We study the problem of fairly allocating indivisible items to agents with different entitlements, which captures, for example, the distribution of ministries among political parties in a coalition government. Our focus is on picking sequences derived from common apportionment methods, including five traditional divisor methods and the quota method. We paint a complete picture of these methods in relation to known envy-freeness and proportionality relaxations for indivisible items as well as monotonicity properties with respect to the resource, population, and weights. In addition, we provide characterizations of picking sequences satisfying each of the fairness notions, and show that the well-studied maximum Nash welfare solution fails resource- and population-monotonicity even in the unweighted setting. Our results serve as an argument in favor of using picking sequences in weighted fair division problems.

翻译：我们研究将不可分割的物品公平分配给享有不同应享权利的代理人的问题,这包括,例如,在联合政府中,各部在政党之间的分配;我们的重点是从共同分配方法,包括五种传统的分割方法和配额方法中挑选出顺序;我们全面描述这些方法,以已知的无嫉妒和相称性放松不可分割物品以及资源、人口和权重方面的单一性能;此外,我们提供了符合每一项公平概念的分选顺序的特征,并表明,即使没有加权,经过良好研究的纳什福利方案在资源和人口流动方面都没有成功。我们的结果是赞成在加权公平分配问题上采用分选顺序的论据。