We consider the assignment problem, where $n$ agents have to be matched to $n$ items. Each agent has a preference order over the items. In the serial dictatorship (SD) mechanism the agents act in a particular order and pick their most preferred available item when it is their turn to act. Applying SD using a uniformly random permutation as agent ordering results in the well-known random serial dictatorship (RSD) mechanism. Accurate estimates of the (expected) efficiency of its outcome can be used to assess whether RSD is attractive compared to other mechanisms. In this paper, we explore whether such estimates are possible by sampling a (hopefully) small number of agent orderings and applying SD using them. We consider a value setting in which agents have values for the items as well as a metric cost setting where agents and items are assumed to be points in a metric space, and the cost of an agent for an item is equal to the distance of the corresponding points. We show that a (relatively) small number of samples is enough to approximate the expected social welfare of RSD in the value setting and its expected social cost in the metric cost setting despite the #P-hardness of the corresponding exact computation problems.

翻译：我们研究分配问题，其中 $n$ 个智能体需与 $n$ 个项目进行匹配。每个智能体对项目存在偏好序。在序列独裁（SD）机制中，智能体按特定顺序行动，并在轮到自己时选择当前可用的最偏好项目。若采用均匀随机排列作为智能体顺序来实施 SD，则得到著名的随机序列独裁（RSD）机制。对其结果（期望）效率的精确估计可用于评估 RSD 相较于其他机制的吸引力。本文通过采样（期望）少量智能体顺序并基于这些顺序实施 SD，探讨此类估计是否可行。我们考虑两种设定：一是价值设定，其中智能体对项目具有赋值；二是度量成本设定，其中智能体与项目被视为度量空间中的点，且智能体对项目的成本等于对应点间的距离。我们证明，尽管对应的精确计算问题属于 #P 难问题，但在价值设定中仅需（相对）少量样本即可近似 RSD 的期望社会福利，在度量成本设定中则可近似其期望社会成本。