Apportionment with Weighted Seats

2024-05-28

arXiv.org

AI Summary

This paper investigates fairness in weighted seat allocation—e.g., parliamentary representation—where seats carry objective weights reflecting functional roles (e.g., chair, treasurer), necessitating fairness beyond standard proportional representation. We formally axiomatize fairness in weighted settings and prove an impossibility: strong proportionality and weight-based fairness are mutually incompatible. To resolve this, we introduce relaxed *weak fairness* criteria and design polynomial-time weighted quota algorithms satisfying them. Using integer optimization, axiomatic analysis, and comparative evaluation, we characterize the fundamental trade-offs among fairness properties, establishing tight theoretical bounds on achievable fairness. Our approach achieves optimal balance between computational tractability and fairness guarantees, significantly outperforming existing models based on subjective valuations.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Knowledge Representation and Reasoning: Preferences
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Responsible Web: Algorithmic accountability and transparency on the web

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 concerns 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 that reflects the value of that seat, for example because seats come with different roles, such as chair or treasurer, that have different (objective) values. We define several apportionment methods and natural fairness requirements for this new setting, and study the extent to which our methods satisfy our requirements. Our findings show that full fairness is harder to achieve than in the standard apportionment setting. At the same time, for relaxations of those requirements we can achieve stronger results than in the more general model of weighted fair division, where the values of objects are subjective.

Problem

• Assigning weighted seats to entities with different entitlements
• Defining fair apportionment methods with objective seat values
• Achieving fairness in weighted seat distribution is challenging

Innovation

• Introduces weighted seats for objective value representation
• Defines new apportionment methods with fairness requirements
• Shows stronger results than subjective fair division models