Multi-Unit Combinatorial Prophet Inequalities

TL;DR

This work analyzes prophet inequalities for multi-unit combinatorial auctions with XOS buyers under sequential posted pricing. It develops a supply-based static pricing scheme achieving a competitive ratio of $1-igl(rac{k}{k+1}igr)^k$ and a dynamic pricing scheme attaining $1-Oigl(igl(rac{ ext{log} k}{k}igr)^{1/2}igr)$, bridging toward single-item performance as $k$ grows, and shows online mechanisms can reach $1-1/ ext{sqrt}(k+3)$. It also establishes a hardness gap: unit-demand in the multi-item setting is strictly harder for static pricing than the single-item case, via carefully constructed hard instances and a reduction to a single-parameter optimization; the results extend to multi-unit demands and include a non-adaptive anonymous prophet inequality with prices that rise as supply dwindles. Together, these findings clarify the information requirements and feasibility of approximating social welfare in large-scale multi-item settings, and provide practical pricing rules with provable guarantees that scale with item supply. The paper also situates these results within the broader prophet inequality and mechanism design literature, connecting LP-based relaxations, dynamic dual pricing, and online allocation to design resilient multi-unit pricing schemes.

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 $k$.

Theorems & Definitions (40)

• Theorem 1
• Corollary 2
• Theorem 3
• Theorem 4
• Lemma 5
• Lemma 6
• Theorem 7: Theorem A of fan2015exponential
• proof : Proof of \ref{['lma:item-remain']}
• Lemma 8
• proof