  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.
