Tight Efficiency Bounds for the Probabilistic Serial and Related Mechanisms
Jugal Garg, Yixin Tao, László A. Végh
TL;DR
This work establishes tight efficiency bounds for the Probabilistic Serial mechanism under cardinal preferences and extends these insights to submodular and chore settings. It shows that PS achieves a logarithmic-factor approximation to both Pareto efficiency and maximum Nash welfare, and that these guarantees extend to the submodular randomized assignment framework. The authors provide a polynomial-time algorithm that yields allocations that are envy-free and approximately Pareto-efficient (within e^{1/e}-ε) for random assignment and submodular cases, and adapt similar approaches to Fisher markets to obtain near-optimal Nash welfare under envy constraints. In the chores setting, PS remains envy-free and ordinally efficient, and, under positive disutilities, yields an asymptotically tight n-approximation to Pareto efficiency, marking a first such guarantee for fair and efficient allocations with chores. Collectively, the results bridge ordinal fairness with cardinal efficiency across goods and chores and offer practical, computable mechanisms with strong theoretical guarantees.
Abstract
The Probabilistic Serial (PS) mechanism -- also known as the simultaneous eating algorithm -- is a canonical solution for the random 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 proving that the PS mechanism guarantees $(\ln n+1)$-approximate Pareto efficiency under cardinal preferences. The key part of our analysis shows that PS achieves a logarithmic $(\ln n + 1)$-approximation to the maximum Nash welfare, in stark contrast to the $O(\sqrt{n})$ loss that can arise in utilitarian social welfare. Our results also extend to the more general submodular setting introduced by Fujishige, Sano, and Zhan (ACM TEAC 2018). 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 random assignment problem with chores under cardinal preferences.