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Procurement Auctions via Approximately Optimal Submodular Optimization

Yuan Deng, Amin Karbasi, Vahab Mirrokni, Renato Paes Leme, Grigoris Velegkas, Song Zuo

TL;DR

It is shown that a submodular optimization algorithm satisfying bi-criteria $(1/2, 1)$-approximation in welfare can be effectively adapted to a descending auction, and a connection between descending auctions and online submodular optimization is established.

Abstract

We study procurement auctions, where an auctioneer seeks to acquire services from strategic sellers with private costs. The quality of services is measured by a submodular function 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, while ensuring incentive compatibility (IC), individual rationality (IR) for sellers, and non-negative surplus (NAS) for the auctioneer. Our contributions are twofold: (i) we provide an improved analysis of existing algorithms for non-positive submodular function maximization, and (ii) we design efficient frameworks that transform submodular optimization algorithms into mechanisms that are IC, IR, NAS, and approximation-preserving. These frameworks apply to both the offline setting, where all sellers' bids and services are available simultaneously, and the online setting, where sellers arrive in an adversarial order, requiring the auctioneer to make irrevocable decisions. We also explore whether state-of-the-art submodular optimization algorithms can be converted into descending auctions in adversarial settings, where the schedule of 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 adapted to a descending auction. Additionally, we establish a connection between descending auctions and online submodular optimization. Finally, we demonstrate the practical applications of our frameworks by instantiating them with state-of-the-art submodular optimization algorithms and empirically comparing their welfare performance on publicly available datasets with thousands of sellers.

Procurement Auctions via Approximately Optimal Submodular Optimization

TL;DR

It is shown that a submodular optimization algorithm satisfying bi-criteria -approximation in welfare can be effectively adapted to a descending auction, and a connection between descending auctions and online submodular optimization is established.

Abstract

We study procurement auctions, where an auctioneer seeks to acquire services from strategic sellers with private costs. The quality of services is measured by a submodular function 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, while ensuring incentive compatibility (IC), individual rationality (IR) for sellers, and non-negative surplus (NAS) for the auctioneer. Our contributions are twofold: (i) we provide an improved analysis of existing algorithms for non-positive submodular function maximization, and (ii) we design efficient frameworks that transform submodular optimization algorithms into mechanisms that are IC, IR, NAS, and approximation-preserving. These frameworks apply to both the offline setting, where all sellers' bids and services are available simultaneously, and the online setting, where sellers arrive in an adversarial order, requiring the auctioneer to make irrevocable decisions. We also explore whether state-of-the-art submodular optimization algorithms can be converted into descending auctions in adversarial settings, where the schedule of descending prices is determined by an adversary. We show that a submodular optimization algorithm satisfying bi-criteria -approximation in welfare can be effectively adapted to a descending auction. Additionally, we establish a connection between descending auctions and online submodular optimization. Finally, we demonstrate the practical applications of our frameworks by instantiating them with state-of-the-art submodular optimization algorithms and empirically comparing their welfare performance on publicly available datasets with thousands of sellers.

Paper Structure

This paper contains 27 sections, 15 theorems, 54 equations, 3 figures, 1 table, 11 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Preliminaries
  4. Submodular Optimization Algorithms
  5. Improved Analysis for Distorted Greedy
  6. A Mechanism Design Framework
  7. Online Mechanism Design Framework
  8. Descending Auctions
  9. Experiments
  10. Running Time Comparison
  11. Welfare Comparison
  12. Conclusion
  13. Related Work
  14. Gains From Trades vs. Utilitarian Objectives.
  15. Budget-feasible Mechanism Design.
  16. ...and 12 more sections

Key Result

Theorem 3.1

Let $\mathcal{N}$ be a universe of $n$ elements, $f: 2^\mathcal{N} \rightarrow \mathbb{R}_{\geq 0}$ be a monotone submodular function, and $c: \mathcal{N} \rightarrow \mathbb{R}_{\geq 0}$ be a cost function. Let $\mathsf{OPT}$ be the optimal solution of the objective $\max_{S \subseteq \mathcal{N},

Figures (3)

  • Figure 1: Average running time for different mechanisms at different sample sizes $n$. Note that both axes are log-scaled.
  • Figure 2: Welfare as a function of the fraction of active agents for $n \in \{100, 200, 500\}$.
  • Figure 3: Welfare as a function of the fraction of active agents for $n \in \{1000, 2000, 4000\}$.

Theorems & Definitions (28)

  • Theorem 3.1
  • Proposition 4.1
  • Theorem 4.3
  • Theorem 4.5
  • Theorem 5.1
  • Theorem 5.2
  • Remark 5.3
  • Theorem B.1: feldman2021guess
  • Lemma B.2: harshaw2019submodular
  • Lemma B.3: harshaw2019submodular
  • ...and 18 more