Envy-Free Allocation of Indivisible Goods via Noisy Queries

TL;DR

This paper studies the problem of fairly allocating indivisible goods when agents' valuations are only accessible through noisy queries, providing bounds on the number of queries needed for envy-free allocations.

Contribution

It introduces a new noisy query model for fair division and derives bounds on query complexity for envy-free allocations in the two-agent case.

Findings

Optimal query complexity scales as m^{2.5}/Δ^2 when Δ ≫ m^{1/4}.

Non-adaptive thresholding algorithms achieve the upper bound.

Lower bounds hold even with adaptive queries and unlimited computation.

Abstract

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, $\Delta$. In particular, when $\Delta$ is not too small (namely, $\Delta \gg m^{1/4}$), we establish that the optimal number of queries scales as $\frac{\sqrt m }{(\Delta / m)^2} = \frac{m^{2.5}}{\Delta^2}$ 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.