Dynamic Pricing and Advertising with Demand Learning

TL;DR

This work studies a monopolist’s joint design of dynamic pricing and fully flexible advertising (information signaling) to shape customer valuations. Using Bayesian persuasion as the information-design backbone, the authors quantify the value of advertising and show a tight universal bound Γ^* = 2 on the revenue gain from signaling, meaning advertising can at most double revenue in the worst case. For settings with unknown demand functions and valuations linear in product quality, they develop a computationally efficient online algorithm achieving a regret of order O$\left(T^{2/3}(m\log T)^{1/3}\right)$, where m is the cardinality of the quality space, without requiring Lipschitz/smoothness assumptions on demand. The algorithm leverages an instance-dependent discretization of the type space and a UCB-based estimate of demand, together with a novel rounding procedure to ensure feasibility of advertising strategies; improved results are obtained for additive valuations. Together, these results justify advertising as a revenue lever and provide practical, provably near-optimal learning strategies for pricing and signaling under demand uncertainty, with clear implications for markets where signaling product characteristics through advertising is feasible and privacy-sensitive pricing is desirable.

Abstract

We consider a novel pricing and advertising framework, where a seller not only sets product price but also designs flexible 'advertising schemes' to influence customers' valuation of the product. We impose no structural restriction on the seller's feasible advertising strategies and allow her to advertise the product by disclosing or concealing any information. Following the literature in information design, this fully flexible advertising can be modeled as the seller being able to choose any information policy that signals the product quality/characteristic to the customers. Customers observe the advertising signal and infer a Bayesian belief over the products. We aim to investigate two questions in this work: (1) What is the value of advertising? To what extent can advertising enhance a seller's revenue? (2) Without any apriori knowledge of the customers' demand function, how can a seller adaptively learn and optimize both pricing and advertising strategies using past purchase responses? To study the first question, we introduce and study the value of advertising - a revenue gap between using advertising vs not advertising, and we provide a crisp tight characterization for this notion for a broad family of problems. For the second question, we study the seller's dynamic pricing and advertising problem with demand uncertainty. Our main result for this question is a computationally efficient online algorithm that achieves an optimal $O(T^{2/3}(m\log T)^{1/3})$ regret rate when the valuation function is linear in the product quality. Here $m$ is the cardinality of the discrete product quality domain and $T$ is the time horizon. This result requires some mild regularity assumptions on the valuation function, but no Lipschitz or smoothness assumption on the customers' demand function. We also obtain several improved results for the widely considered special case of additive valuations.

Figures (6)

• Figure 1: Graphical illustration of binary product quality examples in \ref{['subsec:example']}.
• Figure 2: Graphical illustration for Procedure \ref{['algo_construction_adver_general']}. Given the input price and advertising $(p, \rho)$, fix a posterior mean $q \in\mathsf{supp}(\rho)$ where $\{i'\in[m]: \rho_{i'}(q) > 0\} = \{i, j\}$ (drawn in black dashed line). According to the procedure, we first identify $x = \kappa(p, q)$, and $x^\dagger = \kappa(p^\dagger, q) \in ((z-1)\varepsilon, z\varepsilon)$ where the constructed price $p^\dagger$ is defined as in the procedure. We then find two posterior means $q_L, q_R$ (here $q_L \ge \bar{\omega}_i, q_R \le \bar{\omega}_j$) such that $\kappa(p^\dagger, q_L) = z\varepsilon$ and $\kappa(p^\dagger, q_R) = (z-1)\varepsilon$ (drawn in brown dashed line), and $\kappa(p^\dagger, q_R) < \kappa(p^\dagger, q_L) < \kappa(p, q)$.
• Figure 3: Setting 1: compared to no-information advertising strategy and full-information advertising strategy
• Figure 4: Setting 2: compared to no-information advertising strategy and full-information advertising strategy
• Figure 5: Setting 1: compared to ETC algorithm with different pure exploration rounds

Theorems & Definitions (47)

• Remark 2.2
• Theorem 3.1: Value of advertising
• Definition 4.1: Critical type
• Example 4.2
• Proposition 4.0
• Remark 4.3
• Theorem 5.1: Regret upper bound
• Proposition 5.2: Regret lower bound
• Proposition 5.2
• Lemma 5.3: FTX-22