Picking Sequences and Monotonicity in Weighted Fair Division

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.