We consider the problem of fair allocation of indivisible items to agents that have arbitrary entitlements to the items. Every agent $i$ has a valuation function $v_i$ and an entitlement $b_i$, where entitlements sum up to~1. Which allocation should one choose in situations in which agents fail to agree on one acceptable fairness notion? We study this problem in the case in which each agent focuses on the value she gets, and fairness notions are restricted to be {\em share based}. A {\em share} $s$ is an function that maps every $(v_i,b_i)$ to a value $s(v_i,b_i)$, representing the minimal value $i$ should get, and $s$ is {\em feasible} if it is always possible to give every agent $i$ value of at least $s(v_i,b_i)$. Our main result is that for additive valuations over goods there is an allocation that gives every agent at least half her share value, regardless of which feasible share-based fairness notion the agent wishes to use. Moreover, the ratio of half is best possible. More generally, we provide tight characterizations of what can be achieved, both ex-post (as single allocations) and ex-ante (as expected values of distributions of allocations), both for goods and for chores. We also show that for chores one can achieve the ex-ante and ex-post guarantees simultaneously (a ``best of both world" result), whereas for goods one cannot.

翻译：我们考虑将不可分割物品公平分配给对物品拥有任意权益的代理者的问题。每个代理者$i$具有估值函数$v_i$和权益$b_i$（权益总和为1）。当代理者无法就某个可接受的公平概念达成一致时，应当选择何种分配方案？我们在每个代理者仅关注自身所获价值、且公平概念限定为基于份额的情形下研究该问题。份额$s$是将每个$(v_i,b_i)$映射到数值$s(v_i,b_i)$的函数，表示代理者$i$应获得的最小价值；若总能实现每个代理者$i$至少获得$s(v_i,b_i)$的价值，则称该份额是可行的。我们的主要结果表明：对于物品的加法估值，存在一种分配方案能使每个代理者至少获得其份额值的一半，无论该代理者希望采用何种可行的基于份额的公平概念。此外，该比例是最优的。更一般地，我们针对物品和杂务，分别从事后（作为单次分配）和事前（作为分配分布的期望值）两个维度，给出了可实现性的严格刻画。我们还证明：对于杂务可以同时实现事前与事后的保证（即"两全其美"的结果），而对于物品则无法实现。