We introduce and analyze new envy-based fairness concepts for agents with
weights that quantify their entitlements in the allocation of indivisible
items. We propose two variants of weighted envy-freeness up to one item (WEF1):
strong, where envy can be eliminated by removing an item from the envied
agent's bundle, and weak, where envy can be eliminated either by removing an
item (as in the strong version) or by replicating an item from the envied
agent's bundle in the envying agent's bundle. We show that for additive
valuations, an allocation that is both Pareto optimal and strongly WEF1 always
exists and can be computed in pseudo-polynomial time; moreover, an allocation
that maximizes the weighted Nash social welfare may not be strongly WEF1, but
always satisfies the weak version of the property. Moreover, we establish that
a generalization of the round-robin picking sequence algorithm produces in
polynomial time a strongly WEF1 allocation for an arbitrary number of agents;
for two agents, we can efficiently achieve both strong WEF1 and Pareto
optimality by adapting the adjusted winner procedure. Our work highlights
several aspects in which weighted fair division is richer and more challenging
than its unweighted counterpart.

本文介绍和分析适用于配有权重的代理商在不可分配的物品的分配中的公平概念。我们提出了两种权重嫉妒自由概念：强权重嫉妒自由和弱权重嫉妒自由。我们证明，满足强权重嫉妒自由和帕累托最优的分配总是存在且可以在伪多项式时间内计算。此外，我们还建立了一个轮换选择算法的泛化版本，可以在多项式时间内为任意数量的代理商产生具有强权重嫉妒自由的分配。最后，我们强调了权重公平分配比其无权重对应更为复杂。