Multi-Unit Combinatorial Prophet Inequalities

TL;DR

This paper investigates prophet inequalities in multi-unit combinatorial auctions with subadditive valuations, showing how dynamic pricing can match single-item benchmarks and introducing a non-adaptive pricing scheme with supply-dependent competitive ratios.

Contribution

It demonstrates the increased difficulty of multi-unit settings, develops dynamic and non-adaptive pricing strategies, and analyzes their competitive ratios with minimal distributional information.

Findings

Static pricing is less effective in multi-unit than single-item settings.

Dynamic pricing can asymptotically match single-item competitive ratios.

Non-adaptive prices increase with item supply, requiring minimal distribution info.

Abstract

We consider a combinatorial auction setting where buyers have fractionally subadditive (XOS) valuations over the items and the seller's objective is to maximize the social welfare. A prophet inequality in this setting bounds the competitive ratio of sequential allocation (often using item pricing) against the hindsight optimum. We study the dependence of the competitive ratio on the number of copies, $k$, of each item. We show that the multi-unit combinatorial setting is strictly harder than its single-item counterpart in that there is a gap between the competitive ratios achieved by static item pricings in the two settings. However, if the seller is allowed to change item prices dynamically, it becomes possible to asymptotically match the competitive ratio of a single-item static pricing. We also develop a new non-adaptive anonymous multi-unit combinatorial prophet inequality where the item prices are determined up front but increase as the item supply decreases. Setting the item prices in our prophet inequality requires minimal information about the buyers' value distributions -- merely (an estimate of) the expected social welfare accrued by each item in the hindsight optimal solution suffices. Our non-adaptive pricing achieves a competitive ratio that increases strictly as a function of the item supply $k$.