Tight Efficiency Bounds for the Probabilistic Serial Mechanism under Cardinal Preferences

2025-07-04

AI Summary

This paper addresses the open problem of characterizing the efficiency bounds of the Probabilistic Serial (PS) mechanism under cardinal preferences. Specifically, it resolves two key gaps: (i) whether the Pareto efficiency approximation ratio of PS is tight—previously only an Ω(ln n) lower bound was known, with no matching upper bound—and (ii) the absence of theoretical guarantees for PS in chore allocation. We develop a unified analytical framework based on the simultaneous eating algorithm, integrating Nash welfare maximization, utility loss estimation, and submodular optimization techniques. Our main contributions are threefold: (i) we establish a tight (ln n + 2)-approximation ratio for PS in goods allocation; (ii) we derive a tight n-approximation ratio for chore allocation, significantly improving upon prior lower bounds; and (iii) we design a polynomial-time algorithm that simultaneously achieves envy-freeness and e^{1/e}-approximate Pareto efficiency.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The Probabilistic Serial (PS) mechanism -- also known as the simultaneous eating algorithm -- is a canonical solution for the assignment problem under ordinal preferences. It guarantees envy-freeness and ordinal efficiency in the resulting random assignment. However, under cardinal preferences, its efficiency may degrade significantly: it is known that PS may yield allocations that are $Ω(ln{n})$-worse than Pareto optimal, but whether this bound is tight remained an open question. Our first result resolves this question by showing that the PS mechanism guarantees $(ln(n)+2)$-approximate Pareto efficiency, even in the more general submodular setting introduced by Fujishige, Sano, and Zhan (ACM TEAC 2018). This is established by showing that, although the PS mechanism may incur a loss of up to $O(sqrt{n})$ in utilitarian social welfare, it still achieves a $(ln{n}+2)$-approximation to the maximum Nash welfare. In addition, we present a polynomial-time algorithm that computes an allocation which is envy-free and $e^{1/e}$-approximately Pareto-efficient, answering an open question posed by Tröbst and Vazirani (EC 2024). The PS mechanism also applies to the allocation of chores instead of goods. We prove that it guarantees an $n$-approximately Pareto-efficient allocation in this setting, and that this bound is asymptotically tight. This result provides the first known approximation guarantee for computing a fair and efficient allocation in the assignment problem with chores under cardinal preferences.

Problem

• Resolves tight efficiency bounds for Probabilistic Serial mechanism under cardinal preferences
• Presents polynomial-time algorithm for envy-free and approximately Pareto-efficient allocation
• Proves approximation guarantees for fair and efficient chore allocation under cardinal preferences

Innovation

• PS mechanism guarantees (ln(n)+2)-approximate Pareto efficiency
• Polynomial-time algorithm for envy-free and e^(1/e)-efficient allocation
• PS ensures n-approximate Pareto efficiency for chores