Logarithmic Comparison-Based Query Complexity for Fair Division of Indivisible Goods

TL;DR

This paper introduces a comparison-based query model for fair division of indivisible goods, providing efficient algorithms with logarithmic query complexity for key fairness notions and establishing matching lower bounds.

Contribution

It develops the first logarithmic query algorithms for PROP1, MMS, and EF1 fairness notions in the comparison-based model, and proves lower bounds matching these complexities.

Findings

Algorithms achieve $O(\log m)$ query complexity for PROP1 and $rac{1}{2}$-MMS fairness.

An $O(\log m)$ query algorithm for EF1 fairness in identical, additive valuations.

Lower bounds show $\Omega(\log m)$ queries are necessary for these fairness notions.

Abstract

We study the problem of fairly allocating $m$ indivisible goods to $n$ agents, where agents may have different preferences over the goods. In the traditional setting, agents' valuations are provided as inputs to the algorithm. In this paper, we adopt the query model, which has been widely considered for other similar problems (such as matching [Nis21], graph isomorphism [OS18], and equilibrium in game [Bab16]), and apply it to the fair division problem. In particular, we consider a new \emph{comparison-based query model}, where the algorithm presents two bundles of goods to an agent and the agent responds by telling the algorithm which bundle she prefers. We investigate the query complexity for computing allocations with several fairness notions, including \emph{proportionality up to one good} (PROP1), \emph{envy-freeness up to one good} (EF1), and \emph{maximin share} (MMS). Our main result is an algorithm that computes an allocation that satisfies both PROP1 and $\frac12$-MMS within $O(\log m)$ queries with a constant number of $n$ agents. For identical and additive valuation, we present an algorithm for computing an EF1 allocation within $O(\log m)$ queries with a constant number of $n$ agents. To complement the positive results, we show that the lower bound of the query complexity for any of the three fairness notions is $\Omega(\log m)$ even with two agents.