We introduce a problem of fairly allocating indivisible goods (items) in which the agents' valuations cannot be observed directly, but instead can only be accessed via noisy queries. In the two-agent setting with Gaussian noise and bounded valuations, we derive upper and lower bounds on the required number of queries for finding an envy-free allocation in terms of the number of items, $m$, and the negative-envy of the optimal allocation, $Δ$. In particular, when $Δ$ is not too small (namely, $Δ\gg m^{1/4}$), we establish that the optimal number of queries scales as $\frac{\sqrt m }{(Δ/ m)^2} = \frac{m^{2.5}}{Δ^2}$ up to logarithmic factors. Our upper bound is based on non-adaptive queries and a simple thresholding-based allocation algorithm that runs in polynomial time, while our lower bound holds even under adaptive queries and arbitrary computation time.

翻译：本文研究不可分割物品（物品）的公平分配问题，其中智能体的估值无法直接观测，而只能通过噪声查询获取。在具有高斯噪声和有界估值的双智能体场景下，我们推导了寻找无嫉妒分配所需查询次数的上界与下界，该界以物品数量$m$和最优分配的负嫉妒值$Δ$表示。特别地，当$Δ$不太小（即$Δ\gg m^{1/4}$）时，我们证明最优查询次数在忽略对数因子下按$\frac{\sqrt m }{(Δ/ m)^2} = \frac{m^{2.5}}{Δ^2}$的尺度增长。我们的上界基于非自适应查询和一种在多项式时间内运行的简单阈值分配算法，而下界即使对于自适应查询和任意计算时间仍然成立。