Tight Efficiency Bounds for the Probabilistic Serial and Related Mechanisms

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.