Mathematical Social Sciences 41 2001 39–50 www.elsevier.nl locate econbase
The Condorcet efficiency of Borda Rule with anonymous voters
a , b
William V. Gehrlein , Dominique Lepelley
a
Department of Business Administration , University of Delaware, Newark, DE 19716, USA
b
University of Caen , Caen, France
Received 23 October 1998; received in revised form 10 December 1999; accepted 13 January 2000
Abstract
The Condorcet winner in an election is the candidate that would be able to defeat each of the other candidates in a series of pairwise elections. The Condorcet efficiency of a voting rule is the
conditional probability that it would elect the Condorcet winner, given that a Condorcet winner exists. A closed form representation is obtained for the Condorcet efficiency of Borda Rule in
three candidate elections under the impartial anonymous culture condition.
2001 Elsevier Science B.V. All rights reserved.
Keywords : Condorcet efficiency; Borda Rule
JEL classification : D72
1. Introduction
We wish to consider elections on three candidates [A, B, C]. There are six possible complete preference rankings that voters might have on these candidates:
A A
B C
B C
B C
A A
C B
C B
C B
A A
n n
n n
n n
1 2
3 4
5 6
Since only complete preference rankings are considered, there is no voter indifference between candidates. The n ’s refer to the specific number of voters who have the
i
Corresponding author. Tel.: 11-302-831-1767; fax: 11-302-831-4196. E-mail address
: wvgudel.edu W.V. Gehrlein. 0165-4896 01 – see front matter
2001 Elsevier Science B.V. All rights reserved.
P I I : S 0 1 6 5 - 4 8 9 6 0 0 0 0 0 4 7 - 0
40 W
.V. Gehrlein, D. Lepelley Mathematical Social Sciences 41 2001 39 –50
associated ranking as a representation of their preferences on candidates. We assume that
6
there are a total of n 5 o
n voters, and n is also assumed to be an odd number.
i 51 i
The Condorcet winner in an election is the candidate who would be able to defeat each of the other candidates in a series of pairwise elections. For example, A would be
the Condorcet winner if it would both beat B by majority rule in a pairwise election, with n 1 n 1 n . n 1 n 1 n , and beat C by majority rule in a pairwise election,
1 2
4 3
5 6
with n 1 n 1 n . n 1 n 1 n . It is well known that a Condorcet winner does not
1 2
3 4
5 6
always exist see Condorcet, 1785 in Sommerlad and McLean, 1989. However, if a Condorcet winner does exist, it would seem to be a good candidate for selection as the
winner of the election. Numerous studies have considered the probability that a Condorcet winner exists, along with the Condorcet efficiency of common voting rules.
The Condorcet efficiency of a voting rule is the conditional probability that the voting rule will elect the Condorcet winner, given that a Condorcet winner exists. Much of the
work that has been done in these two areas is summarized in Gehrlein 1997.
In discussing the probability that a Condorcet winner exists, some assumption must be made regarding the likelihood that various combinations of n ’s are observed. A specific
i
combination of n ’s that sum to n is referred to as a profile. Several assumptions
i
regarding the probability that various profiles are observed have become standards in analysis of the probability that a Condorcet winner exists and the analysis of the
Condorcet efficiency of voting rules. The impartial anonymous culture condition IAC is used in the current study. IAC was first used in Kuga and Nagatani 1974, and it has
6
been used in numerous other studies. IAC requires that all profiles with n 5 o
n are
i 51 i
equally likely to be observed. The number of different profiles that exist for a given n is denoted as Nn. Feller 1957 shows that:
5
P
n 1 i n 1 5
i 51
]]]] Nn 5
S D
5 5
120 The IAC condition, in which each of the Nn possible profiles is equally to be observed,
has been described as dealing with situations in which voters’ identities are not retained, so that they remain anonymous Berg and Bjurulf, 1983. Berg 1985 and Stensholt
1999 give various other interpretations of the IAC assumption.
Gehrlein and Fishburn 1976 showed that the probability that a Condorcet winner exists under IAC for n odd voters is given by P
n, IAC, with:
Con 2
15n 1 3 ]]]]]
P n, IAC 5
Con
16n 1 2n 1 4 Gehrlein 1982 developed representations for the Condorcet efficiency of some
common voting rules under the assumption of IAC for three candidate elections. The particular voting rules that were considered in Gehrlein 1982 are plurality rule PR,
negative plurality rule NPR, plurality elimination PER and negative plurality elimination NPER. PR requires voters to vote for their most preferred candidate, and
the winner is the candidate receiving the most votes. NPR requires voters to vote for their two most preferred candidates, and the winner is the candidate receiving the most
votes.
PER and NPER are both two-stage election procedures in which a candidate is
W .V. Gehrlein, D. Lepelley Mathematical Social Sciences 41 2001 39 –50
41
eliminated in the first stage. The remaining two candidates are then carried to a second stage, where the winner is determined by majority rule. PER uses plurality rule in the
first stage, to eliminate the candidate receiving the fewest votes. NPER works in the same fashion as PER, but uses NPR in the first stage. Representations for the Condorcet
efficiency of these voting rules for three candidates under IAC for n [ 9, 21, 33, . . . ,
h 189, . . .
are given by: j
4 3
2
119n 1 1348n 1 5486n 1 10812n 1 10395 ]]]]]]]]]]]]]
CEn, PR, IAC 5
2
135n 1 1n 1 3 n 1 5
3 2
68n 1 501n 1 834n 2 315 ]]]]]]]]]
CEn, NPR, IAC 5 108n 1 1n 1 3n 1 5
4 3
2
523n 1 6191n 1 25117n 1 40749n 1 22140 ]]]]]]]]]]]]]]
CEn, PER, IAC 5
2
540n 1 1n 1 3 n 1 5
4 3
2
131n 1 1542n 1 6144n 1 9018n 1 3645 ]]]]]]]]]]]]]
CEn, NPER, IAC 5
2
135n 1 1n 1 3 n 1 5 Table 1 shows computed values of P
n, IAC and the Condorcet efficiency of the
Con
PR, NPR, PER and NPER using the representations from above. The purpose of the current paper is to develop a closed form representation, like the
ones given above, for the Condorcet efficiency of another important voting rule. The particular voting rule of interest to this study is Borda Rule BR, which is a weighted
scoring rule WSR. WSR’s require voters to rank order the three candidates. Each voter’s first ranked candidate is then given W points, the second ranked candidate is
given 1 point, and the third ranked candidate is given 0 points. Here, we have W 1. The candidate receiving the most total points is the winner of the election. The usual
definition of WSR’s uses weights of 1, l and 0, respectively, for the first, second and third ranked candidates, with 0l1. Then PR is the WSR with l50, and NPR is the
Table 1 P
n, IAC and the Condorcet efficiencies of voting rules with IAC
Con
n P
n,IAC PR
NPR PER
NPER BR
Con
9 0.94406
0.85079 0.53651
0.95238 0.94286
0.88254 21
0.93913 0.86072
0.58537 0.96115
0.95911 0.89433
33 0.93822
0.86667 0.60050
0.96378 0.96335
0.89949 45
0.93791 0.87004
0.60791 0.96503
0.96527 0.90225
57 0.93776
0.87217 0.61231
0.96576 0.96637
0.90395 69
0.93768 0.87364
0.61523 0.96624
0.96708 0.90511
81 0.93763
0.87471 0.61730
0.96658 0.96758
0.90594 93
0.93760 0.87552
0.61885 0.96683
0.96794 0.90658
105 0.93758
0.87616 0.62006
0.96702 0.96822
0.90707 117
0.93757 0.87668
0.62102 0.96717
0.96845 0.90747
129 0.93755
0.87710 0.62181
0.96730 0.96863
0.90779 141
0.93755 0.87746
0.62246 0.96740
0.96878 0.90807
153 0.93754
0.87776 0.62302
0.96749 0.96890
0.90830 `
0.93750 0.88148
0.62963 0.96852
0.97037 0.91111
42 W
.V. Gehrlein, D. Lepelley Mathematical Social Sciences 41 2001 39 –50
WSR with l51. However, the definition that we use, with weights W, 1, 0, is equivalent to the usual definition, and it is particularly useful in simplifying derivations
that follow. BR is the WSR which has W 5 2. Van Newenhizen 1992 showed that BR is the WSR that maximizes Condorcet
efficiency for a class of distributions, not including IAC, that define the probability that each possible profile is observed. Saari 1990 gives a number of other properties of BR
to give strong additional support to its use as a WSR. Given this background, the development of a representation for the Condorcet efficiency of BR under IAC is of
significant interest.
2. A representation for the Condorcet efficiency of Borda Rule
