The Unexpected Behavior of Plurality Rule
William V. Gehrlein
University of Delaware
Dominique Lepelley
Université de la Réunion
The Unexpected Behavior of Plurality Rule
Abstract
When voters’ preferences on candidates are mutually coherent in the sense that
they are at all close to being perfectly single-peaked, perfectly single-troughed or
perfectly polarized there is a large probability that a Pairwise Majority Rule Winner
exists in elections with a small number of candidates. Given this fact, the study develops
representations for Condorcet Efficiency of plurality rule as a function of the proximity
of voters’ preferences on candidates to being perfectly single-peaked, perfectly singletroughed or perfectly polarized.
We find that the widely used plurality rule has
Condorcet Efficiency values that behave in very different ways under each of these three
models of mutual coherence.
1
The Unexpected Behavior of Plurality Rule
Many different criteria can be used to argue as to which candidate should be
selected from a set of m possible candidates in an election. One of the most common
criteria that are discussed in the literature on this topic is the Condorcet Criterion. To
describe this concept, we start by developing the notion of a pairwise majority rule
winner (PMRW).
For the case of elections on three candidates {A,B,C}, there are six possible linear
preference rankings that each voter might have on the candidates:
A
B
C
n1
A
C
B
n2
B
A
C
n3
C
A
B
n4
B
C
A
n5
C
B
A
n6
Here, ni denotes the number of voters that have the corresponding preference ranking on
the candidates, where the total number of voters is n  i 1 ni . These rankings assume
6
that no voter is ever indifferent between candidates, and that voters never have
intransitive preferences, or cyclic preferences, on the candidates. Any given combination
of ni ' s is referred to as a voting situation, denoted by n. A PMRW exists for a given
voting situation if some candidate can defeat each of the other candidates by pairwise
majority rule (PMR) voting on the corresponding pairs of candidates. For example, A
beats B by PMR, denoted by AMB, if n1  n2  n4  n3  n5  n6 , and AMC if
n1  n2  n3  n4  n5  n6 .
Then, A is the PMRW in a given three-candidate voting
situation if both AMB and AMC.
We assume that n is odd throughout this study to avoid the complication of
dealing with the case of ties when using PMR. We also assume that all voters will cast
votes that are in agreement with their actual preferences on the candidates.
The
Condorcet Criterion suggests that the PMRW should be selected as the ultimate winner in
any election.
However, it is well known that some voting situations can result in the
outcome that a PMRW does not exist [Condorcet (1785)]. The concept of the Condorcet
2
Efficiency of a voting rule integrates these two notions by defining the conditional
probability that the voting rule elects the PMRW, given that a PMRW exists.
Many studies have been conducted to consider the veracity of two intuitively
appealing general hypotheses that are related to the probability that a PMRW exists and
to the Condorcet Efficiency of voting rules:
H1 – The probability that a PMRW exists in voting situations tends to increase as the
voters’ preferences show an increased degree of mutual coherence in the way in which
the individual voters developed their preferences on the candidates.
H2 – The Condorcet efficiency of voting rules tends to increase as the voters’ preferences
show an increased degree of mutual coherence in the way in which the individual voters
developed their preferences on the candidates.
Hypothesis H1 has been studied extensively in the literature, and conclusive
evidence has been found to support the ideas behind it. Much less work has been
conducted in attempts to verify H2. The purpose of the current study is to provide
conclusive evidence that the notions behind H2 are generally invalid when plurality rule
is the voting rule under consideration. Plurality rule is the focus of this study since it is
by far the most commonly used election procedure.
Conclusive Support for H1
Gehrlein (2006a) provides an extensive analysis to support the ideas behind H1.
That study finds that the relationship between the probability that a PMRW exists and
measures of the degree of homogeneity, or overall preference agreement, that is present
in voters’ preferences is quite weak, unless these measures reflect the notion that voters’
preferences are obtained by some mutually coherent process. For example, the well
known results from Black (1958) can be used in our case to show that that a PMRW must
exist for odd n when voters’ preferences on candidates are formed according to an
underlying model that is consistent with single-peaked preferences.
Arrow (1963)
provides an alternative definition for single-peaked preferences, such that for the case of
3
three-candidate elections a PMRW must exist for odd n as long as at least one candidate
is never ranked as least preferred by any voter.
Niemi (1969) introduced the notion of the proximity of voting situations to the
condition of perfectly single-peaked preferences, suggesting that voting situations that are
‘close’ to meeting perfect single peakedness should have a high probability of having a
PMRW. In agreement with this concept, Gehrlein (2006a) measures the proximity of a
voting situation to perfectly single-peaked preferences by a parameter b, where
b  Minimumn5  n6 , n2  n4 , n1  n3 .
Parameter b measures the minimum number of times that some candidate is bottom
ranked in a given voting situation. If b  0 for a given voting situation, then the voters’
preferences reflect the condition of perfectly single-peaked preferences, and the value of
b then reflects the relative proximity of a voting situation to the condition of perfect
single-peakedness.
Gehrlein (2006a) uses a procedure called EUPIA2 that is developed in detail in
Gehrlein (2005, 2006b) to obtain a closed form representation for the conditional
probability PPMRW 3,n | IACb k  that a PMRW exists in a randomly selected voting
situation, given that only voting situations are being considered for which parameter b
has some specified value k. This representation is based on the assumption, IACb k  ,
that all voting situations for which b  k are equally likely to be observed for a specified
value of k.
For odd n  7 , this representation is given by
PPMRW 3,n | IACb k 

 


 k 17  21k  11k 2  5  26k  4k 2 n  32  k n2  n3
, for 0  k   n  1 / 4

n  3k n  1n  5  3k 2  k 


 

3 3  2k  6k 3  11  18k 2 n  31  2k n2  n3


, for n  1 / 4  k  n  1 / 3
2k  1n  1n  5  3k 2  k 
 3 / 4 , for k  n / 3 .
4
In the limit that n   , the representation for PPMRW 3,n | IACb k  does not rely
on a specified value of k. Instead, we use the proportion  k of n with  k  k / n . The
identity k   k n is substituted into the representation for PPMRW 3,n | IACb k  and then
the algebraic result is obtained in the limit n   . The resulting limiting representation
is given by PPMRW 3, | IACb  k  with
PPMRW 3, | IACb  k 


11 k3  4 k2  3 k  1
, for 0   k  1 / 4
1  3 k  1  3 k2


 18 k3  18 k2  6 k  1
, for 1 / 4   k  1 / 3.
2ak 1  3 k2


Table 1 lists computed values of PPMRW 3, | IACb  k  for each value of
 k  0.01.02.33 , along with  k  0 and  k  1 / 3 . For example, Table 1 tells us that
when attention is restricted solely to voting situations with  k  .17 , there is probability
.9574 that a randomly selected voting situation will have a PMRW.
Table 1. Computed values of PPMRW 3, | IACb  k  and PPMRW 3, | IACc  k  .
k
0
.01
.03
.05
.07
.09
.11
.13
.15
.17
.19
.21
.23
.25
.27
.29
.31
.33
1/3
PPMRW 3, | IACb  k 
1.0000
.9999
.9991
.9973
.9946
.9907
.9854
.9784
.9693
.9574
.9416
.9203
.8905
.8462
.8009
.7720
.7559
.7501
.7500
PPMRW 3, | IACc  k 
1.0000
.9963
.9888
.9814
.9740
.9665
.9589
.9511
.9431
.9348
.9263
.9174
.9083
.8990
.8903
.8828
.8775
.8751
.8750
5
The results in Table 1 clearly show that PPMRW 3, | IACb  k  decreases as  k
increases, to reflect situations that are more removed from the condition of perfect singlepeakedness. Moreover, PPMRW 3, | IACb  k  remains quite large for relatively large
values of  k and then it decreases quite rapidly for larger values of  k .
Vickery (1960) considers the well known condition of single-troughed
preferences, and the imposition of this assumption will lead to the necessary existence of
a PMRW in the case that we are considering. Vickery asserts that the condition of singletroughed preferences is equivalent to the condition of single-peaked preferences, since
every single-peaked voting situation corresponds to a single troughed-voting situation in
which all voters’ preference rankings are inverted. For a three-candidate election, it
follows from Arrow (1963) that a voting situation with perfectly single-troughed
preferences is one in which at least one candidate is never ranked as most preferred by
any voter. Following previous development of the notion of proximity to perfectly
single-peaked preferences, parameter t measures the proximity of a voting situation to the
condition of perfectly single-troughed preferences, where
t  Minimumn1  n2 , n3  n5 , n4  n6 .
Parameter t measures the minimum number of times that some candidate is top-ranked in
the preference rankings of candidates, so that a voting situation is perfectly singletroughed if t  0 , and the value of t then reflects the relative proximity of a voting
situation to the condition of perfect single-troughedness. Vickery’s assertion turns out to
be true in this situation, since
PPMRW 3,n | IACb k  = PPMRW 3,n | IACt k  .
Ward (1965) develops a result that leads to the conclusion that a PMRW must
exist in a three-candidate election if some candidate is perfectly polarizing, in the sense
that this candidate is never middle ranked, or ranked at the center, of any voter’s
preference ranking. That is, every voter will either consider this candidate to be either
the most preferred or the least preferred. Gehrlein (2006a) develops the parameter c to
reflect the proximity of a voting situation to the condition of perfect polarization, with
c  Minimumn3  n4 , n1  n6 , n2  n5 .
6
It is somewhat surprising to find that PPMRW 3,n | IACc k   P3, n | IACb k  . It is
shown for odd n  3 that
S
3, n | IACc k 
PPMRW






139k 3  472k 2  146k  244 k  4 7 k 3  102k 2  84k  20 n

2
2
3
2
2
2
 6 9k  6k  16 n  16k  1n  3 k 1 6k  24k  1  4k  2n  2n


16k  1n  3k n  1n  5  3k 2  k 

for 0  k  n  1 / 4



 





,

3  39k 4  72k 3  38k 2  76k  1  4 57k 3  54k 2  80k  19 n

2
2
3
4
2
2
2
 2 75k  6k  47 n  48k  5n  n  3 k 1 6k  24k  1  4k  2n  2n

16k  1n  3k n  1n  5  3r 2  k 

for n  1 / 4  k   n  1 / 3








,

7n 2  42n  27
, for k  n / 3.
2
8n  3
In general, we use the notation  xy  1 if x is an integer multiple of y, and  xy  0
otherwise. The notation x  denotes the smallest integer value that is greater than or

equal to x, and x  denotes the smallest integer value that is greater than or equal to x. In

the limit as n   , we find
S
3,  | IACc  k 
PPMRW
139 k3  28 k2  54 k  16

, for 0   k  1 / 4
161  3 k  1  3 k2



39 k3  63 k2  29 k  1
, for 1 / 4   k  1 / 3.
16 k 1  3 k2


Table 1 lists computed values of PPMRW 3, | IACc  k  for each  k  0.01.02.33 , along
with  k  0 and  k  1 / 3 . It is clear that
PPMRW 3, | IACc  k  decreases as  k
increases, despite the obvious fact that PPMRW 3, | IACc  k  is converging to a different
value than PPMRW 3, | IACb  k  as  k increases.
There is a high probability that a PMRW will exist if voters have preferences that
are at all close to being consistent with the underlying models of either single-peaked
7
preferences, single-troughed preferences or polarized preferences.
Moreover, the
conditions of H1 are observed consistently with each of these three models for describing
the internal structural coherence of the way in which voters form preferences on the
available candidates. When models are used to describe voter preference formation that
do not require the same degree of internal structural consistency as the three models that
we have considered, the resulting evidence gives only very weak support for H1.
Previous Conflicting Evidence Related to H2
Many studies have been conducted to estimate the Condorcet Efficiency of voting
rules under various assumptions, typically using Monte-Carlo simulation analysis.
While these studies provide valuable insights, most of them have not focused on the ideas
behind H2, and as a result they are not discussed in the current study. Moreover, the vast
majority of this work has focused H2 has been based on models for obtaining voters’
preferences in voting situations that lack the internal structural consistency of the three
models that were considered in the preceding section. The results typically are based on
models that generate random voter preference profiles. A voter preference profile lists
each individual voter’s preference ranking on candidates. A voting situation can then
easily be obtained by accumulating the number of voters with each of the possible
preference rankings on candidates to obtain the associated ni' s . These models can
therefore easily be used to obtain randomly generated voting situations.
Berg (1985a) generalized several procedures for obtaining random voter
preference profiles by using Pólya-Eggenberger (P-E) models [Johnson and Kotz (1977)].
P-E models are easily described in the context of constructing random voter preference
profiles by drawing colored balls in a classic urn experiment. The experiment starts with
balls of six different colors being placed in the urn.
For each possible individual
preference ranking, there are Ai balls of a color that is associated with the i th possible
individual preference ranking. A ball is then drawn at random and the preference ranking
that is associated with the color of the ball that is drawn is assigned to the first voter. The
ball is then replaced, along with  additional balls of the same color. A second ball is
then drawn, the corresponding preference ranking is assigned to the second voter, and the
8
ball is replaced along with  additional balls of the same color. The process is repeated
n times to obtain a preference ranking on candidates for each of the n voters. When
  0 , the color of the ball that is drawn for the first voter will have an increased
likelihood of representing the color of the ball that is drawn for the second selection, and
so on. This type of model is a contagion model that creates an increasing degree of
statistical dependence among the voters’ preferences as  increases. However, there is
no dependence among voters’ preferences for the particular case with  = 0.
With P-E models, the probability, Pn,  , of observing a given voting situation n
in a three-candidate election is given by
n! 6 Aini , 
.
Pn,   n,  
A
ni !
i 1
Here, A =

6
i 1
Ai and Ak ,  is the generalized ascending factorial with
Ak ,   A A    A  2 .... A  k  1 .
By definition, Ak ,  = A, for k = 0 and k = 1.
We give particular attention to the P-E probability P1 n ,  which has Ai = 1 for all
i = 1,2,3,4,5,6. When we consider the special cases of  = 0 and  = 1, we obtain
P1 n,0 
P1 n,1 
n!
1
n1! n2 ! n3 ! n4 ! n5 ! n6 ! 6n
120
.
n  1n  2n  3n  4n  5
It follows that a P-E probability model with  = 0 is equivalent to an independent voter
model with a multinomial probability representing the probability that a given voting
situation will be observed, with equal likelihood that each preference ranking will be
selected on each draw to obtain a voter preference profile. This situation is identical to a
model that has been named the Impartial Culture Condition (IC) in the literature.
The
form of P1 n,1 leads to the conclusion that each possible voting situation is equally
likely to be observed, given n, for a P-E model with  = 1. This is equivalent to a model
that has been called Impartial Anonymous Culture Condition (IAC) in the literature.
9
Work has been done to develop representations for the Condorcet Efficiency of
various voting rules with different assumptions regarding the probability model that is
used to generate random voting situations. Let CEPR 3,n | PE  denote the Condorcet
Efficiency of plurality rule for three-candidate elections with n voters, given that voting
situations are generated at random with a P-E model that uses parameter  .
Gehrlein and Fishburn (1978a, 1978b) obtain a representation for the limiting
value of CEPR 3,n | PE0 as n   for the special case of IC, with






 3
 4 2


CEPR 3, | PE 0  
1 3

4 4


 Sin 1 






  Li2 






2
2




2  1  1  1  
1  
   Sin     2 Li2  f ,Cos 1   
3   2 
 3 
 3   




1
1
1


f 2 ,Cos 1     Li2 f 4  Li2 f 2

2
 3  4

,
3 3
1
1  

Sin  
4 2
 3
 1  2 
 1 1
1 
  Sin 1 
  Sin 1   
Sin 

 6 2

 3 
 3


 
 
where, f  2  3 and
Li2 a ,   

a
1  log 1  2Cos    2

2 0 

d .


Here, Li2 a  is the dilogarithmic function and Li2 a ,  is the real part of the
dilogarithmic function with a complex argument.
Lewin (1958) describes the
dilogarithmic function in detail and gives an infinite series form for calculating numerical
values of Li2 a  . Quadrature is used to evaluate the Li2 a ,  terms, and we find
CEPR 3, | PE0  .7572. Earlier efforts by Satterthwaite (1972) and Paris (1975) used
non-rigorous techniques to obtain very rough estimates of CEPR 3, | PE0 .
Gehrlein (1982) develops a representation for CEPR 3,n | PE1 by algebraic
techniques for the special case of IAC, and Gehrlein (2002) develops a computer
algorithm (EUPIA) that can be used to obtain the same type of representation, with
10
CEPR 3,n | PE 1 
119n 4  1348n3  5486n 2  10812n  10395
,
2
135n  1n  3 n  5
for n  9,12...
In the limit that n   , we find CEPR 3, | PE1  119 / 135  .8815 . It was noted
above that increasing  in a P-E model increases the degree of statistical dependence
among voters’ preferences. Since CEPR 3, | PE1 > CEPR 3, | PE0 , this all suggests
that increasing the degree of dependence among voters’ preferences will increase the
Condorcet Efficiency of plurality rule, in complete agreement with H2. Berg (1985b)
also obtains the same limiting value for CEPR 3, | PE1 .
Lepelley, et al (2000a) use Monte-Carlo simulation analysis to find that
CEPR m, | PE1 > CEPR m, | PE0 for m  3,4,5,6 . Berg and Bjurulf (1983) argue
that any differences between IC and IAC become insignificant for m  5 , and the
findings of Lepelley, et al (2000) reflect this by showing inconclusive results when
comparing CEPR m, | PE1 and CEPR m, | PE0 for m  6 . Lepelley, et al (2000b)
and Gehrlein (1995, 2003a) obtain computed values of CEPR 3,n | PE  for various odd
n and for numerous values of  to show that CEPR 3,n | PE  consistently increases as
 increases for any given odd n.
Our earlier discussion of P-E models with Ai = 1 for all i = 1,2,3,4,5,6 indicates
that increasing  results in an increase in statistical dependence among voters’
preferences. This observation falls short of suggesting that P-E models imply some form
of “coherence” among voters’ preferences. For example, with a large value of  there is
a high probability that the second voter will have a preference ranking on the candidates
that is identical to those of the first voter. The probability that the second voter has some
preference ranking that is different than that of the first voter is then equally distributed
over all of the other possible rankings on candidates. In a homogeneous population of
voters, it could easily be argued that there is a high probability that the second voter will
have preferences that are identical to the first voter. And, if the second voter does have
different preferences than the first voter, it is not expected that there would be radical
differences in the preference rankings of the two voters. This would suggest that all
preferences rankings that are different than the one chosen for the first voter would not be
11
equally likely to be selected for the second voter.
Other scenarios that describe
coherence among voters’ preferences are also plausible, such as the case of societies with
polarized preferences. The conditions of perfectly single-peaked preferences, perfectly
single-troughed preferences and perfectly polarized preferences would each require the PE model to start with Ai = 0 for two of the rankings. As a result of all of this, the use of
parameter  gives a rather rough measure of coherence among voters’ preferences.
The study of H2 with P-E models that has been presented so far is based on what
Gehrlein (2006a) refers to as using  as a “population specific measure” of agreement in
voters’ preferences. That is, by increasing  for the P-E model that randomly generates
a voting situation, we expect an increase in the Condorcet Efficiency of plurality rule for
voting situations that are generated. However, every possible voting situation could be
generated, with some probability, for any finite  . The relationships that are expressed
in both H1 and H2 should be reflected with greater strength if instead “situation specific
measures” of agreement in voters’ preferences are considered. These measures evaluate
parameters of the specific voting situations that are being considered, rather than using
parameters of the population from which they are selected. Parameters b, t, and c in the
preceding section are examples of situation specific measures of agreement in voters’
preferences, and we continue by examining studies that have considered H2 in the
context of situation specific measures of agreement in voters’ preferences.
Gehrlein and Lepelley (1999) develop a representation for CEPR 3, L | MC  in
three-candidate elections using the Maximal Culture Condition (MC) with parameter L.
MC does not have a fixed number of voters, but uses an integer value of L to define the
maximum number of voters that could have any of the possible complete preference
rankings on candidates. Then, each ni is selected from a uniformly random distribution
over the integers 0,1,2,..., L. This representation was developed by algebraic means,
with
CEPR 3, L | MC  
661L  3216L  6640L  7200L  4264L  1104, for even L  4.
8L  1109L  446L  749L  616L  240
5
4
3
4
3
2
2
Gehrlein (1987) conducts a Monte-Carlo simulation study to examine H2 from
the perspective of MC as L   .
A value of qi was randomly generated from the
12
closed interval 0,1 for each i = 1,2,3,4,5,6 and values of the proportion, pi , of all voters
with each preference ranking in a voting situation was obtained by normalizing the qi
values. That is, each qi was divided by

6
j 1
q j . It was then determined if a PMRW
existed in the voting situation, based on the resulting pi values, and each observation was
discarded for which a PMRW was not found to exist. The process was repeated until
1,000,000 voting situations were found for which a PMRW existed.
Following notions from Fishburn (1973), the agreement among voters’
preferences was calculated for each of the voting situation by using Kendall’s Coefficient
of Concordance, which is a standard measure of agreement between sets of ordinal
rankings. The voting situations were then partitioned into quintiles of increasing levels of
concordance, and the proportion of voting situations within each quintile was found for
which plurality rule elected the PMRW.
The resulting proportion estimates of
CEPR 3, | MC  were found to increase significantly as the measure of concordance
increased. This observation gives strong support to H2 with regard to plurality rule.
Some obstacles to believing the general veracity of H2 started to become visible
in Gehrlein (1973) when some other common voting rules were found to have the exact
opposite results that were observed with plurality rule, behaving in a fashion that was
quite contrary to the general notions in H2. Some explanations were given in an attempt
to explain the observations that were made.
Merrill (1988) summarizes earlier Monte-Carlo simulation studies and presents
some new results that are related to the Condorcet Efficiency of a set of voting rules. In
this study, voters evaluate candidates on the basis of the respective stands that these
candidates take on k different issues. It is assumed that all possible candidate positions
on a given issue can be evaluated on the basis of some measurable characteristic of that
issue. Each voter then has some ideal value for the measurable characteristic for each
issue that expresses the position that the voter would most prefer to see a candidate adopt
on that issue. A given point in a spatial model with a k-dimensional issue space is then
used to identify each voter’s overall ideal point for a candidate over the k issues in that
space. Similarly, each candidate is identified by a point in the k-dimensional issue space,
based on the stands that they actually take on the issues. A voter’s preference ranking on
13
the candidates is then obtained on the basis of the Euclidean distances between the
voter’s ideal point and the points that represent candidates’ stands. That is, the candidate
that has taken a position that is closest to a voter’s ideal point, based on Euclidean
distance, will be that voter’s most preferred candidate, and so on.
The study is based on Monte-Carlo simulation study that generates a set of n
random ideal position points to represent the voters and a set of m random position points
that are taken by the candidates. The distributions from which these two sets of points
are randomly selected both have the same origin, and both are generated from
multivariate normal distributions.
As the parameters of the respective multivariate
normal distributions are changed, the Condorcet Efficiency of all voting procedures that
are considered decreases as m increases.
A reduction in the relative dispersion of candidates’ positions, making them more
IC-like, results in significant reductions in the Condorcet efficiency of Plurality Rule, in
agreement with H2. However, the Condorcet Efficiency of some of the other rules
remains almost unchanged with this change in dispersion. In an extension, an additional
random variable is added to account for voters’ perceptual uncertainty regarding the
position points of candidates.
As the degree of perceptual uncertainty increases,
suggesting more fuzziness in the perception of candidates’ positions, the Condorcet
Efficiency of plurality rule increases significantly, which does not support the general
notions behind H2. When a bipolar multivariate distribution is used to generate ideal
points for voters and for position points of candidates in the issue space, a reduction in
Condorcet Efficiency is observed for the voting rules that are considered.
This
observation does not support H2 with regard to the notion of using proximity to perfectly
polarized preferences to measure group coherence.
Lepelley (1995) began the analysis of the impact of group coherence on the
Condorcet Efficiency of voting rules by considering the impact that restricting voters to
have perfectly single-peaked preferences has on Condorcet Efficiency.
A limiting
representation is developed that can be used to find the value as n   of
CEPR 3, | IACb 0 . Following earlier discussion, IACb 0 assumes that only voting
situations that are perfectly single-peaked can be observed and that each of these
14
perfectly single-peaked voting situations is equally likely to be observed.
Gehrlein
(2003b) uses EUPIA to obtain a representation for CEPR 3,n | IACb 0 , and
CEPR 3,n | IACb 0 
31n3  183n2  153n  81
, for n = 9,15,21,27…
36nn  1n  5
Both results lead to the conclusion that CEPR 3, | IACb 0  31 / 36  .8611. Comparing
this observation with the earlier result that CEPR 3, | PE1  119 / 135  .8815 , we see
that the imposition of the condition of perfectly single-peaked preferences does nothing
to improve the Condorcet Efficiency of plurality rule compared to the case with IAC.
This observation is quite contrary to expectations according to H2, and further
observations in Lepelley (1995) lead to real questions about the veracity of H2. In
particular, while the condition of perfect single-peakedness has no impact on the
Condorcet Efficiency of plurality rule, it is found that perfect single-peakedness does
have a significant impact on the Condorcet Efficiency of some other voting rules.
In conclusion, there has been some substantial support for the idea of H2 in the
literature, but some other observations leave serious concerns about the general veracity
of H2.
A Representation for CEPR 3,n | IACb k 
Given the observation that CEPR 3, | IACb 0 < CEPR 3, | PE1 , it is of
significant interest to consider the general behavior of CEPR 3,n | IACb k  as k increases.
A representation for CEPR 3,n | IACb k  will be very useful in determining the general
veracity of H2 for plurality rule. EUPIA2 was used to obtain such a representation. The
EUPIA2 procedure is explained in detail in Gehrlein (2005, 2006a, 2006b). While the
logic behind the procedure is quite direct, the process becomes very cumbersome to
obtain a representation for CEPR 3,n | IACb k  . The result is presented here without
further development, since it can be verified by computer enumeration. Details of the
derivation will be provided upon request. For odd n  3 :
15
CEPR 3,n | IACb k 


k  1 2216k 3  369k 2  234k  46  922k 2  95k  14n  327k  61n 2  31n 3  16 n63  8 n61 11  3n  


 271   k21 4k  1  n 



,
2
2
2
3
36k  1 k  17  21k  11k   5  26k  4k n  32  k n  n


for 0  k  n  1 / 6


 



 12960k 4  36288k 3  25596k 2  2742k  1249  6696k 3  22248k 2  15768k  229 n  9 12k 2  286k  305 n 2



3
4
2
12
2
12
 300k  311n  2n  648 k 1 k  1  324 n 11 2 k 1  1 n  1  2k   64 n 9 15k  2  5n 


 4 12 864k 2  942k  205  288k  215n  12n 2  162 2 2k  1  n   4 12 402k  113  161n  162 2 2k  1  n  
n7
k 1
n 3
k 1




 16 n121  216k 2  276k  31  236k  37 n  3n 2

,
2
2
2
3
432k  1 k  17  21k  11k  5  26k  4k n  32  k n  n







 








 

for n  1 / 6  k  n  1 / 4




n  3k   144 21k 3  8k 2  6k  8  3 900k 2  256k  363 n  2396k  71n 2  109n 3  324 k21  32 n125 2  15k  5n 


12
2
12
2
2
12
2
 4 n 11 123k  16  41n  162 n 1 n  3k   32 n 1 54k  30k  1  218k  5n  6n  324 n 3 1  2 k 1 n  3k 



12
2
2
2





4

432
k

3
k

8

288
k

1
n

48
n

162

3
k

n
n7
k 1
,

108n  3k  n  1 n 2  2n  9  6 n 2  1 k  18nk 2  18k 3






 




for n  1 / 4  k  n  1 / 3


While this representation for CEPR 3,n | IACb k  has been verified by computer
enumeration for each n = 3(2)245, its extremely complex nature makes it of little direct
use. However, it does provide a basis for considering the limiting representation of
CEPR 3, | IACb  k , following earlier discussion. After algebraic reduction:
CEPR 3, | IACb  k 


432 k3  198 k2  81 k  31
, for 0   k  1 / 6
36 11 k3  4 k2  3 k  1


6480 k4  3348 k3  54 k2  150 k  1
, for 1 / 6   k  1 / 4
216 k 11 k3  4 k2  3 k  1


3024 k3  2700 k2  792 k  109

, for 1 / 4   k  1 / 3
108 18 k3  18 k2  6 k  1


Table 2 lists computed values of CEPR 3, | IACb  k  for each value of
 k  .00.01.33 , and the first observation from these values is that CEPR 3, | IACb  k 
is not monotonic. Of primary interest is the surprising fact that CEPR 3, | IACb  k 
generally tends to increase as
 k increases, particularly for  k  .23 .
Thus, the
Condorcet Efficiency of plurality generally tends to increase as voting situations become
16
farther removed from the condition of perfect single-peakedness, as measured by b. This
observation is completely contrary to the ideas expressed by H2.
Table 2. Computed values of CEPR 3, | IACb  k  , CEPR 3, | IACt  k  and
CEPR 3, | IACc  k  .
k
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
.11
.12
.13
.14
.15
.16
.17
.18
.19
.20
.21
.22
.23
.24
.25
.26
.27
.28
.29
.30
.31
.32
.33
CEPR 3, | IACb  k 
.8611
.8643
.8674
.8702
.8728
.8752
.8774
.8794
.8811
.8826
.8839
.8849
.8857
.8861
.8861
.8858
.8850
.8837
.8818
.8795
.8768
.8741
.8717
.8704
.8710
.8754
.8826
.8897
.8965
.9026
.9080
.9122
.9151
.9166
CEPR 3, | IACt  k 
1.0000
.9963
.9925
.9888
.9850
.9811
.9772
.9731
.9689
.9645
.9598
.9548
.9494
.9435
.9370
.9298
.9218
.9126
.9020
.8896
.8750
.8574
.8357
.8086
.7736
.7273
.6737
.6210
.5698
.5204
.4731
.4281
.3857
.3460
CEPR 3, | IACc  k 
.8611
.8638
.8662
.8683
.8702
.8719
.8733
.8745
.8755
.8763
.8769
.8773
.8776
.8779
.8780
.8781
.8783
.8787
.8793
.8803
.8818
.8838
.8864
.8893
.8924
.8952
.8974
.8989
.9002
.9012
.9023
.9034
.9043
.9047
17
Given earlier observations about the expectation of proximity to single-peaked
preferences having the same impact as proximity to single-troughed preferences, it might
seem unnecessary to develop a representation for CEPR 3,n | IACt k .
However,
intuition leads to the wrong conclusion in this case, as we can see from the following
observation regarding perfectly single-troughed preferences.
Lemma 1. CEPR 3,n | IACt 0  1 for odd n.
Proof. If a voting situation represents perfectly single-troughed preferences with t  0 ,
some candidate is never ranked as most preferred by any voter. The two remaining
candidates must then occupy the first place preferences off all voters. One of these
remaining two candidates must therefore be ranked as most preferred by at least
n  1 / 2
of the voters when n is odd. That candidate must therefore by definition be
both the PMRW and the winner by plurality rule.
QED
This observation starts to suggests that H2 might very well be valid when
proximity of voters’ preferences to the condition of perfectly single-troughed preferences
is used as a measure of underlying group coherence.
A Representation for CEPR 3,n | IACt k 
EUPIA2 was used to develop a representation for CEPR 3,n | IACt k  for odd
n  3 , with:
CEPR 3,n | IACt k 


k  136k  14k 2  7k  7  211k 2  94k  5n  39k  13n 2  8n 3  31   k21 n  1  2k 2k 2  2k  7  2k  3n ,
8k  1k  17  21k  11k 2   5  26k  4k 2 n  32  k n 2  n 3 

for 0  k  n  1 / 4
n  1  2k  36k 3  63k 2  48k  9  57k 2  66k  19n  152k  1n2  5n3  3 k21 2k 2  2k  7  2k  3n ,
 n  1n2  2n  9  6n2  1k  18nk 2  18k 3 
4n  3k 


for n  1 / 4  k  n  1 / 3 .
18
This representation is much more tractable that the earlier representation for
CEPR 3,n | IACb k  , but following previous discussion attention is focused on the
limiting probability CEPR 3, | IACt  k  as n   and we find:
CEPR 3, | IACt  k 
72 k3  22 k2  27 k  8

, for 0   k  1 / 4
8 11 k3  4 k2  3 k  1


2 k  112 k2  15 k

4 18 k3  18 k2  6 k

 5
, for 1 / 4  
 1
k
 1 / 3.
Computed values of CEPR 3, | IACt  k  for each value of  k  .00.01.33 are listed in
Table 2, where results are found that are completely consistent with H2. It is seen that
CEPR 3, | IACt  k  decreases monotonically as  k increases, so that the Condorcet
Efficiency of plurality rule does consistently decrease as voting situations become more
removed from the condition of perfectly single-troughed preferences. Moreover, the
reduction in Condorcet Efficiency is quite dramatic as  k increases, dropping from
1.0000 to 0.3460 over the range of  k values. Since these results are reversed from what
we observed when considering CEPR 3, | IACb  k  , we are interested in obtaining a
similar representation when the degree of group coherence is measured by voting
situation proximity to perfectly polarized preferences.
A Representation for CEPR 3,n | IACc k 
EUPIA2 was used to develop a representation for CEPR 3,n | IACc k  for odd
n  3 with 0  k  n  1 / 6 :



2  16 k65

 
CEPR 3,n | IACc k 

k  1931k  1873k  562k  120  2 27k  118k  35  64 4k  1  54 k  2k  1

8k  1  2 27k  182k  5  6k  1 46k  266k  35  2 9k  4  16  18 k  8  2 9k  8n

 3 63k  50k  122  9 n  62k  1n  16k  12   11  3n 
,
139k  472k  146k  244 k  47 k  102k  84k  20 n  69k  6k  16 n  16k  1n 
9

 3 6k  24k  1  4k  2 n  2n 


3
6
k
2
6
k 2
2
2
3
2
6
k 3
2
2
k 1
2
2
3
3
2
k 1
6
k 4
6
k 2
2
2
6
k 3
6
n 3
2
6
k 4
6
k 5
6
k
6
n 1
2
2
3
2
for 0  k  n  1 / 6 .

19
The complexity of this representation is quite extreme, and the EUPIA2 algorithm was
found to be incapable of obtaining results that would lead to a representation for
CEPR 3,n | IACc k  for k  n  1 / 6 in any reasonable amount of computer processing

time.
Lepelley, et al (2006) observed that the EUPIA2 procedure finds representations
that fall into a much larger class of problems that are the subject of Ehrhart polynomials.
Calculating the probability that some event takes place under variations of IAC is
tantamount to computing the number of points with integer coordinates in a set described
by a finite system of linear constraints (with rational coefficients) depending on one or
more parameters (n and k in our study). Ehrhart’s theory and its later refinements have
shown that this number of points can be represented by a family of polynomials in n and
k with periodic coefficients. Numerous studies exist in applied mathematics and in the
computer science literature that propose different ways for computing Ehrhart
polynomials. The most efficient algorithm is probably the one of Barvinok and
Pommersheim (1999). Lepelley, et al (2006) present a brief description of this algorithm.
This algorithm was used to obtain results that provide representations for
n  1 / 6  k  n  1 / 3
CEPR 3,n | IACc k  over the range
for all odd n  3 .
However, the extreme complexity of these representations resulted in our focusing on a
representation that is valid only for all n  912... , which will be adequate to obtain a
limiting representation CEPR 3, | IACc  k  as n   . After much algebraic reduction,
for all n  912... :
CEPR 3,n | IACc k 







  3 14912k 4  7424k 3  4344k 2  18144k  2565  72 668k 3  444k 2  771k  244 n  18 1380k 2  152k  907 n 2

3
4
6
6
6
2
2
 8597k  443n  77n  1536 k  4 8k  3n  2  768 k  2 8k  3n  5  48 k 5 54k  290k  80  108nk  75n  27n

 1296 k63 2k 2  6k  4nk  n  n 2  48 k61 54k 2  418k  64  108nk  123n  27n 2













 139k 3  472k 2  146k  244 k  4 7k 3  102k 2  84k  20 n  6 9k 2  6k  16 n 2  16k  1n 3 
216

 3 k21 6k 2  24k  1  4k  2n  2n 2





for n  1/ 6  k  n  1/ 4





'

20







 3 34880k 4  23296k 3  4536k 2  21024k  1431  72 1948k 3  804k 2  659k  292 n  18 3972k 2  1688k  793 n 2 


 81821k  785n 3  815n 4  48 k65 486k 2  74k  11  324nk  21n  54n 2  1536 k6 4 8k  3n  2


  1296 6 18k 2  2k  1  12nk  n  2n 2  768 6 8k  3n  1  48 6 486k 2  202k  91  324nk  69n  54n 2 
k

3
k

2
k

1








 




3  39k  72k  38k  76k  1  4 57k  54k  80k  19 n
216
2
2
3
4
2
2
2 
 2 75k  6k  47 n  48k  5n  n  3 k 1 6k  24k  1  4k  2n  2n 

4
3
2

3
2


,

for n  1/ 4  k  3n  8/ 10

 
 



n  3k   9 105k 3  116k 2  40k  150  3 171k 2  232k  294 n  27 k  190 k 2  25n 3  288  k63   k61
2
 9 k2 162k 2  27  108nk  18n 2  36  k6  17 k64  9 k6 2 n  3k 

 


3  39k 4  72k 3  38k 2  76k  1  4 57 k 3  54k 2  80k  19 n
27 
2
2
3
4
2
2
2 
 2 75k  6k  47 n  48k  5n  n  3 k 1 6k  24k  1  4k  2 n  2n 






 





for 3n  7  / 10  k  n  2  / 3 .


The limiting probability representations for CEPR 3, | IACc  k  follow from the
representations above, with:
CEPR 3,  | IACc  k  

 44736


2 931 k3  276 k2  189 k  62
, for 0   k  1 / 6.
9 139 k3  28 k2  54 k  16



 48096 k3  24840 k2  4776 k  77
, for 1 / 6   k  1 / 4
216 k 139 k3  28 k2  54 k  16
104640

4
k



 140256 k3  71496 k2  14568 k  815
, for 1 / 4   k  3 / 10
216 117 k4  228 k3  150 k2  32 k  1

4
k




2  945 k3  513 k2  27 k  25
, for 3 / 10   k  1 / 3.
27 39 k3  63 k2  29 k  1


The first interesting observation that comes from these limiting representations is
that
CEPR 3, | IACc 0  CEPR 3, | IACb 0  31 / 36 .
Computed
values
of
CEPR 3, | IACc  k  for each value of  k  .00.01.33 are listed in Table 2, where it is
seen that CEPR 3, | IACc  k  increases monotonically as  k increases, contrary to the
expectation of H2.
It can be argued that this violation of H2 when using proximity to perfectly
polarized preferences is not very surprising. A society can have preferences that are
highly polarized, while each member of the society forms their preferences in a perfectly
rational manner. For example, liberal and conservative voting blocs might very well
have reversed preference rankings on candidates since they tend to be primarily focused
21



'
on different attributes of candidates’ positions. The voters of such a society are forming
preferences in a mutually coherent fashion, but the preferences of these voters are
definitely not homogeneous, or in mutual agreement. Based on this observation, it can be
argued that H2 should be restated as H2’ with:
H2’ – The Condorcet efficiency of voting rules tends to increase as the voters’
preferences show an increased degree of homogeneity.
Our observations show that proximity to perfectly single-troughed preferences
clearly supports H2’ with voting by plurality rule, and that proximity to perfectly
polarized preferences also does so in the sense that increased polarization reflects
decreased homogeneity. However, results for proximity to perfectly single-peaked
preferences clearly still does not support H2’ for plurality rule.
Conclusion
Earlier work has shown that there is a very large probability of having a PMRW
in voting situations with a small number of candidates when voters’ preferences are
mutually coherent in the sense that they are at all close to being perfectly single-peaked,
perfectly single-troughed or perfectly polarized.
This finding raises interest in the
consideration of the Condorcet Efficiency of voting rules. We find that the widely used
plurality rule has Condorcet Efficiency values that behave in very different ways under
each of these three models of mutual coherence. As a result there is no clear-cut answer
to the question: Is it a good idea to use plurality rule?
In societies in which voters’ preferences are at all close to being perfectly singletroughed, plurality rule has very large values of Condorcet Efficiency. When voters’
preferences are at all close to being perfectly polarized, plurality rule does not have large
values of Condorcet Efficiency. The most surprising result is that plurality rule does not
have large values of Condorcet Efficiency when voters’ preferences are at all close to
being perfectly single-peaked. Extensive analysis is clearly called for to identify the
different scenarios of mutual consistency of preferences for which various voting rules
will tend to have the maximum value of Condorcet Efficiency.
22
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24