Apportionment with Weighted Seats

TL;DR

This paper extends traditional apportionment by assigning weights to seats, exploring new fairness criteria and methods, revealing that achieving full fairness is more complex but attainable under relaxed conditions.

Contribution

It introduces weighted seat apportionment, defines new fairness criteria, and analyzes the feasibility of fair methods in this enriched model.

Findings

Full fairness is harder to achieve with weighted seats.

Certain relaxed fairness conditions can be satisfied more strongly.

Weighted apportionment offers a more nuanced approach to resource allocation.

Abstract

Apportionment is the task of assigning resources to entities with different entitlements in a fair manner, and specifically a manner that is as proportional as possible. The best-known application is the assignment of parliamentary seats to political parties based on their share in the popular vote. Here we enrich the standard model of apportionment by associating each seat with a weight representing the (objective) value of that seat. A seat's weight reflects the fact that different seats might come with different roles, such as chair or treasurer. We define several apportionment methods and natural fairness requirements for this new setting, and we study the extent to which our methods satisfy these requirements. Our findings show that full fairness is harder to achieve than in the standard apportionment setting. Yet, for several natural relaxations of those requirements we can achieve stronger results than in the more expressive model of fair division with entitlements, where the values of objects are subjective.