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This article examines strategic elements of voter behaviour in parliamentary elections where the voting method is a scoring rule other than plurality: the Borda 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 in multi-seat districts. After first examining the general properties of scoring rules, and generating theoretical differences between the two rules, we look at empirical evidence from Nauru and Slovenia. This casts a doubt on predictions based simply on a voting rule's mathematical properties and on the accuracy of assumptions of sincere rank ordering.
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Author(s): Jon Fraenkel
Article Title: The Borda Count and its real-world alternatives: Comparing scoring rules in Nauru and
Article No: CAJP900530
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The Borda Count and its real-world alternatives:
Comparing scoring rules in Nauru and Slovenia
Victoria University of Wellington
University of California
This article examines strategic elements of voter behaviour in parliamentary
elections where the voting method is a scoring rule other than plurality: the
Borda Count, which is used for the election of ethnic minorities in Slovenia,
and the Dowdall rule, which is used in the Pacic island state of Nauru in
multi-seat districts. After rst examining the general properties of scoring rules,
and generating theoretical differences between the two rules, we look at
empirical evidence from Nauru and Slovenia. This casts a doubt on predictions
based simply on a voting rules mathematical properties and on the accuracy of
assumptions of sincere rank ordering.
Keywords: electoral systems; Borda Count; preferential voting systems; Nauru;
Political scientists have long been concerned about principles of social choice, and
have investigated alternative rules governing the translation of individual preferences
into group decisions (Black 1958; McLean and Urken 1995). Many have preferred
methods that allow fuller expression of preferences than is the case with simple cat-
egorical systems such as single-member plurality (rst-past-the-post) or closed-list
proportional representation (PR AQ2). The method of marks, invented by 18th century
Jon Fraenkel is Professor of Political Science and International Relations, Victoria University, Welling-
ton. Bernard Grofman is a professor of Political Science at the University of California, Irvine. We are
indebted to Nauru MP Kieren Keke, University of the South Pacic Centre Director Alamanda Lauti,
and former Nauru Parliamentary Counsel, Katy Le Roy, for assistance in obtaining the electoral data
for Nauru; to the National Electoral Commission of Slovenia and to Jure Toplak for assistance in obtain-
ing electoral data for, and other information regarding, the Hungarian and Italian constituencies in Slo-
venia; and to the Australian National University Cartographic Services for constructing our maps of
Slovenia and Nauru. Work done on this project by the second named author was supported by the
Jack Peltason Chair and the Center for the Study of Democracy at the University of California,
Irvine. During the preparation of an earlier draft he served as a Straus Research Fellow at the Straus Insti-
tute for Advanced Study in the Law, New York University Law School.
Australian Journal of Political Science, 2014
CAJP900530 Techset Composition India (P) Ltd., Bangalore and Chennai, India 3/17/2014
© 2014 Australian Political Studies Association
French mathematician JeanCharles de Borda has aroused particular interest, both
because it permits a full ranking of voter preferences and because of its method of
determining victors (Borda 1781; Saari 1994;1995; Young 1974). Unlike some
other preferential voting systems, Bordas method takes into account all voter prefer-
ences, allotting a value to each, and establishes victors by a simple tallying of the total
each candidate obtains. Despite considerable theoretical interest in such systems,
real-world use of the Borda Count has been restricted in part because of concerns
about potential strategic voting. Electoral reformers more commonly advocate other
electoral 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-transferable
vote (STV).
There are, however, two countries that use sophisticated non-elimination-based
preferential voting systems. The classical Borda method is used for national elections
to special reserved legislative seats for Hungarian and Italian ethnic minorities in Slo-
venia. The small Pacic Island state of Nauru, since 1971, has used a unique points
system named after its inventor the Dowdall method,
that can be considered a rela-
tive of the Borda rule, though it has quite distinctive features and effects.
method has been widely described as a modiedform 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 Borda
Count. 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 rst preference is always worth one, a second
preference 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 how
such systems work in practice. The purpose of this article is to examine how these two
different scoring rules operate in legislative elections in Nauru and Slovenia, and to
compare results under the different systems. We are particularly concerned to ascer-
tain how the potential manipulability of such systems (an issue regularly raised in the
theoretical 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 elections
from 2000 until 2011. We compare actual outcomes under Dowdall (or Borda) to
simulated outcomes with the same preference rankings under the other rule, and
we compare these rules to two other voting methods whose results we can recreate
from aggregate-level preference data on elections in these two countries: plurality
voting in either single-seat constituencies (for Slovenia) or multiple-seat constituen-
cies for Nauru, and the single non-transferable vote (SNTV AQ3) for the multi-seat
constituencies in Nauru.
The countrys Secretary for Justice, an Irishman, Desmond Dowdall, devised the system in 1971. The
Nauru 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).
A similar method was also used (until 2002) for the parliamentary selection of nominees for presidential
elections in another small Pacic Island nation, Kiribati (Reilly 2001a,2001b,2002; Reilly and
Gratschew 2001; Van Trease 1993).
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 the
Borda rule as the most desirable way to make collective choices. Saaris arguments
are several, but arguably the most important is that it guarantees a transitive ordering
of preferences, and thus eliminates voting paradoxes. Others have suggested that the
Borda rule is likely to choose an alternative that is highly regarded by a substantial
number of voters, and that this system reduce[s] the chances of divisive candidates,
and favours candidates occupying moderate positionsor at least ones who do not
evoke either strong hostility or strong support (Dummett 1997: 161; 1998: 290,
292; Emerson 2013).
Moreover, simulations show it to have a high probability of
choosing the Condorcet winner AQ4when one exists,
but it is also without irresoluteness
in the absence of a Condorcet winner (Merrill 1984). And there are formal results spe-
cifying the (relatively plausible) conditions under which 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, JeanCharles de Borda himself famously commented that his system
was only suitable for honest men(Black 1958: 182). In the 18th century, the Académie
Française, of which Borda was a prominent member, experimented with this system in
its deliberations on internal matters. Its archives show that the voters found how to
manipulate the Borda rule: not only by putting their most dangerous rival at the
bottom 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 revive
modern interest in Bordas electoral thought, warned that the practical application of
such systems might prove troublesome: even to the unsophisticated voter, the Borda
count is an invitation to strategic voting It would be exceedingly dangerous to
employ any scheme with this property, in elections to parliament or congress(Black
1976: 15). Even enthusiasts for the Borda Count have acknowledged that there is a
danger in adopting [Bordas] system for parliamentary and local elections when we
cannot be condent 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 prefers
bacd, and under sincere ranking his rst choice bwins. If Voter 1 instead strategically
ranks adcb, by shifting bto last place, he ensures victory for his rst choice candidate.
The reasons for adopting the Borda Count in Slovenia were connected to its potential for generating
election of a more moderate candidate:
There are two or three interest groups in each of the two national minorities (Italian, Hungarian). If
plurality or majority system were used, these groups would confront each other and one would win
over the other. With Borda, usually the winner is the person who is most acceptable to all and who
is not an extremist. (Jure Toplak, personal communication, 28 February 2013)
The Condorect winner, also known as the majority winner, is that alternative, if any, which can defeat
all others in a paired contest (or, for an even number of voters, it is that alternative which cannot be
defeated by any other alternative in a paired contest). When it exists, many authors (Black 1968)
regard the Condorcet winner as the normatively desired choice. Black, however, proposed that the
Borda winner be chosen when no Condorcet winner existed. Nanson (1882) proposed a complex
scheme based on the Borda rule that has the property that it always yields the Condorcet winner
when one exists (see McLean and Urken 1995:5760; Tangian 2013: 203).
In total, aand bhave the same number of both rst and second preferences: with
sincere voting, bs third preference from Voter 4 trumps as fourth preference from
Voter 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 votes
strategically, 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 of
the electoral system.
Other types of strategic manipulation are possible under the Borda Count in
addition to strategic manipulation of revealed preferences. Since the scores attributed
to each preference depend on the number of contestants, political strategists may eld
dummyor red herringcandidates in an effort to manipulate outcomes. In our
hypothetical example above, third and fourth preferences decide between the top
two candidates. Hence, it might be possible to strategically sponsor other candidates
that acquire those preferences and so devalue some points earned by one or other of
these top two contenders by introducing what we might call weak clones. For this
reason, the Marquis de Condorcet, a contemporary of Borda, argued that the Borda
Count relies on irrelevant factors to form its judgmentsand 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,
we need to explore the broader
properties of scoring rules.
Dening weights for scoring rules
The Borda Count and the Dowdall method, along with the far better known plurality
rule (rst-past-the post), are examples of what the social-choice literature calls
scoring rules (Fishburn 1973; Saari 1994;1995). A scoring rule operates over a
set of ballots in which each voter provides a (partial or complete) ranking of a set
of nalternatives. Each scoring rule can be identied with a vector of ranking
weights (w
) where w
is the weight attached to an alternative located
at the ith rank by any given voter. These values must be monotonically
Table 1. Hypothetical scoring under sincere and strategically chosen preference ranking
Preferences Points
Voter 1
a1242 7 1 7
b2123 8 4 6
c3314 5 3 6
d4431 4 2 5
Note: Example entails a rst preference worth 3 and last preference worth 0.
There 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).
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 scoring
rule, the cumulative score for the ith alternative is given by w
, which is the
weighted sum over all voters, j, corresponding to the rank each voter has assigned
to the ith alternative.
Plurality is the simplest scoring rule, since voters only indicate
their rst choice, that is, the rst item in their rank ordering of alternatives. All scoring
rules, including Dowdall as well, are vulnerable to Condorcets criticism (discussed
above) that they fail to satisfy the principle of independence of irrelevant
If we are choosing among nalternatives, a standard way of representing the Borda
rule is as the vector of weights (n1, n2, , 2, 1, 0). With the Dowdall rule, in con-
trast to the Borda rule, a rst preference is worth 1, a second preference ½, a third
preference , a fourth preference ¼ and so on. In other words, for the Dowdall
rule, we have a weighting vector of (1, ½, ,,1/n).
In plurality, we have the
weights vector (w
,0,0,0,, 0), with w
> 0, that is, only rst preferences are
given any weight.
Implementation options
Scoring rules such as Borda and the Dowdall method can be implemented either by
allowing all voters to rank as many candidates as they choose, or by requiring voters
to rank order only a xed number of candidates to cast a xed truncated ballot or
by requiring voters to rank order all candidates. On the island of Nauru, if there are n
candidates, it is compulsory for voters to assign a unique rank to each candidate, from
1ton: that is, to record as many preferences as there are candidates. Otherwise the
ballot 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 Malternatives, M>1, with the M
alternatives receiving the highest weighted vote being chosen. It is most common
to think of plurality in the context of single-member plurality elections, but we can
also have plurality-based outcomes in a multi-seat district, with the most common
form of such an electoral rule called plurality bloc voting, in which each voter
may cast a single vote for M(or up to M) candidates. The potential for a Borda
Count to operate effectively either in single-member or multi-member districts is
one reason why it has been considered for local-government elections in Australias
Northern Territory (Reilly 2011: 23, 28). The Slovenian uses of the Borda rules are
for single-seat elections, but the Dowdall rule in Nauru is used in districts either with
two, three or four seats.
For example, if there were three alternatives and three voters, then if the weight vector for the scoring
rule were (1, ¾, 0) and Voter 1 ranked the alternatives abc, Voter 2 ranked the alternatives abc, and
Voter 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 the
same 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. For
the same example, if the weights were (1, ¼, 0), the scores corresponding to each alternative would
be a=2(2×1+1×0),b= ¾ (3 × ¼) and c= 1 (1 × 1 + 2 × 0), which gives us the ranking of acb.
In its original 1971 formulation, the Nauru system used the fractions 1, ½, , etc., but this was sub-
sequently converted to a decimal basis. For mathematical tractability in deriving exact analytic
results, we begin with the fractional representation, and then report results as decimals.
Theoretical comparisons among scoring rules
For a xed n, there are an innite number of different ways of representing the vector
of 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 (Saari
If we wish to compare weights across scoring rules, we need a common
way to represent these rules. There are two obvious approaches. First, for any given
scoring rule, we might look at the representation of the weights vector for that scoring
rule that has the property that all weights sum to one. Second, following Saari, we
could look at the representation of weights that has the property that the rst-
ranked alternative is scored one and the last-ranked alternative is scored zero.
We can always nd such representations. For example, the plurality rule can be
represented as the vector (1, 0, 0, 0, , 0). This vector has the property that its
entries sum to one and the property that the rst entry is one and the last entry is
zero. Now consider the Borda Count. The sum of the weights in its vector of (n1,
n2, ,2,1,0)is(n)(n1)/2. Consider n= 3, for which the sum of these weights
is 3. We may thus represent the Borda vector for n= 3 as either (2, 1, 0) or as (,
, 0), so that its values sum to one. Moreover, we may take the latter representation
and divide by to get the representation (1, ½, 0), which has the property that the rst
value is one and the last is zero. To nd alternative weights representations for the
Dowdall rule for n=3, we rst nd that 1/n=1+½+= 11/6. Now, instead
of (1, ½, ), 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 get
the new vector (4/11, 1/11, 0), and then normalise by dividing though by 4/11 to
get the vector (1, ¼, 0) in which the rst 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 to
second-place preferences than does the Dowdall rule. This turns out to be true for
all values of n. The Borda Count weights relative preferences differently depending
on the number of candidates, however, according to a principle that ensures that the
gap between each weight is equal. If there are three alternatives, a rst-place vote
under Borda is worth 2 and a second-place vote is worth 1, while if there are four
alternatives a rst-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.
By contrast, as
noted earlier, the Dowdall rule is not dependent on how many candidates there are,
since the weight it gives to any rank is xed.
One consequence of this difference between the two formulas is that nincreases the
relative importance of the weight attached to the rst-ranked alternative relative to
the weight attached lower-ranked alternatives under the Borda rule, but not under
the Dowdall rule. For example, the rst-ranked alternative relative to the weight
attached to the second alternative decreases under the Borda rule from a ratio of 2
to 1 for n= 3 to, for example, a ratio of 5 to 4 for n= 6; with a limiting value of
n/(n1), which goes to 1 as ngoes to innity. In contrast, the ratio between the
A similar non-uniqueness result is found when we look at weight vectors for weighted voting rules (Fel-
senthal and Machover 1998).
As noted earlier, there are many alternative ways to assign Borda weights, and Borda himself used a
different scheme from the moderninstantiation of the Borda Count (Emerson 2013).
weight given by each voter to rst-ranked and second-ranked candidates stays the
same 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) weighting
differences between the two rules as we vary n. That value, along with the average
value 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 increase
with n. In contrast, when we look instead at average (absolute) differences
between the two rules, since we are steadily increasing the divisor, we nd that the
average discrepancy between the two rules will go to zero as nincreases. In other
words, if AQ5we were electing a large enough set of candidates the differences in rankings
between Borda and Dowdall would cease to be important. For the small district mag-
nitudes, we observe in Nauru, however, differences are still quite substantial.
Another important comparison is the following: as nincreases, the relative weight
Borda attaches to rst-place preferences shrinks as compared to the weight the
Dowdall rule attaches. As Figure 1 indicates, where there are three candidates, the
weights attached to the rst- and second-placed positions remain constant (at
1.22). Where there are four candidates or more, this relative weight under Borda
rises 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 of
work 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 used
in 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 tend
to be skewed and often rather sharply peaked (Feld and Grofman 1992; Tsetlin,
Regenwetter, and Grofman 2003). Seminal as such analytic results are, showing
how to nd preference distributions that have the property that they will give any
desired pattern of disagreement across common voting rules, including scoring
rules (Saari 1990a;1994;1995), does not tell us how likely such disagreements
are to be in practice.
Elections in Nauru and Slovenia using the Dowdall rule and the Borda Count
Nauru, 19972013
The Republic of Nauru consists of only a single island, covering eight square miles. It
is located close to the equator in the midst of the Pacic Ocean, roughly equidistant
Table 2. Comparisons of the total and average (absolute) differences between the Borda rule and the
Dowdall rule for various values of n
N3 4 5 6 7 8 9 10 11 12 13 14
Absolute discrepancy .36 .24 .27 .32 .34 .36 .37 .38 .39 .43 .43 .45
Average discrepancy .12 .06 .054 .053 .049 .045 .042 .038 .036 .036 .033 .032
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 territory
under Britain, New Zealand and Australia, but in practice administered by Australia.
It was occupied by the Japanese during the Pacic War and, after the conclusion of
hostilities, became a United Nations Trust territory, again under Australia, until it
obtained independence in 1968. Phosphate exports underpinned the islands
economy 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 asylum
seekers who try to enter Australia by boat.
Nauru has a parliamentary system, with a president selected through the legisla-
ture, rather than by direct popular election. Since independence, voters have cast a
rank-ordered ballot in one of eight multi-member constituencies, based on Naurus
14 traditional districts or (in the case of Ubenide, Anabar and Anetan) amalgamations
of these (Figure 2). Until 2010, seven constituencies returned two members to parlia-
ment, and one constituency, Ubenide, returns four members. As of the 2013 election,
an extra member was added to the two-member Meneng constituency. In total, the
Nauru parliament now has 19 members, up from 18 during the period 19682010.
The rule used to tally these rank-ordered ballots has changed over time. STV was
used for the countrysrst election, in 1968, also in two- and four-member districts.
Figure 1. Ratio of the kth place weight under the Borda scoring rule to the kth place weight under the
Nauru scoring rule for various values of n
Reilly (2002: 154, 366) and Reilly and Gratschew (2001: 699) report use of the AV in the 26 January
1968 election. They draw on the Nauru Supreme Courts1977 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 for
1968 records quotas, however, which are clearly STV quotas calculated for each constituency. The
records also note that the election was conducted in accordance with AustraliasCommonwealth Elec-
toral Act 191866. Since independence came on 31 January 1968, that is, four days after the election, we
believe that the election was conducted under the Australian law using STV, perhaps because of an
awareness that this is a better method of handling elections in constituencies where M>1 than via
usage of multi-member AV. Australia had a negative experience with use of multi-member AV
Since 1971, as noted earlier, the voting rule used in Nauru has been the inverse to the
rank-scoring rule, called the Dowdall rule, and there has been a requirement for full
ranking 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 Nauru
Notes: 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 general
elections of 23 January 1971 (Supreme Court of Nauru 1977).
and 2013.
In these years, as Table 3 indicates, the number of candidates contesting
elections in Nauru ranged from 5 to 22 with a mean per seat of 4.0 (and a standard
deviation of 9.5). There exists no party system, but sharp competition prevails
between 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, 19922011
The Republic of Slovenia is located between Italy, Austria, Hungary and Croatia. It
has a population just over 2 million spread across just over 20,000 km
of land area.
Slovenia broke away from Yugoslavia in 1991, and joined the European Union in
2004. 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 cent
threshold and a four-year electoral cycle.
The remaining two members, representing
Table 3. Correlations between outcomes and rankings under the Dowdall system and simulated
alternative systems in Nauru, 19972013
1997 2000 2003 2004 2007 2008
June 2013 Average
Candidates (No.) 61 72 84 73 79 65 86 60 68 4.0
Invalid votes (%) 3.4 4.0 4.7 4.1 4.4 3.6 4.8 3.3 3.6 4.0
Disagreements 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 30.1
Borda/PBV 33.3 22.2 55.6 27.8 22.2 16.7 38.9 22.2 21.1 28.8
Borda/SNTV 44.4 33.3 55.6 38.9 27.8 22.2 50.0 27.8 26.3 36.2
Dowdall/PBV 16.7 5.6 5.6 0.0 0.0 0.0 0.0 5.6 10.5 4.9
Dowdall/SNTV 5.6 16.7 0.0 5.6 16.7 5.6 11.1 11.1 0.0 8.0
Correlations between outcomes
Dowdall/Borda 0.48 0.70 0.29 0.56 0.71 0.77 0.51 0.76 0.75 0.62
Borda/PBV 0.55 0.70 0.22 0.56 0.71 0.77 0.51 0.76 0.75 0.62
Borda/SNTV 0.40 0.56 0.29 0.48 0.71 0.77 0.51 0.68 0.67 0.56
Dowdall/PBV 0.78 0.93 0.93 0.93 1.00 1.00 1.00 0.84 0.92 0.92
Dowdall/SNTV 0.93 0.78 1.00 0.93 1.00 1.00 1.00 0.92 1.00 0.95
PBV/SNTV 0.70 0.85 0.93 0.85 1.00 1.00 1.00 0.84 0.92 0.90
Correlations between rankings
Dowdall/Borda 0.60 0.73 0.76 0.53 0.77 0.76 0.68 0.81 0.87 0.70
Borda/SNTV 0.53 0.62 0.57 0.43 0.63 0.64 0.44 0.70 0.76 0.57
Dowdall/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 bloc
vote; SNTV, single non-transferable vote.
In 2010, a second election was held in June in an effort to resolve an impasse generated by a 9 versus 9
MP split between government and opposition (Le Roy 2010).
Prior to 2000, the threshold was three parliamentary seats. The system works like a closed listsystem
at the level of the district. The parties eld single candidates in each of the 88 districts (11 for each of the
8 districts), but voters effectively select a party by choosing a candidate.
Mandates are rst allocated within each constituency to the eligible candidate lists using the Droop
quota. After the rst allocation, the overall proportional calculation of the number of mandates
each party is entitled to on a nationwide basis is done, using the dHondt method. The mandates
the Hungarian and Italian minorities, are elected using the Borda Count, although
minority voters are also entitled to cast a second ballot in one of the 88 list PR
seats (Toplak 2006: 8267).
Overall, the population is roughly 88 per cent Slove-
nian, with Hungarians and Italians, respectively, at 0.32 per cent and 0.11 per cent
and enjoying a special protected constitutional position (Republic of Slovenia
2002). Representatives of the two minority groups have regularly held the balance
of 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 Italian
district 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 Hungarian
constituency is located in the Ptuj constituency in eastern Slovenia (Figure 3 AQ6).
In the version of the Borda system used in Slovenia, with ncandidates, a rst pre-
ference is worth n, a second preference is worth n1 and so on with the last preference
worth n(n1): that is, it is worth 1. Of course, we may readily re-normalise to score
the last place candidate zero instead of one, to get the more common Borda scoring
As Table 4 indicates, the Borda system has been used for the minority districts
over six elections (1992, 1996, 2000, 2004, 2008 and 2011), but the scoring details
were not included in the ofcial results for minority constituencies in 1992 and 1996,
and for other years, the Italian constituency was usually uncontested (2000, 2004 and
2011). Hence, we report outcomes only for the Hungarian constituency from 2000
until 2011 and for the Italian constituency in 2008.
In these elections, it was not legally necessary for voters to provide a full ranking.
Many voters mark only a single preference. In the Hungarian constituency, ballots
included a second preference on average in less than half of cases. Because many
ballots were incomplete, in 2008 the Slovenian Electoral Commission decided to
give one point for each unranked position for all candidates (personal communi-
cation, Jure Toplak, 28 February 2013). The outcome of this change is discernible
in the 2008 election results, with numbers with only a single vote correspondingly
inated by around 20 per cent. Several times over the past decade, representatives
of the Hungarian minority have exerted pressure for the Borda method to be
dropped in favour of a plurality system.
not already distributed are then allocated to candidate lists among all the constituencies, in the
order of the highest remainder of votes in proportion to the quota used in each constituency. Man-
dates within a list are assigned to specic candidates on the basis of the percentage of the vote each
received in his/her district. That is, candidates within a list are ranked on the basis of the percen-
tage of votes they received in comparison to the overall total of valid votes in their respective dis-
tricts. The overall equality of the vote is ensured in the Slovenian system because mandates are
awarded to parties proportionately to their nationwide vote total. However, because the allocation
of mandates to candidates within a party list is done on the basis of the percentage of the vote
received by each candidate, the voting weight of voters in a small district can be greater than
that of voters in a large district in allocating mandates within a party list. The Constitutional
Court decided in 2000 that this did not violate the principle of equality of the vote. (OSCE
The Ofce for Democratic Institutions and Human Rights concludes that such dual voting rights
diverge from principles regarding equality of the vote and are at odds with international good practice
(OSCE 2012: 1).
In 2008, unranked candidates were scored with ones rather than zeros.
§73 of the National Assembly Elections Act 1992 (amended 2000) provides that A voter may vote for
only one candidate(Republic of Slovenia 2000).
Figure 3. The Hungarian and Italian minority districts and the eight broader list PR constituencies in
Figure 4. Top-placed victorsshare of preference votes by number of candidates contesting, two-seat
Nauru constituencies 19972013
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 canmdidates.
Invalid voting in Nauru and Slovenia
A common criticism of preferential voting systems is that they encourage a high
number of invalid ballots. In Slovenia, numbers of invalid votes were substantial
in some elections, but this was usually in the Italian constituency in contests where
there 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, 2004
and 2011, all elections when there was only a single candidate standing, whereas
they 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 all
elections between 1992 and 2011, invalid ballots ranged from 1.7 per cent to 3.9 per
cent: 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 invalid
has averaged 2.9 per cent, with very low variation (from slightly below 2 per cent to
slightly above 4 per cent). And there was no statistically signicant change in the pro-
portion of invalid ballots witnessed after the 1971 electoral shifted from the STV
system to the present-day Nauru rule. Some economists have argued that Naurus
voting system is too complex for such a small country (Hughes 2006). Naurus count-
ing system is computerised, however, and the administration of the process has raised
few difculties. Reformers have indicated concerns about the voter-registration
process, about overseas proxy voting and about the usage of a distinct system (the
AV) for single-member by-elections, but not about the usage of the Dowdall
points system at general elections (Cain 2005; Pacic Islands Forum 2010).
Actual and simulated elections in Nauru and Slovenia under alternative
voting rules
Our available data are much more extensive for Nauru than for Slovenia. We have
details of the scoring system that determined the election of 163 candidates from
Table 4. Outcomes in the Hungarian and Italian minority districts, Slovenia, 19922011
1992 1996 2000 2004 2008 2011
Registered voters 6838 6229 N/av 6262 7063 6661
No. candidates 7 4 5552
Invalid votes % 3.9 N/av 4.9 3.6 1.8 1.7
Second preferences lodged (%) N/av N/av 40.9 41.7 44.5 36.5
Victors points 12.8 10.4 11 5.1
Dowdall points (simulated) 23.6 19.4 20 25.7
Registered voters 1913 2604 2750 2737 1385 2712
No. candidates 2 4 1151
Invalid votes % 2.97 N/av 13.2 12.7 3.5 11.0
Second 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.
72 constituency contests (over 9 general elections) in Nauru, but for Slovenia we only
have data indicating the points obtained by 5 candidates in 5 single-member consti-
tuency contests (over 4 general elections). No data are available from either country
to 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 pointers
about how these might work in mass elections.
Naurus 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) for
the Nauru data.
Given differences in the scores attached to preferences under the two electoral
systems, we might expect substantial variations between outcomes under the
Dowdall and Borda rules. This was true for Nauru, but not for Slovenia.
Outcomes under the Dowdall system in Nauru differed from those simulated under
Borda in 30.1 per cent of cases. By contrast, in Slovenia, there were no differences in
outcome between the actually used Borda system and outcomes simulated under
Naurus Dowdall system in any of the ve contests. This difference, as regards Slo-
venia, was not a result of any intrinsic difference between the two scoring rules. It
stemmed 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 rst 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 under
single-member plurality. Indeed, in all the Slovenian results, there was such high
agreement on a single candidate that virtually every set of electoral rules would
have 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 degree
than 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, the
Dowdall rule gets a conicting 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 only
8 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 the
Dowdall system, the Borda Count, SNTV and PBV. Pair-wise correlations
between outcomes under systems other than Borda (Dowdall, PBV and SNTV)
ranged on average between 0.90 and 0.95, whereas those of other systems with
Borda averaged between 0.56 and 0.62.
A similar nding emerges if attention is focused on correlations between rankings
(shown in the bottom part of Table 3), rather than between outcomes (though here the
contrast is necessarily a three-way one). Again, the comparison between Dowdall and
Borda elections in Slovenia are disconnected from regular party competition while Nauru has no pol-
itical parties, only loose and regularly changing parliamentary factions.
SNTV, 0.93, is much closer than that between Dowdall and Borda (0.7) or Borda and
SNTV (0.57).
These ndings cast doubt on the common classication of the Dowdall system as a
modied formof the Borda Count (Reilly 2001b: 154; Reynolds et al. 2005). That
label is earned on account of similarity in ballot structure. If instead electoral systems
were identied by the kind of outcomes they generate, the Dowdall system might be
seen as a more distant relative of Borda. With district magnitudes of 24, the Dowdall
system should be seen as standing somewhere between plurality and the Borda
Count, but as veering more towards plurality.
Strategic voting
Naurus compulsory ranking system may preclude Slovenian-style truncation of pre-
ference ranking, but it does not prevent other forms of strategic voting. As Duncan
Black shows, under Borda, electors may also vote for a favourite candidate rst,
and rank all others in the inverse order to their popularity (Black 1976:1315;
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
rst-preference ballots, but also a signicant share of last preferences. On average,
top-placed victors in Nauru had 30.1 per cent of rst preferences, 20 per cent of
second preferences and 15.1 per cent of last preferences, but only 7.5 per cent of
the other intermediate preferences. Second-placed victors had 22.2 per cent of rst
preferences, 16.8 per cent of second preferences, 15.3 per cent of last preferences,
but only 8.3 per 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 congurations.
There were numerous contests with between 5 and 11 candidates, but there was
only a single case where 12, 13 and 15 candidates contested.
No clear pattern is discernible in the ve-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 on
Nauru remains highly personalised, and closely connected to kinship or clan linkages.
Different factions able to strategically direct voters to place major rivals last usually
contested those constituencies with larger numbers of candidates. Candidates distri-
bute how to votecards at Nauru elections, and citizens are familiar with the tactic of
strategically 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 candidates tended to
contest. Unlike Figure 4, which only shows the top-placed victor and groups
results from different elections, Figure 5 shows all of the victors for each of 10 sep-
arate constituency contests. Separate investigation of those exceptional series in
Figures 4 and 5which are not U-shaped indicates that there was usually a well-
Ben Reilly has remarked that the Nauruan system [is] much more majoritarian than a standard Borda
count(2002: 365). Here we make that statement much more precise.
placed incumbent, expected to win, who obtained a U-shaped preference schedule,
and/or that actual victors were not polarising gures. The data in Figures 4 and 5
are consistent with strategic voting, but similar patterns could potentially arise with
sincere ranking and simply reect polarisation in electorates.
Neither the Slovenian nor the Nauru results provide evidence of intermediate
candidates, promoted up the scale by the practice of ranking major challengers
last, becoming serious contenders, as Dummett (1997: 85) fears may arise under
Borda. In Slovenia, there is no need to allocate preferences in such a way to
avoid strengthening major rivals. Voters can instead more simply avoid submitting
full rankings. In Nauru, factions competing in the two-, three- or four-seat consti-
tuencies put their major rivals as last preference, and thereby potentially promote
middle-ranked contestants. If Nauru used an elimination-based system, such as the
AV or STV, these intermediate candidates might stand a better chance. With the
Dowdall rule, they rarely perform strongly, owing to the low weighting of inter-
mediate preferences.
For the same reason, the incentives for elding red herringor dummycandi-
dates are greater under Naurus compulsory ranking system than under optional
Borda ranking in Slovenia (although not as great as they would be under a Borda
system with compulsory ranking). In Nauru, the factions tend to run two candidates
in the two-member districts but also to encourage buffer candidates’–who are not
expected to win to soak up intermediate preferences, and thereby lower the vote
tallies of their major rivals (personal communication, Roland Kun, Nauru MP, 20
August 2013). These were not difcult-to-engineer clonesintended to split rivals
votes, but irrelevant alternativesin Condorcets sense, designed to diminish the
value of preference votes allocated to arch-rivals. With the Dowdall rule, at least
the scoring of those lower-order alternatives remained xed, irrespective of the
greater number of candidates. Had the Borda Count been in place, the inclusion of
these irrelevant alternativeswould have adjusted the overall vector of ranking
Figure 5. Victorsshare of preference votes by number of candidates contesting, three and four seat
Nauru constituencies 19972013
Notes: as for gure 4. The chart shows the distribution of preferences for all victors in contests for the
four seat Ubenide constituency and for the constituency of Meneng in 2013, which was increased from
two to three members ahead of that election
Our investigation shows major differences between the Slovenian, Nauru and Borda
systems in theory and in practice, but there is one signicant commonality. Both the
Slovenian Borda variant and the Dowdall systems tend to entail ultimate election of
rst-count leaders, but for different reasons. On Nauru, this arose due to relatively
low weighting of lower preferences. In Slovenia, it stemmed from substantial agree-
ment on the strongest candidate and voters choosing to cast truncated ballots, as
allowed 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 the
single-member district plurality. This evidence should not be interpreted as support
for the case for categorical choice-based electoral systems: it only shows that one
type of real-world choice, under rank-order scoring rules, is to express a common
indifference to all but ones most-favoured candidate.
The system used on Nauru is not merely a modied form of Borda, but an impor-
tant rule in its own right. The usual classications of electoral systems tend to derive
from their ballot structure, their methods of determining victors or their broad reper-
cussions. Terms like rst-past-the-postrefer to the method of determining victors;
while the term proportional representationor when we say a system is majoritar-
ian, these refer to the kind of outcomes that are delivered. Dowdall has a similar
ballot structure to Borda, and methods of determining victors closely resemble
each other but, given real-world preference distributions, outcomes can and will
differ between the two methods. As it operates in Nauru, the Dowdall rule is in
fact closer to the SNTV than to the Borda system, at least for the larger (three- and
four-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 rst preferences. By rapidly dis-
counting second and lower preferences, the Dowdall rule meets this objection. In
addition, AV and STV are sometimes criticised because only a restricted share of
voterssecond or lower preferences are ever counted, whereas the Dowdall rule
counts all preferences. It also removes the need for time-consuming or costly elimin-
ations in successive rounds or counts, but without losing sensitivity to lower
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 allvariants.
In the Nauru context, where there are no political parties, that proportionality is not as
apparent as would be the case if that system were used in larger constituencies and in
a party-centered system. On the negative side, the Dowdall system potentially has
some of the features of SNTV, but these are only fully apparent in the three- and
four-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 eld
exactly the right number of candidates with possible heavy penalties for over-
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
such systems may be used strategically to ensure victory of a favoured candidate by
placing 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 tended
to manipulate such systems with the objective of maximising the likelihood of
defeat of a rival to their rst choice. In Slovenia, where lodging full preference rank-
ings was optional, we found that on average, only 41.8 per cent of voters ranked a
second candidate in our ve contests. The majority of voters engaged in a form of
bullet voting, generating in response efforts by the Slovenian Electoral Commission
to enforce a fuller ranking. On Nauru, that difference was maximised by placing
major 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 been
shown to be less susceptible to micro-manipulation than other electoral systems
(Saari 1990b). Manipulation, at least of this type, is only a major problem for
social-choice theorists insofar as the primary justication adopting such systems is
that they better express the felt preferencesof voters (Black 1976: 15; Dummett
1997: 63, 182). If voters submit ballots that record their preferences sincerely
when doing harms their likely ability to achieve results more to their liking, strategic
voting seems desirable rather than problematic (Mckelvey and Niemi 1978). Also, as
Dowding and Van Hess argue, democratic processes are more than mere counting
devices: 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 time
compulsory ranking has not fostered a high rate of invalid voting. Hungarian and
Italian reluctance to specify second or lower preferences may have frustrated the Slo-
venian Electoral Commission, but this was primarily a consequence of voter choices
together 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 the
voters 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 rules
simply by their mathematical properties, and suggests that similarities and differences
across different rules can, and will, vary with political context.
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... This development is connected with changes in energy use [10] and the implementation of new energy solutions [11]. Among the Multi-Criteria Decision-Making (MCDM) or Multi-Criteria Decision Analysis (MCDA) methods, different multi-criteria methods can be employed, considering the primary criteria upon which they are based: utility functions (such as MAUT [12], AHP [13], DEMATEL [14]) relationships outranking (like ARROW-RAYNAUD [15], ELECTRE [16], PROMETHEE [17]), distances (TOPSIS) [18] and decision support (BORDA ranking methods [19] and their modified and COPELAND [20]) (Fig. 1). Methods of aggregating assessments into the form of a utility function are implemented "from top to bottom" in the decision-making process, and these methods are associated with the American school of MCDM [21]. ...
... The next step was the Borda ranking method, which did not use standardized assessments of the criteria [29] and gave them weights to determine their validity [30]. Borda's method considers all criteria preferences [19], allotting a value to each, and establishes the best variant by a simple tallying of the total sum of numbers [31] Borda method, simple criteria numbered according to their importance do not take into account the preferences of the decision-making body, which excludes the subjective character of a decision on the site localization [32]. The problems with the localization of photovoltaic farms can be analysed by considering the many criteria having an essential effect on the realization of a given investment [33]. ...
... Note that both these rules have a bias close to the balanced value of -0.5 and that the polarized PL and consensual BC rules produce significantly different rankings and winners. For example, the official second-placed candidate (2) and joint winner would have come fourth and lost under PL but top under BC. ...
... In comparison to plurality, anti-plurality, the Borda count or the Dowdall method as surveyed in a variety of sources [2,5,[13][14][15], CHPV is a viable and balanced alternative positional voting system. ...
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The Borda count is a positional voting system that favors 'consensual' candidates with broad support while plurality is instead biased towards 'polarizing' ones with strong support. Our article focusses first on developing indices for quantifying system bias and then on vector analysis and design, while seeking to find an intermediate vector evenly balanced between consensus and polarization. The bias indices are based on the preference weightings of a normalized vector that represents a class of affine equivalent ones. The use of weightings that form a geometric progression evolves from this development. Such a 'geometric voting' vector can represent any positional voting vector with three preferences. With its common ratio as the sole variable, this vector can also span the whole spectrum of system bias continuously regardless of the number of preferences it employs; as demonstrated by our case study of the 1860 US presidential election with four candidates. Using this variable vector as an analytical tool, it establishes the 'consecutively halved positional voting' vector as the optimum one for balance. In our case study of the 2019 Nauru general election, this balanced vector is compared to its Dowdall rival that comprises a harmonic progression of weightings and several advantages are identified.
... Thus we use a modification, the Dowdall System, that assigns the first feature a reward of 1, the second a reward of 1/2, the third receives 1/3, and so until, at the end ranking the options from higher reward to lowest. See for example the work of Fraenkel and Grofman [24] on these different voting systems. We chose the Dowdall system because it allows merging the preferences without a big penalisation for being a very distant preference for one of the models, contrary to the Borda Count. ...
In recent years there has been increased use of machine learning (ML) techniques within mathematics, including symbolic computation where it may be applied safely to optimize or select algorithms. This paper explores whether using explainable AI (XAI) techniques on such ML models can offer new insight for symbolic computation, inspiring new implementations within computer algebra systems that do not directly call upon AI tools. We present a case study on the use of ML to select the variable ordering for cylindrical algebraic decomposition. It has already been demonstrated that ML can make the choice well, but here we show how the SHAP tool for explainability can be used to inform new heuristics of a size and complexity similar to those human-designed heuristics currently commonly used in symbolic computation
... Based on the example, we can infer that invoking the function with an array input argument is a more common usage pattern among downstream libraries than invoking the function with a list of floats. As a final step, we ranked each API according to relative usage using the Dowdall positional voting system [29] (a variant of the Borda count [30] that favors candidate APIs having high relative usage). From the rankings, we assigned standardization priorities, with higher ranking APIs taking precedence over lower ranking APIs, and extended the analysis to aggregated API categories (e.g., array creation, manipulation, statistics, etc.). ...
... They are used in group decisions (Dyer and Miles Jr., 1976), group recommender systems (Masthoff, 2015), meta-search engines, multicriteria selection, word association queries (Dwork et al., 2001), sports competitions (Stefani, 2011;Csató, 2021), awarding prizes (Benoit, 1992;Stein et al., 1994;Corvalan, 2018), arbitrator selections (Bloom and Cavanagh, 1986), and even for aggregating results from gene expression microarray studies (Lin, 2010). Many countries use ordinal scoring rules in political elections: most of them use plurality, while Slovenia, Nauru and Kiribati use non-plurality scores (Reilly, 2002;Fraenkel and Grofman, 2014). ...
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Scoring in Multi-Event Tournaments How much is a first place worth? Is the athlete who came first once and third twice better than the athlete who came second three times? Different sports value these positions differently: (60, 54, 48) in IBU biathlon, (25, 18, 15) in F1 racing, (8, 7, 6) in Diamond League athletics. Are these choices based on anything? Is there even a rational way to choose a scoring vector of arbitrary length to rank any number of athletes? In “How should we score athletes and candidates: Geometric scoring rules,” authors A.Y. Kondratev, E. Ianovski, and A.S. Nesterov investigate two approaches to how the problem of choosing a scoring vector for a tournament can be reduced to the choice of a single parameter: an axiomatic approach based on eliminating spoilers and an optimization approach based on maximizing the expected quality of the winner. Intriguingly, the vectors generated by the second approach are uncannily similar to those used in real sporting events.
... Thus we use a modification, the Dowdall System, that assigns the first feature a reward of 1, the second a reward of 1/2, the third receives 1/3, and so until, at the end ranking the options from higher reward to lowest. See for example the work of Fraenkel and Grofman [24] on these different voting systems. We chose the Dowdall system because it allows merging the preferences without a big penalisation for being a very distant preference for one of the models, contrary to the Borda Count. ...
Full-text available
In recent years there has been increased use of machine learning (ML) techniques within mathematics, including symbolic computation where it may be applied safely to optimise or select algorithms. This paper explores whether using explainable AI (XAI) techniques on such ML models can offer new insight for symbolic computation, inspiring new implementations within computer algebra systems that do not directly call upon AI tools. We present a case study on the use of ML to select the variable ordering for cylindrical algebraic decomposition. It has already been demonstrated that ML can make the choice well, but here we show how the SHAP tool for explainability can be used to inform new heuristics of a size and complexity similar to those human-designed heuristics currently commonly used in symbolic computation.
... This enables combining them to get the overall most popular values as shown in Table 7 (full list available under [30]). This problem is similar to a voting problem where multiple individuals can vote with a list of descending preference and can be solved with scoring rules as discussed by Fraenkel et al. [13]. We decided to use the Dowdall rule, which favors parameters with top preferences. ...
Full-text available
Collecting metadata from Transport Layer Security (TLS) servers on a large scale allows to draw conclusions about their capabilities and configuration. This provides not only insights into the Internet but it enables use cases like detecting malicious Command and Control (C &C) servers. However, active scanners can only observe and interpret the behavior of TLS servers, the underlying configuration and implementation causing the behavior remains hidden. Existing approaches struggle between resource intensive scans that can reconstruct this data and light-weight fingerprinting approaches that aim to differentiate servers without making any assumptions about their inner working. With this work we propose DissecTLS, an active TLS scanner that is both light-weight enough to be used for Internet measurements and able to reconstruct the configuration and capabilities of the TLS stack. This was achieved by modeling the parameters of the TLS stack and derive an active scan that dynamically creates scanning probes based on the model and the previous responses from the server. We provide a comparison of five active TLS scanning and fingerprinting approaches in a local testbed and on toplist targets. We conducted a measurement study over nine weeks to fingerprint C &C servers and analyzed popular and deprecated TLS parameter usage. Similar to related work, the fingerprinting achieved a maximum precision of 99 % for a conservative detection threshold of 100 %; and at the same time, we improved the recall by a factor of 2.8.
... Voter awards the first-ranked candidate with 1 point, while the second-ranked candidate receives half of a point, the third-ranked candidate receives onethird of a point, et cetera. This variant is used in the electoral system of the third-smallest country in the world, island nation of Nauru [62]. ...
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Deep learning has contributed greatly to many successes in artificial intelligence in recent years. Today, it is possible to train models that have thousands of layers and hundreds of billions of parameters. Large-scale deep models have achieved great success, but the enormous computational complexity and gigantic storage requirements make it extremely difficult to implement them in real-time applications. On the other hand, the size of the dataset is still a real problem in many domains. Data are often missing, too expensive, or impossible to obtain for other reasons. Ensemble learning is partially a solution to the problem of small datasets and overfitting. However, ensemble learning in its basic version is associated with a linear increase in computational complexity. We analyzed the impact of the ensemble decision-fusion mechanism and checked various methods of sharing the decisions including voting algorithms. We used the modified knowledge distillation framework as a decision-fusion mechanism which allows in addition compressing of the entire ensemble model into a weight space of a single model. We showed that knowledge distillation can aggregate knowledge from multiple teachers in only one student model and, with the same computational complexity, obtain a better-performing model compared to a model trained in the standard manner. We have developed our own method for mimicking the responses of all teachers at the same time, simultaneously. We tested these solutions on several benchmark datasets. In the end, we presented a wide application use of the efficient multi-teacher knowledge distillation framework. In the first example, we used knowledge distillation to develop models that could automate corrosion detection on aircraft fuselage. The second example describes detection of smoke on observation cameras in order to counteract wildfires in forests.
... The Borda count [24] is a classical voting method, which is a sorted voting mechanism, and has been employed in the elections in Nauru [27] and Slovenia [28]. Unlike the plurality voting mechanism, the Borda count not only lets voters choose the best candidates, but also to let voters rank these candidates. ...
Full-text available
Blockchain technology is well known due to the advent of Bitcoin. With the development of recent years, blockchain technology has been widely used in medicine, digital currency, energy, etc. The practical Byzantine fault-tolerant (PBFT) algorithm is a consensus algorithm widely used in consortium blockchains. Aiming to address the problems of the PBFT algorithm, low consensus efficiency due to high communication complexity, and malicious behavior of the primary node leading to consensus failure, an improved PBFT algorithm based on a comprehensive evaluation model (TB-PBFT) is proposed. First, nodes are divided into several groups based on the multi-formation control strategy of an unmanned aerial vehicle (UAV) cluster, which significantly reduces the communication complexity. Second, a comprehensive evaluation model combining the entropy method, TOPSIS method, and Borda count is proposed, which uses the behavior of nodes as an evaluation index, and the comprehensive score of nodes is obtained according to the preferences of other nodes. Finally, the highest ranking node is selected as the primary node through the comprehensive evaluation model to ensure the security and stability of the blockchain network. We analyze TB-PBFT algorithms and compare them with other Byzantine fault tolerance algorithms. Theoretical analysis and simulation results show that the TB-PBFT algorithm can improve node scalability and fault tolerance and reduce communication complexity and view switching probability. We also prove that the comprehensive evaluation model can improve the consensus success rate of the algorithm, and the feasibility and effectiveness of the improved consensus algorithm are verified. Hence, it can be applied to the consortium blockchain system effectively and efficiently.
This chapter briefly presents some of the mathematical and geospatial tools (for geospatial analysis, see, e.g., DeSmith et al., Geospatial Analysis. A Comprehensive Guide to Principles, Techniques and Software Tools, 6th edn, 2018) that are important in the context of multicriteria location decisions (see Malczewski, GIS and Multicriteria Decision Analysis, Wiley, New York, 1999). The first chapter discusses interpolation and curve fitting techniques. These methods are important in case measurements of some attribute have been taken at sites (so-called observation points), whereas the values of these attributes are needed at other points. For instance, seismic tests have been taken at some discrete points that provide a profile of different subsoil strata. Information regarding the thickness of, say, coal seams are needed in areas in which the Federal Government offers land leases for the exploitation of coal. Given that the observation points are reasonably close and there are no major geological faults in between, interpolation of known data can provide important clues regarding the feasibility of exploiting the natural resource.
Popular elections are at the heart of representative democracy. Thus, understanding the laws and practices that govern such elections is essential to understanding modern democracy. In this book, Cox views electoral laws as posing a variety of coordination problems that political forces must solve. Coordination problems - and with them the necessity of negotiating withdrawals, strategic voting, and other species of strategic coordination - arise in all electoral systems. This book employs a unified game-theoretic model to study strategic coordination worldwide and that relies primarily on constituency-level rather than national aggregate data in testing theoretical propositions about the effects of electoral laws. This book also considers not just what happens when political forces succeed in solving the coordination problems inherent in the electoral system they face but also what happens when they fail.
In recent years there has been a marked resurgence of interest in the effects of electoral laws on important aspects of politics such as party competition. In this volume, a distinguished group of scholars looks at the impact of one set of electoral rules--the single non-transferable vote--on electoral competition in Japan, Korea and Taiwan. Under this plan citizens are allowed one vote even though there is more than one seat to be filled. In comparative studies of the adoption and rejection of the single nontransferable vote and the consequences of its use across different settings, the contributors explore the differences in the operation and effects of the application of the same rule in different countries. Arguing that any single feature of a political system is embedded in a political structure and cannot be understood in isolation, the authors demonstrate how the same rule can have different consequences depending on the context in which it operates. The contributors offer fresh insights into the comparative study of political institutions as well as into the operation of particular electoral rules. In addition to the editors, the contributors include Kathleen Bawn, John Boland, Jean-Marie Bouissou, Gary Cox, John Fu-Sheng Hsieh, Arend Lijphart, Emerson Niou, Steven R. Reed, and Frances Rosenbluth, among others. Bernard Grofman is Professor of Political Science, University of California at Irvine. Edwin A. Winckler is at the East Asian Institute, Columbia University. Brian Woodall is Assistant Professor in the School of International Affairs, Georgia Institute of Technology. Sung-Chull Lee is Assistant Professor of Political Science, University of California at Irvine.
Over two centuries of theory and practical experience have taught us that election and decision procedures do not behave as expected. Instead, we now know that when different tallying methods are applied to the same ballots, radically different outcomes can emerge, that most procedures can select the candidate, the voters view as being inferior, and that some commonly used methods have the disturbing anomaly that a winning candidate can lose after receiving added support. A geometric theory is developed to remove much of the mystery of three-candidate voting procedures. In this manner, the spectrum of election outcomes from all positional methods can be compared, new flaws with widely accepted concepts (such as the "Condorcet winner") are identified, and extensions to standard results (e.g. Black's single-peakedness) are obtained. Many of these results are based on the "profile coordinates" introduced here, which makes it possible to "see" the set of all possible voters' preferences leading to specified election outcomes. Thus, it now is possible to visually compare the likelihood of various conclusions. Also, geometry is applied to apportionment methods to uncover new explanations why such methods can create troubling problems.
I The Theory of Committees and Elections.- I. A Committee and Motions.- II. Independent Valuation.- III. Can a Motion be Represented by the same Symbol on Different Schedules?.- IV. A Committee using a Simple Majority: Single-peaked Preference Curves.- V. A Committee using a Simple Majority: other Shapes of Preference Curves.- 1. Curves either single-peaked or single-peaked with a plateau on top.- 2. Other classes of curves.- VI. A Committee using a Simple Majority: any Shapes of Preference Curves, Number of Motions Finite.- VII. Cyclical Majorities.- VIII. When the Ordinary Committee Procedure is in use the Members' Scales of Valuation may be Incomplete.- IX. Which Candidate ought to be Elected?.- X. Examination of some Methods of Election in Single-member Constituencies.- XI. Proportional Representation.- XII. The Decisions of a Committee using a Special Majority.- 1. When the members' preference curves are single-peaked.- 2. When the members' preference curves are subject to no restriction.- XIII. The Elasticity of Committee Decisions with an Altering Size of Majority.- 1. When the members' preference curves are single-peaked.- 2. When the members' preference curves are subject to no restriction.- XIV. The Elasticity of Committee Decisions with Alterations in the Members' Preference Schedules.- 1. When the members' preference curves are single-peaked.- 2. When the members' preference curves are subject to no restriction.- XV. The Converse Problem: the Group of Schedules to Correspond to a Given Voting Matrix.- XVI. A Committee using a Simple Majority: Complementary Motions.- XVII. International Agreements, Sovereignty and the Cabinet.- II History of the Mathematical Theory of Committees and Elections (Excluding Proportional Representation).- XVIII. Borda, Condorcet and Laplace.- 1. Jean-Charles de Borda (1733-1799).- 2. Marie Jean Antoine Nicolas Caritat, Marquis de Condorcet (1743-1794).- 3. Pierre-Simon, Marquis de Laplace (1749-1827).- 4. Conclusions.- XIX. E. J. Nanson and Francis Galton.- XX. The Circumstances in which Rev. C. L. Dodgson (Lewis Carroll) wrote his Three Pamphlets.- Appendix. Text of Dodgson's Three Pamphlets and of 'The Cyclostyled Sheet'.- A Discussion of the Various Methods of Procedure in Conducting Elections (1873).- Suggestions as to the Best Method of Taking Votes, Where More than Two Issues are to be Voted on (1874).- A Method of Taking Votes on More than Two Issues (1876) 'The Cyclostyled Sheet' (7 Dec. 1877).- Notes on Dodgson's Third Pamphlet 'A Method...' (1876).
A variety of electoral systems for single-winner, multicandidate elections are evaluated according to their tendency to (a) select the Condorcet candidate--the candidate who could beat each of the others in a two-way race--if one exists, and (b) select a candidate with high social (average) utility. The proportion of Condorcet candidates selected and a measure of social-utility efficiency under either random society or spatial model assumptions are estimated for seven electoral systems using Monte Carlo techniques. For the spatial model simulations, the candidates and voters are generated from multivariate normal distributions. Numbers of candidates and voters are varied, along with the number of spatial dimensions, the correlation structure, and the relative dispersion of candidates to voters. The Borda, Black, and Coombs methods perform well on both criteria, in sharp contrast to the performance of the plurality method. Approval voting, plurality with runoff, and the Hare system (preferential voting) provide mixed, but generally intermediate results. Finally, the results of the spatial model simulations suggest a multicandidate equilibrium for winning-oriented candidates (under plurality, runoff, and Hare) that is not convergent to the median.