Optimal Pricing for Bundles: Using Submodularity in Offline and Online Settings

ICLR 2026 Conference Submission21460 Authors

Keywords: pricing, submodular, regret minimization, choice model

Abstract:

We study revenue-maximizing bundle pricing under a cardinality constraint: in each offer the seller chooses a bundle $S\subseteq[n]$ with $|S|\le k$ and posts a single price $p(S)$. Buyers have unknown valuations $V:2^{[n]}\to\mathbb{R}$ drawn from a population distribution and purchase with probability given by a choice model that depends on the surplus $V(S)-p(S)$.
Based on the seller's ability to collect data on bundles at various price points, we analyze two data regimes:

**Offline.** Given a dataset of past purchases (e.g., receipts) and assuming a logit choice model for customers, we identify near-optimal bundle candidates for bundling. We demonstrate, both theoretically and empirically, that submodularity in the valuation of bundles serves as an effective and sample-efficient criterion for determining promising bundles.

**Online.** With sequential interaction and bandit feedback (sale/no sale) under a more general nonparametric smooth choice model, we design an algorithm with regret of $T^{3/4}$ against an $\alpha$-approximation of the optimal revenue in hindsight, where $\alpha \leq 1 - e^{-1}$ is determined by the supermodular curvature of the expected revenue function.

Primary Area: other topics in machine learning (i.e., none of the above)

Submission Number: 21460