Optimal Pricing is Hard

TL;DR

This paper proves that computing the revenue-optimal deterministic auction in various Bayesian settings is computationally hard, highlighting the complexity of optimal pricing even in seemingly simple single-buyer scenarios.

Contribution

It establishes the computational intractability of optimal deterministic auction design in unit-demand and multi-item settings with independent values, using complexity and reduction proofs.

Findings

Computing optimal auction prices is #P-hard.

Optimal pricing problems are reducible from SQRT-SUM complexity.

Simple pricing schemes can approximate revenue but differ significantly in structure.

Abstract

We show that computing the revenue-optimal deterministic auction in unit-demand single-buyer Bayesian settings, i.e. the optimal item-pricing, is computationally hard even in single-item settings where the buyer's value distribution is a sum of independently distributed attributes, or multi-item settings where the buyer's values for the items are independent. We also show that it is intractable to optimally price the grand bundle of multiple items for an additive bidder whose values for the items are independent. These difficulties stem from implicit definitions of a value distribution. We provide three instances of how different properties of implicit distributions can lead to intractability: the first is a #P-hardness proof, while the remaining two are reductions from the SQRT-SUM problem of Garey, Graham, and Johnson. While simple pricing schemes can oftentimes approximate the best scheme in revenue, they can have drastically different underlying structure. We argue therefore that either the specification of the input distribution must be highly restricted in format, or it is necessary for the goal to be mere approximation to the optimal scheme's revenue instead of computing properties of the scheme itself.