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Dynamic Pricing and Advertising with Demand Learning

This paper establishes that flexible advertising can increase a seller's revenue by at most a factor of two compared to pricing alone, and proposes an efficient online algorithm that achieves an optimal $O(T^{2/3} (m\log T)^{1/3})$ regret rate for jointly learning and optimizing pricing and advertising strategies under demand uncertainty.

Original authors: Shipra Agrawal, Yiding Feng, Wei Tang

Published 2026-08-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Shipra Agrawal, Yiding Feng, Wei Tang

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

In the modern marketplace, a seller faces a constant challenge: how to set the right price for a product when they do not fully know what their customers are willing to pay. This uncertainty is a fundamental problem in economics. If a seller sets a price too high, customers walk away; if they set it too low, they leave money on the table. For decades, researchers have studied how sellers can learn the true value of their goods by watching how people react to different prices over time. This process is known as demand learning. However, sellers possess another powerful tool that is often overlooked in these studies: advertising. While traditional economic models often treat advertising as a way to simply remind people a product exists, real-world experience shows that advertising does something more subtle. It shapes how customers perceive the quality of a product before they even see it. A seller might choose to reveal every detail about a service, or they might choose to hide certain aspects, creating a sense of mystery or trust. The question is whether this ability to control information can actually make a seller more money, and if so, how they can figure out the perfect balance between what to say and what to charge without knowing their customers' preferences in advance.

A team of researchers has tackled this problem by building a new framework that combines the science of pricing with the art of information design. They imagined a scenario where a seller offers a product with varying levels of quality, such as a box of rescued produce where the specific fruits and vegetables might change from week to week, or a ride-sharing service where the driver's reliability is unknown until the match is made. In this setting, the seller knows the true quality of the product at the moment of sale, but the customer does not. The seller can then choose to send a signal to the customer—essentially an advertisement—that reveals some, all, or none of the product's true qualities. The customer, seeing this signal, forms a belief about the product and decides whether to buy it at the posted price. The researchers wanted to know two things: first, how much extra revenue can a seller generate by using these flexible advertising strategies compared to just setting a price and saying nothing? Second, if the seller does not know the customers' preferences at the start, how can they learn the best combination of price and advertising strategy through trial and error?

The study reveals a clear and surprising limit to the power of advertising. When the seller is allowed to design the perfect advertising campaign and pair it with the perfect single price, they can increase their revenue, but only up to a point. The researchers proved mathematically that no matter how clever the advertising strategy, it can never more than double the revenue a seller would get from simply posting the best possible price without any advertising. This finding is significant because it sets a realistic expectation for businesses. It suggests that while advertising is a valuable lever to pull, it is not a magic wand that can create infinite wealth. The value of advertising is bounded, and once a seller optimizes their price, the additional gain from manipulating information has a ceiling. Furthermore, the researchers found that this bound holds even if the seller were allowed to change the price based on the specific information revealed in the ad. A simple, single price that does not change based on the ad's content is nearly as effective as a complex system of variable pricing, provided the advertising strategy is chosen carefully.

The second part of the research addresses the practical reality that sellers rarely know their customers' preferences perfectly. In the real world, a seller must learn the demand curve while simultaneously trying to make money. The researchers developed a new computer algorithm that allows a seller to learn the optimal price and advertising strategy over time. This algorithm works by observing whether customers buy the product or not after seeing a specific price and a specific type of advertisement. By analyzing these responses, the algorithm gradually builds a picture of what customers value. The researchers showed that this learning process is highly efficient. The algorithm can find the best strategy with a level of accuracy that improves steadily as time goes on, reaching a point where the loss in potential revenue is very small compared to what a perfect, all-knowing seller would achieve. This result holds true even without making strong assumptions about how smooth or predictable customer behavior is. The algorithm is robust enough to handle complex situations where the relationship between product quality and customer desire is not straightforward.

To test their ideas, the researchers ran simulations using different types of customer behaviors and product qualities. In one scenario, they looked at a situation where customers have a continuous range of preferences, and they found that a carefully designed partial disclosure strategy could increase revenue by more than thirty percent compared to simply hiding all information or revealing everything. In another test, they compared their new learning algorithm against older methods that either ignored advertising or used a fixed advertising strategy. The new algorithm consistently outperformed these baselines, especially as the number of sales rounds increased. The simulations confirmed that the algorithm successfully learns to balance the trade-off between exploring new strategies to gather information and exploiting known strategies to make money. It also demonstrated that misspecifying the customer's preferences—assuming they are different from who they really are—can lead to significant revenue losses, reinforcing the need for adaptive learning.

The implications of this work extend beyond theoretical economics. It provides a roadmap for businesses operating in markets with opaque products or uncertain quality, such as online marketplaces, subscription boxes, or on-demand services. The research suggests that the most effective approach is not to rely on a single static strategy, but to use data to continuously refine both the price and the information shared with customers. While the study confirms that advertising can significantly boost revenue, it also offers a cautionary note: the benefits are finite, and the strategy must be tailored to the specific distribution of customer types. A strategy that works for a market of high-end buyers might fail completely for a market of budget-conscious shoppers. By understanding the precise limits of advertising and using efficient learning tools, sellers can navigate these uncertainties with greater confidence, ensuring they extract the maximum possible value from every transaction without overestimating the power of their marketing messages.

Technical Summary: Dynamic Pricing and Advertising with Demand Learning

Problem Formulation

This work addresses the joint optimization of pricing and advertising strategies for a monopolist seller facing demand uncertainty. The setting involves a seller offering a product with varying quality $\omega \in \Omega$ to a market of heterogeneous customers with private types $\theta \sim F$ .

Information Asymmetry and Advertising:
The seller observes the realized product quality $\omega$ , while customers do not. The seller can influence customer beliefs by designing an advertising strategy, modeled as an information policy (or signaling scheme) $\phi$ . This policy prescribes a distribution over signals $\sigma$ sent to customers conditional on the realized quality $\omega$ . Upon observing a signal, customers form a Bayesian posterior belief $\mu$ over the quality and decide to purchase if their expected utility is non-negative.

Valuation and Decision:
Customers have a valuation function $v(\theta, \omega)$ . The paper assumes $v(\theta, \omega)$ is linear in quality $\omega$ and increasing and Lipschitz in type $\theta$ . A customer of type $\theta$ purchasing at price $p$ after observing signal $\sigma$ (inducing posterior mean $q = E[\omega|\sigma]$ ) buys if $v(\theta, q) \geq p$ .

Objectives:
The paper investigates two primary questions:

Static Value: What is the revenue value of advertising compared to pricing alone? How much revenue is lost by restricting the seller to a single quality-independent price?
Dynamic Learning: How can a seller, who does not know the customer type distribution $F$ a priori, adaptively learn and optimize both pricing and advertising strategies to maximize cumulative revenue over a time horizon $T$ ?

The dynamic problem is formulated as an online learning task where the seller minimizes regret, defined as the difference between the total expected revenue of the optimal static policy (knowing $F$ ) and the revenue achieved by the learning algorithm.

Methodology

Static Analysis

The authors characterize the value of advertising by comparing three benchmarks:

$OPT_{PO}$ : Optimal revenue with a single quality-independent price and no advertising (no information disclosed).
$OPT$: Optimal revenue with a single quality-independent price and an optimally designed advertising strategy.
$OPT_{SC}$ : Optimal revenue with an advertising strategy and a price menu that can vary with the realized signal (signal-contingent pricing).

The analysis leverages the linearity of the valuation function in quality. This property implies that the customer's purchase decision depends on the advertising signal only through the posterior mean quality $q$ . Consequently, the seller's advertising strategy can be reformulated as choosing a distribution over posterior means $\rho$ subject to Bayes-consistency constraints (i.e., the distribution must be a mean-preserving contraction of the prior).

Dynamic Learning Algorithm

To address the unknown demand distribution $F$ , the authors propose a model-based online learning algorithm (Algorithm 1).

Key Challenges and Solutions:

High-Dimensional Decision Space: The space of feasible advertising strategies (distributions over posterior means) is infinite-dimensional.
- Solution: The algorithm reduces the problem to learning the demand function $D(\theta) = 1 - F(\theta)$ over a discretized type space.
Feasibility of Discretization: Standard uniform discretization of the type space may render the Bayes-consistency constraints infeasible (i.e., no valid advertising strategy exists for the restricted grid).
- Solution: The authors introduce a novel quality-and-price-dependent discretization scheme. The discretized type space $S$ is constructed not just as a uniform grid, but by including all critical types $\kappa(p, \omega)$ generated by the discretized price grid $P$ and the discrete quality space $\Omega$ . This ensures that for any price in $P$ , there exists a feasible advertising strategy whose induced critical types lie within $S$ .
Lack of Smoothness: The demand function $D(\cdot)$ $D (\cdot)$ is not assumed to be Lipschitz or smooth.
- Solution: The algorithm uses an Upper Confidence Bound (UCB) approach on the discretized type space. It constructs a piecewise-constant UCB estimate of the demand function and solves an optimization problem (analogous to the static problem but with the estimated demand) to select the price and advertising strategy in each round.

Algorithm Mechanics:

Exploration Phase: The algorithm initially explores the discretized type space to gather data.
Optimization Phase: In each round, it computes $D^{UCB}_t(x)$ for all $x \in S$ . It then solves a convex program to find the price $p \in P$ and advertising distribution $\rho$ (supported on $Q_p = \{q : \kappa(p, q) \in S\}$ ) that maximizes the estimated revenue using the UCB demand.
Rounding: A critical technical component is a rounding procedure that maps any optimal continuous strategy to a feasible strategy on the discretized grid with bounded revenue loss ( $2\epsilon$ ).

Key Contributions and Results

1. The Value of Advertising (Static)

The paper provides a tight quantitative characterization of the value of advertising:

Revenue Bound: Under standard assumptions, the optimal revenue with joint advertising and pricing ($OPT$) is at most twice the revenue from pricing alone without advertising ( $OPT_{PO}$ ).
$OPT_{PO} \leq OPT \leq OPT_{SC} \leq 2 \cdot OPT_{PO}$
Tightness: The factor of 2 is tight for all three comparisons ( $OPT/OPT_{PO}$ , $OPT_{SC}/OPT_{PO}$ , and $OPT_{SC}/OPT$ ).
Implication: Advertising can nearly double revenue, but it cannot improve revenue by more than a factor of two once the price is optimized. Furthermore, a single quality-independent price is a robust approximation, achieving at least half the revenue of the more flexible signal-contingent pricing benchmark.

2. Dynamic Learning Regret

The main algorithmic contribution is a computationally efficient online algorithm for the dynamic setting:

Regret Bound: The algorithm achieves a regret of $O(T^{2/3}(m \log T)^{1/3})$ , where $T$ is the time horizon and $m$ is the cardinality of the discrete quality space.
Optimality: This rate matches the known lower bound of $\Omega(T^{2/3})$ for dynamic pricing without advertising (the case where $m=1$ ), up to logarithmic factors. This indicates that learning the optimal advertising policy incurs no additional asymptotic cost compared to learning demand for pricing alone, provided the quality space is small.
Assumptions: The result holds without Lipschitz or smoothness assumptions on the demand function, relying only on the linearity of valuation in quality and mild regularity conditions on the valuation function.

3. Special Cases and Extensions

Additive Valuations: For the widely studied case of additive valuations ( $v(\theta, \omega) = \theta + \omega$ ), the authors derive improved regret bounds. If the quality space is equally spaced, the regret improves to $O(T^{2/3}(\log T)^{1/3} + \sqrt{mT \log T})$ .
Large/Continuous Quality Spaces: For additive valuations with arbitrarily large or continuous quality spaces, a pooling-based reduction allows achieving a regret of $\tilde{O}(T^{3/4})$ independent of the quality space size.

Significance and Claims

The paper claims to be the first to integrate Bayesian persuasion (information design) with dynamic pricing and demand learning.

Theoretical Justification: It provides a rigorous theoretical justification for the common practice of using a single posted price. Even when advertising is used to shape beliefs, a single price is within a factor of two of the optimal signal-contingent pricing strategy.
Practical Relevance: The work addresses the practical challenge where sellers lack prior knowledge of customer preferences. It demonstrates that sellers can learn to jointly optimize information disclosure (advertising) and pricing to maximize revenue, even without knowing the demand curve.
Algorithmic Innovation: The proposed discretization and rounding techniques overcome the high-dimensional and non-smooth nature of the advertising design problem, offering a computationally tractable solution that achieves optimal regret rates.

The authors note that while the 2-factor revenue bound extends to valuations non-linear in quality, the demand-learning results critically rely on the linearity assumption to reduce the state space to posterior means. They identify extending these learning techniques to non-linear valuations as a direction for future work.