A generalized parametric divisor method for political apportionment

• Date: 01 January 2021
• Author: Tatsuo Oyama,Nicholas G. Hall,Kazuhiro Kobayashi
• DOI: http://doi.org/10.1111/itor.12622
• Published date: 01 January 2021

Intl. Trans. in Op. Res. 28 (2021) 327–355

DOI: 10.1111/itor.12622

INTERNATIONAL

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IN OPERATIONAL

RESEARCH

A generalized parametric divisor method

for political apportionment

Tatsuo Oyamaa, Nicholas G. Hallband Kazuhiro Kobayashic,∗

aNational Graduate Institute for PolicyStudies, 7-22-11 Roppongi, Minato-ku, Tokyo 106-8677, Japan

bDepartment of Management Sciences, Fisher College of Business, The Ohio State University,

658 Fisher Hall, 2100 Neil Avenue, Columbus, OH 43210-1144, USA

cDepartment of Industrial Administration, Faculty of Science and Technology, Tokyo University of Science,

2641 Yamazaki, Noda-shi, Chiba 278-8510, Japan

E-mail: oyamat@grips.ac.jp[Oyama]; hall.33@osu.edu [Hall]; kkoba@rs.tus.ac.jp [Kobayashi]

Received 1 November2017; received in revised form 19 July 2018; accepted 13 November 2018

Abstract

The political apportionment problem has been studied for more than 200 years. In this paper, we introduce

a generalized parametric divisor method (GPDM), which generalizes most of the classical and widely used

apportionment methods from the literatureas special cases. Moreover,it allows for very flexible interpolation

between previous methods by appropriately setting two parameters in the GPDM. We identify an inequity

measure that the GPDM globally optimizes. We also identify two natural inequity measures for which an

apportionment given by the GPDM is locally optimal. These results generalize similar results for classical

apportionment methods, and justify the use of a large class of new apportionment methods given by the

GPDM. From this class, we identify and recommend specific new methods. Our numerical experiments

compare the apportionments given by the new methods with those givenby existing methods using real data

for the United States,Germany, Canada, Australia,England, and Japan. Explicit definition of the GPDM has

enabled us to perform computational experiments for evaluating the unbiasedness of the GPDM using two

standard measures while comparing with other traditional methods.Based upon our generalization technique

and numerical experiments,we show that the GPDM outperforms all the traditional apportionment methods

by selecting appropriateparameter values. Thus, we can conclude that the GPDM is the most “unbiased” and

fairer if parameters can be agreed ex ante, and the GPDM is applicable to actual electoral voting systems.

Keywords:political apportionment problem; generalized parametric divisor method; optimization; unbiasedness

1. Introduction

The political apportionment problem arises in a federal election system where a given total number

of seats in a representativeassembly need to be allocated among constituencies in proportion to each

∗Corresponding author.

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328 T. Oyamaet al. / Intl. Trans. in Op. Res. 28 (2021) 327–355

constituency’spopulation. It also arises in a proportional representation system where the seatsneed

to be allocated among political parties in proportion to the votes they receive. In either case, such

an allocation is called an apportionment. For simplicity, weadopt the notation of a federal election

system throughout this paper. A typical constraint on the apportionment is that each constituency

must receive at least a specified minimum number of seats, no matter how small its population.

The most desirable apportionment is one that is in some sense “fairest” to all the constituencies.

We know, however, that defining the fairness, and moreover, measuring the fairness, is not easy as

there are various types of definitions. Each individual or each country may have different criteria

for fairness.In this paper, we try to find a most “fair” and “unbiased” apportionment method based

on the divisor method.

The apportionment problem can be stated as follows. Let N={1,...,n}denote the set of con-

stituencies. Let the population of constituency i∈Nbe denoted by pi,whereP=n

i=1pi. Finally,

let the total number of available seats, or house size, be denoted byK. Then the apportionment prob-

lem is to find numbers {di|i∈N}, where direpresentsthe number of seats allocated to constituency

i, such that



i∈N

di=K:di≥0 and integer,i∈N.

The U.S. House of Representatives allocates one seat to each state, whereas the Japanese Lower

House allocates one seat to each prefecture, before they allocate seats proportionally based on

population. This problem can be modeled using our formulation by changing the total number

of seats to be allocated from Kto K−n. That is, the remaining K−nseats are allocated to

each constituency in proportion to its population. When we apply the apportionment method to

allocate seats among political parties for a proportional representation system, the problem can be

formulated as above. Thus, we assume that di≥0 instead of di≥1 in our formulation.

For g iv en p1,...,pnand K, the exact fair share of seats or “quota”, qi, of constituency iis given

by

qi=piK

P,i∈N,

where n

i=1qi=K. Assuming that given a total number of seats as K, the fair allocation of political

seats to each constituency should be based upon the criteria that number of seats per capita is equal

for all constituencies; we can find that an exact fair allocation of seats is given as above. We know

of course that exact fair allocation of seats cannot necessarily be given by integers.What makes this

problem mathematically challenging is that the quotas {qi|i∈N} usually have fractional parts.

Therefore, the apportionment problem can be thought of as finding a way to adjust the fractional

quotas {qi|i∈N} to nearby integers, while keeping their sum equal to a given value K.

Balinski and Young (1982) provide a comprehensive and entertaining introduction to the ap-

portionment problem. Lucas (1983) provides a more concise and less technical exposition. Both

these works discuss the considerablevariety of apportionment methods, which have been developed.

However, several of those methods are fundamentally flawed, in that for some data sets they allow

one or more of the following paradoxes to occur:

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2018 The Authors.

International Transactionsin Operational Research C

2018 International Federation ofOperational Research Societies