Envy-Free Allocation of Indivisible Goods via Noisy Queries

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.