Buy-Many Mechanisms for Many Unit-Demand Buyers

TL;DR

This work extends the buy-many mechanism design paradigm to multi-buyer settings by introducing an ex-ante relaxation that captures multiple extensions of the buy-many constraint. The authors prove that, for unit-demand or additive buyers, a simple sequential item-pricing scheme with buyer-specific prices achieves an $O(\log m)$-approximation to the ex-ante BuyMany revenue, matching the single-buyer benchmark and independent of value distributions. The analysis combines a supply-constrained single-buyer buy-many bound, a Lagrangian relaxation, and a multi-dimensional online contention-resolution style argument to relate ex-ante buy-many revenue to ex-ante and standard item-pricing revenues. These results yield a practical and robust mechanism design approach for multi-buyer, multi-item markets with production costs, and they highlight both the power and the limitations of simple pricing schemes in complex multi-agent environments.

Abstract

A recent line of research has established a novel desideratum for designing approximately-revenue-optimal multi-item mechanisms, namely the buy-many constraint. Under this constraint, prices for different allocations made by the mechanism must be subadditive, implying that the price of a bundle cannot exceed the sum of prices of individual items it contains. This natural constraint has enabled several positive results in multi-item mechanism design bypassing well-established impossibility results. Our work addresses the main open question from this literature of extending the buy-many constraint to multiple buyer settings and developing an approximation. We propose a new revenue benchmark for multi-buyer mechanisms via an ex-ante relaxation that captures several different ways of extending the buy-many constraint to the multi-buyer setting. Our main result is that a simple sequential item pricing mechanism with buyer-specific prices can achieve an $O(\log m)$ approximation to this revenue benchmark when all buyers have unit-demand or additive preferences over m items. This is the best possible as it directly matches the previous results for the single-buyer setting where no simple mechanism can obtain a better approximation. From a technical viewpoint we make two novel contributions. First, we develop a supply-constrained version of buy-many approximation for a single buyer. Second, we develop a multi-dimensional online contention resolution scheme for unit-demand buyers that may be of independent interest in mechanism design.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 3.1
• Theorem \ref{cor:eabuymany-vs-srev}
• Theorem \ref{cor:eabuymany-vs-srev}
• Theorem \ref{cor:eabuymany-vs-srev}
• proof : Proof of Theorem \ref{['thm:constrained-alloc-approx']}
• proof : Proof of Theorem \ref{['thm:cost-gap-example']}
• Theorem \ref{thm:seq-pricing-approx-exante}