Inverse optimization for divisor methods of apportionment
Abstract
The apportionment problem is to allocate a fixed number of seats among regions in proportion to their populations. In this paper, we propose an inverse optimization approach to evaluate how fairly an observed apportionment treats each region. It is well known that every divisor method can be formulated as a separable discrete convex minimization problem. We introduce a weighted divisor method, in which a nonnegative weight is assigned to each region, as the forward problem. Then we formulate an inverse optimization problem that finds the weight vector closest to the all-one vector in the ℓ1-norm such that the observed apportionment is optimal for the forward problem. We show that the optimal weight vector is always unique and has a closed-form expression as the median of three values that depend on a single parameter. Based on this result, the inverse optimization problem is reduced to a one-dimensional monotone equation, and it can be solved in O(n log n) time, where n is the number of regions. We also apply the proposed approach to the apportionment of the single-member districts of the House of Representatives of Japan.
Related
- Probability of Quota Violations in Divisor Apportionment Methods with Nonzero Allocations
- A higher order family of iterative methods for approximating inverse and generalized inverse matrices
- Reconstruction methods for inverse scattering problems with phaseless data
- A pedestrian's approach to Bayesian methods for inverse problems
- Optimization Inverse Spectral Problem for the Schrödinger Operator on a Compact Graph
BibTeX
@misc{kawane2026inverse,
title = {{Inverse optimization for divisor methods of apportionment}},
author = {Daiki Kawane and Hokuto Nagano and Yoichi Izunaga},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02720},
eprint = {2611.02720},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02720},
}
RIS
TY - UNPB
TI - Inverse optimization for divisor methods of apportionment
AU - Daiki Kawane
AU - Hokuto Nagano
AU - Yoichi Izunaga
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02720
UR - https://arxiv.org/abs/2611.02720
AB - The apportionment problem is to allocate a fixed number of seats among regions in proportion to their populations. In this paper, we propose an inverse optimization approach to evaluate how fairly an observed apportionment treats each region. It is well known that every divisor method can be formulated as a separable discrete convex minimization problem. We introduce a weighted divisor method, in which a nonnegative weight is assigned to each region, as the forward problem. Then we formulate an inverse optimization problem that finds the weight vector closest to the all-one vector in the ℓ_1-norm such that the observed apportionment is optimal for the forward problem. We show that the optimal weight vector is always unique and has a closed-form expression as the median of three values that depend on a single parameter. Based on this result, the inverse optimization problem is reduced to a one-dimensional monotone equation, and it can be solved in O(n log n) time, where n is the number of regions. We also apply the proposed approach to the apportionment of the single-member districts of the House of Representatives of Japan.
ER -