On Approximate Welfare- and Revenue-Maximizing Equilibria for Size-Interchangeable Bidders

In a Walrasian equilibrium (WE), all bidders are envy-free (EF), meaning that their allocation maximizes their utility; and the market clears (MC), meaning that the price of unallocated goods is zero. EF is desirable to ensure the long-term viability of the market. MC ensures that demand meets supply. Any allocation that is part of a WE is also welfare-maximizing; however, it need not be revenue-maximizing. Furthermore, WE need not exist, e.g., in markets where bidders have combinatorial valuations. Œe traditional approach to simultaneously addressing both existence and low revenue is to relax the MC condition and instead require the price of unallocated goods be some, positive reserve price. Œe resulting solution concept, known as Envy-Free Pricing (EFP), has been studied in some special cases, e.g., single-minded bidders. In this paper, we go one step further; we relax EF as well as MC. We propose a relaxation of the EF condition where only winners are envy-free, and further relax the MC condition so that unallocated goods are priced at least at the reserve. We call this new solution concept Restricted Envy-Free Pricing (REFP). We investigate what REFP entails for single-minded bidders, and show that for size-interchangeable bidders (a generalization of single-minded introduced in this paper) we can compute a REFP in polynomial time, given a €xed allocation. As in the study of EFP, we remain interested in maximizing seller revenue. Instead of computing an outcome that simultaneously yields an allocation and corresponding prices, one could €rst solve for an allocation that respects a reserve price, and then solve for a corresponding set of supporting prices, each one at least the reserve. Œis two-step process fails in the case of EFP since, given a €xed allocation, envy-free prices need not exist. However, restricted envy-free prices always exist. We derive necessary and sucient conditions for €nding them in the case of size-interchangeable bidders. Ours is a linear characterization and thus, coupled with natural greedy approximation algorithms for €nding allocations, we propose ecient computational methods to €nd REFP, which we then use within a heuristic to €nd seller-revenue maximizing REFP outcomes. We provide theoretical bounds for our algorithms where possible, and run extensive experiments to evaluate their performance in practice. Compared to other benchmarks in the literature, they perform well on the metrics of revenue and eciency, without incurring too many violations of the true WE conditions.