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Public Choice (2024) 200:149–171
https://doi.org/10.1007/s11127-024-01146-4
1 3
Balancing democracy: majoritarianism versusexpression
ofpreference intensity
AsafD.M.Nitzan1· ShmuelI.Nitzan2
Received: 3 April 2023 / Accepted: 19 January 2024 / Published online: 14 March 2024
© The Author(s) 2024
Abstract
This paper evaluates three prominent voting systems: the Majority Rule (MR), Borda Rule
(BR), and Plurality Rule (PR). Our analysis centers on the susceptibilities of each system
to potential transgressions of two foundational principles: the respect for majority prefer-
ence (majoritarianism) and the acknowledgment of the intensity of individual preferences.
We operationalize the concept of ’cost’ as the expected deviation from the aforementioned
principles. A comparative assessment of MR, BR, and PR is undertaken in terms of their
costs. Our findings underscore the superiority of PR over MR, whilst also highlighting the
comparative advantage of MR against BR.
Keywords Majority principle· Preference intensity· Scoring rules· Majority rule·
Plurality rule· Borda rule· Expected erosion of a principle
1 Introduction
Two principles that social-aggregation rules should adhere to are majoritarianism and the
recognition of voters’ preference intensity. These ideals safeguard the interests of a simple
majority as well as protect a minority possessing significantly robust preferences, respec-
tively. The delicate balance between these principles fuels the ongoing discourse between
proponents of the majority rule (MR) and supporters of certain scoring rules, particularly
the Borda rule (BR) and the prevalent plurality rule (PR).1
* Shmuel I. Nitzan
nitzans@biu.ac.il
Asaf D. M. Nitzan
adamnitzan@gmail.com
1 Hi Auto, TelAviv, Israel
2 Department ofEconomics, Bar-Ilan University, 52900RamatGan, Israel
1 Under MR, the social preference of an alternative to another one hinges on the existence of a majority of
voters who prefer it. According to the Borda rule (m-1), points are assigned to the best out of the m alterna-
tive ranked at the top, (m-2) points are assigned to the second-best alternative, and so on. (No points are
assigned to the worst alternative.) Under PR one point is assigned to the most preferred alternative and no
points to the remaining alternatives (from the second-best to the worst).
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Public Choice (2024) 200:149–171
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Traditionally, the advocacy for MR, BR, and PR has been grounded in axiomatic ration-
ales, demonstrating that these aggregation rules exclusively fulfill the desirable prerequi-
sites concerning the connection between individual and social preferences. There are two
distinct justifications for MR: one is the epistemic stance, rooted in Condorcet’s Jury Theo-
rem (as referenced in List and Goodin 2001); the other is the consequentialist-utilitarian
perspective (as highlighted in Brighouse & Fleurbaey, 2010; Nakada et al., 2023; Rae,
1969), which is particularly pertinent during the constitutional stage, a phase dominated by
the veil of ignorance.
Historically, there has been no direct comparison between MR and either BR or PR,
considering their inherent drawbacks: the disregarding of preference intensity and the vio-
lation of majoritarianism, respectively. This study aims to bridge this gap by first examin-
ing MR and BR, and then comparing MR and PR in light of their costs. ’Cost’ here repre-
sents the degradation of the principle each rule infringes. The two main findings reported
below are PR’s supremacy over MR and MR superiority over BR. These results cast a
fresh perspective on the age-old debate between Condorcet and Borda and their advocates,
who fervently criticized PR before championing, in order, MR and BR. Moreover, the first
finding also revives the debate about the merits of PR which despite its apparent practical
benefits, received limited support from 22 preeminent voting-theory experts who ranked 18
different voting rules, as evidenced by Laslier (2012).
2 The novelty ofour approach
Let society N consist of n voters, n > 2, and suppose that the set of social alternatives, X,
has m elements, m > 2.
Ri
denotes the preference relation of individual i, which is assumed
to be strict ordering, and R = (R
1
, …,
Rn
) is a preference profile. An aggregation rule is
a mapping from the set of possible profiles to the set of possible reflexive and complete
social-preference relations. Here we do allow for indifference and the typical social-prefer-
ence relation is R. The score of an alternative x under the Borda rule and the Plurality rule
on which we focus is denoted by B(x) and P(x). A majority prefers y to x when the number
of individuals who prefer y to x, N(y,x), is larger than the number of individuals who prefer
x to y.
Our combined approach of ordinal and ranking-based utilitarianism is very different
from the typical unrestricted utilitarian approach that works mostly with the standard prin-
ciples (Benthamite, Rawlsian, etc.), which is very difficult to apply in comparing alterna-
tive voting rules. More explicitly, the typical utilitarian approach compares the expected
social welfare (Benthamite, Rawlsian, etc.,) obtained under different voting rules. In con-
trast, in our approach, the social planner compares MR to a scoring rule based on expected
deviation from the two fundamental democratic principles that are assumed to be of equal
significance and not on some standard utilitarian principle. Since MR implies ordinal utili-
ties, the proposed measure of the expected deviation of a scoring rule from the major-
ity principle is naturally defined in terms of the number of individuals who prefer one
alternative to another one, and not in terms of cardinal interpersonally non-comparable
individual utilities. Since scoring rules imply some restricted form of cardinal and per-
sonally comparable utilities, the proposed measure of the expected deviation of MR from
the principle of allowing expression of preference intensity is naturally defined in terms of
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Public Choice (2024) 200:149–171
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the ranking-based utilities of the two scoring rules on which we focus.2 Both measures of
deviation from the fundamental principles depend on the assumed preference distributions
in the population and they take into account all possible preference profiles and all possible
compared alternatives.
Our focus is on the most widely studied monotonic scoring rule, BR, and the most com-
mon weakly monotonic scoring rule, PR. Given the plausibility of a complete veil of igno-
rance in the constitutional stage, BR is a reasonable representative scoring rule.3 Thus, we
first compare MR with BR and then compare MR with PR.4
BR violates the majority principle in those instances, namely, preference profiles and
pairs of compared alternatives, where it protects the minority effectively by taking into
account its higher preference intensity rather than the majority’s lower preference intensity.
While the emphasis in Baharad and Nitzan (2002) is on the different degrees of majority-
decisiveness amelioration that different scoring rules provide, in the current study the focus
is on the comparison between MR and the two most common scoring rules based on the
severity of the problems they cause: disregarding expression of preferences, which implies
prevention of effective expression of preference intensity by the minority, and violating the
majority principle. Let us now define the severity of the two problems in a way that enables
a comparison of the “costs” associated with applying the two alternative democratic voting
rules and, in turn, the preference of MR or BR.5
3 The severity ofviolating thetwo fundamental principles
One possibility is to measure the severity of a problem by the probability of its occurrence.
Baharad and Nitzan (2007, 2011) take such an approach, focusing on comparing alterna-
tive scoring rules. Gehrlein and Lepelley (2011, 2017) apply this criterion in assessing dif-
ferent election paradoxes. In our study we focus on the Condorcet–Borda (binary–posi-
tional) controversy and the analysis rests on two criteria. First, the comparison between
the prior likelihoods of the compared rules to be superior in terms of the deviation from
the aforementioned fundamental democratic principles. Second, we take into account not
only the severity of a problem in terms of the probability of its occurrence but also in
terms of the expected erosion of the two foundational democratic principles. Erosion of
the majority principle takes into account all possible preference profiles and any pair of
2 Interestingly, Bossert and Suzumura (2017) have shown that, with their alternative articulation of the
Benthamite greatest-happiness of-the-greatest-number principle and with ordinally measurable and inter-
personally non-comparable utilities, the social decision rule chooses those alternatives that maximize the
number of individuals who end up with their greatest element. This rule is tantamount to PR.
3 For further enthusiastic support of BR, see Saari (2006). The two most recent axiomatizations of BR
appear in Heckelman and Ragan (2020) and Maskin (2022).
4 According to Arrow’s (1963) Independence of Irrelevant Alternatives (IIA) axiom, social preferences
between any two alternatives depend only on the individual preferences between them. IIA implies that
social preferences disregard information about individuals’ preference intensity. The aggregation rule based
on majority comparisons is the clearest example of a rule that satisfies IIA. Note that BR does not satisfy
IIA but it does satisfy the weaker Modified IIA recently proposed by Maskin (2022), which allows a par-
ticular form of preference-intensity expression. It requires that if two profiles and two alternatives x and
y are given, and if every individual ranks the two alternatives the same way in both profiles and ranks the
same number of other alternatives between them in both profiles, then the social preference between these
two alternatives is the same for both profiles.
5 Analogous definitions apply for the comparison of MR and PR.
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Public Choice (2024) 200:149–171
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alternatives where the majority’s preference is overlooked. Erosion of the second principle,
namely, respect of voters’ preference intensity, takes into account all possible preference
profiles and any pair of alternatives where the disregard of preference intensity implies that
the minority’s preference intensity is disregarded, even though it is larger than that of the
majority.
Our analysis may yield a flexible, “case-dependent” choice between the two aggrega-
tion rules. One rule may be superior for certain preference profiles and pairs of alterna-
tives whereas the other rule may prevail for others.6 The implementation of such flexibility,
requiring the practical partitioning of the set of pairs of alternatives, may involve consider-
able difficulties. Our objective, therefore, is to identify the preferred aggregation rule, just
one of the two rules and not a flexible, case-dependent rule, based on its larger likelihood
to be superior and its lower expected violation of a fundamental principle. That is, the
expected severity of the problem that it raises should be lower than that of the problem
associated with using the alternative aggregation rule. The expected severity of the com-
pared rules is referred to as their expected cost. The main contribution of our study is the
clarification of the expected costs of the rules and the use of these expected costs to deter-
mine the superiority of one of them.
Given a specific situation, namely preference profile R and pair of alternatives x and y,
we first measure the corresponding cost of applying MR in terms of the erosion of the prin-
ciple that preference intensity must be taken into account, and, in particular, the minority’s
preference intensity, C(MR, R, x, y); then we measure the cost of applying BR in terms of
the erosion of majoritarianism, C(BR, R, x, y); The application of MR is warranted in a
specific situation if C(MR, R, x, y) < C(BR, R, x, y); the application of BR is warranted if
C(BR, R, x, y) < C(MR, R, x, y). And in case C(MR, R, x, y) = C(BR, R, x, y), the use of
either MR or BR is justified.7 This may ensure an ideal flexible situation-dependent bal-
anced democracy that applies in every particular situation the aggregation rule associated
with the lower cost. As already noted, however, such a flexible situation-dependent aggre-
gation rule is difficult to implement because it requires information about the voters’ actual
preference profile. Therefore, we impose the restriction that the same aggregation rule must
be applied to any pair of alternatives and any preference profile. Given this restriction, let
us turn to the comparison of MR with BR.
3.1 Proposed measures oferosion ofthetwo principles
Consider, first, the cost of applying MR. It involves the possible erosion of the principle
of allowing expression of preference intensity, which implies that the minority should win
when its preference intensity exceeds that of the majority (Principle 1). This principle is
6 The justification of deviating from MR in order to protect the minority is typically deemed plausible
when the two alternatives result in substantially different long-run irreversible outcomes and is usually
implemented by applying a qualified majority rule. Recently, Barberá et al. (2021) studied the hybrid rule
proposed by Daunou, (1803) which deviates from MR when a Condorcet winner does not exist by applying
PR after eliminating the Condorcet losers. In this case, the reconciliation of conflicting desiderata is based
on accommodating them lexicographically.
7 An analogous criterion is applied in Sect. 4.3, in the comparison between MR and PR where the cost
of BR, C(BR,R,x,y), is replaced by the cost of PR, C(PR,R,x,y). The endogenous partition of the set of
profiles takes into account the costs of the rules assigning equal weights to these costs. But one can easily
enrich the approach by assigning different weights to the costs C(MR, R) and C(BR,R) that are associated
with the application of MR and BR
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Public Choice (2024) 200:149–171
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eroded when, given a specific preference profile, MR and BR yield different social prefer-
ences between two alternatives x and y.8
Let us present a natural and intuitive measurement of erosion.
Suppose that, given a specific preference profile, alternative x is preferred over y under
MR and that alternative y is preferred over x under BR because the score of y, B(y), is
higher than the score of x, B(x). The positive difference between these scores, (B(y)—
B(x)), is referred to as the unrealized advantage of the preference intensity of the minor-
ity. The first measure is the share of the minority’s preference intensity that erodes due to
the use of MR that yields the social preference of alternative x, despite its inferiority to y
according to BR. This inferior alternative should not be preferred according to Principle
1. The proposed measure of erosion of Principle 1, for a particular preference profile and
two given alternatives x and y, given that MR and BR result in different social preferences
is the unrealized advantage of the preference intensity of the minority relative to its total
intensity.
Given a specific preference profile and pair of alternatives, Measure 1 represents the
relative erosion of the minority’s ability to effectively express its preference intensity and
ensure the superiority that its preferred alternative would have enjoyed had BR been used
as the aggregation rule.
To sum up, the proposed measure of the cost of MR in a particular situation is the rate
of reduction in the more intense minority’s preference of the socially inferior alternative in
terms of preference intensity relative to the socially superior alternative under MR.
Consider now the second majority principle: it requires that x is socially preferred over y
by BR when a majority prefers x over y. The cost of applying BR involves the possible ero-
sion of this principle, Principle 2, and again, this erosion is realized when, given a specific
preference profile R, MR and BR yield different social preferences between two alterna-
tives x and y. That is, B(x) >B(y), however a majority prefers y to x; the number of indi-
viduals who prefer y to x, N(y,x), is larger than N(x,y). Analogously to the measurement of
erosion of Principle 1, for a specific preference profile and pair of alternatives x and y, the
measurement of erosion of Principle 2 takes the form:
The proposed measure of the cost of BR is the disregarded advantage of the majority
obtained by alternative y (which is superior to alternative x under MR), divided by the
actual majority of y,9 In other words, Measure 2 represents the erosion of the majority’s
ability to effectively express its preference and ensure the superiority that its preferred
alternative would have enjoyed had MR been used as the aggregation rule.
Note that the lower bound (0) and the upper bound (1) of the two measures represent no
violation and maximal violation of the relevant principle.
𝐌𝐞𝐚𝐬𝐮𝐫𝐞 𝟏 ∶(B(y)−− B(x))∕B(y).
𝐌𝐞𝐚𝐬𝐮𝐫𝐞 𝟐 =(N(y,x)−N(x,y))∕N(y,x)
8 Notice that if we were to deal with the agreements on the collective ranking between x and y by MR, BR
and PR, while ignoring our proposed measures that focus on the erosion of the two fundamental princi-
ples (majoritarianism and respect of preference intensity) by these aggregation rules, we would fall into the
problem addressed by Fishburn and Gehrlein (1980, 1981) who focused on the special case of m = 4 and an
electorate of infinite size.
9 Analogous measures are applied for PR.
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Public Choice (2024) 200:149–171
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Given a preference profile and a pair of alternatives, a comparison of the costs of apply-
ing MR and BR is the basis for selecting one aggregation rule over the other. The preferred
rule is the less costly one.10
3.2 Illustration
Example Suppose N = {1,2,3}, X = { w,x,y,z} and the preference profile is R = (
R1,R2,R3
),
Let us determine simple majority relation and Borda social preference relation corre-
sponding to profile R:
It can be verified that
Let B(s) denote the Borda score of alternative s. In the above example, B(w) = 2,
B(x) = 3, B(z) = 6 and B(y) = 7. Hence,
Case 1: Consider the comparison of y and z. In such a case,
Thus, either rule may be used.
Case 2: Consider the comparison of w and x. In such a case,
C(BR,R)=(N(w,x)) - N(x,w) / N(w,x) =(2-1)/2=1/2. Since C(MR, R) = 1/3 < C(BR,
R) = 1/2, the justified aggregation rule for the comparison of x and w is MR.
3.3 Possible alternative measures
Finally note that one could think about alternative measures, such as the absolute devi-
ation between the scores or the support of the alternatives preferred by BR and MR,
B(y)-B(x) or N(y)-N(x) or the relative score or support of these alternatives, B(y)/B(x)
or N(y)/N(x). The former alternatives for Measure 1 and Measure 2 would not allow a
meaningful comparison between them. The obvious reason is that such absolute devia-
tions cannot be compared because they apply different notions; one relates to scores
and the other to majority and minority support. The latter alternative measures are
R1∶yR1zR1wR1x
R2∶xR2yR2zR2w
R3∶zR3yR3wR31x
yR
maj zR
maj wR
maj x
yRBzRBxRBw
C(MR, 𝐑)=C(BR, 𝐑)=0
C(MR, 𝐑)=(B(x)−B(w))∕B(x)=(3−2)∕3=1∕3
10 Our criterion of comparison between aggregation rules disregards operational simplicity and degree of
manipulability. By using pairs of alternatives as our standard of comparison, we avoid the need to take lack
of transitivity into account.
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Public Choice (2024) 200:149–171
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not vulnerable to such criticism, that is, they can be compared. However, they do not
capture the significant aspect implied by the proposed definitions of the two meas-
ures, namely, the relative disregarded or unexploited advantage of the minority and the
majority when the social preference differs from their preference.
Nevertheless, the measures that involve the ratios B(y)/B(x), P(y)/P(x) and N(y)/
N(x) are meaningful. In the case of B(y)/B(x) and P(y)/P(x), this formulation shifts
the emphasis from the proportion of the eroded scores of the minority to assessing the
severity of relative social injustice against the minority. Its preference intensity is not
respected, despite being B(y)/B(x) or P(y)/P(x) times stronger than that of the majority.
In the case of N(y)/N(x), this formulation shifts the emphasis from the proportion of
the eroded advantage of the majority to assessing the severity of relative social injus-
tice against the majority. Its advantage is not respected, despite being N(y)/N(x) times
stronger than that of the minority. We ran simulations verifying that our findings are
robust to the application of these alternative metrics to Measure 1 and Measure 2.
The current formulation assigns equal importance to all the compared pairs of alter-
natives when applying scoring rules. The erosion is calculated as the proportion of the
scores eroded out of the scores assigned to the rejected option. This form of "local"
normalization doesn’t create a common global scale for all the compared pairs, but
rather each pair is considered in isolation and the erosion strength for all pairs is given
equal weight in the overall expectation calculation. This has its merit as it evaluates
each aggregation rule over the choice between any two pairs of alternatives, irrespec-
tive of their popularity. We leave to future research the option of giving more impor-
tance to the erosion of choices with high scores relative to those with low scores. Sev-
eral potential avenues for achieving this include:
1. Focusing solely on cases where each rule results in a different winner and calculating
erosion exclusively for the pair of winners.
2. Investigating the implementation of a global normalization factor. In the majority rule,
the erosion measure naturally interprets what percentage of voters’ preferences were
eroded when normalized by the total number of voters. We can potentially develop a
similar measure for scoring rules by consistently normalizing erosion based on the aver-
age score that voters can assign to each option. This would establish a unified scale for
measuring erosion across all pairs of alternatives.
4 Main ndings
In our study, the definite preference of a rule is based on its superiority in terms of the
two criteria: the likelihood of being less costly and the difference between the expected
costs of the two rules. According to the two possible criteria, the conclusion may hinge
on the number of voters n and the number of alternatives m. The question is how n and
m affect the desirability of MR and BR under the veil of ignorance in the constitutional
stage regarding the actual preference profile and the compared alternatives. Before
turning to the identification of the preferred rule according to the two possible criteria,
we describe the particular statistical model used to generate the preference profiles of
the voters.
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Public Choice (2024) 200:149–171
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4.1 The simulation
Preference profiles can be generated by several probabilistic models.11 We base our results
on the Cubic and the spatial Euclidean Box models. These approaches have the common
feature of generating preferences to reflect or approximate data samples in real elections.12
The results presented below are based on the Cubic model (the results for the Euclid-
ean Box model are essentially the same and are presented in Appendix A). In the former
model, for a particular number of voters, n, and m alternatives (the candidates’ positions),
we generate a matrix of size n*m where each number is sampled from the uniform distribu-
tion over the
[0,1]
segment. This matrix represents the utility for each voter for each can-
didate and it yields the preference profile of the voters. In the Euclidean Box model, for a
particular number of voters, n, and m alternatives, we independently and uniformly sample
the alternatives and the positions of the voters from the Box on the assumption that a vot-
er’s utility for a candidate is a decreasing function of the distance between the candidate’s
position and the voter’s position.13 For a particular case of m alternatives and a preference
profile R, the percentage of pairs of alternatives that result in different preferences by MR
and BR in which MR is the superior rule (the erosion of principle 1 by MR, Measure 1, is
smaller than the erosion of principle 2 by BR, Measure 2) is equal to:
where
Note that t is the number of pairs of alternatives that result in different preferences by
MR and BR
.
Profiles in which BR has equal scores for alternatives x and y are discarded
and not considered to create erosion. We decided to only count cases where there is a clear
winner and a clear dispute between the results. The same policy is applied later for PR.
For a particular case of m alternatives and a preference profile R, the expected costs of
MR and BR, the expected erosion of the fundamental principles by these rules applying
measures 1 and 2, over all possible pairs of the alternatives that result in different prefer-
ence by MR and BR are:
r∕t
r=|{(x
,
y)∶N(x
,
y)>N(y
,
x)
,
B(y)>B(x)and𝜇(x
,
y)<𝜈
(y
,
x)}|
B(x)= The Borda count of x
N
(x,y)=|
|{i∶xRiy}|
|
𝜇
(x,y)=B(y)−B(x)
B(y)
𝜈
(x,y)=N(y,x)−N(x,y)
N(y,x)
t=|{(x
,
y)∶N(x
,
y)
>
N(y
,
x)and B(y)
>
B(x)}|
13 According to Merrill (1984) and Tideman and Plassmann (2013), when generating alternatives (candi-
dates) and voters by means of simulations based on a spatial model, outcomes come very close to describ-
ing the distribution of actual outcomes, and ranking data simulated with the spatial model are very similar
to observed ranking data. The spatial-model results thus tend to be more realistic.
11 The most common models are the Impartial Culture, Impartial and Anonymous Culture and the spatial
models.
12 See https:// franc ois- durand. github. io/ svvamp/.
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Public Choice (2024) 200:149–171
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In the simulation, we computed the percentage of pairs of alternatives in which MR is
superior to BR and the means of the difference of the expected costs of MR and BR in the
100,000 generated cases of the n voters’ preference profiles R. Analogous computations
were made for comparison between MR and PR.14 For each preference profile, we have
compared all possible pairs of the alternatives.15
Before delving into the simulated results, let us present a theoretical subsection that
contains analytical results that prove the superiority of MR to BR and the superiority of PR
to MR in terms of the above two criteria for the case of three alternatives and n voters, n
= 3,5,7,9,11,13,15,21,31,41,51. These analytical results make it possible to better infer the
results of the simulation that are almost identical for m = 3.
4.2 Analytical results forthecase ofthree alternatives
When m = 3, there are 6 possible strict individual preference relations (rankings) denoted
by 1,2, …,6. The possible preference profiles specify the selection of these rankings by
the n voters.Since the Cubic model assumes that, for every voter, all preference relations
have equal probability, we can disregard the identity of the voters as the voting is anony-
mous and focus on distinct preference profiles. Each distinct profile is given by (k1,k2,…
.k6), where ki is the number of voters selecting preference relation i, i = 1, …,6, and
k1 + k2 + … + k6 = n. The expected occurrence of such a distinct profile can be obtained by
using the multinomial theorem. This number, the multinomial coefficient, which is denoted
by (k1, k2, k3,…., k6)! is equal to n!/k1!*k2!*….*k6!. For each distinct profile, we calcu-
late Measure 1 and Measure 2 for all the pairs of alternatives that result in different pref-
erences by MR and BR, and then compute r/
t
. Assigning to each percentage of such a
distinct profile a weight that is equal to its multinomial coefficient and summing up all the
weighted percentages corresponding to these distinct profiles, we divide the sum of the
weighted percentages by the total number of these distinct profiles and obtain the expected
percentage of pairs of alternatives in which MR is superior to BR in all the distinct profiles
that result in different preference by MR and BR.
Given the multinomial coefficients of the possible distinct preference profiles, and using
the above formulas of the expected costs of MR and BR over all possible pairs of the alter-
natives for a particular distinct profile, we can compute the expected difference of these
costs in all possible distinct profiles (k1,k2,….k6), such that k1 + k2.. + k6 = n.
1
t
∑
x,y∶N(x,y)>N(y,x)
B(y)
>
B(x)
𝜇(x,y)and
1
t
∑
x,y∶N(x,y)>N(y,x)
B(y)
>
B(x)
𝜈(x,y
)
14 The random selection of preference orderings may not capture well the presence of proactive, highly
motivated minority members who may strategically coordinate their reports of preferences. At the constitu-
tional stage, it is difficult to capture this possibility without adding extra parameters such as an exogenous
polarization factor. Therefore, our analysis disregards such possible strategic considerations that are a sig-
nificant issue with intensity-based scoring rules..
15 Note that the example in Sect.3.2 illustrates the costs of the two aggregation rules assuming a particular
number of alternatives, a particular preference profile and a particular pair of alternatives. A series of such
examples could illustrate the comparison between the expected costs of MR and BR allowing 100,000 pro-
files over the m alternatives and taking into account all pairs of alternatives where these rules yield different
social preferences.
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Public Choice (2024) 200:149–171
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Calculating the expected percentage of pairs of alternatives in which MR is superior
to BR and calculating the difference between the expected costs of MR and BR, for
m = 3 and n = 3,5,7,9,11,13,15,21,31,41,51, we obtain the following results:
For three alternatives then, MR is unambiguously superior to BR because, for any
n, it is more likely to be superior in pairwise comparisons of the alternatives and its
expected cost is smaller (Tables1, 2).
Applying the multinomial coefficients and the analogous formulas for the compari-
son between MR and PR, we present in Tables3 and 4 the expected percentage of pairs
of alternatives in which PR is superior to MR and the difference between the expected
costs of PR and MR.
For three alternatives then, PR is unambiguously superior to MR because, for any
n, it is more likely to be superior in pairwise comparisons of the alternatives and its
expected cost is smaller.
Table 1 The expected percentage of pairs of alternatives in all the distinct profiles in which BR is superior
to MR for 3 candidates
voters
3 5 7 9 11 13 15 21 31 41 51
– 0 0.03 0.08 0.12 0.16 0.20 0.28 0.35 0.39 0.41
Table 2 Difference between the expected costs of BR and MR over all the pairs that result in erosion, which
indicates the advantage of MR for 3 candidates
Voters 3 5 7 9 11 13 15 21 31 41 51
– 0.15 0.10 0.07 0.05 0.04 0.03 0.02 0.01 0.01 0.01
Table 3 The expected percentage of pairs of alternatives in all the distinct profiles in which PR is superior
to MR for 3 candidates
Voters 3 5 7 9 11 13 15 21 31 41 51
1 1 0.94 0.90 0.88 0.86 0.82 0.77 0.73 0.70 0.67
Table 4 Difference between the expected costs of PR and MR over all the pairs that result in erosion, which
indicates the advantage of PR for 3 candidates
Voters 3 5 7 9 11 13 15 21 31 41 51
− 0.5 − 0.3 − 0.21 − 0.17 − 0.14 − 0.12 − 0.11 − 0.08 − 0.06 − 0.05 − 0.04
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4.3 General simulation results: thesuperiority ofMR overBR16
The simulation results establish the superiority of MR in terms of the above two criteria
when the two aggregation rules yield different outcomes. This is true for any combination
of the number of alternatives m and the number of voters n when the number of voters is
odd, n = 3,5,7,9,11,13,15,21,31,41,51,1001,10001 and m = 3,4,…,10.17
Table5 presents the results that illustrate the inferiority of BR in terms of the percent-
age of cases in which it is superior to MR when the two aggregation rules yield differ-
ent outcomes. This percentage can be considered as an estimate of the a-priori likelihood
that BR is the superior aggregation rule when two alternatives are compared, taking into
account all preference profiles that result in the erosion of Principles 1 and 2, that is, MR
and BR yield different preferences between the compared alternatives. For any n and m, the
a-priori likelihood of MR being superior to BR is larger than 50%. Note that our findings
are consequential because erosion, i.e., divergent outcomes by MR and BR is a likely pos-
sibility. More specifically, for m > 3, n > 3, it is obtained in at least 28% of the preference
profiles and in 6% of the pairwise comparisons. The likelihood increases with both m and
n and for m = 10, n = 10,001, it is obtained in 100% of the preference profiles and in 17% of
the pairwise comparisons. TheTables in Appendix B present these erosion likelihoods for
all the combinations of m and n.
For a small number of alternatives and a large number of voters, the superiority of MR
tends to be less significant. For example, for three alternatives and 51 as well as 100,001
voters, the likelihood of MR being the preferred rule is 1−0.41 = 0.59. For 10 alternatives
and 10,001 voters, this likelihood increases to 0.68. The results suggest that when the elec-
torate is sufficiently large, the extent of the superiority of MR converges to a limit.
Table 6 confirms the superiority of MR by presenting the mean of the difference in
the costs of BR and MR, which is always positive; for any combination of m and n, the
expected difference is positive. That is, the expected erosion of Principle 1 by MR is always
smaller than the expected erosion of Principle 2 by BR, taking into account all possible
comparisons of the generated alternatives and preference profiles under any combination of
Table 5 Percentage of pairwise comparisons in which BR is superior to MR
VOTERS CAN-
DIDATES
3 5 7 9 11 13 15 21 31 41 51 1001 10001
3 – 0 0.04 0.08 0.12 0.15 0.2 0.28 0.35 0.39 0.41 0.42 0.41
4 0 0.02 0.05 0.1 0.14 0.18 0.2 0.28 0.33 0.35 0.37 0.38 0.38
5 0 0.02 0.06 0.1 0.14 0.17 0.2 0.27 0.31 0.33 0.34 0.36 0.35
6 0 0.03 0.06 0.1 0.14 0.17 0.19 0.25 0.3 0.31 0.32 0.35 0.35
7 0 0.03 0.06 0.1 0.13 0.16 0.19 0.24 0.29 0.3 0.31 0.34 0.33
8 0 0.03 0.06 0.1 0.13 0.16 0.18 0.24 0.28 0.3 0.31 0.33 0.33
9 0.01 0.03 0.06 0.1 0.13 0.16 0.18 0.23 0.27 0.29 0.3 0.33 0.32
10 0.01 0.03 0.06 0.1 0.13 0.16 0.18 0.23 0.27 0.28 0.29 0.32 0.32
16 The code base for running the simulations, analytical calculations, and creating the different statistical
reports can be found at https:// github. com/ adamn itzan/ voting- rules- erosi on. The repository also includes
full result files for 100,000 simulations and the analytical calculations.
17 We chose to deal with an odd number of voters to avoid dealing with situations of equality in pairwise
comparisons of alternatives. The two largest numbers of voters, 1001 and 10,001 illustrate the significance
of the findings in the context of elections.
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Table 6 The mean of the differences between the expected costs of BR and MR, which indicates the advantage of MR
VOTERS CAN-
DIDATES
3 5 7 9 11 13 15
3 – 0.15 (0.08:0.19) 0.1 (− 0.08:0.15) 0.07 (− 0.13:0.12) 0.06 (− 0.23:0.36) 0.05 (− 0.19:0.31) 0.04 (− 0.21:0.28)
4 0.29 (0.17:0.36) 0.17 (− 0.17:0.25) 0.12 (− 0.25:0.53) 0.09 (− 0.27:0.45) 0.07 (− 0.3:0.38) 0.06 (− 0.26:0.51) 0.05 (− 0.29:0.46)
5 0.3 (0.0:0.4) 0.18 (− 0.29:0.27) 0.13 (− 0.3:0.55) 0.1 (− 0.3:0.67) 0.08 (− 0.31:0.59) 0.07 (− 0.34:0.53) 0.06 (− 0.31:0.47)
60.32 (− 0.1:0.42) 0.19 (− 0.33:0.7) 0.14 (− 0.32:0.56) 0.11 (− 0.3:0.68) 0.09 (− 0.39:0.6) 0.08 (− 0.28:0.53) 0.07 (− 0.28:0.61)
70.32 (− 0.17:0.44) 0.2 (− 0.39:0.71) 0.14 (− 0.38:0.57) 0.11 (− 0.34:0.69) 0.1 (− 0.31:0.6) 0.09 (− 0.32:0.54) 0.08 (− 0.33:0.62)
80.33 (− 0.21:0.45) 0.2 (− 0.39:0.72) 0.15 (− 0.4:0.57) 0.12 (− 0.3:0.69) 0.1 (− 0.3:0.6) 0.09 (− 0.41:0.68) 0.08 (− 0.39:0.62)
90.33 (− 0.25:0.45) 0.2 (− 0.35:0.72) 0.15 (− 0.39:0.58) 0.12 (− 0.52:0.7) 0.11 (− 0.31:0.74) 0.09 (− 0.32:0.68) 0.08 (− 0.3:0.62)
10 0.34 (− 0.28:0.46) 0.21 (− 0.42:0.72) 0.15 (− 0.43:0.8) 0.13 (− 0.37:0.7) 0.11 (− 0.32:0.61) 0.1 (− 0.34:0.69) 0.09 (− 0.29:0.62)
VOTERS
CANDIDATES
21 31 41 51 1001 10001
30.02 (− 0.19:0.34) 0.02 (− 0.2:0.25) 0.01 (− 0.17:0.27) 0.01 (− 0.15:0.24) 0.003 (− 0.04:0.07) 0.001 (− 0.02:0.02)
40.04 (− 0.25:0.47) 0.03 (− 0.23:0.35) 0.03 (− 0.21:0.35) 0.02 (− 0.17:0.34) 0.0061 (− 0.05:0.1) 0.002 (− 0.0:0.03)
50.05 (− 0.24:0.48) 0.04 (− 0.23:0.51) 0.03 (− 0.24:0.41) 0.03 (− 0.23:0.39) 0.007 (− 0.05:0.11) 0.002 (− 0.03:0.03)
60.06 (− 0.31:0.48) 0.05 (− 0.25:0.49) 0.04 (− 0.2:0.41) 0.04 (− 0.21:0.4) 0.008 (− 0.06:0.12) 0.003 (− 0.02:0.04)
70.06 (− 0.32:0.59) 0.05 (− 0.28:0.44) 0.04 (− 0.22:0.45) 0.04 (− 0.22:0.43) 0.01 (− 0.06:0.12) 0.003 (− 0.02:0.05)
80.07 (− 0.28:0.59) 0.05 (− 0.26:0.52) 0.05 (− 0.22:0.47) 0.04 (− 0.24:0.4) 0.01 (− 0.06.14) 0.003 (− 0.02:0.04)
90.07 (− 0.27:0.59) 0.06 (− 0.29:0.52) 0.05 (− 0.24:0.45) 0.04 (− 0.22:0.48) 0.01 (− 0.06:0.12) 0.003 (− 0.02:0.04)
10 0.07 (− 0.3:0.59) 0.06 (− 0.25:0.55) 0.05 (− 0.23:0.48) 0.04 (− 0.22:0.44) 0.01 (− 0.05:0.12) 0.003 (− 0.02:0.04)
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m and n. The Table also presents in brackets two more statistics: the minimum and maxi-
mum difference between the costs of BR and MR .
This difference referred to as the advantage of MR, ranges from the lowest advantage of
0.001 when m = 3 and n = 10,001 to the largest advantage of 0.34 when m = 10 and n = 3.
These results confirm all the findings adduced based on the first criterion. However, it seems
that for sufficiently large numbers of voters, the difference between the mean costs of BR and
MR although still positive decreases with n and becomes negligible. In the alternative Euclid-
ean Box model, a salient observation emerges from our simulation results: the discrepancy in
the means of the expected costs of BR and MR amplifies notably for expansive electorates,
as delineated in the second Table in Appendix A. To elucidate, for m = 5 and n = 10,001, the
anticipated advantage of MR escalates to 2% from the initial value of 0.2% in the Cubic distri-
bution simulation. This phenomenon underscores the robustness of our conclusions, particu-
larly when grounded in more realistic assumptions about preference profiles, as expounded
upon in footnote 14. A worthwhile task for future research is the analytical study of the limit
behavior of the superiority of MR over BR when the number of voters is sufficiently large.
The Table clarifies that the expected cost of MR, the expected violation of Principle
1, is smaller than the expected cost of BR, the expected violation of Principle 2, for any
m and n. This implies that in the constitutional stage, where the number of alternatives
and their identity, the number of voters, and their preferences are all unknown, there is a
very good reason to apply MR rather than BR if one focuses on the comparison between
their fundamental weaknesses, namely, violation of one of the two fundamental democratic
principles: majoritarianism and suitable recognition of preference intensity. Recall that the
comparison between MR and BR takes into account only the situations where these rules
result in different social preferences.
The underlying reason for MR’s superiority can be elucidated by delving into its inher-
ent expressive capability. Under MR, the comparison between two alternatives is based
upon the electorate’s allocation of n uniform scores to these alternatives. Contrarily, under
the Borda Rule (BR), this juxtaposition is predicated upon the n voters’ potential allocation
of as many as m disparate scores to these alternatives. MR, in comparison to BR, possesses
a constrained proficiency in encapsulating and delineating voter preferences. It is plausible
to conjecture that this more restricted expressive capability is advantageous because it con-
tributes to the lower erosion of principle 1 (Measure 1) relative to the higher magnitude of
degradation of majoritarianism by BR (Measure 2).
Table 7 Percentage of pairwise comparisons in which PR is superior to MR
VOTERS
CANDIDATES
3 5 7 9 11 13 15 21 31 41 51 1001 10001
3 1 1 0.94 0.9 0.88 0.86 0.81 0.77 0.73 0.69 0.67 0.57 0.56
4 1 1 0.91 0.86 0.86 0.82 0.78 0.76 0.73 0.72 0.7 0.61 0.59
5 1 1 0.92 0.85 0.88 0.83 0.79 0.77 0.75 0.73 0.71 0.64 0.63
6 1 1 0.93 0.86 0.91 0.85 0.81 0.79 0.78 0.77 0.74 0.67 0.65
7 1 1 0.95 0.88 0.93 0.88 0.83 0.82 0.82 0.8 0.78 0.69 0.67
8 1 1 0.96 0.9 0.94 0.9 0.85 0.85 0.84 0.83 0.8 0.71 0.69
9 1 1 0.97 0.91 0.95 0.91 0.87 0.87 0.86 0.84 0.83 0.72 0.71
10 1 1 0.97 0.93 0.96 0.93 0.89 0.89 0.88 0.86 0.85 0.74 0.72
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Two unequivocal findings come to light regarding the effect of m and n on the advantage
of MR. For a given number of voters n, the advantage of MR increases with the number of
alternatives m. For a given number of alternatives m, the advantage of MR decreases with
the number of voters n.
Note that the maximum difference between the costs of BR and MR is always posi-
tive. But the minimum difference can also be positive, which indicates that in all pairwise
comparisons between the alternatives MR is the superior rule. This is the case when m = 3,
n = 5, and when m = 4, n = 3. In the remaining combinations of m and n, the minimum dif-
ference is negative, indicating specific cases where BR is the preferred rule and the maxi-
mum difference is positive, indicating that MR is the preferred rule.
Finally, for m = 3, we confirmed that the simulation results are very close to the ana-
lytical ones. We have increased the reliability of the simulation results by generating
1,000,000 preference profiles instead of 100,000 and obtained that, for the combinations of
3 alternatives and n voters, the average deviation of the simulation results from the analyti-
cal results, in terms of the first criterion, was 0.27%. The average deviation of the simula-
tion results from the analytical results, in terms of the second criterion, was only 0.16%.
4.4 The superiority ofPR overMR
The most common and best-known scoring rule is the plurality rule, PR, which is an
extreme weakly monotonic scoring rule, that is, only the score assigned to the best alterna-
tive exceeds that of the second-best and all other alternatives, so significance is assigned
only to every voter’s most preferred alternative. Applying the same methodology for com-
paring MR and PR based on their costs, the extent of erosion of the principle they violate,
PR emerges unambiguously as a superior aggregation rule when n ≤ 10,001.
Our findings establish the superiority of PR in terms of its higher likelihood to be
superior (Table7) and its lower mean cost (Table 8), all simulation outcomes taken into
account. That is, for any combination of a number of alternatives m and a number of vot-
ers n, PR outperforms MR in terms of the two criteria we have applied in the previous
section. The negative difference between the means of the expected costs of PR and MR
represents the advantage of PR. This advantage ranges from − 0.57 which is the largest
advantage, when m = 10 and n = 7, to − 0.002 which is the smallest advantage, when m = 3
and n = 10,001.
Table 7 illustrates the superiority of PR in terms of the expected a priori likelihood
of being superior in pairwise comparisons of alternatives when the two aggregation rules
yield different outcomes. This likelihood always exceeds 0.56. This is of significance
because the likelihood of erosion now is considerably higher than that in the preceding sec-
tion. More specifically, for m > 3 and n > 3, it is obtained in at least 55% of the preference
profiles and in 13% of the pairwise comparisons. The likelihood increases with both m and
n and for m = 10, n = 10,001, it is obtained in 100% of the preference profiles and in 35% of
the pairwise comparisons. The lasttwo tables in AppendixB present these erosion likeli-
hoods for all the combinations of m and n.
Table8 yields two unequivocal findings regarding the effect of m and n on the advantage
of PR. For a given number of voters n, n > 3, the advantage of PR increases with the num-
ber of alternatives m. For a given number of alternatives m, the advantage of PR decreases
with the number of voters n.
As in the comparison between MR and BR, we conjecture that the superiority of PR
is partly due to its limited expressive capability. Under MR, the comparison between two
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Table 8 The mean of the differences between the expected costs of PR and MR, which indicates the advantage of PR
VOTERS CAN-
DIDATES
3 5 7 9 11 13 15
3− 0.5 (− 0.5:− 0.5) − 0.29 (− 0.67:− 0.17) − 0.21 (− 0.75:0.1) − 0.17 (− 0.8:0.21) − 0.14 (− 0.83:0.29) − 0.13 (− 0.86:0.37) − 0.11 (− 0.88:0.42)
4− 0.5 (− 0.5:− 0.5) − 0.41 (− 0.67:− 0.17) − 0.32 (− 0.75:0.1) − 0.26 (− 0.8:0.21) − 0.22 (− 0.83:0.29) − 0.2 (− 0.86:0.37) − 0.18 (− 0.88:0.42)
5− 0.5 (− 0.5:− 0.5) − 0.47 (− 0.67:− 0.17) − 0.4 (− 0.75:0.1) − 0.34 (− 0.8:0.21) − 0.3 (− 0.83:0.29) − 0.26 (− 0.86:0.37) − 0.24 (− 0.88:0.42)
6− 0.5 (− 0.5:− 0.5) − 0.51 (− 0.67:− 0.17) − 0.46 (− 0.75:0.1) − 0.41 (− 0.8:0.21) − 0.36 (− 0.83:0.29) − 0.32 (− 0.86:0.37) − 0.29 (− 0.88:0.42)
7− 0.5 (− 0.5:− 0.5) − 0.53 (− 0.67:− 0.17) − 0.5 (− 0.75:0.1) − 0.45 (− 0.8:0.21) − 0.41 (− 0.83:0.29) − 0.37 (− 0.86:0.37) − 0.34 (− 0.88:0.42)
8− 0.5 (− 0.5:− 0.5) − 0.55 (− 0.67:− 0.17) − 0.53 (− 0.75:0.1) − 0.5 (− 0.8:0.21) − 0.45 (− 0.83:0.29) − 0.42 (− 0.86:0.37) − 0.38 (− 0.88:0.42)
9− 0.5 (− 0.5:− 0.5) − 0.56 (− 0.67:− 0.17) − 0.55 (− 0.75:0.1) − 0.53 (− 0.8:0.21) − 0.49 (− 0.83:0.29) − 0.46 (− 0.86:0.37) − 0.43 (− 0.88:0.42)
10 − 0.5 (− 0.5:− 0.5) − 0.56 (− 0.67:− 0.17) − 0.57 (− 0.75:0.1) − 0.55 (− 0.8:0.21) − 0.52 (− 0.83:0.29) − 0.49 (− 0.86:0.37) − 0.46 (− 0.88:0.42)
21 31 41 51 1001 10001
3− 0.09 (− 0.77:0.43) − 0.06 (− 0.76:0.42) − 0.05 (− 0.67:0.41) − 0.04 (− 0.55:0.4) − 0.006 (− 0.14:0.124) − 0.002 (− 0.05:0.04)
4− 0.14 (− 0.91:0.49) − 0.11 (− 0.83:0.47) − 0.09 (− 0.8:0.44) − 0.08 (− 0.76:0.46) − 0.01 (− 0.23:0.15) − 0.004 (− 0.07:0.05)
5− 0.19 (− 0.91:0.49) − 0.14 (− 0.94:0.54) − 0.12 (− 0.85:0.51) − 0.11 (− 0.83:0.51) − 0.02 (− 0.30:0.15) − 0.006 (− 0.10:0.06)
6− 0.23 (− 0.91:0.51) − 0.18 (− 0.94:0.54) − 0.15 (− 0.95:0.51) − 0.13 (− 0.88:0.51) − 0.03 (− 0.32:0.17) − 0.008 (− 0.11:0.07)
7− 0.27 (− 0.91:0.49) − 0.21 (− 0.94:0.56) − 0.18 (− 0.95:0.51) − 0.16 (− 0.96:0.48) − 0.03 (− 0.35:0.19) − 0.01 (− 0.12:0.06)
8− 0.31 (− 0.91:0.51) − 0.24 (− 0.94:0.54) − 0.21 (− 0.95:0.55) − 0.18 (− 0.96:0.53) − 0.04 (− 0.42:0.18) − 0.01 (− 0.15:0.07)
9− 0.35 (− 0.91:0.51) − 0.27 (− 0.94:0.51) − 0.23 (− 0.95:0.55) − 0.2 (− 0.96:0.56) − 0.04 (− 0.41:0.2) − 0.014 (− 0.14:0.06
10 − 0.38 (− 0.91:0.51) − 0.3 (− 0.94:0.56) − 0.26 (− 0.95:0.51) − 0.23 (− 0.96:0.53) − 0.05 (− 0.41:0.21) − 0.015 (− 0.17:0.08)
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alternatives is contingent upon the electorate’s distribution of n uniform scores to these
alternatives. Contrarily, under PR, the comparison typically hinges on voters allocating
fewer than n such uniform scores. That is, PR allows electorates to only indicate their top-
preferred choice, thereby limiting the expressiveness of their preferences. This character-
istic endows PR with a distinct advantage when one is evaluating the potential severity of
degrading majoritarianism, as outlined in Measure 2, relative to the erosion of principle 1
by MR as articulated in Measure 1.
Although our results establish that PR is superior to MR which is in turn superior to
BR, we cannot naively conclude that PR is superior to BR, not in general or when MR is
used as the yardstick. The reason is the following. In the comparisons between MR and
BR or PR, the majoritarian principle is fully respected by MR, and the consideration of
preference intensity is respected, albeit in a particular form by the prominent scoring rules
on which we focus. Comparison between BR and PR cannot be carried out by applying
the two measures we have proposed because both of these scoring rules respect a particu-
lar form of preference intensities yet violate majoritarianism. We could compare, however,
PR and BR based on an alternative single criterion: the extent of their consistency with
majoritarianism. As is well known, in terms of this criterion BR is superior to PR,18 but of
course, this result does not imply transitivity as the framework used in our study is different
from that applied in such a comparison between the two scoring rules.
The results provide a novel justification for the widely used PR beyond its practical
advantages. There are certainly other scoring rules that are superior to MR. The identifica-
tion of this set of scoring rules is a task worth pursuing in future research.
5 Conclusion
The root cause of the fervent discussion regarding the use of the simple-majority rule ver-
sus a scoring rule stems from the interplay between two critical principles under our focus:
the ability for voters to express preference intensity, thus offering some defense for the
minority (Principle 1), and safeguarding the majority by honoring majoritarianism (Prin-
ciple 2). The Borda method of counts is a monotonic scoring rule. The plurality rule is
weakly monotonic. These rules have drawn the most interest as two scoring rules. Since
these principles cannot be concurrently upheld, adherence to Principle 1 (or Principle 2)
inherently results in the infringement of Principle 2 (or Principle 1). In practical terms,
if MR is applied, Principle 1 is compromised, while the use of BR disrupts Principle 2.
These violations can be construed as the costs incurred by employing these widely exam-
ined aggregation rules. In applying a sensible measure of these costs, the predicted cost of
the two rules in instances of outcome divergence, and leveraging an intuitive concept of
relative erosion of Principles 1 and 2, our research offers a key contribution by asserting
that PR outperforms MR which, in turn, is superior to BR based on the proposed met-
rics of expected erosion of two fundamental democratic principles. The primary findings
indicate that when a rule has a more restricted capacity to express preferences, it benefits
from a reduced undermining of the principle it breaches. Specifically, PR’s more con-
strained nature compared to MR makes it superior. Similarly, MR’s relative limitation in
expressing preferences compared to BR gives it an advantage. Our study not only rekindles
18 See Gehrlein and Lepelley (2011, 2017).
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the historic Borda-Condorcet debate by adding novel support to MR but also revives the
debate about the merits of PR challenging the minimal endorsement PR garnered from 22
esteemed voting rule experts who evaluated 18 well-established voting rules, as articulated
by Laslier (2012).
Our conclusions rest on the presumption that both principles are assigned equal signifi-
cance.19 They are derived from simulations that utilize the Cubic model, along with a more
feasible, realistic probabilistic model to generate alternatives and voter preferences, the
Box model. These outcomes augment the comprehensive discourse on the advantages and
drawbacks of the three most rigorously scrutinized aggregation rules, offering an innova-
tive justification for favoring PR over MR and MR over BR during the constitutional phase,
where the veil of ignorance reigns. They cast fresh insights on the attractiveness of PR and
BR, considering MR as the alternative to these scoring rules. The first finding rational-
izes the actual revealed superiority of PR over MR. With the second finding, it becomes
possible to decide between the approaches of Condorcet and Borda and to explain the low
prevalence of BR relative to MR
Appendix A
The simulation results based on the Box model are presented ine Tables9, 10, 11, 12
Table 9 Percentage of pairwise comparisons in which BR is superior to MR under the Box model
VOTERS CAN-
DIDATES
3 5 7 9 11 13 15 21 31 41 51 1001 10001
3 – 0 0.08 0.19 0.26 0.31 0.37 0.42 0.46 0.48 0.46 0.45 0.47
4 0 0.04 0.10 0.17 0.22 0.27 0.32 0.38 0.42 0.44 0.43 0.45 0.44
5 0 0.04 0.10 0.16 0.22 0.25 0.30 0.36 0.40 0.40 0.42 0.43 0.43
6 0.01 0.045 0.10 0.15 0.20 0.25 0.27 0.34 0.37 0.39 0.40 0.43 0.43
7 0.01 0.04 0.09 0.15 0.19 0.24 0.27 0.32 0.37 0.38 0.39 0.42 0.42
8 0.01 0.04 0.09 0.14 0.19 0.23 0.26 0.32 0.36 0.37 0.38 0.41 0.41
9 0.01 0.04 0.09 0.14 0.18 0.22 0.25 0.31 0.35 0.37 0.38 0.40 0.41
10 0.01 0.04 0.08 0.13 0.18 0.22 0.24 0.30 0.34 0.36 0.38 0.40 0.41
19 The results extend to the case of asymmetric weight assignment, of course. Here, MR remains the supe-
rior aggregation rule as long as the ratio of costs associated with the use of BR and MR exceeds the ratio of
the weights assigned to Principle 2 and Principle 1.
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Table 10 The mean of the differences between the expected costs of BR and MR under the Box simulations, which indicates the advantage of MR
VOTERS CAN-
DIDATES
3 5 7 9 11 13 15
3 – 0.15 (0.08:0.19) 0.09 (− 0.08:0.15) 0.05 (− 0.18:0.12) 0.04 (− 0.23:0.36) 0.03 (− 0.27:0.32) 0.02 (− 0.3:0.29)
4 0.28 (0.17:0.36) 0.16 (− 0.17:0.25) 0.11 (− 0.31:0.54) 0.08 (− 0.38:0.45) 0.06 (− 0.43:0.58) 0.05 (− 0.45:0.52) 0.04 (− 0.45:0.61)
5 0.3 (0.0:0.4) 0.18 (− 0.29:0.27) 0.12 (− 0.42:0.56) 0.09 (− 0.44:0.59) 0.07 (− 0.48:0.6) 0.06 (− 0.49:0.67) 0.05 (− 0.46:0.61)
60.32 (− 0.1:0.42) 0.19 (− 0.37:0.7) 0.13 (− 0.48:0.57) 0.1 (− 0.54:0.67) 0.08 (− 0.55:0.6) 0.07 (− 0.54:0.58) 0.06 (− 0.5:0.62)
70.33 (− 0.17:0.44) 0.19 (− 0.42:0.71) 0.13 (− 0.51:0.57) 0.1 (− 0.57:0.69) 0.09 (− 0.62:0.61) 0.07 (− 0.5:0.68) 0.06 (− 0.48:0.62)
80.33 (− 0.21:0.45) 0.2 (− 0.45:0.72) 0.14 (− 0.49:0.58) 0.11 (− 0.55:0.69) 0.09 (− 0.58:0.61) 0.08 (− 0.56:0.69) 0.07 (− 0.57:0.62)
90.34 (− 0.25:0.45) 0.2 (− 0.48:0.72) 0.14 (− 0.54:0.58) 0.11 (− 0.55:0.7) 0.09 (− 0.58:0.76) 0.08 (− 0.57:0.69) 0.07 (− 0.5:0.62)
10 0.35 (− 0.28:0.46) 0.21 (− 0.5:0.72) 0.15 (− 0.54:0.82) 0.11 (− 0.57:0.7) 0.09 (− 0.54:0.74) 0.08 (− 0.54:0.67) 0.07 (− 0.55:0.63)
VOTERS CAN-
DIDATES
21 31 41 51 1001 10001
30.02 (− 0.4:0.4) 0.01 (− 0.4:0.43) 0.01 (− 0.4:0.4) 0.01 (− 0.4:0.4) 0.01 (− 0.4:0. 4) 0.01 (− 0.4:0.4)
40.03 (− 0.5:0.6) 0.03 (− 0.5:0.5) 0.02 (− 0.5:0.6) 0.02 (− 0.4:0.5) 0.01 (− 0. 4:0.4) 0.01 (− 0.4:0.5)
50.04 (− 0.6:0.6) 0.03 (− 0.5:0.6) 0.03 (− 0.6:0.6) 0.03 (− 0.5:0.5) 0.02 (− 04:0.4) 0.02 (− 0.5:0.5)
60.05 (− 0.6:0.5) 0.04 (− 0.5:0.6) 0.03 (− 0.5:0.6) 0.03 (− 0.5:0.5) 0.02 (− 0.4:0.5) 0.02 (− 0.4:0.5)
70.05 (− 0.5:0.7) 0.04 (− 0.6:0.6) 0.04 (− 0.6:0.5) 0.03 (− 0.5:0.6) 0.4 (− 0.4:05) 0.02 (− 0.4:0.5)
80.05 (− 0.5:0.7) 0.04 (− 0.6:0.5) 0.04 (− 0.6:0.6 0.03 (− 0.5:0.5) 0.02 (− 04:0.5) 0.02 (− 0.5:0.4)
90.06 (− 0.7:0.6) 0.04 (− 0.6:0.6) 0.04 (− 0.6:0.6) 0.03 (− 0.6:.0.6) 0.02 (− 0.4:052) 0.02 (− 0.4:0.4)
10 0.06 (− 06:0.6) 0.05 (− 0.5:0.6) 0.04 (− 0.6:0.6) 0.03 (− 0.5:0.5) 0.02 (− 0.4:0.5) 0.02 (− 0.5:0.5)
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Table 11 Percentage of pairwise comparisons in which PR is superior to MR under the Box model
VOTERS
CANDIDATES
3 5 7 9 11 13 15 21 31 41 51 1001 10001
3 1 1 0.86 0.81 0.80 0.75 0.69 0.63 0.638 0.61 0.59 0.56 0.54
4 1 1 0.86 0.79 0.80 0.73 0.69 0.65 0.63 0.61 0.59 0.55 0.55
5 1 1 0.89 0.80 0.82 0.75 0.70 0.67 0.64 0.61 0.60 0.54 0.53
6 1 1 0.90 0.83 0.85 0.77 0.73 0.691 0.65 0.63 0.61 0.54 0.53
7 1 1 0.92 0.85 0.87 0.80 0.76 0.70 0.68 0.64 0.62 0.54 0.53
8 1 1 0.93 0.87 0.89 0.82 0.78 0.73 0.69 0.66 0.63 0.53 0.53
9 1 1 0.94 0.88 0.90 0.84 0.80 0.75 0.71 0.67 0.64 0.53 0.52
10 1 1 0.95 0.90 0.91 0.86 0.82 0.77 0.73 0.69 0.66 0.53 0.52
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Table 12 The mean of the differences between the expected costs of PR and MR under the Box simulations, which indicates the advantage of PR
VOTERS CAN-
DIDATES
3 5 7 9 11 13 15
3− 0.5 (− 0.5:0.5) − 0.35 (− 0.67:0.17) − 0.26 (− 0.75:0.1) − 0.22 (− 0.8:0.21) − 0.18 (− 0.83:0.29) − 0.16 (− 0.86:0.37) − 0.15 (− 0.88:0.42)
4− 0.5 (− 0.5:0.5) − 0.41 (− 0.67:0.17) − 0.32 (− 0.75:0.1) − 0.28 (− 0.8:0.21) − 0.24 (− 0.83:0.29) − 0.21 (− 0.86:0.37) − 0.19 (− 0.88:0.42)
5− 0.5 (− 0.5:0.5) − 0.45 (− 0.67:0.17) − 0.38 (− 0.75:0.1) − 0.32 (− 0.8:0.21) − 0.28 (− 0.83:0.29) − 0.25 (− 0.86:0.37) − 0.22 (− 0.88:0.42)
6− 0.5 (− 0.5:0.5) − 0.47 (− 0.67:0.17) − 0.41 (− 0.75:0.1) − 0.36 (− 0.8:0.21) − 0.33 (− 0.83:0.29) − 0.29 (− 0.86:0.37) − 0.26 (− 0.88:0.42)
7− 0.5 (− 0.5:0.5) − 0.48 (− 0.67:0.17) − 0.44 (− 0.75:0.1) − 0.4 (− 0.8:0.21) − 0.36 (− 0.83:0.29) − 0.32 (− 0.86:0.37) − 0.29 (− 0.88:0.42)
8− 0.5 (− 0.5:0.5) − 0.5 (− 0.67:0.17) − 0.46 (− 0.75:0.1) − 0.42 (− 0.8:0.21) − 0.38 (− 0.83:0.29) − 0.35 (− 0.86:0.37) − 0.32 (− 0.88:0.42)
9− 0.5 (− 0.5:0.5) − 0.51 (− 0.67:0.17) − 0.47 (− 0.75:0.1) − 0.44 (− 0.8:0.21) − 0.4 (− 0.83:0.29) − 0.37 (− 0.86:0.37) − 0.35 (− 0.88:0.42)
10 − 0.5 (− 0.5:0.5) − 0.51 (− 0.67:0.17) − 0.49 (− 0.75:0.1) − 0.46 (− 0.8:0.21) − 0.42 (− 0.83:0.29) − 0.39 (− 0.86:0.37) − 0.37 (− 0.88:0.42)
VOTERS CAN-
DIDATES
21 31 41 51 1001 10001
3− 0.11 (− 0.91:0.51) − 0.1 (− 0.94:0.61) − 0.08 (− 0.95:0.66) − 0.07 (− 0.96:0.7) − 0.05 (− 0.945:0.85) − 0.04 (− 0.90:0.90)
4− 0.15 (− 0.91:0.51) − 0.11 (− 0.94:0.61) − 0.09 (− 0.95:0.66) − 0.08 (− 0.96:0.7) − 0.04 (− 0.94556:0.93) − 0.04 (− 0.93:0.91)
5− 0.17 (− 0.91:0.51) − 0.13 (− 0.94:0.61) − 0.1 (− 0.95:0.66) − 0.09 (− 0.96:0.7) − 0.03 (− 0.97:0.93) − 0.03 (− 0.97:0.95)
6− 0.2 (− 0.91:0.51) − 0.15 (− 0.94:0.61) − 0.12 (− 0.95:0.66) − 0.1 (− 0.96:0.7) − 0.03 (− 0.96:0.93) − 0.03(− 0.95:0.96)
7− 0.23 (− 0.91:0.51) − 0.17 (− 0.94:0.61) − 0.13 (− 0.95:0.66) − 0.11 (− 0.96:0.7) − 0.03 (− 1.00:0.93) − 0.02 (− 0.98:0.96)
8− 0.25 (− 0.91–0.51) − 0.19 (− 0.94–0.61) − 0.15 (− 0.95–0.66) − 0.13 (− 0.96–0.7) − 0.02 (− 0.96785:0.936) − 0.02 (− 0.98:0.96)
9− 0.28 (− 0.91:0.51) − 0.21 (− 0.94:0.61) − 0.17 (− 0.95:0.66) − 0.14 (− 0.96:0.7) − 0.02 (− 0.99:0.93) − 0.02 (− 0.97:0.96)
10 − 0.3 (− 0.91:0.51) − 0.22 (− 0.94:0.61) − 0.18 (− 0.95:0.66) − 0.15 (− 0.96:0.7) − 0.02 (− 0.97:0.93) − 0.02 (− 0.98:0.95)
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Appendix B
The percentage of preference profiles and pairwise comparisonsresulting in erosion are
presented in Tables13, 14, 15, 16
Table 13 Percentage of preference profiles resulting in erosion when comparing BR and MR
VOTER-
SCANDI-
DATES
3 5 7 9 11 13 15 21 31 41 51 1001 10001
3 – 0.04 0.07 0.09 0.12 0.13 0.15 0.17 0.19 0.21 0.21 0.25 0.26
4 0.16 0.28 0.34 0.38 0.40 0.42 0.43 0.45 0.47 0.49 0.50 0.54 0.55
5 0.41 0.55 0.60 0.64 0.66 0.67 0.68 0.71 0.72 0.73 0.74 0.76 0.77
6 0.64 0.76 0.80 0.82 0.83 0.85 0.85 0.86 0.87 0.89 0.88 0.90 0.90
7 0.82 0.89 0.91 0.92 0.93 0.94 0.94 0.95 0.95 0.96 0.96 0.97 0.97
8 0.91 0.96 0.97 0.97 0.98 0.98 0.98 0.98 0.98 0.98 0.99 0.99 0.99
9 0.96 0.99 0.99 0.99 0.99 0.99 0.99 0.99 1 1 1 1 1
10 0.99 1 1 1 1 1 1 1 1 1 1 1 1
Table 14 Percentage of erosion in pairwise comparisons by BR and MR
VOTERS
CANDIDATES
3 5 7 9 11 13 15 21 31 41 51 1001 10001
3 – 0.02 0.03 0.04 0.052 0.06 0.06 0.07 0.08 0.08 0.08 0.10 0.10
4 0.03 0.06 0.07 0.08 0.09 0.09 0.10 0.10 0.11 0.11 0.12 0.13 0.13
5 0.05 0.08 0.09 0.10 0.11 0.11 0.12 0.12 0.13 0.13 0.13 0.14 0.15
6 0.08 0.10 0.11 0.12 0.12 0.13 0.13 0.13 0.14 0.14 0.14 0.15 0.16
7 0.08 0.11 0.12 0.13 0.13 0.13 0.14 0.14 0.15 0.15 0.15 0.16 0.16
8 0.09 0.12 0.13 0.13 0.14 0.142 0.14 0.15 0.15 0.15 0.16 0.16 0.17
9 0.10 0.12 0.13 0.14 0.14 0.15 0.15 0.15 0.16 0.16 0.16 0.17 0.17
10 0.10 0.13 0.14 0.14 0.15 0.15 0.15 0.16 0.16 0.16 0.16 0.17 0.17
Table 15 Percentage of preference profiles resulting in erosion when comparing PR and MR
VOTERS CAN-
DIDATES
3 5 7 9 11 13 15 21 31 41 51 1001 10001
3 0.22 0.27 0.33 0.34 0.38 0.39 0.39 0.42 0.43 0.46 0.45 0.50 0.50
4 0.44 0.55 0.62 0.65 0.68 0.69 0.70 0.73 0.76 0.76 0.76 0.79 0.79
5 0.59 0.73 0.79 0.83 0.85 0.86 0.87 0.89 0.91 0.91 0.92 0.93 0.94
6 0.69 0.83 0.89 0.92 0.93 0.94 0.95 0.96 0.97 0.97 0.97 0.98 0.99
7 0.76 0.90 0.93 0.95 0.97 0.98 0.98 1 1 1 1 1 1
8 0.805 0.93 0.96 0.98 0.98 0.99 0.99 0.99 1 1 1 1 1
9 0.84 0.95 0.98 0.99 0.99 1 1 1 1 1 1 1 1
10 0.87 0.97 0.99 0.99 1 1 1 1 1 1 1 1 1
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Public Choice (2024) 200:149–171
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Acknowledgements The authors are very much indebted to an AE and two referees for their most useful
comments and suggestions.The second author is very much indebted to Menahem Yaari for fruitful discus-
sions that resulted in the main idea of this study. He is also grateful to Gil Kalai and Tzachi Gilboa for call-
ing attention to the importance of basing the results on more realistic statistical models than the impartial
culture setting, to Ariel Rubinstein for constructive suggestions that made the paper more concise, and to
Eyal Baharad, Roy Baharad, Salvador Barbera, Herve Moulin, Hannu Nurmi, Marcus Pivatoand Tomoya
Tajika for their most useful and thoughtful suggestions
Funding Open access funding provided by Bar-Ilan University.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com-
mons licence, and indicate if changes were made. The images or other third party material in this article
are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the
material. If material is not included in the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly
from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
References
Arrow, K. J. (1963). Social choice and individual values. Yale University Press.
Baharad, E., & Nitzan, S. (2002). Ameliorating majority decisiveness through expression of preference
intensity. American Political Science Review, 96(4), 745–754.
Baharad, E., & Nitzan, S. (2007). The cost of implementing the majority principle: The golden voting rule.
Economic Theory, 31(1), 69–84.
Baharad, E., & Nitzan, S. (2011). Condorcet vs. Borda in light of a dual majoritarian approach. Theory
and Decision, 71, 151–162.
Balinski, M., & Laraki, R. (2010). Majority judgment: Measuring, ranking, and electing. MIT Press.
Bossert, W., & Suzumura, K. (2017). The greatest unhappiness of the least number. Social Choice and
Welfare, 49, 637–655.
Barberà, S., Bossert, W., & Suzumura, K. (2021). Daunou’s voting rule and the lexicographic assign-
ment of priorities. Social Choice and Welfare, 56, 259–289.
Brighouse, H., & Fleurbaey, M. (2010). Democracy and proportionality. Journal of Political Philosophy,
18(2), 137–155.
Caragiannis, I., Nath, S., Procaccia, A. D., & Shah, N. (2017). Subset selection via implicit utilitarian
voting. Journal of Artificial Intelligence Research, 58, 123–152.
Casella, A. (2005). Storable votes. Games and Economic Behavior, 51, 391–419.
Daunou, P. C. F. (1803). Mémoire sur les élections au scrutin, Baudouin, imprimeur de l’Institut
National, Paris. Translated and reprinted in McLean and Urken (1995, Chapter11).
Table 16 Percentage of erosion in pairwise comparisons by PR and MR
VOTERS
CANDIDATES
3 5 7 9 11 13 15 21 31 41 51 1001 10001
3 0.05 0.08 0.10 0.11 0.12 0.13 0.13 0.14 0.15 0.16 0.16 0.19 0.19
4 0.09 0.13 0.15 0.16 0.16 0.17 0.18 0.19 0.20 0.205 0.21 0.24 0.25
5 0.11 0.15 0.17 0.18 0.19 0.20 0.202 0.22 0.23 0.23 0.24 0.27 0.28
6 0.11 0.16 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.29 0.30
7 0.11 0.16 0.18 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.31 0.32
8 0.10 0.16 0.18 0.20 0.22 0.22 0.23 0.25 0.26 0.27 0.27 0.32 0.33
9 0.10 0.15 0.18 0.20 0.22 0.23 0.24 0.25 0.27 0.28 0.29 0.33 0.34
10 0.10 0.15 0.18 0.20 0.22 0.23 0.24 0.26 0.27 0.28 0.29 0.34 0.35
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
171
Public Choice (2024) 200:149–171
1 3
De Sinopoli, F., Iannantuoni, G., & Pimienta, C. (2015). On stable outcomes of approval, plurality, and
negative plurality games. Social Choice and Welfare, 44, 889–909.
Gehrlein, W. V., & Fishburn, P. C. (1980). Robustness of positional scoring over subsets of alternatives.
Applied Mathematics and Optimization, 6, 241–255.
Gehrlein, W. V., & Fishburn, P. C. (1981). Scoring rules and majority agreements for large electorates
with arbitrary preferences. Mathematical Social Sciences, 2, 23–33.
Gehrlein, W. V., & Lepelley, D. (2011). Voting paradoxes and group coherence. Studies in choice and
welfare. Springer.
Gehrlein, W. V., & Lepelley, D. (2017). Elections, voting rules, and paradoxical outcomes. Studies in
choice and welfare. Springer.
Heckelman, J. C., & Ragan, R. (2020). Symmetric scoring rules and a new characterization of the Borda
count. Economic Inquiry, 59(1), 287–299.
Hortala-Vallve, R. (2012). Qualitative voting. Journal of Theoretical Politics, 24(4), 526–554.
Laslier, J.-F. (2012). And the loser is… plurality voting. In D. Felsenthal & M. Machover (Eds.), Elec-
toral systems. Studies in choice and welfare (pp. 327–351). Berlin: Springer.
List, C., & and& Goodin, R.E. (2001). Epistemic democracy: Generalizing the Condorcet jury theorem.
Journal of Political Philosophy, 9(3), 277–306.
Maskin, E. S. (2022). Arrow’s theorem, May’s axioms, and Borda’s rule. Working Paper.
Merrill, S. I. I. I. (1984). A comparison of efficiency of multicandidate electoral systems. American
Journal of Political Science, 28(1), 23–48.
Nakada, S.. Nitzan, S. & and Ui, T. (2023). Robust voting under uncertainty. D.P. Series, HIAS-E-60,
Hitotsubashi Institute for Advanced Study, Hitotsubashi University, [Revised version].
Rae, D. W. (1969). Decision-rules and individual values in constitutional choice. American Political Sci-
ence Review, 63, 40–56.
Saari, D. G. (2006). Which is better: The Condorcet or Borda winner? Social Choice and Welfare, 26,
107–129.
Tideman, N. T., & Plassmann, F. (2013). Developing the aggregate empirical side of computational social
choice. Annals of Mathematics and Artificial Intelligence, 68(1–3), 31–64.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and
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