An O(log log n)-approximate budget feasible mechanism for subadditive valuations

2025-06-05

AI Summary

This paper studies budget-feasible mechanism design: designing a polynomial-time mechanism for buyers with subadditive valuation functions that ensures incentive compatibility, total payment within budget $B$, and maximizes the value of the allocated item set. The best prior approximation ratio was $O(log n / log log n)$, stagnant for years. We break this long-standing barrier, achieving the first $O(log log n)$ approximation ratio. Our approach leverages a demand oracle and integrates threshold pricing, sampling-based calibration, and recursive filtering—advanced combinatorial optimization techniques tailored to subadditive valuations. The resulting mechanism is strictly truthful, budget-feasible, and computationally efficient. To our knowledge, this is the strongest approximation guarantee known for general subadditive valuations under budget feasibility, representing a significant theoretical advance in algorithmic mechanism design.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In budget-feasible mechanism design, there is a set of items $U$, each owned by a distinct seller. The seller of item $e$ incurs a private cost $overline{c}_e$ for supplying her item. A buyer wishes to procure a set of items from the sellers of maximum value, where the value of a set $Ssubseteq U$ of items is given by a valuation function $v:2^U o mathbb{R}_+$. The buyer has a budget of $B in mathbb{R}_+$ for the total payments made to the sellers. We wish to design a mechanism that is truthful, that is, sellers are incentivized to report their true costs, budget-feasible, that is, the sum of the payments made to the sellers is at most the budget $B$, and that outputs a set whose value is large compared to $ ext{OPT}:=max{v(S):overline{c}(S)le B,Ssubseteq U}$. Budget-feasible mechanism design has been extensively studied, with the literature focussing on (classes of) subadditive valuation functions, and various polytime, budget-feasible mechanisms, achieving constant-factor approximation, have been devised for the special cases of additive, submodular, and XOS valuations. However, for general subadditive valuations, the best-known approximation factor achievable by a polytime budget-feasible mechanism (given access to demand oracles) was only $O(log n / log log n)$, where $n$ is the number of items. We improve this state-of-the-art significantly by designing a budget-feasible mechanism for subadditive valuations that emph{achieves a substantially-improved approximation factor of $O(loglog n)$ and runs in polynomial time, given access to demand oracles.}

Problem

• Design truthful budget-feasible mechanism for subadditive valuations
• Achieve O(log log n) approximation for maximum value procurement
• Improve upon previous O(log n / log log n) approximation results

Innovation

• O(log log n)-approximate budget feasible mechanism
• Polynomial time with demand oracles
• Handles general subadditive valuations