Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and
Majorities
Author(s): Michael Gallagher
Source: British Journal of Political Science, Vol. 22, No. 4 (Oct., 1992), pp. 469-496
Published by: Cambridge University Press
Stable URL: http://www.jstor.org/stable/194023
Accessed: 20/02/2010 02:47
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B.J. Pol. S. 22,469-496
Printed in Great Britain 469
Comparing Proportional Representation Electoral
Systems: Quotas, Thresholds, Paradoxes and
Majorities
MICHAEL GALLAGHER*
The relationship between electoral systems can be examined on a number of dimensions. Seat
allocation methods are conveniently divided into two groups: those based on largest remainders
and those based on highest averages. The single transferable vote has its own distinct character-
istics. Focusing on certain elements - the quota, thresholds, paradoxes and the conditions
under which a majority of seats can be won - enables comparisons to be drawn between
seat allocation methods. Certain seat allocation methods conventionally seen as variants of
proportional representation (PR) cannot be regarded as such. PR methods can be rank ordered
according to whether, when complete proportionality is not attainable, they display electoral
bias towards larger or smaller parties. However, a definitive ordering is elusive, since some
methods that are generally more favourable to larger parties can in some circumstances set
lower thresholds of representation than methods generally favourable to smaller parties.
Electoral systems can be compared on the basis of many criteria, such as their
proportionality or the characteristics of political systems (two-party or multi-
party systems, stable government and so on) with which they seem to be associ-
ated. This article takes a slightly more abstract approach and seeks instead
to establish an ordering of seat allocation methods. It will rank the formulae
in order of the degree to which, when some disproportionality is unavoidable,
they tend to be favourable or unfavourable to large parties, comparing the
methods on the basis of the quotas they use and the thresholds they entail.
It will also compare the methods in terms of their vulnerability to paradox
and the circumstances under which parties can win or be deprived of a majority
of parliamentary seats. It will look at eleven seat allocation methods, including
all of those employed in Western Europe. These consist of three largest remain-
ders methods (using the Hare, Droop and Imperiali quotas), seven highest
averages methods (d'Hondt, Sainte-Lague, modified Sainte-Lague, Imperiali,
the Danish method, equal proportions and smallest divisors/Adams), and the
single transferable vote. An overview of the eleven methods is presented in
Figure 1. We shall start by illustrating the methods, thereby establishing the
basis for comparison.
* Department of Political Science, Trinity College, University of Dublin. I am grateful to Arend
Lijphart, Ivor Crewe and the anonymous referees for comments on an earlier draft. The usual
disclaimer applies.