Envy-Free Allocation of Indivisible Goods via Noisy Queries

TL;DR

This work studies envy-free allocation of $m$ indivisible items between two agents when valuations are only accessible through noisy per-item Gaussian queries. The authors develop a non-adaptive, thresholding-based allocation algorithm and prove a tight query complexity of $\widetilde{\Theta}( \frac{m^{2.5}}{\Delta^{2}} )$ in the regime $\Delta \gg m^{1/4}$, with an upper bound of $q = {\widetilde{O}}( \frac{m^{2.5}}{\Delta^{2}} )$ and a matching algorithm-independent lower bound. The analysis leverages Gaussian concentration, careful thresholding, and information-theoretic arguments (e.g., Assouad’s lemma) to handle the noise and the gap parameter $\Delta$, and generalizes to other noise levels $\sigma^2$. The results illuminate the sample complexity of fair division under imperfect information and suggest extensions to more agents and broader fairness notions.

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, $Δ$. 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.

Theorems & Definitions (48)

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