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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

n1 n2 n3 n4 n5 n6

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 thei

*Corresponding author. Tel.:1_{1-302-831-1767; fax:}1_{1-302-831-4196.}

E-mail address: wvg@udel.edu (W.V. Gehrlein).

associated ranking as a representation of their preferences on candidates. We assume that

6

there are a total of n5_{o n voters, and n is also assumed to be an odd number.} i51 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 specifici

combination of n ’s that sum to n is referred to as a profile. Several assumptionsi

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 n5_{o n are} i51 i

equally likely to be observed. The number of different profiles that exist for a given n is denoted as N(n). Feller (1957) shows that:

5

P

(n1_{i )}

n1₅ i51 ]]]]

N(n)5

S

D

5

5 120

The IAC condition, in which each of the N(n) 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 PCon(n, IAC), with:

2

15(n1₃₎ ]]]]]

P (n, IAC)5

Con _16(n12)(n14)

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.

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[h9, 21, 33, . . . , 189, . . .j are given by:

4 3 2

(119n 1_1348n 1_5486n 1_10812n1₁₀₃₉₅₎ ]]]]]]]]]]]]]

CE(n, PR, IAC)5

2

135(n1_1)(n1_{3) (n}1₅₎

3 2

(68n 1_501n 1_834n2₃₁₅₎ ]]]]]]]]]

CE(n, NPR, IAC)5

108(n1_1)(n1_3)(n1₅₎

4 3 2

(523n 1_6191n 1_25117n 1_40749n1₂₂₁₄₀₎ ]]]]]]]]]]]]]]

CE(n, PER, IAC)5

2

540(n1_1)(n1_{3) (n}1₅₎

4 3 2

(131n 1_1542n 1_6144n 1_9018n1₃₆₄₅₎ ]]]]]]]]]]]]]

CE(n, NPER, IAC)5

2

135(n1_1)(n1_{3) (n}1₅₎

Table 1 shows computed values of PCon(n, IAC) and the Condorcet efficiency of the 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 0#l#_{1. Then PR is the WSR with} l5_{0, and NPR is the}

Table 1

PCon(n, IAC) and the Condorcet efficiencies of voting rules with IAC

n PCon(n,IAC) PR NPR PER NPER BR

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

WSR with l5_{1. 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 W5_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

A

Let J (n) denote the number of IAC profiles with n voters for which A is both the

A

Condorcet winner and the winner by BR. To develop a representation for J (n), we first

B.A

let J (n) denote the number of IAC profiles with n voters for which A is the

A

Condorcet winner and for which B beats A by Borda Rule. Let Con (n) denote the number of IAC profiles with n voters for which A is the Condorcet winner. Lhuilier (1793) (in Sommerlad and McLean, 1989) showed that the Condorcet winner can never be beaten by both of the other candidates with BR. It then follows that:

A A B.A C.A

J (n)5_{Con (n)}2_{J (n)}2_{J (n)}

By the symmetry of IAC with respect to candidates:

A A B.A

J (n)5_{Con (n)}2_{2J (n)}

A

We know the representation for Con (n) from Gehrlein and Fishburn (1976):

3

(n1_1)(n1_{3) (n}1₅₎

A _]]]]]]

Con (n)5

384

B.A

so we only need to develop a representation for J (n).

The conditions to have A as the Condorcet winner, with B the BR winner over A are given by:

n 1_n 1_n ._n 1_n 1_n 1 2 3 4 5 6 n 1_n 1_n ._n 1_n 1_n

1 2 4 3 5 6

2(n 1_{n )}1_n 1_n ._2(n 1_{n )}1_n 1_n 3 5 1 6 1 2 3 4

The restrictions that are needed to require consistency among the n ’s, given all threei

n2₁

When A is the Condorcet winner, it is easily shown that we must have:

2n 1_n 1_2n 2_n,_n2_n 2_n 2_n

3 5 56 3 4 56

Thus, the Min argument in the n summation counter can be removed. In addition, to2

have consistency between the upper and lower summation limits of the n counter, we2

must be assured that:

2n 1_n 1_2n 2_n$₀

3 5 56

So, we must add the restriction that:

1

]

n $ _[n2_n 2_{2n ]} 3 2 5 56

To then have consistency between the upper and lower summation limits of the n3

counter, we must be assured that:

n2_{1 1}

The resulting limitations on the n ’s after accounting for all of these consistencyi

requirements are given by:

It simplifies matters in later discussion to treat the specific case of n 5_{1 as a special} 56

case. For this special case, the n ’s must have: n 5_{0, n} 5_{1, n} 5_(n2_{3) / 2, n} 5_0,

i 6 5 3 2

and 0#_n #_(n2_{3) / 2. It is easily shown that there are (n –1) / 2 possible profiles of this} 4

n2₁

The [n2_n 2_{2n ] term in the Max argument on the n counter poses a particular}

5 56 3

2

problem, since it is constrained to have an integer value that is consistent with that limit. To account for this, it is necessary to deal with odd and even values of n as separate5

cases. The first step in the procedure is to account for odd and even values of n56 as separate cases. We assume that the term (n2_{1) / 4 is integer valued, so that n[h9, 21,}

33, . . . , 189, . . . . These values on n will then allow for direct comparison with thej

Condorcet efficiency values that are reported in Table 1 for the other voting rules. For any given (n2_{1) / 2 upper limit of n , there are (n}2_{1) / 4 different even values}

shown above is then partitioned into two subspaces to account for the differences that result from the odd and even n56 terms.

*

The case of even values of n defines n 5_{2n with:} 56 56 56

Consider the subspace that accounts for even values of n , and the corresponding56

established, the associated integer value of the [n2_n 2_{2n ] term to be used in the} 5 56

2

Max argument on the n counter can be determined. The summation that is given above3

for even n56is then partitioned into two subspaces to account for odd and even values of

n , when n5 56 is even.

*

The case of even n with even values of n , with n 5_{2n [denoted as Subspace}

56 5 5 5

Even–Even] has limits given by:

n2₁

Even–Odd] has limits given by:

n2₁

The same general procedure can then be used for the subspace that accounts for odd values of n , and the corresponding upper limit of the n index. For each possible odd56 5

upper limit value of n , there are (n 2_{1) / 2 even values of n to be enumerated, with}

odd–even status of both of the n56 and n5 indexes have been established, the

1

]

corresponding integer value of the [n2_n 2_{2n ] term to be used in the Max} 5 56

2

argument on the n counter can be determined. The summation that is given above for3

odd n56 is partitioned into two subspaces to account for odd and even values of n when5 n56 is odd.

*

The case of odd n with even values of n , with n 5_{2n [denoted as Subspace}

56 5 5 5

n2₅

Odd–Odd] has limits given by:

n2₅

At this point, the odd–even factor has been removed from each of the subspaces, and it is now possible to use standard procedures to further partition the subspaces, in order to remove the Max argument from the n index limit, as in Gehrlein (1982).3

Subspace [Even–Even] is partitioned into three subspaces:

Subspace No. 1 Subspace No. 2

To describe the partitioning of Subspace [Even–Even], suppose that we eliminate the

* *

Max argument in the n counter by requiring 0._(n1_{1) / 2}2_n 2_{2n . This then leads}

3 5 56

to the additional consistency restriction that:

n1₃

For consistency between the upper and lower limits of this modified n5 counter, we

* *

this leads to the summation limits of Subspace No. 1.

* *

The remaining partition of Subspace [Even–Even], with 0#_(n1_{1) / 2}2_n 2_{2n to} 5 56

eliminate the Max argument in the n3 counter, leads to the summation limits for Subspace No. 2 and Subspace No. 3.

In the same fashion, Subspace [Even–Odd] is partitioned into three subspaces:

Subspace No. 4 Subspace No. 5

n1_{3 n}2_{1 n}1_{3 n}2₁

Subspace No. 7 Subspace No. 8

Subspace [Odd–Odd] is partitioned into three subspaces:

Subspace No. 10 Subspace No. 11

*

the upper and lower limits on the n56index in these subspaces would simply be modified to account for the restriction that each must have an integer value.

By sequentially using known algebraic relations for sums of powers of integers, a simple representation,[S , can be obtained for the number of IAC profiles in each of thei

twelve subspaces. Including the special case with n 5_{1, we can then obtain a} 56

A

representation for J (n) from:

12 n2₁

A A _]]

J (n)5_{Con (n)}2₂

H

1

O

_[S

J

i

2 i51

After significant algebraic reduction, we obtain the representation:

5 4 3 2

123n 1_1785n 1_9970n 1_27270n 1_38547n1₂₄₇₀₅

A _{]]]]]]]]]]]]]]]]}

J (n)5

51840

A

This representation for J (n) has been verified by computer enumeration.

It follows from the symmetry of IAC with respect to candidates that the joint probability that a candidate is both the Condorcet winner and the BR winner is

A

3J (n) /N(n). This joint probability is then divided by PCon(n, IAC) to obtain the Condorcet efficiency, CE(n, BR, IAC), of BR for n voters with three candidates under IAC. After substitution and reduction we obtain:

4 3 2

123n 1_1416n 1_5722n 1_10104n1₈₂₃₅ ]]]]]]]]]]]]

CE(n, BR, IAC)5

2

135(n1_1)(n1_{3) (n}1₅₎

with n[h9, 21, 33, . . . , 189, . . . . Computed values of CE(n, BR, IAC) are given inj

Table 1. The computed values of CE(n, BR, IAC) in Table 1 are consistent with results obtained from computation by enumeration in Gehrlein (1995).

3. Conclusion

Different types of voting rules require different amounts of input from the voters. PR and NPR simply require voters to report their most preferred candidates, without ranking them. WSR’s, like BR, require that voters report their complete preference rankings on candidates. Two-stage elimination rules require that voters must go through the election procedure in two steps. It is natural to consider the expected benefits that are obtained as we increase the complexity of the election procedure that is used.

Acknowledgements

The authors are indebted to Sven Berg and an anonymous reviewer for helpful suggestions on an earlier version of this manuscript.

References

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formula. Public Choice 40, 307–316.

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