Karpov. Measurement Of Disproportionality In Proportional Representation Systems

Centre for Advanced Studies Research Notes D

Alexander Karpov

Measurement of Disproportionality in PR Systems RN #2007/09

State University - Higher School of Economics Moscow

Centre for Advanced Studies Russia, Moscow http://www.cas.hse.ru E-mail: [email protected]

The Centre for Advanced Studies was created in 2006 at Higher School of Economics (HSE) in cooperation with the New Economic School. Its aim is to promote international standards of research in the socio-economic sciences in close collaboration with foreign academics to be published in English in high-level peer-reviewed journals. It involves the active participation of young Russian researchers working with their counterparts. Its approach to multidisciplinary academic studies at the frontiers of modern social sciences is organized in the framework of its Centre for Economic Policy, its International Research Program and its Program of Visiting Academics at the Higher School of Economics.

The paper was presented at the International Conference "Frontiers of Macroeconomics and International Economics" organized by the Centre for Advanced Studies (HSE-NES), May 24-26, 2007, Moscow.

Measurement of Disproportionality in PR Systems Alexander Karpov State University - Higher School of Economics, Moscow [email protected] Nineteen indices characterizing the disproportionality in a parliament are studied. A classification of these indices is suggested, and computational experiments and an axiomatic approach are used to study their properties. Effects of the number of parties and the degree of disproportionality are considered. The indices are evaluated for parliamentary elections (1995-2007) in Russia.

Keywords: PR systems, disproportionality indices.

1. Introduction In the 20th century, the wide appearance of PR systems was one of the reasons for the development of various methods for measuring the quality of electoral systems. M. Balinski and P. Young [3] gained one of the main results. They proved the impossibility of constructing a PR system that allocates seats in an exactly proportional way. The impossibility of creating an ideal electoral system forced researchers to search for quantitative indices that would reflect the degree to which the system satisfies certain conditions. Such indices contain quantitative information and allow researchers to conduct empirical research and compare various electoral systems. One of the most extensive investigations of this sort was conducted by A. Lijphart [8]. Up to now, many indices describing disproportionality have been constructed. Some of them were developed for studying some particular electoral system, while others were adapted from other areas of science. Thus, there is no agreement concerning what indices are better in a particular situation. A survey of such indices can be found in [2]. Until now, there has been almost no research that studies the properties of disproportionality indices. This paper studies the properties of disproportionality indices. It is organized as follows. Section 2 introduces the main concepts. Sections 3 to 7 describe the main indices. Section 8 presents an axiomatic approach. Section 9 provides the results of computational experiments using a simulation program. Section 10 concludes with the results of calculating indices for the parliament elections in Russia from 1995 to 2007.

1

2. Main concepts Consider a PR election with n parties participating. Let (V1 ,V2 ,...,Vn ) be the vector of votes and ( S1 , S 2 ,..., S n ) be the vector of seats each party receives. Then n

∑V i =1

i

n

∑S i =1

i

=V , =S.

The problem of proportional representation consists in allotting a fixed number of seats between parties proportional to the number of votes they received. There are several methods for distributing seats, which are based on different principles and which lead to different outcomes. A description of these methods can be found in [2]. The purpose of the election is to represent voters’ preferences as closely as possible. According to the principle “one person - one vote,” each ballot should have ‘equal force’ in the sense of the share of seats in the parliament: Si S = , Vi V

Let vi =

i = 1, n .

Vi S S , si = i be the vote and seat shares that party i receives. We call yi = i the V S Vi

representation of party i. Party i is overrepresented when

Si S < , and party i is underrepresented when the opposite Vi V

inequality holds. Using representations, it is possible to compare various parties with each other. In the ideal case, each vote has equal force and each party obtains a share of seats equal to the share of votes,

vi = s i ,

i = 1, n .

A minimal requirement for apportionment is the monotonicity of seat assignment: if

v1 ≥ v2 ≥ v3 L ≥ vn , then s1 ≥ s 2 ≥ s3 L ≥ s n . We do not use any other conditions for the analysis of disproportionality indices, since this paper does not deal with specific apportionment methods. The deviation from the exact equality is not only a mathematical problem, but also a political one, because it is a distortion of citizens’ true preferences.

2

Real political systems cannot achieve equality. First of all, the number of seats is an integer. Furthermore, there is a threshold of vote shares below which a party does not get seats. For this reason, small parties cannot have their own representatives in parliament. By introducing a higher threshold, electoral laws attempt to exclude small parties. The extent to which parliament does not represent citizens’ preferences can be examined by using disproportionality indices. In this paper, the consequences of participation restrictions for some parties and voters in the election are not considered. The focus of this paper is on election results1. Disproportionality indices measure the deviation between the real apportionment and the exact one. We do not give references to the original articles in which the indices were introduced; these references can be found in [5]. The various approaches for measuring the quality of the electoral system can be divided into several groups.

3. Absolute deviation indices The first group of indices characterizes apportionment by means of absolute deviations between vote shares and seat shares. Indices are equal to zero if and only if vi = si for each party; this corresponds to the ideal representation. Maximum deviation and a sort of mean can be calculated. 3.1 The Maximum Deviation index: MD = max s i − vi . i =1, n

(1)

This index shows the size of distortion of the most inaccurately represented party. The maximum value is equal to 1; this is the case when a party that did not receive any votes obtains all seats. This case is impossible if the monotonicity property of seat assignment holds. If the number of parties is very high, then the value of the index could reach values that are arbitrarily close to 1, when all parties have a very low share of the votes and one of them has all the seats. 3.2 The Rae index. This index is the arithmetic mean of absolute deviations: I Rae =

1

1 n ∑ si − vi . n i =1

(2)

A representation index that takes into account the absence of voters can be found in [9].

3

This index has a clear interpretation: how much each party deviates from the exact representation on average. However, this index has a significant disadvantage: its value depends on the number of parties. When the number of parties which are not elected is very large, the value of the index is very low. It also tends to have very low values, describing the system to be more proportional than it actually is. This is closely related with the following property: n

max ∑ s i − vi = 2 and max Rae = 2 / n . i =1

The lower values of the index do not correspond to a more accurate representation. Only those parties that have more than 0.5% of votes should be considered in order to avoid misinterpretation of the low values of the index. Nevertheless, this measure does not resolve the problem of potential misinterpretation. 3.3 The Loosmore-Hanby index, unlike the Rae index, always takes values from 0 to 1: I LH =

1 n ∑ s i − vi . 2 i =1

(3)

Although the Loosmore-Hanby index seems to be similar to the Rae index, it has a completely different meaning. The value of the Loosmore-Hanby index gives the total excess of seat shares of overrepresented parties over the exact quota and the total shortage accruing to other parties. 3.4 The Grofman index [2]. Although this index implies calculating the mean of absolute deviations, their sum is divided by the effective number of parties rather than by the total number of parties: IG =

where E =

1 n

∑v i =1

1 n ∑ si − vi , E i =1

(4)

is the effective number of parties. 2 i

The Grofman index does not completely correct the disadvantages of the Rae index, e.g. the upper limit remains changeable. Moreover, this index can be greater than 1. 3.5 The Lijphart index is calculated in the same way as the Rae index, but only the two largest parties are considered:

IL =

s i − vi + s j − v j 2

.

(5) 4

The Lijphart index is evaluated for the two largest parties. Since the largest parties usually have the most significant deviations from their exact quota, this measure can be used to evaluate the disproportionality of the whole system. These indices are closely interconnected. The following inequalities hold:

I Rae ≤ MD ≤ I LH , I Rae ≤ I G , I L ≤ MD . Note that the Rae index is not necessarily the smallest. It can exceed the values of the Lijphart index when small parties have high deviations.

4. Quadratic indices The previous group of indices is based on various versions of the arithmetic mean. Linearity in the deviations results in an insensitivity to changes in the seat assignment, because these indices would consider a big deviation and a group of small deviations to be the same. Consider the example given in Table 4.1, where there are four parties with given vote and seat distributions. Table 4.1. Example 1. Parties Vote Shares

Seat Shares

A

0.1

0.05

B

0.2

0.15

C

0.3

0.3

D

0.4

0.5

Table 4.2 presents an example in which the distribution of seats between parties A and B is changed, but the other shares remain as they were. Table 4.2. Example 2. Parties Vote Shares

Seat Shares

A

0.1

0

B

0.2

0.2

C

0.3

0.3

D

0.4

0.5

5

In both cases the absolute deviation indices remain unchanged, and they have the following values. Table 4.3. The values of the absolute deviation indices. MD

0.1

Rae index

0.05

LH index

0.2

Grofman index

0.06

Lijphart index

0.05

Note that the values of the indices remain unchanged in any seat distribution between parties A and B, when party B has a larger share of seats than party A. However, the seat distribution given in Table 4.1 is more proportional than the distribution with a zero seat share for party A. Quadratic indices can be used to compare distributions with an equal sum of absolute deviations. 4.1 The Gallagher Index (often called the least squares index):

Lsq =

1 n (s i − v i ) 2 . ∑ 2 i =1

(6)

This index has a different sensitivity to large and small deviations between vote and seat shares. Small differences have less influence on the index than big ones, which increase the index significantly. Small deviations are generally not eliminated. Big deviations imply that the distribution is less proportional. It is important to note that the disproportionality index considers various deviations differently. This property can be strengthened as suggested in [2]:

Hk = k

1 n (si − vi )k . ∑ k i =1

(7)

This index is not monotone with respect to k, and this complicates the interpretation of the index. In addition, the maximum possible value depends on k: max H k = k

2 . k

A small change in the form of the index can correct the latter defect:

6

~ Hk =

k

1 n (si − vi )k . ∑ 2 i =1

(8)

In both cases, the indices take values from zero to one and become less sensitive to small deviations as k increases. In the limit, the index tends to the maximum deviation: ~ lim H k = lim H k = MD . k →∞

k →∞

Note that the Gallagher index can be higher than the maximum deviation. An example is given in Table 4.4. Table 4.4. Example 3. Parties Vote Shares

Seat Shares

A

0.20

0.25

B

0.20

0.25

C

0.30

0.25

D

0.30

0.25

The Gallagher index and the maximum deviation for this distribution are equal: MD=0.05 and Lsq=0.07. The values of both indices change when identical parties are merged. Table 4.5. Example 4. Parties

Vote Share

Seat Share

A

0.4

0.5

B

0.6

0.5

Here MD=0.10 and Lsq=0.10. These indices do not satisfy the property of independence from split, which can be described as follows. If all parties can be separated into several equal groups (for instance, in Table 4.4 there are two equal groups of parties: A, C and B,D), then the value of the index calculated for all parties should be equal to the value of the index for one group considered as a whole. We can test the independence of the index from the split as follows. If every vote and seat share of each party is divided into k equal parts, then the index should remain equal to its initial value. 7

4.2 The Monroe index [7] is a small modification of the Gallagher index: n

∑ (s

I Monroe =

i =1

i

− vi ) n

1+ ∑v i =1

2

(9) 2 i

The sum of the squares of vote shares characterizes the number of parties. The denominator decreases as the number of parties increases. The next three indices are borrowed from socio-economic statistics. This science considers, among other things, the problem of measuring structural distinctions. For example, it may be necessary to compare economic structures in different regions, or compare the structure of actual output with what may be expected. A variety of indices have been developed in this field, which can be used to measure disproportionality in parliament. Unlike the Gallagher index, these indices satisfy the property of independence from split. 4.3 The Gatev index [10]. This index is calculated according to the formula: n

∑ (s

I Gatev =

i =1 n

i

∑ (s i =1

− vi )

2

. 2 i

(10)

+v ) 2 i

This index is higher when parties are approximately equal in size than if there is significant inequality between parties or if there is a higher number of parties. Thus, this index is more sensitive to small parties than the Gallagher index. This index does not change after a division of each vote and seat share into k equal parts:

I Gatev =

2

1 ⎞ ⎛1 k ∑ ⎜ s i − vi ⎟ k ⎠ i =1 ⎝ k = 2 2 n ⎛ ⎞ 1 1 ⎛ ⎞ ⎛ ⎞ k ∑ ⎜ ⎜ s i ⎟ + ⎜ vi ⎟ ⎟ ⎜ ⎠ ⎝ k ⎠ ⎟⎠ i =1 ⎝ ⎝ k n

n

∑ (s i =1 n

i

∑ (s i =1

− vi )

2

. 2 i

+v ) 2 i

The property of independence from split is useful in comparing election results with different numbers of parties. 4.4 The Ryabtsev index [12] insignificantly differs from the Gatev index, and it has lower values:

8

n

∑ (s

I Ryabtsev =

i =1 n

∑ (s i =1

i

− vi )

i

+ vi )

2

.

(11)

2

4.5 The Szalai index [9] is used in time-use research for comparing activity profiles: 2

⎛ s i − vi ⎞ ⎜⎜ ⎟⎟ ∑ i =1 ⎝ s i + v i ⎠ . n n

I Szalai =

(12)

This index differs from all others considered in this group. As party size increases, the value of (si + vi ) also increases. This reduces the value of the index and increases the effect of 2

small parties. When some party fails to obtain even one seat, the following condition holds:

(si − vi )2 = (si + vi )2 = si2 + vi2 . The value of the Szalai index is close to one when a significant number of parties do not have any representation. Thus, this index is very sensitive to the incorrect representation of small parties. Consider the example in Table 4.6, where one small party does not have any representation. Table 4.6. Example 5. Parties Vote Shares

Seat Shares

A

0.99

1

B

0.01

0

In this example, the absolute deviation indices and the Gallagher index are close to zero, implying a good representation. The Szalai index, however, differs, with a value close to 0.7. If this property is not desirable, the weighted Szalai index, proposed in [9], can be used: ~ I Szalai =

2

⎛ si − vi ⎞ ⎟⎟ ⋅ ⎜⎜ ∑ i =1 ⎝ s i + v i ⎠ n

si + vi n

∑ (s j =1

j

1 n (s i − v i ) . ∑ 2 i =1 si + vi 2

=

+ vj)

(13)

In some respects, this index is closer to the absolute deviation indices. The index can be considered to be the weighted sum of absolute deviations:

9

~ I Szalai =

1 n si − vi ⋅ s i − vi . ∑ 2 i =1 si + vi

The main problem of using indices from socio-economic statistics is an absence of intuitive understanding of them, and therefore choosing this kind of index can be a complex task. The Ryabtsev and Gatev indices differ only in their denominators, but the lack of a clear interpretation hinders selecting the best.

5. The Aleskerov-Platonov index The indices considered above are all based on measuring the difference between vote and seat shares, but equal deviations give different effects in terms of proportions. The significance of a deviation for large and small parties can vary greatly. Consider the following example: Table 5.1. Parties

Vote Shares

Seat Shares

A

0.5

0.6

B

0.01

0.11

Party B has a more significant overrepresentation in spite of the equality of deviations. This problem can be solved by using ratios. Differences and ratios between vote and seat shares are given in Table 5.2. Table 5. 2. Parties

|v-r|

r/v

A

0.1

1.2

B

0.1

11

Comparing the relative representations with 1, the problem of measuring disproportionality can be considered from a new point of view. The Aleskerov-Platonov index [2] is calculated only for overrepresented parties: R=

1 k si ∑ . k i =1 vi

(14)

When some parties are not represented, the other parties will obtain on average more than one percent of seats for each percent of votes. This index shows the average excess of seat share 10

over the vote share for overrepresented parties. The index equals 1 for the ideal representation. Note that we use this index only for overrepresented parties. If there is no electoral threshold and we take overrepresented and underrepresented parties for calculation, the value of the index can be equal to 1, because this value is a mean of some values that are greater than 1 and of some values that are less than 1. Therefore, use of this index should be limited only to overrepresented parties.

6. Inequality indices At the beginning of the XX century, welfare economics faced the problem of measuring inequality, a problem that is quite similar to the problem of measuring disproportionality. Several methods were developed to solve this problem. A voter receives an 'electoral income' in terms of party representation. It is possible to consider yi =

Si to be electoral income. Since equality of Vi

party representation is never achieved, the following indices can be used to measure the inequality. 6.1 The Gini index. This is one of the first indices proposed in welfare economics. The Gini index is calculated using the Lorenz curve. The curve passes through the points with cumulative shares of the income. Vote shares are located on the horizontal axis, and income shares are located on the vertical axis.

1

Lorenz curve

1

Fig. 6.1 Lorenz curve. If wealth is distributed equally among individuals, then the curve is a straight line. If not, the curve will lie under this line and be convex. The cumulative shares of electoral income are calculated by the following formula: 11

h

h

Th =

∑ yi i =1 n

∑y i =1

i

=

si

∑v i =1 n

.

i

si ∑ j =1 v i

(10)

The Gini index is the ratio of the area between the Lorenz curve and the perfect equality line to the area of the triangle under the straight line. 6.2 The Atkinson index [4]. This index uses a parameter ε, which characterizes the attitude of a society to inequality. A negative attitude to inequality is strengthened by an increase in ε. The Atkinson index is as follows: ⎡ n ⎛y A = 1 − ⎢∑ vi ⎜⎜ i ⎣⎢ i =1 ⎝ µ

where µ is the representation of parliament,

µ=

⎞ ⎟⎟ ⎠

1

1−ε

⎤ 1−ε ⎥ , ⎦⎥

(16)

S , V

⎡ n ⎛s A = 1 − ⎢∑ vi ⎜⎜ i ⎢⎣ i =1 ⎝ vi

⎞ ⎟⎟ ⎠

1−ε

⎤ ⎥ ⎥⎦

1 1−ε

.

6.3 Generalized entropy [4]. The index of generalized entropy can be computed as follows: 1 GE = 2 α −α

⎡ n ⎛y ⎢∑ vi ⎜⎜ i ⎢⎣ i =1 ⎝ µ

α ⎤ ⎞ ⎟⎟ − 1⎥ , ⎠ ⎥⎦

1 ⎡ n ⎛ si ⎢∑ vi ⎜ GE = 2 α − α ⎢ i =1 ⎜⎝ vi ⎣

(17)

α ⎤ ⎞ ⎟⎟ − 1⎥ . ⎥⎦ ⎠

Varying the value of the parameter α produces a class of indices with similar properties. The Atkinson index and the generalized entropy are very similar not only in form, but also in properties. Generalized entropy is widely used to measure income inequality, because it satisfies many desirable properties such as decomposability. An analysis of the other properties of these indices is given below.

12

7. Objective functions For each apportionment method, an objective function can be defined such that its optimization gives the best seat allocation. This very function can be used to measure disproportionality. For example, the Rae index, the Loosmore-Hanby index, Lsq can be viewed as objective functions for the method of largest remainders. 7.1 The d’Hondt index equals the maximum excess of seat share over vote share: H = max i =1, n

si . vi

(18)

7.2 The Sainte-Lague index is a weighted sum of squares of relative deviation: 2

⎞ ⎛r SL = ∑ vi ⋅ ⎜⎜ i − 1⎟⎟ . i =1 ⎠ ⎝ vi n

(19)

These indices are objective functions for the corresponding apportionment methods. The D’Hondt and Sainte-Lague systems satisfy the necessary axiomatic properties and are widely applied in various countries. These indices have no upper limit, making their interpretation more difficult. The SainteLague index has the same form as the χ2 statistic, which is used in goodness-of-fit tests. This feature makes the Sainte-Lague index significantly different from the others [6].

8. Axiomatic approach Disproportionality indices must have certain properties for them to be applied in practice. Indices should measure disproportionality in every possible distribution. There are axiomatic approaches in income distribution studies [4], where there is a similar problem of measuring inequality. Certain principles need to be formulated for the study of PR. 8.1 Axioms 1.

Anonymity.

Any permutation of party labels does not change the value of the index. 2.

Principle of transfers.

13

If we transfer seats from an overrepresented party to an underrepresented party the value of the index should not increase. 3.

Independence from split.

Suppose there are many parties with equal vote and seat shares, and these parties are grouped into one. If the value of the index calculated for all of the parties in the group is equal to the value of the index for the group considered as a whole, then the property of independence from split holds. 4.

Scale invariance.

The index should not depend on any proportional change in the number of votes or seats in the parliament. 5.

An index i is zero normalized if I (v1 ,..., v n ; s1 ,..., s n vi = s i ∀i = 1, n) = 0 .

All indices satisfy properties 1 and 4. Table 8.1. Axiomatic properties of the indices. Maximum deviation Rae index LH index Grofman index Lijphart index Lsq Hk Gatev index Ryabtsev index Szalai index Szalai weighted index Aleskerov-Platonov index* Gini index Atkinson index Generalized entropy D’Hondt index Sainte-Lague index

2 + + + + + + + + + + + + +

3 + + + + + + + + + + +

+ The index satisfies the property. - The index does not satisfy the property. * This index is considered only for overrepresented parties.

Seat share can change only by discrete values, because the number of seats is an integer. Violation of Property 2 appears only for very small changes in seat distribution that are unrealistic, given the usual size of parliament. 14

Violation of property 3 means that the index depends on the number of parties. Thus, these indices are better for elections with an equal number of parties. Choosing the correct index for measuring disproportionality requires a clear interpretation of the index.

9. Simulation experiment Analysis of the analytical form does not reveal all features of the indices. Some properties appear only after analyzing a large sample. The number of real election results is limited and, in addition, the apportionment method, number of parties and other parameters are all different for different elections. Therefore, we cannot compare the indices directly. On the other hand, we can use indices in real data only after studying their properties on the homogeneous set. 9.1 Statement of the experiment. Vote shares and seat shares are simulated by random variables. The difficulty of modeling various distributions of the variables vi and si arises from the condition n

∑v i =1

i

= 1,

i

= 1.

n

∑s i =1

The variables are linearly dependent. The following method is used to eliminate absolute r correlation. At first, the random vector X , which consists of n non-negative independent identically distributed random variables, is generated. The parameter n is the number of parties. Vote share is defined as the ratio of the i-th random variable to the total sum, vi =

xi

.

n

∑x i =1

i

By construction, all vi are identically distributed. Naturally, correlations are not equal to zero, but at least there is no absolute dependence. The vector of shares of seats is modeled based on the vote vector. Note that the purpose is not to model a concrete apportionment method, because some indices are closely related to such methods. It is better to generate a random allocation of seats to analyze indices. The values si

15

are generated using random deviations from vi , which are independent normal random variables with zero expectation and variance σ 2 : si = 1+ εi , vi

where ε i ~ N (0, σ 2 ) ,

si = vi (1 + ε i ) . Each element is divided by the sum of all values, so that values are constrained to the interval from 0 to 1. 8 experiments are conducted, for different numbers of parties n={4,5,7,10} and for different degrees of disproportionality σ = {0.1,0.5} . Each simulation is repeated 10000 times. We took xi ~ abs( N (0,1)) , the absolute value of a normal random variable. The distribution of

vi depends on the choice of the distribution parameters of xi very slightly, because vote share expectation is a function only of the number of parties, and the variance of vi does not depend on the variance of xi if it has a normal distribution. Thus, homogeneous data can be used to analyze the influence of the number of parties and of the degree of disproportionality. 9.1 Results Each index creates a unique ordering on the election results set. We can measure the distance between orderings using rank correlations between sets of the index values, which are calculated for random vote and seat vectors. The Spearman rank correlation coefficient is used for this purpose. The values of the Spearman correlation coefficient are given in Table 9.1. The LoosmoreHanby index is not reported, since it gives an ordering that is equivalent to that given by the Rae index. Table 9.1. The Spearman rank correlation coefficients for absolute deviation indices. The number of parties is n=4 and the degree of disproportionality is σ=0.1. MD Rae index Grofman index Lijphart index MD

1

0.974

0.930

0.918

0.974

1

0.935

0.913

Grofman index 0.930

0.935

1

0.913

Rae index

16

Lijphart index 0.918

0.913

0.913

1

High rank correlations reflect considerable similarity and homogeneity among the group of absolute deviation indices. The following figures demonstrate the means and standard deviations for various numbers of parties at degree of disproportionality σ = 0.1 .

Mean

Standard Deviation

0,04

0,018

0,035

0,016

0,03

0,014 0,012

0,025

0,01

0,02 0,008

0,015

0,006

0,01

0,004

0,005

0,002

n

0 4

6

8

MD LH index Lijphart index

10

Rae index Grofman index

n

0 4

5

6

MD LH index Lijphart index

Fig. 9.1.

7

8

9

10

Rae index Grofman index

Fig. 9.2.

The mean of the majority of the indices decreases with an increase in n, since the average size of the parties and of their deviations are also decreasing. In the given experiment, an increase in the number of parties can be viewed as a party split. There are more parties with a similar rule for the distribution of seats. The behavior of the indices that satisfy the property of independence from split significantly differs from the others. The means of the first group of indices increase, while the means of the others decrease. Standard deviation decreases considerably. This makes the ordering more random and weakens rank correlations. An increase in disproportionality has a reverse effect. The choice of index is less important when the degree of disproportionality is high. Orderings became closer in terms of rank-order correlations. The mean and standard deviation increase for a 0.5 degree of disproportionality. The second group of indices demonstrates a closer relationship in terms of Spearman rank correlations. Table 9.2. The Spearman rank correlation coefficients for absolute deviation indices. The number of parties is n=4 and the degree of disproportionality is σ=0.1. LSQ Hk k=5Gatev indexRyabtsev index Szalai index 17

LSQ

1

0.996

0.987

0.987

0.736

Hk k=5

0.996

1

0.980

0.980

0.726

Gatev index

0.987

0.980

1

1.000

0.695

Ryabtsev index 0.987

0.980

1.000

1

0.695

Szalai index

0.726

0.695

0.695

1

0.736

Modifications of the Gallagher index do not result in noteworthy changes. The Gatev and Ryabtsev indices differ insignificantly, as could be expected given the similarity in their analytical form. The Monroe index, which was not considered in the experiment, would be very close to the indices given in the table. The Szalai index demonstrates a markedly different behavior and will be considered in more detail.

Mean

Standard Deviation

0,07

0,03

0,06

0,025

0,05

0,02

0,04

0,015 0,03

0,01 0,02

0,005 0,01

n

0 4

5 6 LSQ Gatev index Szalai index

7

8

9

Hk Ryabtsev index

Fig. 9.3.

10

n

0 4

5

6

7

LSQ Gatev index Szalai index

8

9

10

Hk Ryabtsev index

Fig. 9.4.

The means of Lsq and Hk diminish, because the average differences between vote and seat share are reduced. The Gatev, Ryabtsev, and Szalai indices are higher for a higher number of parties, because they are more sensitive to small parties. This property was considered above in the analysis of their analytical form. We again notice that indices that satisfy the property of independence from split differ from the others. The histograms of index values in most cases are very similar. They are asymmetric onepeaked distributions. For the degree of disproportionality σ = 0.5 and the number of parties n=4, the Gallagher index histogram is shown in figure 9.5.

18

750

Count

500

250

0 0,200

0,400

0,600

lsq

Fig. 9.5. For the degree of disproportionality σ = 0.1 , the value of the Gallagher index drops, but the histogram’s shape does not change. Almost all indices have similar distributions. The Szalai index histogram looks like the histogram given above at σ = 0.1 . Parties that have no seats appear for larger degrees of disproportionality. This has considerable impact on the value of the index, and therefore on the shape of the histogram as well. There are two separate groups of elections results. On one hand, there are election results with parties that did not win any seats, and on the other hand there are election results without any such parties. Two distributions will result. The first distribution will have one peak; the second distribution will depend on the number of unrepresented parties. Joint distributions have many peaks. The following figures show histograms of the Szalai index with the various numbers of parties n={4,5,7,10}. n=4

n=5

600

500

400

Count

Count

400

300

200

200

100

0 0,20 0

0,40 0

ISzalai

Fig. 9.6.

0,60 0

0,80 0

0,200

0,400

0,600

0,800

ISzalai

Fig. 9.7. 19

n=7

n=10

500

400

400

Count

Count

300

300

200

200

100

100

0,200

0,400

0,100

0,600

0,200

0,300

0,400

0,500

0,600

ISzalai

ISzalai

Fig. 9.8.

Fig. 9.9.

At σ = 0.5 the Szalai index distribution is bimodal, because parties without any representation appear. This can be explained using the analytical form of the index:

I Szalai =

2

⎛ s i − vi ⎞ ⎜⎜ ⎟⎟ ∑ i =1 ⎝ s i + v i ⎠ . n n

If si = 0 , the sum will include 1; this considerably increases the index, because the other values are rather small. This effect is more significant for a small number of parties. At n=4 the histogram demonstrates a sharp increase after the value I Szalai =

1 = 0.5 . At n=10 the values 4

1 and more are reached when there are parties that did not obtain a seat, and therefore this 10

effect is less evident. It is therefore incorrect to compare election results with a different number parties that received no seats. Inequality indices and objective functions greatly differ in analytical form. Therefore, they have smaller rank-order correlations. The d’Hondt index has the smallest Spearman correlation coefficient, at about 0.6-0.8. The d’Hondt index measures the maximum discrepancy, but the other indices are closer to measuring the average dissimilarity. The rank correlations for the Atkinson index and for generalized entropy significantly depend on the parameters ε and α, respectively. The disproportionality indices describe the quality of apportionment. They are strongly correlated, though they provide different results. This experiment can be used to consider the unique features of the indices and explain how the values of the indices change in real situations. 20

This understanding of the structure of the indices and of their properties improves the analysis of election results.

10. Computed disproportionality indices for the elections to the Russian Parliament, 1995-2003 A mixed election system was used in 1995-2003 for elections to the State Duma (Russian Parliament). Half of the 450 seats were elected based on the majoritarian method; to be exact, the relative majority in single-member districts was used. The other half was elected by the proportional representation system with a 5% threshold. Using indices, it is possible to estimate how strongly the real seat assignment deviated from the ideal apportionment. Table 10.1. Computed disproportionality indices for elections to the State Duma, 19952003. 1995

1999

2003

0.495

0.186

0.282

Maximum deviation

0.217

0.055

0.152

Rae index

0.022

0.013

0.023

LH index

0.495

0.186

0.282

Grofman index

0.091

0.055

0.108

Lijphart index

0.164

0.053

0.101

Lsq

0.210

0.073

0.136

Hs

0.191

0.049

0.133

Gatev index

0.473

0.170

0.258

Ryabtsev index

0.355

0.121

0.186

Szalai index

0.960

0.888

0.919

Aleskerov-Platonov index 1.984

1.232

1.392

Generalized entropy

1.158

0.392

0.611

D’Hondt index

2.000

1.263

1.406

Sainte-Lague index

0.990

0.479

0.627

The vote share of parties which did not obtain seats

21

43 parties participated in the 1995 elections. Only 4 of them achieved the necessary fivepercentage minimum. 49.5% votes did not obtain any representation in the parliament. As a result, each party that was in the State Duma received a seat share almost double that of the vote share. The Aleskerov-Platonov index reflects this phenomenon. The Communist party received the highest vote share and had the maximum deviation from the exact quota. 26 parties participated in the 1999 elections. 6 of them passed the threshold into parliament. In 2003, 4 parties out of 23 received seats. This led to a significant reduction in the share of parties that did not get into parliament in comparison with 1995. Almost all the indices create equal ordering. This is closely connected to the considerable difference in the number and size of parties. On the set of indices, the parliament of 1999 is the most proportional and the parliament of 1995 is the least. The Grofman and Rae indices differ. Their value quickly decreases, and they become very close for a large number of parties. It is very difficult to construct a correct ordering using these indices, because their values for different election results are very close. In 2007 the election of members of parliament took place under a new law, therefore it is considered separately. The major change is the use of the proportional representation method for all seats. The threshold is increased to 7%. The line ”Against all” is eliminated. Table 10.2 Computed disproportionality indices for elections to the State Duma 2007 2007 The vote share of parties which did not obtain seats

0.083

Maximum deviation

0.057

Rae index

0.014

Lsq

0.048

Szalai index

0.817

Aleskerov-Platonov index

1.092

D’Hondt index

1.095

The parliament of 2007 is the more proportional than others. We show two sources of the disproportionality reduction. The vote share of parties which did not obtain seats is the main factor of disproportionality in Russian elections. This share includes “against all” votes. The line “against all” was excepted in last election. Probably because of this some people did not participate in elections. By public opinion polls main 4 parties obtained the vast majority of votes. Other parties had very little chance to pass the threshold. The vote given to the small party

22

could not change seat allocation, but in other case it had some influence. This is the reason why some voters might be followed by insincere preferences.

Acknowledgements The author is much indebted to Professor Fuad Aleskerov for invaluable advice during the work with disproportionality measurement problems. For useful comments and suggestions, the author would like to thank the seminar participants of the Fifth Workshop “Voting and Power Indices” at the Higher School of Economics in Moscow. The author would like to thank Jeffrey Lockshin for help with translation from Russian to English. The paper was partially supported by the HSE Centre for Advanced Studies and the Scientific Foundation of the State University – Higher School of Economics under the grant No. 06-04-0052.

23

Appendix 1

Optimization of quadratic indices The influence of changes in vote and seat shares is studied here. We limit our analysis to quadratic indices. The Gallagher index can be written in the following form:

LSq =

1 n (s i − v i ) 2 , ∑ 2 i =1

subject to n

∑v i =1

i

= 1,

i

= 1.

n

∑s i =1

Taking the restrictions into account, the index can be written as

LSq =

n −1 n −1 1 n −1 2 ( ∑ (s i − v i ) + ( − ∑ s i + ∑ v i ) 2 ) . 2 i =1 i =1 i =1

Consider how the index depends on the share of seats: n −1 n −1 ∂ 1 LSq = ⋅ ( s j − v j + ∑ s i − ∑ vi ) . ∂s j 2 ⋅ LSq i =1 i =1

The necessary condition for an extreme is n −1

n −1

i =1

i =1

s j − v j + ∑ si − ∑ vi = 0 .

Analogously, we find the condition for the vote share: n −1 n −1 ∂ 1 LSq = ⋅ ( − s j + v j − ∑ s i + ∑ vi ) , ∂v j 2 ⋅ LSq i =1 i =1 n −1

n −1

i =1

i =1

− s j + v j − ∑ si + ∑ vi = 0 .

Note that the derived conditions for vote and seat shares are equivalent. Changing the share of one party by changing the share of some other party, the index attains its minimum at the point where si ≠ vi . 24

If we minimize the index with respect to all variables, we will have 2n-2 conditions. They can be written as follows: n −1

n −1

i =1

i =1

− si + vi = ∑ si − ∑ vi ,

where

n −1

n −1

i =1

i =1

∑ si − ∑ vi = const for all 2n-2 conditions, hence si − vi = −const .

Summing with respect to i = 1, n − 1 , we obtain n −1

n −1

i =1

i =1

∑ si − ∑ vi = −(n − 1) ⋅ const , which is correct if and only if const = 0 ,

si = vi , i = 1, n . The Hessian matrix confirms that the critical point is a minimum. Proof that the first-order conditions for the Hk index are the same is shown below. The Hk index takes seat and vote shares into account as follows:

Hk = k

1 n (si − vi )k . ∑ k i =1

Using the restrictions, the index can be written as

Hk = s

n −1 n −1 1 n −1 k ( ∑ (s i − v i ) + ( − ∑ s i + ∑ v i ) k ) . k i =1 i =1 i =1

Consider how the index depends on the share of seats: n −1 n −1 ∂ 1 ⎞ 1− k ⎛ H k = (H k ) ⋅ ⎜ ( s j − v j ) k −1 − (−∑ si + ∑ vi ) k −1 ⎟ . ∂s j k i =1 i =1 ⎝ ⎠

The first-order condition is n −1

n −1

i =1

i =1

( s j − v j ) k −1 = ( − ∑ s i + ∑ vi ) k −1 , n −1

n −1

i =1

i =1

s j − v j + ∑ s i − ∑ vi = 0 .

25

Analogously, we find the condition for the vote share. This result shows that these indices demonstrate similar behavior for any k. Consider the example in Table A.1, where there are 4 parties. Table A.1 Parties

Vote Shares

Seat Shares

A

0,1

0

B

0,2

0,2

C

0,3

0,3

D

0,4

0,5

Suppose the share of seats of party B is transferred to party A, and all other seat and vote shares in Table A.1 remain constant. Lsq

0,2

Hs

Index's value

0,18 0,16 0,14 0,12 0,1 0,08 0,06 0,04 0,02 0 0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

0,20

Vote share of party A

Fig. A.1 Figure A.1 shows that the indices are minimally different from each other. This implies that the strengthening of the Gallagher index is unnecessary.

26

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Stewart J. Assessing alternative dissimilarity indexes for comparing activity profiles.

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