning their bias doesn't always push in just one direction), the Sender can gain by using a multi-step strategy.

The Limits of the Trick

It is important to note what this paper does not say. It does not claim that you can convince a biased person to do anything you want. There are limits.

The authors show that even with an infinite amount of time and information, a Sender cannot always achieve their "first-best" outcome (the absolute perfect result). For example, if the Receiver is stubborn, there is a mathematical ceiling on how far their belief can drift in a specific direction. The paper provides a specific example where the Sender's maximum possible payoff is strictly lower than what they would get if the Receiver were a perfect, unbiased calculator. So, while sequential persuasion is a powerful tool, it is not a magic wand that breaks all the rules of logic.

Why This Matters

This research is a constructive proof, meaning the authors didn't just say "it's possible"; they showed how to do it. They demonstrated that the interaction between the Receiver's specific type of bias and the Sender's strategy creates a new dynamic.

The key takeaway is that information structure matters when people are imperfect. In a world of perfect robots, the timing of information is irrelevant. But in a world of humans, who update their beliefs with a mix of logic and stubbornness, the story you tell matters just as much as the facts you tell. By breaking a big truth into smaller, sequential pieces, a persuader can navigate around a person's biases and guide them to a decision that a single, overwhelming fact could never achieve. The paper confirms that in many common economic situations, the old rule that "timing doesn't matter" is broken, and the art of the slow burn is a mathematically proven strategy for success.

Technical Summary: Sequential Non-Bayesian Persuasion

1. Problem Statement

This paper investigates a model of strategic information transmission (persuasion) where the receiver's belief updating rule systematically deviates from Bayes' rule. The authors build upon the framework of De Clippel and Zhang [2022], which established that the standard Bayesian Persuasion (BP) concavification method applies even when receivers hold biased beliefs, provided the bias is a function of the prior and the correct Bayesian posterior.

The central question addressed is whether sequential persuasion—where a sender can provide information over multiple periods, conditioning future signals on past realizations—offers value to the sender when the receiver is non-Bayesian. In the classic Bayesian setting (Kamenica and Gentzkow [2009]), sequential persuasion is redundant; any sequential strategy can be replicated by a single information structure inducing the same distribution over posteriors. The authors argue that a departure from Bayesianism can explain why sequential persuasion might be more effective than a one-shot release of information.

2. Methodology and Model Setup

The Environment

The model involves a Sender ( $S$ ) and a Receiver ( $R$ ) with a finite state space $\Omega$ and a common full-support prior $\mu_0$ .

Sender: Chooses an information structure (experiment) to influence $R$ 's action. $S$ 's utility is $v(\omega, a)$ .
Receiver: Chooses an action $a \in A$ to maximize expected utility $u(\omega, a)$ .
Bias: $R$ does not update via Bayes' rule. Instead, a bias function $D: \Delta_{op}(\Omega) \times \Delta(\Omega) \to \Delta(\Omega)$ maps the prior $\nu$ and the correct Bayesian posterior $\nu^1$ to the biased posterior $D(\nu, \nu^1)$ .

Key Properties of the Bias

The analysis relies on two specific properties of the bias function $D$ :

Interiority: The receiver never completely rules out any state. Formally, $\text{Supp}(D(\nu, \nu^1)) = \Omega$ for all priors and posteriors. This distinguishes the model from standard Bayesian updating where extreme posteriors can be reached.
Eventual Learning: If the receiver is repeatedly presented with information inducing a fixed Bayesian posterior $\nu^1$ , their belief converges to $\nu^1$ . Formally, $\lim_{n \to \infty} D^n(\nu, \nu^1) = \nu^1$ .

Sequential Persuasion Mechanism

In a sequential strategy, $S$ chooses an information structure in period 1. Upon observing a signal, $R$ updates their belief using $D$ . In subsequent periods, $S$ can choose a new information structure based on the history of signals. Crucially, the updating process is recursive: the belief at the end of period $n-1$ serves as the prior for period $n$ .

Failure of Concavification: Unlike the static case, sequential persuasion breaks the one-to-one mapping between the sender's Bayesian posterior and the receiver's biased posterior. Different sequences of signals can lead to the same Bayesian posterior for $S$ but different biased posteriors for $R$ . Consequently, the standard concavification method (used to solve static BP) is not applicable for characterizing the optimal sequential payoff.

Specific Bias Example: Conservative Bayesianism

To illustrate results, the authors utilize a parametric family of biases known as "conservative Bayesianism":
$D_\alpha(\nu, \nu^1) = \alpha \nu + (1-\alpha)\nu^1$
where $\alpha \in (0,1)$ represents the strength of the bias (under-updating). This rule is interior and satisfies eventual learning.

3. Key Contributions and Results

The paper provides constructive proofs demonstrating that the sender gains from sequential persuasion (i.e., achieves a strictly higher expected payoff than the optimal one-shot strategy) in several classic environments, provided the bias satisfies specific properties.

A. Two-Action Environments (Proposition 1)

Context: The receiver has two actions, and the sender has state-independent preferences (transparent motives), such as the "Prosecutor-Judge" example.
Condition: The sender gains from sequential persuasion if and only if the bias is weakly Bayes plausible. This property requires that for any set of Bayesian posteriors whose convex hull contains the prior, the convex hull of the corresponding biased posteriors also contains the prior.
Result: Conservative Bayesianism is weakly Bayes plausible. Thus, in these environments, the sender can exploit the fact that the receiver's belief remains interior. Even if a single signal induces a posterior that leads the receiver to choose the sender's less-preferred action, the sender can continue providing information to "drift" the receiver's belief toward the preferred action.

B. Linear Preferences (Proposition 2)

Context: The receiver's optimal action is the mean of their belief (e.g., $A = \mathbb{R}^k$ ), and the sender has linear, state-independent preferences.
Condition: The sender gains from sequential persuasion if the bias is interior, satisfies eventual learning of states, and is balanced.
Balancedness: This property restricts the directions of the bias. It requires that the vectors pointing from Bayesian posteriors to biased posteriors (given a prior) are not one-sided; their convex hull must contain the zero vector. Conservative Bayesianism is balanced.
Result: In linear environments, one-shot persuasion yields no gain for the sender if the receiver is Bayesian (due to Bayes plausibility). However, with a balanced, interior bias, sequential persuasion allows the sender to manipulate the distribution of actions to increase payoff, as the sender and receiver diverge in beliefs after the first signal.

C. Common Preferences (Proposition 3)

Context: The sender and receiver have identical preferences (benevolent sender).
Condition: The sender gains from sequential persuasion if and only if the optimal one-shot payoff is strictly less than the full-information payoff.
Result: Since the bias satisfies eventual learning, the sender can approach the full-information payoff by repeatedly sending fully revealing signals. If the one-shot constraint prevents reaching this payoff, sequential persuasion is strictly valuable.

D. Crawford-Sobel Preferences (Proposition 4)

Context: Preferences are misaligned by a constant bias term $b$ (standard Crawford-Sobel model).
Condition: The sender gains from sequential persuasion if the bias is interior, satisfies eventual learning of states, and is balanced.
Result: Starting from an optimal one-shot strategy, the sender can improve their payoff by sending additional information if a specific posterior is realized, effectively pushing the receiver's belief further in the direction that benefits the sender.

4. Mechanisms Driving the Results

The paper identifies three primary mechanisms through which sequential persuasion outperforms static persuasion in the presence of bias:

Belief Drift: With a single signal, the receiver's biased posterior cannot deviate too far from the prior. By repeating information, the sender can induce a "drift" where the belief moves further away, eventually crossing the threshold required for the receiver to choose the sender's preferred action.
Conditional Continuation: Even if a single signal induces the preferred action, the sender can increase the probability of this action by providing additional information only when the initial signal realization leads to a non-preferred action.
Violation of Bayes Plausibility: In static persuasion, the distribution of biased posteriors must be Bayes plausible (the prior is in the convex hull). In sequential persuasion, because the sender and receiver hold different beliefs after the first signal, the distribution of biased posteriors over the entire sequence is no longer constrained by the initial prior in the same way, allowing for new, sender-preferred action distributions.

5. Significance and Limitations

The authors claim that their work demonstrates the value of dynamic information design in economically relevant environments where receivers exhibit systematic biases. The proofs are constructive, revealing how specific properties of the bias (interiority, eventual learning, balancedness) interact with payoff structures to make dynamic persuasion beneficial.

Limitations and Scope:

Optimality: The paper does not generally solve for the optimal sequential persuasion strategy. Due to the failure of the concavification method, finding the optimum is computationally difficult even in simple environments. The results focus on proving that some sequential strategy yields a strictly higher payoff than the best static strategy.
First-Best Payoff: The authors note that sequential persuasion does not typically allow the sender to achieve their "first-best" payoff (where the receiver chooses the sender's favorite action in every state). Limits exist; for example, in linear environments with conservative Bayesianism, the sender's payoff is bounded strictly below the first-best due to supermartingale properties of the belief process.
Comparison to Correlation Neglect: The paper distinguishes its model from "correlation neglect" (Levy et al. [2022]). In correlation neglect, the sender can achieve the first-best by correlating signals. In this model, the bias is intrinsic to the updating rule itself, and the sender's power is more limited.
Updating Procedures: The results rely on a specific recursive updating procedure where the bias is applied at every step using the current posterior as the new prior. The authors briefly discuss alternative procedures (e.g., anchoring to the original prior) but note these may yield different results and are left for future work.

In summary, the paper establishes that in the presence of systematic, non-Bayesian belief updating, the ability to provide information sequentially is a powerful tool for senders, offering gains in classic settings where static persuasion would be ineffective or suboptimal.