Pricing Ordered Items

TL;DR

This paper analyzes the revenue guarantees and computational complexity of item pricing when items are partitioned into categories, showing improved approximation bounds and a PTAS for constant categories, with NP-hardness results for the single-category case.

Contribution

It introduces a refined analysis of item pricing based on item categories, providing new approximation guarantees and algorithms, and establishing hardness results.

Findings

Item-pricing guarantees an O(log k) approximation when items are partitioned into k categories.

A PTAS exists for computing optimal item prices when k is constant.

The problem is strongly NP-hard even when there is only one category.

Abstract

We study the revenue guarantees and approximability of item pricing. Recent work shows that with $n$ heterogeneous items, item-pricing guarantees an $O(\log n)$ approximation to the optimal revenue achievable by any (buy-many) mechanism, even when buyers have arbitrarily combinatorial valuations. However, finding good item prices is challenging -- it is known that even under unit-demand valuations, it is NP-hard to find item prices that approximate the revenue of the optimal item pricing better than $O(\sqrt{n})$. Our work provides a more fine-grained analysis of the revenue guarantees and computational complexity in terms of the number of item ``categories'' which may be significantly fewer than $n$. We assume the items are partitioned in $k$ categories so that items within a category are totally-ordered and a buyer's value for a bundle depends only on the best item contained from every category. We show that item-pricing guarantees an $O(\log k)$ approximation to the optimal (buy-many) revenue and provide a PTAS for computing the optimal item-pricing when $k$ is constant. We also provide a matching lower bound showing that the problem is (strongly) NP-hard even when $k=1$. Our results naturally extend to the case where items are only partially ordered, in which case the revenue guarantees and computational complexity depend on the width of the partial ordering, i.e. the largest set for which no two items are comparable.