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Revenue Maximization and Learning in Products Ranking

Ningyuan Chen, Anran Li, Shuoguang Yang

TL;DR

This paper studies revenue-maximizing product ranking under cascade-style consumer behavior with random attention spans. It reveals a nested optimal-ranking structure for fixed attention and develops Best-x, a $1/e$-approximation algorithm under IFR for random spans, along with a clairvoyant upper bound for benchmarking. It further develops RankUCB, an online learning approach that handles feature-based conditional purchase probabilities and censored observations, achieving $\tilde{O}(\sqrt{T})$ regret relative to the approximation. Numerical experiments validate the effectiveness of both Best-x and RankUCB, showing substantial practical gains in revenue and learning efficiency in simulated environments.

Abstract

We consider the revenue maximization problem for an online retailer who plans to display in order a set of products differing in their prices and qualities. Consumers have attention spans, i.e., the maximum number of products they are willing to view, and inspect the products sequentially before purchasing a product or leaving the platform empty-handed when the attention span gets exhausted. Our framework extends the well-known cascade model in two directions: the consumers have random attention spans instead of fixed ones, and the firm maximizes revenues instead of clicking probabilities. We show a nested structure of the optimal product ranking as a function of the attention span when the attention span is fixed. \sg{Using this fact, we develop an approximation algorithm when only the distribution of the attention spans is given. Under mild conditions, it achieves $1/e$ of the revenue of the clairvoyant case when the realized attention span is known. We also show that no algorithms can achieve more than 0.5 of the revenue of the same benchmark. The model and the algorithm can be generalized to the ranking problem when consumers make multiple purchases.} When the conditional purchase probabilities are not known and may depend on consumer and product features, we devise an online learning algorithm that achieves $\tilde{\mathcal{O}}(\sqrt{T})$ regret relative to the approximation algorithm, despite the censoring of information: the attention span of a customer who purchases an item is not observable. Numerical experiments demonstrate the outstanding performance of the approximation and online learning algorithms.

Revenue Maximization and Learning in Products Ranking

TL;DR

This paper studies revenue-maximizing product ranking under cascade-style consumer behavior with random attention spans. It reveals a nested optimal-ranking structure for fixed attention and develops Best-x, a -approximation algorithm under IFR for random spans, along with a clairvoyant upper bound for benchmarking. It further develops RankUCB, an online learning approach that handles feature-based conditional purchase probabilities and censored observations, achieving regret relative to the approximation. Numerical experiments validate the effectiveness of both Best-x and RankUCB, showing substantial practical gains in revenue and learning efficiency in simulated environments.

Abstract

We consider the revenue maximization problem for an online retailer who plans to display in order a set of products differing in their prices and qualities. Consumers have attention spans, i.e., the maximum number of products they are willing to view, and inspect the products sequentially before purchasing a product or leaving the platform empty-handed when the attention span gets exhausted. Our framework extends the well-known cascade model in two directions: the consumers have random attention spans instead of fixed ones, and the firm maximizes revenues instead of clicking probabilities. We show a nested structure of the optimal product ranking as a function of the attention span when the attention span is fixed. \sg{Using this fact, we develop an approximation algorithm when only the distribution of the attention spans is given. Under mild conditions, it achieves of the revenue of the clairvoyant case when the realized attention span is known. We also show that no algorithms can achieve more than 0.5 of the revenue of the same benchmark. The model and the algorithm can be generalized to the ranking problem when consumers make multiple purchases.} When the conditional purchase probabilities are not known and may depend on consumer and product features, we devise an online learning algorithm that achieves regret relative to the approximation algorithm, despite the censoring of information: the attention span of a customer who purchases an item is not observable. Numerical experiments demonstrate the outstanding performance of the approximation and online learning algorithms.

Paper Structure

This paper contains 21 sections, 24 theorems, 120 equations, 4 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Consumer Choice Models and Assortment Optimization
  4. Online Learning
  5. The Model
  6. Revenue Maximization for the Retailer
  7. Best-x Algorithm for Optimal Ranking
  8. Revenue Maximization for Fixed Attention Spans
  9. Random Attention Spans: The Best-x Algorithm and $1/e$ Approximation Ratio
  10. Clairvoyant upper bound for the expected revenue.
  11. Special Cases with Solvable Optimal Ranking
  12. Personalized Ranking via Online Learning
  13. Preliminaries
  14. The RankUCB Algorithm
  15. Numerical Experiments
  16. ...and 6 more sections

Key Result

Lemma 1

Fix attention span at $x$. Suppose the assortment $S$ with size $|S|=x$ is given and $\sigma^x$ maximizes $R(\sigma,x)$ among $\sigma\in P(S)$. We have that the products in $\sigma^{x}$ are displayed in increasing order of their product indices, i.e., $\sigma^{x}(i) \leq \sigma^{x}(i+1)$ for $i=1,\d

Figures (4)

  • Figure 1: A graph illustrating the sequence of how $Z_{t,k}$ and $Y_{t,k}$ are observed.
  • Figure 2: The histogram of 1000 instances for the performance of Best-x and other benchmarks relative to the clairvoyant upper bound under $G^{(1)} = (1,0.95,0.9,\cdots, 0.1,0.05)$, $G^{(2)} = (1,0.9,0.9^2,\cdots, 0.9^{19})$, and $G_k^{(3)} = \prod_{i=1}^{k-1}(1-h_i)$ where $h_i = 0.1 - 0.05i/M$.
  • Figure 3: Performance ratio of RankUCB over 10000 rounds.
  • Figure 4: Estimated failure rate in the learning process.

Theorems & Definitions (32)

  • Example 1
  • Lemma 1
  • Proposition 1
  • Definition 1
  • Proposition 2
  • Theorem 1
  • Proposition 3
  • Theorem 2
  • Proposition 4
  • Remark 1: Rank the remaining products.
  • ...and 22 more