Abstract: We study the problem of procurement auctions, in which an auctioneer seeks to acquire services from a group of strategic sellers with private costs. The quality of the services is measured through some submodular function that is known to the auctioneer. Our goal is to design computationally efficient procurement auctions that (approximately) maximize the difference between the quality of the acquired services and the total cost of the sellers, in a way that is incentive compatible (IC) and individual rational (IR) for the sellers, and generates non-negative surplus (NAS) for the auctioneer. {Our contribution is twofold: \textbf{i)} we provide an improved analysis of existing algorithms for non-positive submodular function maximization and \textbf{ii)} we design computationally efficient frameworks that transform submodular function optimization algorithms to mechanisms that are IC and IR for the sellers, NAS for the auctioneer, and approximation-preserving.} Our frameworks are general and work both in the offline setting where the auctioneer can observe the bids and the services of all the sellers simultaneously, and in the online setting where the sellers arrive in an adversarial order and the auctioneer has to make an irrevocable decision whether to purchase their service or not. We further investigate whether it is possible to convert state-of-art submodular optimization algorithms into descending auctions. We focus on the adversarial setting, meaning that the schedule of the descending prices is determined by an adversary. We show that a submodular optimization algorithm satisfying bi-criteria $(1/2,1)$-approximation in welfare can be effectively converted to a descending auction in this setting. We further establish a connection between descending auctions and online submodular optimization. Finally, we demonstrate the practical applications of our frameworks by instantiating them with different state-of-the-art submodular optimization algorithms and comparing their welfare performance through empirical experiments on publicly available datasets that consist of thousands of sellers.

Lay Summary:

In this paper, we study procurement auctions where an auctioneer aims to purchase services from strategic sellers who each have a private cost. The value derived from procuring a subset of sellers is captured by a known submodular function, which naturally models diminishing returns. Our goal is to design mechanisms that approximately maximize the difference between the total value obtained and the total cost of the sellers, while ensuring that the mechanism is incentive compatible, individually rational, and guarantees non-negative surplus for the auctioneer. We also require that the mechanism be computationally efficient.We begin by revisiting the problem of maximizing a submodular function minus a modular cost function and provide improved guarantees for existing algorithms, most notably the distorted greedy algorithm. Our analysis shows that this algorithm satisfies a continuum of bi-criteria approximation guarantees, which we later leverage in mechanism design.We then introduce a general framework that transforms any suitable submodular maximization algorithm into a mechanism satisfying all our desired properties. This framework applies both in an offline setting, where sellers submit bids simultaneously, and in an online setting, where sellers arrive sequentially and decisions must be made irrevocably. The key idea is to preserve the allocation behavior of the original algorithm while carefully constructing payments to ensure truthfulness.We also explore the design of descending auctions, where prices start high and decrease over time until sellers accept. In adversarial scenarios, we show that using a perfectly optimal demand oracle can lead to poor welfare outcomes, whereas approximate greedy oracles can sometimes perform better. We further establish a formal connection between descending auctions and online submodular optimization, showing that limitations in one setting imply limitations in the other.Finally, we evaluate our framework empirically on large-scale instances derived from real-world graphs. We compare various mechanisms in terms of welfare and runtime, and demonstrate that our approaches strike a practical balance between economic guarantees and computational efficiency.