Determining Total Favoring Apportionments Using the General Linear Divisor Method

Aspects of total favouring of large or small beneficiaries in proportional apportionments of entities using linear divisor methods (LDM) are discussed in the paper. In this aim, the requirements of apportionments compliance with linear divisor methods’ solution were defined and the conditions of LDM apportionments compliance with the requirements of total favouring large or small beneficiaries were determined. Subsequently, the A1 algorithm for determining the LDM apportionments which totally favour beneficiaries was elaborated. Using A1, calculations for three examples were performed, obtaining: a d’Hondt method’s apportionment which totally favours large beneficiaries, a Sainte-Laguë method’s apportionment which totally favours large beneficiaries and a Dependent linear divisor method’s apportionment which totally favors small beneficiaries. Obtained results confirm the opportunity of using the algorithm A1 for the determining of LDM apportionments which totally favour large beneficiaries or, on the contrary, the small ones.

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.