In a setting where $m$ items need to be partitioned among $n$ agents, we evaluate the performance of mechanisms that take as input each agent's \emph{ordinal preferences}, i.e., their ranking of the items from most- to least-preferred. The standard measure for evaluating ordinal mechanisms is the \emph{distortion}, and the vast majority of the literature on distortion has focused on worst-case analysis, leading to some overly pessimistic results. We instead evaluate the distortion of mechanisms with respect to their expected performance when the agents' preferences are generated stochastically. We first show that no ordinal mechanism can achieve a distortion better than $e/(e-1)\approx 1.582$, even if each agent needs to receive exactly one item (i.e., $m=n$) and every agent's values for different items are drawn i.i.d.\ from the same known distribution. We then complement this negative result by proposing an ordinal mechanism that achieves the optimal distortion of $e/(e-1)$ even if each agent's values are drawn from an agent-specific distribution that is unknown to the mechanism. To further refine our analysis, we also optimize the \emph{distortion gap}, i.e., the extent to which an ordinal mechanism approximates the optimal distortion possible for the instance at hand, and we propose a mechanism with a near-optimal distortion gap of $1.076$. Finally, we also evaluate the distortion and distortion gap of simple mechanisms that have a one-pass structure.

翻译：在需要将$m$个物品分配给$n$个智能体的场景中，我们评估了以每个智能体的序数偏好（即其对物品从最偏好到最不偏好的排序）作为输入的机制性能。评估序数机制的标准度量是失真度，现有失真度研究文献绝大多数聚焦于最坏情况分析，这导致了一些过于悲观的结果。我们转而基于智能体偏好随机生成时的期望性能来评估机制的失真度。首先证明即使每个智能体恰好需要获得一个物品（即$m=n$），且每个智能体对不同物品的价值评估均从同一已知分布独立同分布抽取，任何序数机制也无法实现优于$e/(e-1)\approx 1.582$的失真度。随后通过提出一种序数机制来补充这一负面结论，该机制即使当每个智能体的价值评估来自机制未知的智能体特定分布时，仍能达到$e/(e-1)$的最优失真度。为完善分析，我们还优化了失真间隙（即序数机制对当前实例可能达到的最优失真度的近似程度），并提出具有$1.076$接近最优失真间隙的机制。最后，我们还评估了具有单次遍历结构的简单机制的失真度与失真间隙。