Apportionment with Weighted Seats

TL;DR

This work extends classical apportionment to a weighted-seat setting where each seat $t$ carries a weight $w_t$, and the total weight is $oldsymbol{ ext{ω}}= extstyle\sum_{t=1}^k w_t$. Quotas are redefined as $q(p)= extstyleoldsymbol{ ext{ω}} rac{v_p}{n}$, with party representation $r_{oldsymbol{s}}(p)= extstyle\sum_{t:s_t=p} w_t$, leading to weighted-divisor rules (Adams$_ ext{ω}$, D'Hondt$_ ext{ω}$) and a Greedy variant that extend standard apportionment methods. The authors introduce relaxations of lower and upper quota—WLQ$^o$, WLQ-X, WLQ-1, WLQ-X-r and WUQ$^o$, WUQ-X, WUQ-1—with links to envy-freeness (WEFX) and explore house monotonicity (full-HM and min-HM). Key findings show that full proportionality is harder under weights, but meaningful guarantees are achievable for mild relaxations; two-party WLQ$^o$ is tractable, while general WLQ$^o$ is NP-hard to decide, and empirical Bundestag analyses show that weighted methods such as D'Hondt$_ ext{ω}$ and Greedy perform well on fairness relaxations and empirical distance to quotas. The results highlight practical relevance for real-world settings like Bundestag committees and bankruptcies, and point to fruitful directions for extending weighted apportionment to broader multiwinner and resource-allocation contexts.

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.

Theorems & Definitions (67)

• Definition 1: Divisor methods
• Definition 2: $\text{Adams}_{\omega}$ and $\text{D'Hondt}_{\omega}$
• Definition 3: Greedy Method
• Example 1
• Example 2
• Definition 4: Obtainable Weighted-seat Lower Quota, $\text{WLQ}^o$
• Proposition 1
• Proposition 2
• proof
• Proposition 3