Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many Buyers

TL;DR

This paper introduces a versatile framework for designing approximate multi-buyer Bayesian combinatorial auction mechanisms by leveraging single buyer mechanisms, applicable under broad assumptions about buyer independence and linear separability.

Contribution

The authors develop a general framework that reduces multi-buyer auction design to single buyer problems without strong valuation assumptions, providing approximation guarantees and extending prophet inequalities.

Findings

Framework achieves at least 50% approximation for single buyer case.

Multi buyer mechanisms approximate optimal with factor depending on buyer distribution.

Generalizes prophet inequalities for broader auction settings.

Abstract

For Bayesian combinatorial auctions, we present a general framework for approximately reducing the mechanism design problem for multiple buyers to single buyer sub-problems. Our framework can be applied to any setting which roughly satisfies the following assumptions: (i) buyers' types must be distributed independently (not necessarily identically), (ii) objective function must be linearly separable over the buyers, and (iii) except for the supply constraints, there should be no other inter-buyer constraints. Our framework is general in the sense that it makes no explicit assumption about buyers' valuations, type distributions, and single buyer constraints (e.g., budget, incentive compatibility, etc). We present two generic multi buyer mechanisms which use single buyer mechanisms as black boxes; if an $\alpha$-approximate single buyer mechanism can be constructed for each buyer, and if no buyer requires more than $\frac{1}{k}$ of all units of each item, then our generic multi buyer mechanisms are $\gamma_k\alpha$-approximation of the optimal multi buyer mechanism, where $\gamma_k$ is a constant which is at least $1-\frac{1}{\sqrt{k+3}}$. Observe that $\gamma_k$ is at least 1/2 (for $k=1$) and approaches 1 as $k \to \infty$. As a byproduct of our construction, we present a generalization of prophet inequalities. Furthermore, as applications of our framework, we present multi buyer mechanisms with improved approximation factor for several settings from the literature.