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 study a new 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 proportionality up to one good (PROP1), envy-freeness up to one good (EF1), and maximin share (MMS). Our main result is an algorithm that computes an allocation satisfying 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.

翻译：我们研究了将 $m$ 个不可分物品公平分配给 $n$ 个代理人的问题，其中各代理人对物品可能有不同偏好。在传统模型中，代理人的估值作为算法输入。本文研究了一种新的基于比较的查询模型，该模型向代理人展示两组物品包，代理人通过告知算法其更偏好的物品包来响应。我们研究了计算满足若干公平性概念（包括至多一个物品的比例性(PROP1)、至多一个物品的无嫉妒性(EF1)和最大最小份额(MMS)）的分配方案的查询复杂度。主要成果是提出了一种算法，在代理人数量为常数 $n$ 时，通过 $O(\log m)$ 次查询即可计算出同时满足 PROP1 和 $\frac12$-MMS 的分配方案。针对同质可加性估值，我们提出了一种算法，在代理人数量为常数 $n$ 时，通过 $O(\log m)$ 次查询即可计算出满足 EF1 的分配方案。作为对正面结果的补充，我们证明了即使仅有两个代理人，针对上述三种公平性概念中的任意一种，查询复杂度的下界均为 $\Omega(\log m)$。