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The Borda Count and its real-worldalternatives: Comparing scoring rules inNauru and SloveniaJon Fraenkela & Bernard Grofmanb

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The Borda Count and its real-world alternatives:Comparing scoring rules in Nauru and Slovenia

JON FRAENKEL

Victoria University of Wellington

BERNARD GROFMAN

University of California

This article examines strategic elements of voter behaviour in parliamentaryelections where the voting method is a scoring rule other than plurality: theBorda Count, which is used for the election of ethnic minorities in Slovenia,and the Dowdall rule, which is used in the Pacific island state of Nauru inmulti-seat districts. After first examining the general properties of scoring rules,and generating theoretical differences between the two rules, we look atempirical evidence from Nauru and Slovenia. This casts a doubt on predictionsbased simply on a voting rule’s mathematical properties and on the accuracy ofassumptions of sincere rank ordering.

Keywords: electoral systems; Borda Count; preferential voting systems; Nauru;Slovenia

Introduction

Political scientists have long been concerned about principles of social choice, andhave investigated alternative rules governing the translation of individual preferencesinto group decisions (Black 1958; McLean and Urken 1995). Many have preferredmethods that allow fuller expression of preferences than is the case with simple cat-egorical systems such as single-member plurality (first-past-the-post) or closed-list

Jon Fraenkel is a Professor of Comparative Politics in the School of History, Philosophy, PoliticalScience and International relations, Victoria University, Wellington. Bernard Grofman is a Professorof Political Science at the University of California, Irvine. We are indebted to Nauru MP KierenKeke, University of the South Pacific Centre Director Alamanda Lauti, and former Nauru ParliamentaryCounsel, Katy Le Roy, for assistance in obtaining the electoral data for Nauru; to the National ElectoralCommission of Slovenia and to Jure Toplak for assistance in obtaining electoral data for, and other infor-mation regarding, the Hungarian and Italian constituencies in Slovenia; and to the Australian NationalUniversity Cartographic Services for constructing our maps of Slovenia and Nauru. Work done on thisproject by the second named author was supported by the Jack Peltason Chair and the Center for theStudy of Democracy at the University of California, Irvine. During the preparation of an earlier drafthe served as a Straus Research Fellow at the Straus Institute for Advanced Study in the Law,New York University Law School.

Australian Journal of Political Science, 2014http://dx.doi.org/10.1080/10361146.2014.900530

© 2014 Australian Political Studies Association

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proportional representation (PR). The ‘method of marks’, invented by 18th centuryFrench mathematician Jean–Charles de Borda has aroused particular interest, bothbecause it permits a full ranking of voter preferences and because of its method ofdetermining victors (Borda 1781; Saari 1994; 1995; Young 1974). Unlike someother preferential voting systems, Borda’s method takes into account all voter prefer-ences, allotting a value to each, and establishes victors by a simple tallying of the totaleach candidate obtains. Despite considerable theoretical interest in such systems,real-world use of the Borda Count has been restricted – in part because of concernsabout potential strategic voting. Electoral reformers more commonly advocate otherelectoral systems making use of ranked preferences, such as the alternative vote (AV)(in the USA, promoted under the name ‘the instant runoff’) or the single transferablevote (STV).There are, however, two countries that use sophisticated non-elimination-based

preferential voting systems. The classical Borda method is used for national electionsto special reserved legislative seats for Hungarian and Italian ethnic minorities in Slo-venia. The small Pacific Island state of Nauru, since 1971, has used a unique pointssystem named after its inventor – the Dowdall method,1 that can be considered a rela-tive of the Borda rule, though it has quite distinctive features and effects.2 Thismethod has been widely described as a ‘modified’ form of the Borda Count(Golder 2005: 109; IFES 2013; Reilly 2002; Reynolds et al. 2005), but it has distinc-tive properties and, in practice on Nauru, generates different outcomes to the BordaCount. One key difference between the Borda Count and the Dowdall method is that,for Borda, the scores allotted to each candidate vary with the number of candidates,whereas with the Dowdall method a first preference is always worth one, a secondpreference half, a third preference one-third and so on.Others have pointed to the existence of these forms of preferential voting in Nauru

and Slovenia, but until now no data has been available to permit investigation of howsuch systems work in practice. The purpose of this article is to examine how these twodifferent scoring rules operate in legislative elections in Nauru and Slovenia, and tocompare results under the different systems. We are particularly concerned to ascer-tain how the potential manipulability of such systems (an issue regularly raised in thetheoretical discussions) plays out in practice, and whether real-world resort to stra-tegic voting amounts to a formidable argument against such rules.We examine results from nine legislative elections on Nauru, occurring between

1997 and 2013, and outcomes in the two minority districts in four Slovenian electionsfrom 2000 until 2011. We compare actual outcomes under Dowdall (or Borda) tosimulated outcomes with the same preference rankings under the other rule, andwe compare these rules to two other voting methods whose results we can recreatefrom aggregate-level preference data on elections in these two countries: pluralityvoting in either single-seat constituencies (for Slovenia) or multiple-seat constituen-cies for Nauru, and the single non-transferable vote (SNTV) for the multi-seatconstituencies in Nauru.

1The country’s Secretary for Justice, an Irishman, Desmond Dowdall, devised the system in 1971. TheNauru government in 1971 indicated a preference that its system be described as the ‘Dowdall system’

instead of the previously used term ‘exhaustive ballot system’ (cited in Supreme Court of Nauru 1977).2A similar method was also used (until 2002) for the parliamentary selection of nominees for presidentialelections in another small Pacific Island nation, Kiribati (Reilly 2001a, 2001b, 2002; Reilly andGratschew 2001; Van Trease 1993).

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Theoretical work on the Borda Count and the potential for manipulation

Social scientists have differed about the merits of practical use of the Borda Count.Saari (1994; 1995) has been the leading contemporary academic exponent of theBorda rule as the most desirable way to make collective choices. Saari’s argumentsare several, but arguably the most important is that it guarantees a transitive orderingof preferences, and thus eliminates voting paradoxes. Others have suggested that theBorda rule is likely to choose an alternative that is highly regarded by a substantialnumber of voters, and that this system ‘reduce[s] the chances of divisive candidates’,and favours candidates ‘occupying moderate positions’ or at least ones who do notevoke either strong hostility or strong support (Dummett 1997: 161; 1998: 290, 292;Emerson 2013; Tabarrok and Spector 1999).3 Moreover, simulations show it tohave a high probability of choosing the Condorcet winner when one exists,4 but it isalso without irresoluteness in the absence of a Condorcet winner (Merrill 1984).And there are formal results specifying the (relatively plausible) conditions underwhich the Borda winner and the Condorcet winner will coincide (Tangian 2013).On the other hand, several scholars have worried about the potential for strategic

manipulation under the Borda Count by deliberate misrepresentation of preferences.For this reason, Jean–Charles de Borda himself famously commented that his systemwas only suitable for ‘honestmen’ (Black 1958: 182). In the 18th century, theAcadémieFrançaise, of which Borda was a prominent member, experimented with this system inits deliberations on internal matters. Its archives show that ‘the voters found how tomanipulate the Borda rule: not only by putting their most dangerous rival at thebottom of their lists, but also by truncating their lists’ (McLean and Urken 1995: 40).Duncan Black, whose 1958 Theory of Committees and Elections did much to revivemodern interest in Borda’s electoral thought, warned that the practical application ofsuch systems might prove troublesome: ‘even to the unsophisticated voter, the Bordacount is an invitation to strategic voting… It would be exceedingly dangerous toemploy any scheme with this property, in elections to parliament or congress’ (Black1976: 15). Even enthusiasts for the Borda Count have acknowledged that ‘there is adanger in adopting [Borda’s] system for parliamentary and local elections when wecannot be confident that sincere voting will be the general rule’ (Dummett 1997: 86).Consider the following example. Table 1 assumes four voters and four candidates:

a, b, c and d. Voter 1 backs the four candidates in the order abcd. Voter One prefersbacd, and under sincere ranking his first choice bwins. If Voter 1 instead strategicallyranks adcb, by shifting b to last place, he ensures victory for his first choice candidate.

3The reasons for adopting the Borda Count in Slovenia were connected to its potential for generatingelection of a more moderate candidate:

There are two or three interest groups in each of the two national minorities (Italian, Hungarian). Ifplurality or majority system were used, these groups would confront each other and one would winover the other. With Borda, usually the winner is the person who is most acceptable to all and whois not an extremist. (Jure Toplak, personal communication, 28 February 2013)

4The Condorect winner, also known as the majority winner, is that alternative, if any, which can defeatall others in a paired contest (or, for an even number of voters, it is that alternative which cannot bedefeated by any other alternative in a paired contest). When it exists, many authors (Black 1958)regard the Condorcet winner as the normatively desired choice. Black, however, proposed that theBorda winner be chosen when no Condorcet winner existed. Nanson (1882) proposed a complexscheme based on the Borda rule that has the property that it always yields the Condorcet winnerwhen one exists (see McLean and Urken 1995: 57–60; Tangian 2013: 203).

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In total, a and b have the same number of both first and second preferences: withsincere voting, b’s third preference from Voter 4 trumps a’s fourth preference fromVoter 3, but that advantage vanishes if Voter 1 votes strategically. Under the non-strategic ranking, the Borda Count is a = 7 and b = 8, whereas if Voter 1 votesstrategically, the outcome is a = 7 and b = 6. To achieve this end result deliberately,Voter 1 must be aware of the preferences of all other voters, and of the workings ofthe electoral system.Other types of strategic manipulation are possible under the Borda Count in

addition to strategic manipulation of revealed preferences. Since the scores attributedto each preference depend on the number of contestants, political strategists may field‘dummy’ or ‘red herring’ candidates in an effort to manipulate outcomes. In ourhypothetical example above, third and fourth preferences decide between the toptwo candidates. Hence, it might be possible to strategically sponsor other candidatesthat acquire those preferences and so devalue some points earned by one or other ofthese top two contenders by introducing what we might call ‘weak clones’. For thisreason, the Marquis de Condorcet, a contemporary of Borda, argued that the BordaCount ‘relies on irrelevant factors to form its judgments’ and was consequently‘bound to lead to error’ (Condorcet cited in Young 1995: 57).Before we examine the operation of such electoral systems in real-world settings,

and how strategic voting operates in these contexts,5 we need to explore the broaderproperties of scoring rules.

Defining weights for scoring rules

The Borda Count and the Dowdall method, along with the far better known pluralityrule (first-past-the post), are examples of what the social-choice literature callsscoring rules (Fishburn 1973; Saari 1994; 1995). A scoring rule operates over aset of ballots in which each voter provides a (partial or complete) ranking of a setof n alternatives. Each scoring rule can be identified with a vector of rankingweights (w1, w2,… , wn) where wi is the weight attached to an alternative locatedat the ith rank by any given voter. These values must be monotonically

Table 1. Hypothetical scoring under sincere and strategically chosen preference ranking

Voters

1 2 3 4

Preferences PointsVoter 1(strategic)

Adjustedpoints

a 1 2 4 2 7 1 7b 2 1 2 3 8 4 6c 3 3 1 4 5 3 6d 4 4 3 1 4 2 5

Note: Example entails a first preference worth 3 and last preference worth 0.

5There has been a large body of empirical work measuring strategic voting in real-world single-seat plur-ality contests in countries such as Canada and the UK, and for some elections, including party primaries,in the USA, but little empirical work has been done measuring strategic voting in multi-seat systems(see, however, Irwin and Van Holsteyn 2012).

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non-decreasing if ties are allowed and monotonically decreasing if ties are not per-mitted, which is the case for the rules used in Slovenia and Nauru. For any scoringrule, the cumulative score for the ith alternative is given by ∑wij, which is theweighted sum over all voters, j, corresponding to the rank each voter has assignedto the ith alternative.6 Plurality is the simplest scoring rule, since voters only indicatetheir first choice, that is, the first item in their rank ordering of alternatives. All scoringrules, including Dowdall as well, are vulnerable to Condorcet’s criticism (discussedabove) that they fail to satisfy the principle of ‘independence of irrelevantalternatives’.If we are choosing among n alternatives, a standard way of representing the Borda

rule is as the vector of weights (n–1, n–2,… , 2, 1, 0). With the Dowdall rule, in con-trast to the Borda rule, a first preference is worth 1, a second preference ½, a thirdpreference ⅓, a fourth preference ¼ and so on. In other words, for the Dowdallrule, we have a weighting vector of (1, ½, ⅓,… , 1/n).7 In plurality, we have theweights vector (w1, 0, 0, 0,… , 0), with w1 > 0, that is, only first preferences aregiven any weight.

Implementation options

Scoring rules such as Borda and the Dowdall method can be implemented either byallowing all voters to rank as many candidates as they choose, or by requiring votersto rank order only a fixed number of candidates – to cast a fixed truncated ballot – orby requiring voters to rank order all candidates. On the island of Nauru, if there are ncandidates, it is compulsory for voters to assign a unique rank to each candidate, from1 to n: that is, to record as many preferences as there are candidates. Otherwise theballot is discarded as invalid.Borda, the Dowdall rule and plurality can each be used either for selecting a single

alternative from among several, or for selecting M alternatives, M>1, with the Malternatives receiving the highest weighted vote being chosen. It is most commonto think of plurality in the context of single-member plurality elections, but we canalso have plurality-based outcomes in a multi-seat district, with the most commonform of such an electoral rule called plurality bloc voting, in which each votermay cast a single vote for M (or up to M ) candidates. The potential for a BordaCount to operate effectively either in single-member or multi-member districts isone reason why it has been considered for local-government elections in Australia’sNorthern Territory (Reilly 2011: 23, 28). The Slovenian uses of the Borda rules arefor single-seat elections, but the Dowdall rule in Nauru is used in districts either withtwo, three or four seats.

6For example, if there were three alternatives and three voters, then if the weight vector for the scoringrule were (1, ¾, 0) and Voter 1 ranked the alternatives abc, Voter 2 ranked the alternatives abc, andVoter 3 ranked the alternatives cba, the scores corresponding to each alternative would be a = 2 (2 ×1 + 1 × 0), b = 9/4 (3 × ¾) and c = 1 (1 × 1 + 2 × 0), which gives us the ranking of bac. For thesame example, if the weights were (1, ½, 0), the scores corresponding to each alternative would be a= 2 (2 × 1 + 1 × 0), b = 1.5 (3 × ½) and c = 1 (1 × 1 + 2 × 0), which gives us the ranking of abc. Forthe same example, if the weights were (1, ¼, 0), the scores corresponding to each alternative wouldbe a = 2 (2 × 1 + 1 × 0), b = ¾ (3 × ¼) and c = 1 (1 × 1 + 2 × 0), which gives us the ranking of acb.7In its original 1971 formulation, the Nauru system used the fractions 1, ½, ⅓, etc., but this was sub-sequently converted to a decimal basis. For mathematical tractability in deriving exact analyticresults, we begin with the fractional representation, and then report results as decimals.

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Theoretical comparisons among scoring rules

For a fixed n, there are an infinite number of different ways of representing the vectorof weights for any given scoring rule, since any two sets of weights that may be rep-resented as linear transformations of one another can be shown to be equivalent (Saari1994; 1995).8 If we wish to compare weights across scoring rules, we need a commonway to represent these rules. There are two obvious approaches. First, for any givenscoring rule, we might look at the representation of the weights vector for that scoringrule that has the property that all weights sum to one. Second, following Saari, wecould look at the representation of weights that has the property that the first-ranked alternative is scored one and the last-ranked alternative is scored zero.We can always find such representations. For example, the plurality rule can be

represented as the vector (1, 0, 0, 0,… , 0). This vector has the property that itsentries sum to one and the property that the first entry is one and the last entry iszero. Now consider the Borda Count. The sum of the weights in its vector of (n–1,n–2,… , 2, 1, 0) is (n)(n–1)/2. Consider n = 3, for which the sum of these weightsis 3. We may thus represent the Borda vector for n = 3 as either (2, 1, 0) or as (⅔,⅓, 0), so that its values sum to one. Moreover, we may take the latter representationand divide by⅔ to get the representation (1, ½, 0), which has the property that the firstvalue is one and the last is zero. To find alternative weights representations for theDowdall rule for n = 3, we first find that ∑1/n = 1 + ½ + ⅓ = 11/6. Now, insteadof (1, ½, ⅓), we can specify the weights vector for the Dowdall rule as (6/11, 3/11, 2/11), which sums to one. Then, we can subtract 2/11 from all entries to getthe new vector (4/11, 1/11, 0), and then normalise by dividing though by 4/11 toget the vector (1, ¼, 0) in which the first entry is one and the last entry zero.In this latter representation, comparing the vectors (1, ½, 0) and (1, ¼, 0), it is

apparent that, at least for n = 3, the Borda rule gives more relative weight tosecond-place preferences than does the Dowdall rule. This turns out to be true forall values of n. The Borda Count weights relative preferences differently dependingon the number of candidates, however, according to a principle that ensures that thegap between each weight is equal. If there are three alternatives, a first-place voteunder Borda is worth 2 and a second-place vote is worth 1, while if there are fouralternatives a first-place vote is worth 3 and a second-place alternative is worth 2,and so on, but the last-place vote is always given as zero weight.9 By contrast, asnoted earlier, the Dowdall rule is not dependent on how many candidates there are,since the weight it gives to any rank is fixed.One consequence of this difference between the two formulas is that n increases the

relative importance of the weight attached to the first-ranked alternative relative tothe weight attached lower-ranked alternatives under the Borda rule, but not underthe Dowdall rule. For example, the first-ranked alternative relative to the weightattached to the second alternative decreases under the Borda rule from a ratio of 2to 1 for n = 3 to, for example, a ratio of 5 to 4 for n = 6; with a limiting value ofn/(n–1), which goes to 1 as n goes to infinity. In contrast, the ratio between the

8A similar non-uniqueness result is found when we look at weight vectors for weighted voting rules (Fel-senthal and Machover 1998).9As noted earlier, there are many alternative ways to assign Borda weights, and Borda himself used adifferent scheme from the ‘modern’ instantiation of the Borda Count (Emerson 2013).

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weight given by each voter to first-ranked and second-ranked candidates stays thesame under the Dowdall rule: 2 to 1.There are several useful ways to look at the hypothetical comparisons between the

two rules. One thing we can do is to calculate the sum of the (absolute) weightingdifferences between the two rules as we vary n. That value, along with the averagevalue of the discrepancy is shown in Table 2.For the sum of the total (absolute) differences between the two rules, except for the

comparisons between value for n = 3 and n = 4, differences monotonically increasewith n. In contrast, when we look instead at average (absolute) differencesbetween the two rules, since we are steadily increasing the divisor, we find that theaverage discrepancy between the two rules will go to zero as n increases. Substantialdifference between the Borda rule and the Dowdall rule persist as long as the numberof voters is much greater than the number of alternatives. For the small district mag-nitudes, we observe in Nauru, however, differences are still quite substantial.Another important comparison is the following: as n increases, the relative weight

Borda attaches to first-place preferences shrinks as compared to the weight theDowdall rule attaches. As Figure 1 indicates, where there are three candidates, theweights attached to the first- and second-placed positions remain constant (at1.22). Where there are four candidates or more, this relative weight under Bordarises but then falls.Having established these theoretical differences, we are now interested in how

these rules might differ when applied to actual elections. There is a great deal ofwork done showing the extent to which, under particular distribution assumptions,voting rules are likely to yield similar or identical outcomes (Merrill 1984;Tangian 2013: Chap 4). We are concerned, however, that assumptions often usedin such analytic or simulation results, the uniform distribution, for example, or sym-metric distributions such as the univariate or multivariate normal, bear little resem-blance to the distribution of ballots we observe in real-world elections, which tendto be skewed and often rather sharply peaked (Feld and Grofman 1992; Tsetlin,Regenwetter, and Grofman 2003). Seminal as such analytic results are, showinghow to find preference distributions that have the property that they will give anydesired pattern of disagreement across common voting rules, including scoringrules (Saari 1990a; 1994; 1995), does not tell us how likely such disagreementsare to be in practice.

Elections in Nauru and Slovenia using the Dowdall rule and the Borda Count

Nauru, 1997–2013

The Republic of Nauru consists of only a single island, covering eight square miles. Itis located close to the equator in the midst of the Pacific Ocean, roughly equidistant

Table 2. Comparisons of the total and average (absolute) differences between the Borda rule and theDowdall rule for various values of n

N 3 4 5 6 7 8 9 10 11 12 13 14

Absolute discrepancy .36 .24 .27 .32 .34 .36 .37 .38 .39 .43 .43 .45Average discrepancy .12 .06 .054 .053 .049 .045 .042 .038 .036 .036 .033 .032

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from Hawaii and Australia. The island was brought under German control in 1888,but with the onset of World War I, became a League of Nations-mandated territoryunder Britain, New Zealand and Australia, but in practice administered by Australia.It was occupied by the Japanese during the Pacific War and, after the conclusion ofhostilities, became a United Nations Trust territory, again under Australia, until itobtained independence in 1968. Phosphate exports underpinned the island’seconomy for most of the 20th century, but the industry is now close to exhaustion.Over recent years, Nauru has become the location of a detention centre for asylumseekers who try to enter Australia by boat.Nauru has a parliamentary system, with a president selected through the legislature,

rather than by direct popular election. Since independence, voters have cast a rank-ordered ballot in one of eight multi-member constituencies, based on Nauru’s 14 tra-ditional districts or (in the case of Ubenide, Anabar and Anetan) amalgamations ofthese (Figure 2). Until 2010, seven constituencies returned two members to parliament,and one constituency, Ubenide, returned four members. As of the 2013 election, anextra member was added to the two-member Meneng constituency. In total, theNauru parliament now has 19 members, up from 18 during the period 1968–2010.The rule used to tally these rank-ordered ballots has changed over time. STV was

used for the country’s first election, in 1968, also in two- and four-member districts.10

Figure 1. Ratio of the kth place weight under the Borda scoring rule to the kth place weight under theNauru scoring rule for various values of n

10Reilly (2002: 154, 366) and Reilly and Gratschew (2001: 699) report use of the AV in the 26 January1968 election. They draw on the Nauru Supreme Court’s 1977 decision which mentions the 1965 Elec-toral Act, which provided for use of the AV both in single-member by-elections and, at general elections,in the two- and four-member constituencies (Supreme Court of Nauru 1977). Available electoral data for1968 records quotas, however, which are clearly STV quotas calculated for each constituency. Therecords also note that the election was conducted in accordance with Australia’s Commonwealth Elec-toral Act 1918–66. Since independence came on 31 January 1968, that is, four days after the election, webelieve that the election was conducted under the Australian law using STV, perhaps because of anawareness that this is a better method of handling elections in constituencies where M>1 than viausage of multi-member AV. Australia had a negative experience with use of multi-member AV

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Since 1971, as noted earlier, the voting rule used in Nauru has been the inverse to therank-scoring rule, called the Dowdall rule, and there has been a requirement for fullranking in order to cast a valid ballot. We will report data from elections held in 1997,2000, 2003, 2004, 2007, 2008, April 2010 and a second election in June of that year

Figure 2. Electoral and traditional districts on NauruNotes: Figures in brackets show the number of seats per district.

during the interwar years (Lijphart 1997). The Dowell system was introduced by way of the Electoral(Electoral System) Regulations 1971, gazetted 22 January. The new system was used for the generalelections of 23 January 1971 (Supreme Court of Nauru 1977).

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and 2013.11 In these years, as Table 3 indicates, the number of candidates contestingelections in Nauru ranged from 5 to 22 with a mean per seat of 4.0 (and a standarddeviation of 9.5). There exists no party system, but sharp competition prevailsbetween the parliamentary factions over the capture of government.Whereas Nauru uses the Dowdall rule in multi-member constituencies, the Slove-

nian system uses a classical Borda system in single-member districts.

Slovenia, 1992–2011

The Republic of Slovenia is located between Italy, Austria, Hungary and Croatia. Ithas a population just over 2 million spread across just over 20,000 km2 of land area.Slovenia broke away from Yugoslavia in 1991, and joined the European Union in2004. The country has a parliamentary system, but with a popularly elected president.Since 1992, the National Assembly has had 90 elected members, 88 of whom are

selected by a list PR system from eight 11-member constituencies using a 4 per centthreshold and a four-year electoral cycle.12 The remaining two members, representing

Table 3. Correlations between outcomes and rankings under the Dowdall system and simulatedalternative systems in Nauru, 1997–2013

1997 2000 2003 2004 2007 20082010April

2010June 2013 Average

Candidates (No.) 61 72 84 73 79 65 86 60 68 4.0Invalid votes (%) 3.4 4.0 4.7 4.1 4.4 3.6 4.8 3.3 3.6 4.0Disagreements between outcomes (percentages of total)Dowdall/Borda 38.9 22.2 55.6 33.3 22.2 16.7 38.9 16.7 26.3 30.1Borda/PBV 33.3 22.2 55.6 27.8 22.2 16.7 38.9 22.2 21.1 28.8Borda/SNTV 44.4 33.3 55.6 38.9 27.8 22.2 50.0 27.8 26.3 36.2Dowdall/PBV 16.7 5.6 5.6 0.0 0.0 0.0 0.0 5.6 10.5 4.9Dowdall/SNTV 5.6 16.7 0.0 5.6 16.7 5.6 11.1 11.1 0.0 8.0Correlations between outcomesDowdall/Borda 0.48 0.70 0.29 0.56 0.71 0.77 0.51 0.76 0.75 0.62Borda/PBV 0.55 0.70 0.22 0.56 0.71 0.77 0.51 0.76 0.75 0.62Borda/SNTV 0.40 0.56 0.29 0.48 0.71 0.77 0.51 0.68 0.67 0.56Dowdall/PBV 0.78 0.93 0.93 0.93 1.00 1.00 1.00 0.84 0.92 0.92Dowdall/SNTV 0.93 0.78 1.00 0.93 1.00 1.00 1.00 0.92 1.00 0.95PBV/SNTV 0.70 0.85 0.93 0.85 1.00 1.00 1.00 0.84 0.92 0.90Correlations between rankingsDowdall/Borda 0.60 0.73 0.76 0.53 0.77 0.76 0.68 0.81 0.87 0.70Borda/SNTV 0.53 0.62 0.57 0.43 0.63 0.64 0.44 0.70 0.76 0.57Dowdall/SNTV 0.97 0.96 0.90 0.96 0.94 0.91 0.88 0.92 0.94 0.93

Notes: Ties for second place under Borda in Buada in April 2010 and under PBV. PBV, plurality blocvote; SNTV, single non-transferable vote.

11In 2010, a second election was held in June in an effort to resolve an impasse generated by a 9 versus 9MP split between government and opposition (Le Roy 2010).12Prior to 2000, the threshold was three parliamentary seats. The system works like a ‘closed list’ systemat the level of the district. The parties field single candidates in each of the 88 districts (11 for each of the8 districts), but voters effectively select a party by choosing a candidate.

Mandates are first allocated within each constituency to the eligible candidate lists using theDroopquota. After the first allocation, the overall proportional calculation of the number of mandateseach party is entitled to on a nationwide basis is done, using the d’Hondt method. The mandates

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the Hungarian and Italian minorities, are elected using the Borda Count, althoughminority voters are also entitled to cast a second ballot in one of the 88 list PRseats (Toplak 2006: 826–7).13 Overall, the population is roughly 88 per cent Slove-nian, with Hungarians and Italians, respectively, at 0.32 per cent and 0.11 per centand enjoying a special protected constitutional position (Republic of Slovenia2002). Representatives of the two minority groups have regularly held the balanceof power, and decided election outcomes in favour of left-leaning governments(Toplak 2006: 827). Figure 4 shows the location of the Borda-using Italian and Hun-garian constituencies, as well as the eight broader list PR constituencies. The Italiandistrict covers the three Adriatic port towns of Koper, Isola and Piran in the consti-tuency of Postojna in the southeastern part of the country, whereas the Hungarianconstituency is located in the Ptuj constituency in eastern Slovenia (Figure 3).In the version of the Borda system used in Slovenia, with n candidates, a first pre-

ference is worth n, a second preference is worth n–1 and so on with the last preferenceworth n–(n–1): that is, it is worth 1. Of course, we may readily re-normalise to scorethe last place candidate zero instead of one, to get the more common Borda scoringrule.14 As Table 4 indicates, the Borda system has been used for the minority districtsover six elections (1992, 1996, 2000, 2004, 2008 and 2011), but the scoring detailswere not included in the official results for minority constituencies in 1992 and 1996,and for other years, the Italian constituency was usually uncontested (2000, 2004 and2011). Hence, we report outcomes only for the Hungarian constituency from 2000until 2011 and for the Italian constituency in 2008.In these elections, it was not legally necessary for voters to provide a full ranking.15

Many voters mark only a single preference. In the Hungarian constituency, ballotsincluded a second preference on average in less than half of cases. Because manyballots were incomplete, in 2008 the Slovenian Electoral Commission decided togive one point for each unranked position for all candidates (personal communi-cation, Jure Toplak, 28 February 2013). The outcome of this change is discerniblein the 2008 election results, with numbers with only a single vote correspondinglyinflated by around 20 per cent. Several times over the past decade, representativesof the Hungarian minority have exerted pressure for the Borda method to bedropped in favour of a plurality system.

not already distributed are then allocated to candidate lists among all the constituencies, in theorder of the highest remainder of votes in proportion to the quota used in each constituency. Man-dates within a list are assigned to specific candidates on the basis of the percentage of the vote eachreceived in his/her district. That is, candidates within a list are ranked on the basis of the percentageof votes they received in comparison to the overall total of valid votes in their respective districts.The overall equality of the vote is ensured in the Slovenian system because mandates are awardedto parties proportionately to their nationwide vote total. However, because the allocation of man-dates to candidates within a party list is done on the basis of the percentage of the vote received byeach candidate, the voting weight of voters in a small district can be greater than that of voters in alarge district in allocating mandates within a party list. The Constitutional Court decided in 2000that this did not violate the principle of equality of the vote. (OSCE 2012: 4)

13The Office for Democratic Institutions and Human Rights concludes that such dual voting rightsdiverge from principles ‘regarding equality of the vote and are at odds with international good practice’(OSCE 2012: 1).14In 2008, unranked candidates were scored with ones rather than zeros.15§73 of the National Assembly Elections Act 1992 (amended 2000) provides that ‘A voter may vote foronly one candidate’ (Republic of Slovenia 2000), but there is some debate about whether this appliessolely to the PR districts or also to the minority districts.

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Invalid voting in Nauru and Slovenia

A common criticism of preferential voting systems is that they encourage a highnumber of invalid ballots. In Slovenia, numbers of invalid votes were substantialin some elections, but this was usually in the Italian constituency in contests wherethere was only one candidate, suggesting some deliberate spoiling of ballots.Invalid ballots stood at 13.2 per cent, 12.7 per cent and 11 per cent in 2000, 2004and 2011, all elections when there was only a single candidate standing, whereasthey stood at 3 per cent and 3.5 per cent in 1992 and 2008 when more than one can-didate contested. In the Hungarian constituency, which had multiple candidates at allelections between 1992 and 2011, invalid ballots ranged from 1.7 per cent to 3.9 percent: that is, close to the lower end seen in the competitive Italian elections (Table 4).On Nauru, despite the requirement to rank order all candidates, the share of invalid

votes was small. Over the 21 elections since 1968, the share of ballots ruled invalidhas averaged 2.9 per cent, with very low variation (from slightly below 2 per cent toslightly above 4 per cent). And there was no statistically significant change in the pro-portion of invalid ballots witnessed after the 1971 electoral shifted from the STVsystem to the present-day Nauru rule. Some economists have argued that Nauru’svoting system is too complex for such a small country (Hughes 2006). Nauru’s

Figure 3. The Hungarian and Italian minority districts and the eight broader list PR constituencies inSlovenia

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counting system is computerised, however, and the administration of the process hasraised few difficulties. Reformers have indicated concerns about the voter-regis-tration process, about overseas proxy voting and about the usage of a distinctsystem (the AV) for single-member by-elections, but not about the usage of theDowdall points system at general elections (Cain 2005; Pacific Islands Forum 2010).

Actual and simulated elections in Nauru and Slovenia under alternativevoting rules

Our available data are much more extensive for Nauru than for Slovenia. We havedetails of the scoring system that determined the election of 163 candidates from72 constituency contests (over 9 general elections) in Nauru, but for Slovenia weonly have data indicating the points obtained by 5 candidates in 5 single-member con-stituency contests (over 4 general elections). No data are available from either countryto illuminate how either Borda or the Dowdall rule might operate in a political party-based system (for example, as regards votes/seats disproportionality), though empiri-cal scrutiny of the practical operation of both systems offers some useful pointersabout how these might work in mass elections.16 Nauru’s use of multi-member con-stituencies, with a single vote as compared to the single-seat contests in Slovenia,limits the comparisons we can make between countries. So we focus on within-country comparisons across rules, not just between Borda and the Dowdall rule,but also other electoral rules, such as SNTV and plurality bloc voting (PBV) forthe Nauru data.Given differences in the scores attached to preferences under the two electoral

systems, we might expect substantial variations between outcomes under theDowdall and Borda rules. This was true for Nauru, but not for Slovenia.

Table 4. Outcomes in the Hungarian and Italian minority districts, Slovenia, 1992–2011

1992 1996 2000 2004 2008 2011

HungarianRegistered voters 6838 6229 N/av 6262 7063 6661No. candidates 7 4 5 5 5 2Invalid votes % 3.9 N/av 4.9 3.6 1.8 1.7Second preferences lodged (%) N/av N/av 40.9 41.7 44.5 36.5Victors points 12.8 10.4 11 5.1Dowdall points (simulated) 23.6 19.4 20 25.7

ItalianRegistered voters 1913 2604 2750 2737 1385 2712No. candidates 2 4 1 1 5 1Invalid votes % 2.97 N/av 13.2 12.7 3.5 11.0Second preferences lodged (%) N/ap N/ap N/ap N/ap 45.5 N/ap

Correlations between rankings 100% 100% 100% 100%Correlations between outcomes 100% 100% 100% 100%

Note: N/av, not available and N/ap, not applicable.

16Borda elections in Slovenia are disconnected from regular party competition while Nauru has no pol-itical parties, only loose and regularly changing parliamentary factions.

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Outcomes under the Dowdall system in Nauru differed from those simulated underBorda in 30.1 per cent of cases. By contrast, in Slovenia, there were no differences inoutcome between the actually used Borda system and outcomes simulated underNauru’s Dowdall system in any of the five contests. This difference, as regards Slo-venia, was not a result of any intrinsic difference between the two scoring rules. Itstemmed primarily from the use of optional ranking in the Slovenian minority dis-tricts, coupled with the reluctance of voters to indicate lower-order preferences. Slo-venian minority voters tend to cast truncated ballots indicating only a first preference,but no subsequent choices, whereas Nauru voters are compelled to lodge full prefer-ences. Dowdall (simulated) and Borda may have yielded identical outcomes in Slo-venia, but these were also the exact same outcomes that would have arisen undersingle-member plurality. Indeed, in all the Slovenian results, there was such highagreement on a single candidate that virtually every set of electoral rules wouldhave delivered the same result.More broadly, and based on the larger Nauru data set alone, simulated outcomes

under Borda varied from those under PBV or the SNTV to a much greater degreethan do the Dowdall results. Table 3 shows the percentage share of outcomes that dif-fered between Dowdall and other systems at the nine elections. On average, theDowdall rule gets a conflicting outcome to PBV only 4.9 per cent of the time, as com-pared to 28.8 per cent for the Borda Count, and a different outcome to SNTV in only8 per cent of cases, as compared to 36.2 per cent of cases with the Borda Count. Next,Table 3 shows correlations between actual and simulated outcomes under theDowdall system, the Borda Count, SNTV and PBV. Pair-wise correlationsbetween outcomes under systems other than Borda (Dowdall, PBV and SNTV)ranged on average between 0.90 and 0.95, whereas those of other systems withBorda averaged between 0.56 and 0.62.A similar finding emerges if attention is focused on correlations between rankings

(shown in the bottom part of Table 3), rather than between outcomes (though here thecontrast is necessarily a three-way one). Again, the comparison between Dowdall andSNTV, 0.93, is much closer than that between Dowdall and Borda (0.7) or Borda andSNTV (0.57).These findings cast doubt on the common classification of the Dowdall system as a

‘modified form’ of the Borda Count (Reilly 2001b: 154; Reynolds et al. 2005). Thatlabel is earned on account of similarity in ballot structure. If instead electoral systemswere identified by the kind of outcomes they generate, the Dowdall system might beseen as a more distant relative of Borda. With district magnitudes of 2–4, the Dowdallsystem should be seen as standing somewhere between plurality and the BordaCount, but as veering more towards plurality.17

Strategic voting

Nauru’s compulsory ranking system may preclude Slovenian-style truncation of pre-ference ranking, but it does not prevent other forms of strategic voting. As DuncanBlack shows, under Borda, electors may also vote for a favourite candidate first,and rank all others in the inverse order to their popularity (Black 1976: 13–15;

17Ben Reilly has remarked that ‘the Nauruan system [is] much more majoritarian than a standard Bordacount’ (2002: 365). Here we make that statement much more precise.

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Cox 1997; Ludwin 1978: 85; Saari 1990b). If strategic voting of this type took place,we would expect top-placed candidates not only to obtain large numbers of first-pre-ference ballots, but also a significant share of last preferences. On average, top-placedvictors in Nauru had 30.1 per cent of first preferences, 20 per cent of second prefer-ences and 15.1 per cent of last preferences, but only 7.5 per cent of the other inter-mediate preferences. Second-placed victors had 22.2 per cent of first preferences,16.8 per cent of second preferences, 15.3 per cent of last preferences, but only 8.3per cent of intermediate preferences.Figure 4 examines the distribution of preferences for top-placed victors in each of

the Nauru two-seat constituencies from 1997 until 2013. Between 5 and 15 candi-dates contested these seats, and there is one panel for each of these configurations.There were numerous contests with between 5 and 11 candidates, but there wasonly a single case where 12, 13 and 15 candidates contested.No clear pattern is discernible in the five-seat contexts. As the number of candi-

dates rises, however, a U-shaped trajectory becomes increasingly apparent. Why?Less polarised constituencies tended to have fewer candidates, and the same parlia-mentary faction tended to win both seats. In many cases, electoral support onNauru remains highly personalised, and closely connected to kinship or clan linkages.Different factions able to strategically direct voters to place major rivals last usuallycontested those constituencies with larger numbers of candidates. Candidates distri-bute ‘how to vote’ cards at Nauru elections, and citizens are familiar with the tactic ofstrategically placing key adversaries in the last position (personal communication,Kieren Keke, Nauru MP, 23 August 2013).Figure 5 shows that this characteristic U-shaped trajectory was also evident in

the three- and four-seat constituencies, which a larger number of candidatestended to contest. Unlike Figure 4, which only shows the top-placed victor andgroups results from different elections, Figure 5 shows all of the victors foreach of 10 separate constituency contests. Separate investigation of those excep-tional series in Figures 4 and 5 which are not U-shaped indicates that there wasusually a well-placed incumbent, expected to win, who obtained a U-shaped

Figure 4. Top-placed victors’ share of preference votes by number of candidates contesting, two-seatNauru constituencies 1997–2013

Notes: Y-axis is percentage share of preferences. X-axis is the ranking order. The chart shows all two-seat Nauru contests between 1997 and 2013. There were no contests with fourteen candidates.

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preference schedule, and/or that actual victors were not polarising figures. Thedata in Figures 4 and 5 are consistent with strategic voting, but similar patternscould potentially arise with sincere ranking and simply reflect polarisation inelectorates.Neither the Slovenian nor the Nauru results provide evidence of intermediate

candidates, promoted up the scale by the practice of ranking major challengerslast, becoming serious contenders, as Dummett (1997: 85) fears may arise underBorda. In Slovenia, there is no need to allocate preferences in such a way toavoid strengthening major rivals. Voters can instead more simply avoid submittingfull rankings. In Nauru, factions competing in the two-, three- or four-seat consti-tuencies put their major rivals as last preference, and thereby potentially promotemiddle-ranked contestants. If Nauru used an elimination-based system, such as theAV or STV, these intermediate candidates might stand a better chance. With theDowdall rule, they rarely perform strongly, owing to the low weighting of inter-mediate preferences.For the same reason, the incentives for fielding ‘red herring’ or ‘dummy’ candi-

dates are greater under Nauru’s compulsory ranking system than under optionalBorda ranking in Slovenia (although not as great as they would be under a Bordasystem with compulsory ranking). In Nauru, the factions tend to run two candidatesin the two-member districts but also to encourage ‘buffer candidates’ – who are notexpected to win – to soak up intermediate preferences, and thereby lower the votetallies of their major rivals (personal communication, Roland Kun, Nauru MP, 20August 2013). These were not difficult-to-engineer ‘clones’ intended to split rivals’votes, but ‘irrelevant alternatives’ in Condorcet’s sense, designed to diminish thevalue of preference votes allocated to arch-rivals. With the Dowdall rule, at leastthe scoring of those lower-order alternatives remained fixed, irrespective of thegreater number of candidates. Had the Borda Count been in place, the inclusion ofthese ‘irrelevant alternatives’ would have adjusted the overall vector of rankingweights.

Figure 5. Victors’ share of preference votes by number of candidates contesting, three and four seatNauru constituencies 1997–2013

Notes: as for figure 4. The chart shows the distribution of preferences for all victors in contests for thefour seat Ubenide constituency and for the constituency of Meneng in 2013, which was increased from

two to three members ahead of that election

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Discussion

Our investigation shows major differences between the Slovenian, Nauru and Bordasystems in theory and in practice, but there is one significant commonality. Both theSlovenian Borda variant and the Dowdall systems tend to entail ultimate election offirst-count leaders, but for different reasons. On Nauru, this arose due to relativelylow weighting of lower preferences. In Slovenia, it stemmed from substantial agree-ment on the strongest candidate and voters choosing to cast truncated ballots, asallowed by an optional ranking scheme. Since so few indicated second or lower pre-ferences, outcomes were often identical to those that would have arisen under thesingle-member district plurality. This evidence should not be interpreted as supportfor the case for categorical choice-based electoral systems: it only shows that onetype of real-world choice, under rank-order scoring rules, is to express a commonindifference to all but one’s most-favoured candidate.The system used on Nauru is not merely a ‘modified form of Borda’, but an impor-

tant rule in its own right. The usual classifications of electoral systems tend to derivefrom their ballot structure, their methods of determining victors or their broad reper-cussions. Terms like ‘first-past-the-post’ refer to the method of determining victors;while the term ‘proportional representation’ or when we say a system is ‘majoritar-ian’, these refer to the kind of outcomes that are delivered. Dowdall has a similarballot structure to Borda, and methods of determining victors closely resembleeach other but, given real-world preference distributions, outcomes can and willdiffer between the two methods. As it operates in Nauru, the Dowdall rule is infact closer to the SNTV than to the Borda system, at least for the larger (three- andfour-member) districts.The Dowdall rule can be viewed as a corrective to preferential voting systems such

as the AV or the STV which, it is sometimes argued accord second preferences exces-sive weight by counting these as of equal value to first preferences. By rapidly dis-counting second and lower preferences, the Dowdall rule meets this objection. Inaddition, AV and STV are sometimes criticised because only a restricted share ofvoters’ second or lower preferences are ever counted, whereas the Dowdall rulecounts all preferences. It also removes the need for time-consuming or costly elimin-ations in successive rounds or counts, but without losing sensitivity to lowerpreferences.By introducing multiple members per constituency, the Dowdall system increases

the likelihood that second or lower preferences will determine at least some out-comes, since voters cast only a single (ranked) ballot. This adds an element of poten-tial proportionality that would be absent in single-member ‘winner takes all’ variants.In the Nauru context, where there are no political parties, that proportionality is not asapparent as would be the case if that system were used in larger constituencies and ina party-centered system. On the negative side, the Dowdall system potentially hassome of the features of SNTV, but these are only fully apparent in the three- andfour-member constituencies, not the more widely used two-member districts. If pol-itical parties were to operate under such a system, as with SNTV (see Grofman et al.1999), they would potentially have to gauge their voter support accurately, and fieldexactly the right number of candidates – with possible heavy penalties for over-nomination.Supporters of the Borda Count, or other ranking-based methods, have often been

attracted to such systems because of their potential to express preferences fully. That

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such systems may be used strategically to ensure victory of a favoured candidate byplacing his or her most popular challengers last is often counted as a major defect(Black 1976; Ludwin 1978). In Nauru and Slovenia, we found that voters tendedto manipulate such systems with the objective of maximising the likelihood ofdefeat of a rival to their first choice. In Slovenia, where lodging full preference rank-ings was optional, we found that on average, only 41.8 per cent of voters ranked asecond candidate in our five contests. The majority of voters engaged in a form of‘bullet voting’, generating in response efforts by the Slovenian Electoral Commissionto enforce a fuller ranking. On Nauru, that difference was maximised by placingmajor rivals as last preference and/or by introducing weak candidates to mop up inter-mediate preferences.The Borda Count and similar scoring rules cannot be simply dismissed on account

of susceptibility to strategic manipulation. All electoral systems are potentially vul-nerable to voter-expressed insincerity about true preferences (Gibbard 1973; Sat-terthwaite 1975). Where there are only three candidates, the Borda Count has beenshown to be less susceptible to micro-manipulation than other electoral systems(Saari 1990b). Manipulation, at least of this type, is only a major problem forsocial-choice theorists insofar as the primary justification adopting such systems isthat they better express the ‘felt preferences’ of voters (Black 1976: 15; Dummett1997: 63, 182). Instead of submitting ballots that record preferences sincerely in cir-cumstances when doing so harms the chance of achieving desired results, voters mayget better results when they engage in strategic voting. Under some circumstancesstrategic voting may be normatively desirable (Mckelvey and Niemi 1978). Also,as Dowding and Van Hess argue, democratic processes are more than mere countingdevices: ‘the fact that aggregative systems are manipulable encourages the very delib-eration that makes politics what it is’ (2007: 11).Strategic use of these scoring rules in Nauru and Slovenia has not generated

chaotic or unrepresentative outcomes that might warn against their adoption else-where. Nauru has used the Dowdall system for over 40 years, during which timecompulsory ranking has not fostered a high rate of invalid voting. Hungarian andItalian reluctance to specify second or lower preferences may have frustrated the Slo-venian Electoral Commission, but this was primarily a consequence of voter choicestogether with the law on optional ranking, not of the Borda Count itself. Those reac-tions to the real-world presence of such rank-order preferential systems, both by thevoters and the responsible institutions, must be an important focus of inquiry by pol-itical scientists. The evidence that we report here warns against judging electoral rulessimply by their mathematical properties, and suggests that similarities and differencesacross different rules can, and will, vary with political context.

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