The Complexity of Optimal Multidimensional Pricing

TL;DR

This paper investigates the computational complexity of finding revenue-optimal deterministic pricing strategies in a single-buyer setting with independent item values, revealing polynomial solvability for small support sizes and NP-completeness in more general cases.

Contribution

It characterizes the complexity landscape of the optimal item pricing problem, providing polynomial algorithms for support size 2 and NP-completeness results for larger or identical distributions.

Findings

Polynomial-time solution for support size 2 distributions

NP-completeness for support size 3 distributions

NP-completeness persists for identical distributions

Abstract

We resolve the complexity of revenue-optimal deterministic auctions in the unit-demand single-buyer Bayesian setting, i.e., the optimal item pricing problem, when the buyer's values for the items are independent. We show that the problem of computing a revenue-optimal pricing can be solved in polynomial time for distributions of support size 2, and its decision version is NP-complete for distributions of support size 3. We also show that the problem remains NP-complete for the case of identical distributions.