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.

翻译：概率序列（PS）机制——亦称同步消耗算法——是序数偏好下随机分配问题的经典解法。该机制能确保所得随机分配的无嫉妒性与序数有效性。然而，在基数偏好下，其效率可能出现显著退化：已知PS机制可能产生较帕累托最优分配劣$Ω(\ln{n})$倍的分配方案，但该界限是否紧致始终是悬而未决的问题。我们的首要成果通过证明PS机制在基数偏好下能保证$(\ln n+1)$-近似帕累托效率，彻底解决了该问题。分析的核心部分表明，PS机制对最大纳什福利可实现对数级$(\ln n + 1)$近似，这与功利主义社会福利可能出现的$O(\sqrt{n})$损失形成鲜明对比。我们的研究成果还可推广至藤重、佐野与詹（ACM TEAC 2018）提出的更广义次模设定。此外，我们提出一种多项式时间算法，该算法能计算出无嫉妒且$e^{1/e}$-近似帕累托有效的分配方案，从而回答了Tröbst与Vazirani（EC 2024）提出的开放性问题。PS机制同样适用于杂务（而非物品）的分配场景。我们证明在此设定下该机制能保证$n$-近似帕累托有效分配，且该界限在渐近意义下是紧致的。该成果为首个在基数偏好下计算杂务随机分配问题中公平高效分配的近似保证提供了理论依据。