Fast Revenue Maximization

Abstract

Problem definition: We study a data-driven pricing problem in which a seller offers a price for a single item based on demand observed at a small number of historical prices. Our goal is to derive precise evaluation procedures of the value of the historical information gathered by the seller, along with prescriptions for more efficient price experimentation. Methodology/results: Our main methodological result is an exact characterization of the maximin ratio (defined as the worst-case revenue garnered by a seller who only relies on past data divided by the optimal revenue achievable with full knowledge of the distribution of values). This result allows to measure the value of any historical data consisting of prices and corresponding conversion rates. We leverage this central reduction to provide new insights about price experimentation. Managerial implications: Motivated by practical constraints that impede the seller from changing prices abruptly, we first illustrate our framework by evaluating the value of local information and show that the mere sign of the gradient of the revenue curve at a single point can provide significant information to the seller. We then showcase how our framework can be used to run efficient price experiments. On the one hand, we develop a method to select the next price experiment that the seller should use to maximize the future robust performance. On the other hand, we demonstrate that our result allows to considerably reduce the number of price experiments needed to reach preset revenue guarantees through dynamic pricing algorithms.

Figures (5)

• Figure 1: Illustration of the implications of the bounds derived in \ref{['lemma:singlecross']}.
• Figure 2: Value of a gradient measurement for various $q_1$ ($\pazocal{C} = \pazocal{F}$). The maximin ratio is computed by evaluating $\underline{\pazocal{L}}(\pazocal{C}(\mathcal{I}_{\mathbf{p}, \mathbf{q}}), \mathbb{A})$ with a discretization involving $M = 2500$ points.
• Figure 3: Value of the gradient-sign for various $q_1$. The maximin ratio with gradient-sign information is computed by modifying the evaluation of $\underline{\pazocal{L}}(\pazocal{C}(\{(p_1,q_1)\}), \mathbb{A})$ where we only consider the constraints in \ref{['eq:constraint_lb']} for $i$ such that $a_i \leq p_1$ (resp. $a_i \geq p_1$) when the gradient-sign is negative (resp. positive). ($p_1=10$)
• Figure 4: Value of one additional global experiment for various $(p_1, q_1)$.
• Figure 5: Distribution of the number of price queries needed to achieve 99% worst-case ratio performance. ($[\underline{v}, \overline{v}] = [1, 100]$)

Theorems & Definitions (23)

• Theorem 1: Reduction for Policy Evaluation
• Lemma 1: Local Bounds
• proof : Proof of \ref{['lemma:singlecross']}
• Proposition 1: Worst-case Revenue
• Lemma 2
• proof : Proof of \ref{['lem:lower_revenue']}
• Lemma 3
• proof : Proof of \ref{['lem:L_belongs']}
• Lemma 4
• proof : Proof of \ref{['lem:max_L']}