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：我们研究在不可直接观测到代理人估值，仅能通过含噪声查询获取信息的情况下，对不可分割物品（商品）进行公平分配的问题。在具有高斯噪声和有界估值的双代理人场景中，我们推导出寻找无嫉妒分配所需查询次数的上下界，该上下界以物品数量 $m$ 和最优分配的负嫉妒程度 $\Delta$ 表示。特别地，当 $\Delta$ 不小（即 $\Delta\gg m^{1/4}$）时，我们证明在忽略对数因子的情况下，最优查询次数约为 $\frac{\sqrt m }{(\Delta/ m)^2} = \frac{m^{2.5}}{\Delta^2}$。我们的上界基于非适应性查询和一种可在多项式时间内运行的简单阈值分配算法，而下界即使考虑适应性查询和任意计算时间仍然成立。