Uniform Budgets and the Envy-Free Pricing Problem

Abstract

We consider the unit-demand min-buying pricing problem, in which we want to compute revenue maximizing prices for a set of products $\mathcal{P}$ assuming that each consumer from a set of consumer samples $\mathcal{C}$ will purchase her cheapest affordable product once prices are fixed. We focus on the special uniform-budget case, in which every consumer has only a single non-zero budget for some set of products. This constitutes a special case also of the unit-demand envy-free pricing problem.

We show that, assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas, the unit-demand min-buying pricing problem with uniform budgets cannot be approximated in polynomial time within $\mathcal{O}(\log ^{\varepsilon} |\mathcal{C}|)$ for some ε> 0. This is the first result giving evidence that unit-demand envy-free pricing, as well, might be hard to approximate essentially better than within the known logarithmic ratio.

We then introduce a slightly more general problem definition in which consumers are given as an explicit probability distribution and show that in this case the envy-free pricing problem can be shown to be inapproximable within $\mathcal{O}(|\mathcal{P}|^{\varepsilon})$ assuming NP $\nsubseteq \bigcap _{\delta >0}$ BPTIME($2^{\mathcal{O}(n^{\delta})}$). Finally, we briefly argue that all the results apply to the important setting of pricing with single-minded consumers as well.