Classifying electoral systems by input rules: Building on Blais and Rae through Australian experience

Abstract

By revisiting the work of Blais and Rae, this article develops a new classification of electoral systems focused on input rules. An Unknown Winning Number family is distinguished from a Quota family with known winning numbers for most of the counting process. Branching family trees are developed and used to help explain some Australian experience with accentuated disproportionality in two electoral systems which have been omitted from otherwise path-breaking recent analysis (Taagepera, R. and Shugart, M.S. 1989. Seats and votes: The effects and determinants of electoral systems. New Haven, CT: Yale University Press). These omitted systems are identified as effectively giving electors as many votes as seats available in a district. The input-rule family trees remind us that number of votes is an important component of electoral systems, although elusive and somewhat forgotten in much recent analysis. More conventional groupings of electoral systems are identified as output peer groups.

本文通过重读布莱斯和雷的著作，提出了一种强调输入规则的选举制度的新分类，即未知获胜票数系，以区别于为大多数计票过程所知的获胜票数配额系。该系及其分支有助于解释澳大利亚所经历的两种选举体制的某种不均衡性。这种不均衡性被本来可能具有开创性的最新分析（塔格佩拉与舒噶特的《席位与投票：选举制度的效果和成因》）所忽略。被忽略的制度有效地给予就一个选区既定席位而言尽可能多的票数。输入规则系告诉我们投票数是选举制度的重要组成部分，只是在最新的分析中没被正视，甚至被忽视了。选举制度更传统的分类被定为输出同类组。

Keywords:

• electoral formulae
• quota
• unknown winning number

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 This use of the terms categorical and ordinal ballots is not in line with Rae's original use, but is I argue clearer and more defensible. Rae suggested that categorical ballots force the voter ‘to say that he prefers one party … He cannot equivocate … he cannot divide his mandate among parties or among candidates of different parties’ (Rae 1967: 17). By contrast, ordinal votes ‘allow the voter to express a more complex, equivocal preference by rank-ordering the parties’ or even sometimes the candidates of different parties’ (1967: 17–8). This distinction was unclear and indeterminate on ballots which do not use rank ordering, but in which a voter can ‘distribute his numerous mandates among party lists in proportions of his own choosing’ (Rae 1967: 18). Rae called these ordinal ballots because they ‘produce an ordinal effect by allowing the voter to award single mandates to candidates appearing on various party lists’ (1967: 18). This is unnecessarily complex and mixes issues of vote marking, vote number and whether votes are cast for candidates or parties (i.e., the three components of ballot structure). A simpler distinction can be made between ordinal/preferential vote marking, using ranked numbers and categorical vote marking using unranked approval indicators, no matter how many of these marks or how they are apportioned between candidates and/or parties. This simpler distinction referring only to the method of vote marking is the usage of ‘categorical’ and ‘ordinal/preferential’ adopted in this article.

2 Massicote and Blais (1999) focused on electoral systems which, through having two tiers of districts, ‘mixed’ elements of proportional representation and majority/plurality formulae. They divided these ‘mixed’ electoral systems into two types (Independent and Dependent) and five sub-types (Coexistence, Superposition, Fusion, Correction and Conditional).

3 Linkage and adjustment equates to ‘dependent’ as defined by Massicote and Blais (1999).

4 In both these types of systems there can be thresholds below which candidates or parties are excluded at various points in the count. I am unsure at this stage whether thresholds are best seen analytically as a refinement of this component of electoral formula, or possibly as an additional separate component.

5 Beyond these two options, there are also other possible answers to this question which do not start with a simple tally, such as the weighting and summation of preferences expressed in ordinal votes.

6 Harrop and Miller opted for a vertical rather than horizontal presentation. They suggested two family groupings by including in the ‘proportional’ category systems which others separated out as semi-proportional.

7 Harrop and Miller (1987: 49–50) described the use of this system in Japan and placed it graphically among ‘proportional’ systems, but called it the ‘limited ballot’ rather than SNTV. Different word usage in referring to particular electoral systems is another symptom of classification indeterminacy.

8 An anonymous referee noted the absence of ‘two-round’ electoral systems in Figures 1 and 2 and correctly observed that in this approach each round might need to be categorised separately. Many two-round systems have unknown winning numbers in the first round and then, like the Alternative Vote, a known winning number of a majority in the second round. Some however, have unknown winning numbers in each round and are therefore simply a two-round elaboration of Single Member Plurality.

9 Blais (1988: 110) cited this work in manuscript form.

10 I here focus on what Taagepera and Shugart refer to in that table as ‘Single-round elections in Districts’.

11 The Cumulative Vote and Limited Vote within the Unknown Winning Number family could also be seen, on one interpretation, to be covered by Taagepera and Shugart's omission. These are briefly discussed later.

12 This similarity of naming suggests that past scholars have recognised the peer relationship between these systems. That peers come from different families has been the missing element of analysis.

13 These experiments were not the major focus of 19th century electoral reform debates which, under the influence of British lawyer Thomas Hare, were, in the terminology of this article, pushing for a move to the Quota family with large M. See Hart (1992) for historical details, though not these conceptual categories.

14 Actually Dodgson imagined districts in which there were 100 electors; so the ‘necessary and sufficient’ figures in this table are not quite ‘percentages’ so much as numbers of electors required out of 100. Were these ‘necessary and sufficient’ numbers to be changed to percentages of any number of electors, they would drop a little and require > signs. Hence all the 51s would become >50 per cent. All the 34s and 67s would become >33.333 per cent and >66.666 per cent. All the 21s, 41s, 61s and 81s would become >20 per cent, >40 per cent, >60 per cent and >80 per cent. These necessary and sufficient numbers/percentages occurring in the bottom lines of each section of Dodgson's table, when electors only have one vote, should then be recognised by many readers as the Droop quotas for the election of one or more candidates from a group. Droop quotas in a district electing M members, with single votes per elector, are the lowest whole number of votes which M candidates can attain, but M + 1 candidates cannot.

15 This table is on p. 310 of Dodgson ([1884] 1995). Under his pseudonym Lewis Carroll, as author of Alice in Wonderland, Dodgson often played with absurd, but philosophically informed ideas (see McLean, McMillan and Monroe 1996).

16 In workshops and social situations, I find that a significant proportion of people quickly understand this dynamic, but there is still a proportion whose eyes glaze over. I use a real-life example from local government in which one locality with 60 per cent of electors in a district won all four available seats, while two other localities each with about 20 per cent of electors kept missing out (Sanders 2009). I have also constructed a hypothetical example using an array of 11 votes, five of which are cast for one candidate. The other six votes, while initially cast for three or even four candidates, can be easily made with preferences continually to win, while the five votes for the first candidate continually miss out (Sanders 2011).

17 Gallagher and Mitchell (2005: 596) finish their account of ‘the mechanics of electoral systems’ by noting that these two systems are ‘analogous’ in their respective relationships to Single Member Plurality and the Alternative Vote and that both ‘compound’ disproportionality of vote/seat outcomes rather than ‘making the result more proportional’. This is perhaps as close as the existing literature comes to the analysis in this article. Gallagher and Mitchell effectively identify the Block Vote and Preferential Block as ‘peer’ electoral systems, but without developing this article's complementary idea that they are both N = M vote systems within the two input-rule ‘families’.

18 See note 1.

19 Like Blais (1988), Taylor and Johnston (1979: 40) was another instance where authors tried to bring clear attention back to number of votes.

20 See note 16.

21 Taagepera and Shugart also effectively left aside 1 < N < M systems, such as the Limited Vote which they mentioned in passing as being restricted in modern times at the national level to the Spanish Senate (Taagepera and Shugart 1989: 28–9). As Taagepera and Shugart's analysis focused on national lower house elections, even this one modern instance of the Limited Vote effectively disappeared from view. A peer equivalent of the Limited Vote within the Quota family has not, to my knowledge, been developed, as suggested by the absence of any 1 < N < M electoral system in Figure 2.

22 Towards the end of their work, Taagepera and Shugart (1989: 221–24) briefly discussed two instances of plurality in multi-member electorates, but these were outside their graphical and mathematical analysis. These instances in Mauritius and the Seychelles were discussed for their ‘overamplification’, another term for accentuated vote/seat disproportionality. Including these systems in the graphical and mathematical analysis of proportionality would indeed have broadened Taagepera and Shugart's continuum.

23 In line with Rae's usage of categorical and ordinal, they suggested that ‘cumulative voting’ had the ‘degree of choice of ordinal voting’ (Taagepera and Shugart 1989: 13). In the usage adopted in this article which focuses just on vote marking (Q2), cumulative voting is categorical rather than ordinal. See note 2.