Multi-Unit Combinatorial Prophet Inequalities

2025-05-21

AI Summary

This paper studies social welfare maximization in multi-unit combinatorial auctions with buyers possessing XOS valuations, focusing on competitive ratio analysis of sequential allocation mechanisms. We first uncover a fundamental hardness gap between multi-unit and single-unit prophet inequalities for items with $k$ copies per type. We propose the first non-adaptive, anonymous, supply-aware dynamic pricing mechanism achieving a competitive ratio of $1 - O(1/sqrt{k})$, strictly improving upon the single-unit bound. Additionally, we design a progressive static pricing scheme that attains the same asymptotic guarantee using only estimates of each item’s expected contribution to the optimal social welfare. Unlike conventional sequential mechanisms whose performance is insensitive to $k$, our mechanism establishes the first theoretically proven improvement in competitive ratio as $k$ increases—providing the first $k$-dependent tight bound for multi-unit combinatorial auctions.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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 $k$.

Problem

• Maximizing social welfare in multi-unit combinatorial auctions with XOS valuations
• Comparing competitive ratios of static vs dynamic pricing in multi-unit settings
• Developing non-adaptive pricing strategies with minimal distributional information

Innovation

• Dynamic item pricing improves competitive ratio
• Non-adaptive pricing increases with item supply
• Minimal buyer value distribution information required