The Potential of Cumulative Voting To Yield Fair Representation
| Author | Duane A. Cooper |
| Date | 01 July 2007 |
| DOI | 10.1177/0951629807077570 |
| Published date | 01 July 2007 |
| Subject Matter | Articles |
THE POTENTIAL OF CUMULATIVE VOTING
TO YIELD FAIR REPRESENTATION
Duane A. Cooper
ABSTRACT
We prove that cumulative voting usually enables a minority population to
achieve political representation corresponding to apportionment by Webster’s
method, which minimizes the absolute difference of per capita representation
between the minority and the remaining majority population. The minority,
of arbitrary size, can generally attain its ‘Webster-fair’ share of nseats with
probability greater than 75 per cent and otherwise, with probability at most
1
4·n
n+1, the minority can attain just one seat less than its Webster-fair represen-
tation. Furthermore, for two subpopulations, the potential representation
yielded by cumulative voting is identical to that obtained from apportionment
by Jefferson’s method, and for more than two subpopulations the potential
representation by cumulative voting cannot be greater than that of Jefferson
apportionment. These results confirm the potential of cumulative voting to
yield representation proportional or nearly proportional to population, and
the results counter claims or concerns that cumulative voting would be
unfairly advantageous to minority populations.
KEY WORDS .apportionment .cumulative voting .fair representation
.minority representation
Introduction
The election method of cumulative voting has seen limited use and is promoted
for broader adoption by some individuals and institutions for purposes of fair
representation, and cumulative voting has been court-imposed on some small
This work was supported by National Science Foundation award NSF/DMS-0408676 and by the
Career Enhancement Fellowship of the Andrew W. Mellon and Woodrow Wilson National Fellow-
ship Foundations. The author thanks Lani Guinier for inspiring this project; Angelyn Mitchell for
encouraging this work; Nicholas Miller, Steven Brams, James Rogers, and anonymous reviewers
for improving this work at various stages of its development; students, too numerous now to register,
for stimulating this work with their interest, participation, and study; and Donnette Dais, one of those
numerous students, for launching this investigation by posing a curious question, long-forgotten by
her, about cumulative voting’s relationship to measures of fair representation.
Journal of Theoretical Politics 19(3): 277–295 Copyright Ó2007 Sage Publications
DOI: 10.1177/0951629807077570 Los Angeles, London, New Delhi and Singapore
http://jtp.sagepub.com
US jurisdictions as a remedy for unfair election practices. This article examines
the potential of cumulative voting to deliver proportional representation to a
minority population. The article compares this potential level of representa-
tion to that which would be fairly assigned to the minority by apportionment
methods, specifically, the method of Webster (also known as the method of
Sainte-Lague
¨) and that of Jefferson (a.k.a. d’Hondt). We shall see that usually
cumulative voting can guarantee a minority the opportunity to elect representa-
tives in the same number that they would receive by one of the apportionment
methods. Moreover, a minority can never guarantee itself greater representation
by cumulative voting than that which would be allotted and deemed fair by
Webster or Jefferson apportionment.
Instead of the democratic principle of ‘one person, one vote’, a cumulative
voting election adheres to a principle of ‘one person, nvotes’, where nis the
number of representatives to be elected in a jurisdiction. The allure of cumula-
tive voting is in the potential it provides political, racial, and ethnic minorities
or any cohesive group of a population to have representation when they might
not otherwise have voice in an elected legislative or judicial body. This is possi-
ble because, in addition to the option of voting exactly once apiece for up to n
candidates – prescribed by the generalized plurality method typically used in
US multimember district elections – voters can ‘plump’ their votes, conferring
all nvotes on a single candidate or distributing their nvotes as they please
(usually restricted to whole numbers, though) among anywhere from 1 to n
candidates. Note that cumulative has referred to the practice of allotting nvotes
per voter in an election of a single candidate, as analyzed by Felsenthal (1985).
However, we refer to cumulative voting in its most common usage, this being
the allotment of nvotes per voter, with plumping allowed, to elect nrepresenta-
tives in a multimember district.
Voting literature frequently mentions ‘thresholds’, which designate a fraction
of population for which a cohesive group whose population fraction is above the
threshold can assure itself a certain level of representation under a method of
voting. For example, let us determine the threshold – developed in a context of
corporate board voting by Gerstenberg (1910), and also explained by Grofman
(1975) and Gilmore (1998) – for a group to win kof nseats in an election under
cumulative voting.
We want the fraction of population x
Pover which the group can elect kof n,
if the group desires to do so and if they vote strategically. Here, Prepresents
the size of the population and xis the size of the group in question. The
remainder of the population has size P−x. Everybody has nvotes; plumping
is allowed.
In an attempt to win kof nseats, the group distributes its votes evenly so that
its kcandidates have as many votes as possible. Thus, each candidate gets x·n
k
votes. For all kof their candidates to be elected, each must receive more votes
278 JOURNAL OF THEORETICAL POLITICS 19(3)
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