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The probability that all weighted scoring rules elect the same

winner

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 9 February 1999; accepted 20 April 1999

Abstract

Monte Carlo simulation is used to obtain estimates of the probability that all weighted scoring rules elect the same winner for large electorates under the impartial culture condition. While this probability is relatively large for three candidate elections (0.535), it decreases significantly as the number of candidates increases. The same general observations are made when considering the probability that all weighted scoring rules elect the

Keywords: Weighted scoring rules; Condorcet winner

JEL classification: D72

1. Introduction

Weighted scoring rules (WSRs) are used in elections to determine a winner. A weight of l _is

i

assigned to each voter’s ith ranked candidate, and the winner is determined as the candidate receiving the most total points. It is assumed, without loss of generality, that l 5_{1 and that} l 5_{0 in an}

1 m

m-candidate election. In addition, we assume l $l _{, for all 1}#_i#_m2_{1, and that each voter’s} i i11

preferences form a complete ranking, so that no indifference between candidates is allowed. For three-candidate elections, a weighted scoring rule assigns 1, l_{, and 0 points respectively to each}

voter’s first, second and third most preferred candidates. Rule l _{then denotes the WSR on three}

candidates in which l _{points are assigned to the second ranked candidate for voters.}

Some specific WSR’s are of particular interest in the three-candidate case. Rule 0 is the commonly used plurality rule, Rule 1 is negative plurality rule, and Rule 1 / 2 is Borda Rule. Many different

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

E-mail address: [email protected] (W.V. Gehrlein)

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studies have been conducted to determine the properties of various WSRs. Borda Rule has received particular attention (Saari, 1990). There have been studies to examine the likelihood that the winner of an election changes, if the l_{that is used in a WSR changes (Gehrlein and Fishburn, 1980, 1983), and}

to examine the probability that Rulel_{will select a candidate that is different than the one selected by}

Borda Rule (Gehrlein, 1998).

Merlin et al. (1997) present a particularly interesting result to suggest that there is a relatively large likelihood that all WSRs will elect the same winner for three-candidate elections with large electorates. Their procedure uses a geometric approach to the problem, as developed in Saari (1994). Gehrlein (1999) recreates the main result from Merlin et al. (1997) by using classical probabilistic arguments. This result could minimize the importance of research involving the significance of selecting particular l _{values for a WSR, if it extends to more than three candidate elections. Let}

i

Q(m,`_{) denote the probability that all weighted scoring rules elect the same winner for m-candidate}

elections for the limiting case of voters. The purpose of the current study is to extend the approach in Gehrlein (1999) to obtain estimates of Q(4,`_{) and Q(5,}`_{). The results are based on simulation}

analysis, and they suggest that the probability that all WSRs elect the same candidate decreases significantly as the number of candidates tends to increase.

2. A representation for the probability

The procedure used in the current study follows the same general notions that were used in Gehrlein (1985a,b). In particular, it is not tractable to directly perform simulation analysis to obtain an estimate for Q(m,`_{). We can however obtain a representation for Q(m,}`_{) as a multivariate normal}

orthant probability. While a closed form solution of this orthant probability is not obtainable, we can use Monte Carlo simulation to obtain an estimate of the orthant probability. We begin by developing a representation for Q(m,`_{) as a multivariate normal orthant probability.}

m

Let Rule C_k denote a constant scoring rule for m candidate elections. The set of candidates is

m

denoted byhX , X , X , . . . ,X₁ ₂ ₃ _mj. Rule C_k assigns 1 point to each of the k most preferred candidates for each voter, and assigns 0 points to the remaining candidates. Thus, a constant scoring rule is a particular class of WSRs in which alll _{values are 0 or 1. Merlin et al. (1997) use a result proved in}

i

Saari (1994) to find the probability that all WSRs select the same candidate in three-candidate

3 3

elections as the joint probability that both Rule C (plurality rule) and Rule C (negative plurality₁ ₂ rule) select the same winner. Merlin et al. (1997) also give a generalization of this result for more than three candidates, following the work of Saari (1992). In particular, the same candidate will be elected

m

by all WSRs in an m-candidate election if and only if it is the winner by all Rule C , for_k k5_{1,2,3, . . . ,m}2_1.

There are m! complete rankings that could represent a randomly selected voter’s preferences on candidates. The impartial culture condition (IC) assumes that each of these rankings is equally likely to be observed, and it is used in the current study. We define discrete variables to reflect events that are observed in the preference ranking of a randomly selected voter over the m candidates with IC:

k m

Y 5_1, _{if candidate X beats candidate X by Rule C}

i, j i j k

k

Y 5_0, _otherwise.

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k

Thus, we have Y 5_{1 for a randomly selected voter only when candidate X is among the top}

i, j i

k-ranked candidates, and candidate is X among the bottom (m2_{k)-ranked candidates in a voter’s} j

preference ranking._]

k k

Let Y_{i, j} denote the average value of Y_{i, j} over a set of n randomly selected voters. Following earlier discussion, the probability, P(1,m,n), that all WSRs will select candidate X as the winner is given by_] 1

k

number of voters becomes very large, with n→`_{, the probability that Y} _{will take on any particular} i, j

value, such as 0, goes to 0. Given the discussion above, P(1,m,`_{) can be defined as the joint}

] ]

k _Œ] k _Œ]

probability that Y n$_E[Y _{n], for all 2}#_j#_{m and all 1}#_k#_m2_{1, which defines a positive}

1, j 1, j _]

k _Œ]

orthant probability. The Central Limit Theorem requires that this joint distribution of the Y_{1, j} n variables must be multivariate normal as n→`_{, with correlation terms obtained from the definitions}

k

of the Y variables. By the symmetry of IC, Q(m,`₎5_mP(1,m,`_). i, j

In order to develop the correlation terms for the multivariate normal positive orthant probability P(1,m,`_{), we need to compute some expected values. All of these derivations are based on counting}

k 2

arguments. We begin with a derivation of the expected value E Y

fs d g

1, j . We begin with a list of the

k

(m2_{2)! possible rankings on the set of candidates, with X and X excluded. To obtain Y} 5_{1, there}

1 j 1, j

are k positions in each of the (m2_{2)! possible rankings in which X can be positioned. Then, there are} 1

k(m2_{2)! possible rankings on the set of candidates with X excluded, and there are (m}2_{k) positions} j

in each where X can be positioned. With IC, each of these k(m2_k)(m2_{2)! possible rankings occurs} j

k

with probability 1 /m!. The same number of rankings exists for which we have Y 5 2_{1. Both of} 1, j

these parts can be combined and reduced to obtain

2k(m2_k)

k 2 _]]]

E Y

fs d g

5 _.

1, j _m(m₂₁₎

k l

To obtain E Y

f

Y

g

, with k,_{l, we note that the cross-product cannot be negative since this would} 1, j 1, j

require a voter’s preference ranking to simultaneously have X1 ranked over X for the positivej

component and X ranked over X for the negative component. For both terms to be 1_{1, X must be}

j 1 1

ranked in among the first k candidates in a ranking and X must be ranked among the bottom (m2_{l )} j

candidates in the ranking. Mirroring the discussion above, there are k(m2_{l )(m}2_{2)! such rankings,}

and a similar number in which both terms are 2_{1. It then follows directly that}

2k(m2_{l )}

k l _]]]

E Y

f

Y

g

5 _. 1, j 1, j _m(m₂₁₎

With the use of similar counting arguments, we obtain

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Table 1

The correlation matrix, R, with terms R Y

f

1, j,Y1,t

g

is obtained for the multivariate normal distribution from the variance and covariance representations as

]]]

P(1,m,`_{) has (m}2_{1) dimensions, and closed form representations for these orthant probabilities}

only exist for special cases. However, it is possible to obtain Monte Carlo simulation estimates of multivariate normal positive orthant probabilities. Simulation estimates were obtained by using a

2

procedure that is outlined in Naylor et al. (1966). In particular, an (m2_{1) dimensional normal} 2

observation was generated at random from an (m2_{1) dimensional multivariate normal distribution}

with correlation matrix R by using a ‘square root method’ based on 48 random observations per dimension, using multiple precision computation.

The process was repeated one million times. In each case, it was determined whether or not all 2

(m2_{1) dimensions of the random observation were positive. Then P(1,m,}`_{) was estimated as the}

proportion of randomly generated observations for which all dimensions were positive. An estimate of Q(m,`_{) was obtained as mP(1,m,}`_).

The number of observations that was used in each sample (one million) accounted for the degree of accuracy that could be expected as the number of dimensions increased. A discussion of the determination of the sample size to be used will be developed later. Table 1 lists simulation estimates for Q(m,`_{) for m}5_3,4,5.

The simulation results in Table 1 can be partially verified. The entry for Q(3,`_{) is very close to an}

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3. The existence of a condorcet winner

The Condorcet winner in an election is the candidate who could defeat each of the other candidates in a series of pairwise elections by majority rule. It is well known that such a candidate does not always exist, but such a candidate would seem to be a good candidate for selection when there is one. Merlin et al. (1997) consider the conditional probability that all WSRs elect the Condorcet winner, given that there is a Condorcet winner. Let Q*(m,`_{) denote this conditional probability for} m-candidate elections in the limit of voters under IC. Merlin et al. (1997) compute an exact value of Q*(3,`₎5_{0.5475. We wish to extend the procedures that were used above to obtain simulation}

estimates for Q*(4,`_{) and Q*(5,}`_).

To begin, we develop a representation for the joint probability, P*(1,m,`_{), that candidate X is both} 1

elected by all WSRs and the Condorcet winner in the limit of voters under IC. To have X elected by_] ₁

k

all WSRs, we need to have Y ._{0 for all 2}#_j#_{m and all 1}#_k#_m2_{1, as above. To have candidate} 1, j

X be the Condorcet winner, we define an additional set of m2_{1 discrete variables:} 1

Z 5_1, _{if candidate X beats candidate X in a voter’s preference ranking}

1, j 1 j

Z 5 2_1, _{if candidate X beats candidate X in a voter’s preference ranking.}

1, j j 1

Following earlier discussion, candidate X will be the Condorcet winner if X ._{0 for all}

1 1, j

2#_j#_{m, and P*(1,m,}`_{) can be represented as an m(m}2_{1) dimensional multivariate normal orthant}_] k _Œ]

Niemi and Weisberg (1968) show that

1

] R Z

f

,Z

g

5 _.

1, j 1,t ₃

Gehrlein and Fishburn (1981) show that

]]]

Monte Carlo simulation was used, as described above, with samples of one million random observations, to obtain estimates for P*(1,m,`_{) with m}5_{3,4,5. To obtain the conditional probability} Q*(m,`_{), we need values for P} _(m,`_{) that there is a Condorcet winner as n}→` _{under IC. Niemi}

gives simulation estimates of Q*(m,`_{) for m}5_{3,4,5. The simulation estimate for Q*(3,}`_{) is very}

close to the known exact result in Merlin et al. (1997). In addition, we observe that values of Q*(m,`₎

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Table 2

It is important to consider the accuracy of these simulation estimates. It would be expected that the probability estimates would have greater relative variation as the number of dimensions in the multivariate normal distribution increases. To determine some measure of the accuracy that we can expect, we use the process described above to estimate orthant probabilities for which exact results are known, for the number of dimensions used in the study.

Niemi and Weisberg (1968) presented a general result related to the probability that a Condorcet winner exists, with P (m,`₎5_mF s_{1 / 3 . Here,}d F s_{1 / 3 is the multivariate normal orthant}d

Con m21 m21

probability on m2_{1 dimensions where all correlation terms equal 1 / 3. Known values of} Fs_{1 / 3 are}d

K

given in Ruben (1954). We test the simulation technique used in this study by using the K5_m(m2₁₎

9

dimension case from computing P*(1,m,`_{) to obtain estimates,} F _s_{1 / 3 , of}_d F _s_{1 / 3 with samples of}_d

K K

one million observations, as used throughout the study. Results of this test of our simulation procedure are summarized in Table 2.

The results of Table 2 indicate that we have very reasonable percent error measurements, with one million observations, in our samples for the numbers of dimensions on the orthant probabilities used for m5_{3,4,5. Obviously, the sample size is not nearly adequate to estimate probabilities for m equal}

to six, which explains why we only report probabilities in Table 1 for m5_3,4,5.

References

Gehrlein, W.V., 1985a. The Condorcet criterion and committee selection. Mathematical Social Sciences 10, 199–209. Gehrlein, W.V., 1985b. Condorcet efficiency of constant scoring rules for large electorates. Economics Letters 19, 13–15. Gehrlein, W.V., 1998. The sensitivity of weight selection on the Condorcet efficiency of weighted scoring rules. Social Choice

and Welfare 15, 351–358.

Gehrlein, W.V., 1999. On the probability that all weighted scoring rules elect the condorcet winner. Quality and Quantity (in press).

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. Constant scoring rules for choosing one among many alternatives. Quality and Quantity 15, 203–210.

Gehrlein, W.V., Fishburn, P.C., 1983. Scoring rule sensitivity to weight selection. Public Choice 40, 249–261.

Merlin, V., Tataru, M., Valognes, F., 1997. On the probability that all the rules select the same winner (unpublished manuscript).

Naylor, T.H., Balintfy, J.L., Burdich, D.S., Chu, K., 1966. Computer simulation. In: Techniques, Wiley, New York. Niemi, R.G., Weisberg, H.F., 1968. A mathematical solution for the probability of the paradox of voting. Behavioral Science

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Ruben, H., 1954. On the moments of order statistics in samples from normal populations. Biometrika 41, 200–227. Saari, D.G., 1990. The Borda dictionary. Social Choice and Welfare 7, 279–317.