A simple and fast linear-time algorithm for divisor methods of apportionment

Proportional apportionment is the problem of assigning seats to states (resp. parties) according to their relative share of the population (resp. votes), a field heavily influenced by the early work of Michel Balinski, not least his influential 1982 book with Peyton Young (Fair representation, 2nd e...

Full description

Saved in:
Bibliographic Details
Published in:Mathematical programming Vol. 203; no. 1-2; pp. 187 - 205
Main Authors: Reitzig, Raphael, Wild, Sebastian
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2024
Springer
Subjects:
Algorithms
Analysis
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
d’Hondt method
68Q25
91B12
Proportional apportionment
Selection algorithms
68Q17
Divisor methods
Fair division
Rounding percentages
ISSN:0025-5610, 1436-4646
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Proportional apportionment is the problem of assigning seats to states (resp. parties) according to their relative share of the population (resp. votes), a field heavily influenced by the early work of Michel Balinski, not least his influential 1982 book with Peyton Young (Fair representation, 2nd edn. Brookings Institution Press, Washington, D.C., 2001). In this article, we consider the computational cost of divisor methods (also known as highest averages methods), the de-facto standard solution that is used in many countries. We show that a simple linear-time algorithm can exactly simulate all instances of the family of divisor methods of apportionment by reducing the problem to a single call to a selection algorithm. All previously published solutions were iterative methods that either offer no linear-time guarantee in the worst case or require a complex update step that suffers from numerical instability.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-023-01929-5