Measuring Election Fairness: A New Approach

© Research Journal of Internatıonal Studıes - Issue 21 (October, 2011) 40

Measuring Election Fairness: A New Approach

Basel N Asmar

2050 Consulting Ltd, 48 Imperial Hall, 104-122 City Road, London EC1V 2NR, United Kingdom

Email: [email protected]

Abstract

Measures or indices to quantify elections’ disproportionality have been widely

proposed in literature and used extensively to analyse and compare election results.

This study introduces the Unfairness Index (UI) as a generalised versatile easy-

to-use disproportionality index. UI satisfies 11 out of 11 criteria proposed by Taapegara

and Grofman (2003), which are considered desirable for any disproportionality index,

and thus it scores higher than the scores of Loosemane-Hanby Index (LHI) and Galleger

Index (GhI). UI is bound between 0 and 1 (0 and 100%), with 0 indicates no

disproportionality. Applying UI to elections in the UK and ten other countries for the

period 1979-2011 established that UI not only agrees with LHI and GhI in its

assessment regarding election disproportionality, it also interprets better the

unfairness in an election compared to LHI and GhI.

Keywords: electoral system, disproportionality, disproportionality index, unfairness,

British general elections

1. Introduction

An Electoral system is an extremely important institution affecting the way in which a country's system

of government works. Even with each voter casting exactly the same vote and with exactly the same

number of votes for each party, one electoral system may lead to a coalition government or a

minority government, while another may allow a single party to assume majority control.

There are a large number of different electoral systems currently in use. They are generally

categorized into three broad families (Reynolds et al., 2005):

1. plurality/majority systems (subfamilies: First Past The Post (FPTP), Block Vote (BV),

Party Block Vote (PBV), Alternative Vote (AV), and the Two-Round System (TRS)),

2. proportional systems (subfamilies: List Proportional Representation (LPR) and the Single

Transferable Vote (STV))),

3. mixed systems (subfamilies: Mixed Member Proportional (MMP) and Parallel systems (PS)).

In addition, a handful of countries use other systems such as the Single Non-

Transferable Vote (SNTV), the Limited Vote (LV), and the Borda Count (BC) which do not fit neatly

into any particular category.

In an ideal election the composition of any elected body should reflect exactly the wishes

of the voters. However, deviation from the ideal, referred to as disproportionality, is the

norm and most electoral systems aspire to minimise the deviation between party vote shares and

party seat shares. Generally speaking plurality/majority systems lead to higher disproportionality

than proportional representation systems with mixed systems scoring in between, while aiming to

lower disproportionality as well. It is worth noting though that there are exceptions to these

generalisations reported all over the world.

With the rapid increase in number of emerging democracies in Africa, Asia, Eastern Europe,


Latin America and the former Soviet Union in 1990s, and the repeated loud calls to reform

disproportional electoral systems in many established democracies such as the United Kingdom and

Canada, a need to find a generalised tool to evaluate and compare electoral systems is increasingly

necessary.

Measures or indices to quantify disproportionality have been widely proposed in literature

and used extensively to analyse and compare election results. To date there is a debate concerning

which measure is better. The objective of this study is to present an index to quantify the fairness of

an election and demonstrate its application.

2. Disproportionality Measures

Several measures were proposed to quantify disproportionality usually in the form of a number

designed to measure the degree of deviation from proportionality in the allocation of seats to parties

or groupings which participated in an election. The choice of the measure used in any study may

affect its conclusion, since two different measures may give conflicting results.

Taking into consideration the reviews performed by Monroe (1994), Lijphart (1994) and

Pennisi (1998), Taapegara and Grofman (2003) reviewed disproportionality indices evaluating 19

published indices against set criteria, and showed that none satisfied all these criteria. They ranked

all indices and recommended the use of the Gallager Index (GhI) (Gallager, 1991) as the favoured

disproportionality index, with the Loosemore-Hanby Index (LHI) (Loosemore and Hanby, 1971) a

close second. Kestelman (2005) cited two additional measures and concluded that LHI remained the

most convenient index. Furthermore, Karpov (2007) also reviewed disproportionality measures and

classified them into several groups. Koppel and Diskin (2009) put forward lists of desirable criteria

to evaluate disproportionality indices that roughly subsume those of Taapegara and Grofman

(2003).

LHI is also called the "Index of Distortion". It is mathematically defined as:

LHI = ½ ∑ |V𝑖 − S𝑖|P𝑖=1 (1)

where for each party i, Vi is party vote share and Si is party seat share. P is the number of parties

participating in the elections. Σ denotes sum (over all parties).

GhI is usually referred to as the "least square measure" is mathematically defined as:

GhI = [½ ∑ (V𝑖 − S𝑖)P𝑖=1

2]0.5

(2)

For both LHI and GhI, Vi and Si can be taken as fractions or percentages. Lower LHI and GhI values

point to better proportionality.

It is worth noting that disproportionality measures indicate if the electoral system is fair not if

the election conduct or process is fair. The latter requires evaluating whether the election law is

respected, the correct procedures implemented, the freedom to campaign allowed, no intimidation

detected, etc. As such, disproportionality indices do not determine the legitimacy of a government. The

measures give a verdict over an electoral system assuming the process has been implemented

according to the laws of that system. Thus it cannot detect if irregularities are observed in conducting

the elections. For example the 2005 elections in Zimbabwe indicated highly proportional results,

however, the elections were far from free and fair as witnessed by neutral observers. Another example

is the 1983 elections in Turkey where even though the results show a highly proportional electoral

system, it concealed the fact that the major political parties which governed Turkey prior to the military

coup of 1980 were all banned from taking part.

3. Fairness Index (FI) and Unfairness Index (UI)

Since all indices reviewed by Taapegara and Grofman (2003) did not satisfy all set criteria, a new

disproportionality index, referred to as the Unfairness Index (UI) is presented. As other

disproportionality indices it quantifies the deviation between the vote and seat shares of a specific

election from an ideal baseline where vote and seat shares should be equal.

In any election, the advantage ratio of party i, denoted Ai, is defined as:

Ai = Si / Vi (3)


Obviously the ideal value for Ai is 1, which indicates there is no bias and the party i got its fair

share of the seats. However in practice this can hardly happen and any election result will reward or

penalise each party. Ai < 1 means the party is penalised while Ai > 1 means the party is rewarded by

the system. If a party wins no seats then its corresponding Ai is zero irrespective of its share of the

votes. Depending on the electoral system some parties will be always more rewarded or penalised than

others. For example some systems reward larger parties on the expense of smaller ones.

The fairness in an election, denoted F, can be measured as

F = [ ∑ (S𝑖 A𝑖)P𝑖=1 ] / VR (4)

where VR is the fraction of voters who voted for parties who won seats in the elected body and is

defined as:

VR = ∑ V𝑖 Q𝑖=1

(5)

where Q is the number of parties which won seats in the elected body.

F values range from 1 to infinity. Similar to the party advantage ratio, the ideal value for F is

also 1. Values of F > 1 indicate the electoral system has unfairness, the larger the value the more

unfairness is in the system.

It has to be noted that if independent candidates participate in an election each candidate is

treated separately as a single party and independents are not lumped together.

To ensure the value of the fairness index ranges between 0 and 1 (or 0 and 100%), the Fairness

Index, denoted FI, is defined as:

FI = 1 / F (6a)

FI (percent) = 100 / F (6b)

The Unfairness Index, UI, is therefore complements the Fairness Index, FI, and is defined as:

UI = 1 - FI (7a)

UI (percent) = 100 - FI (7b)

FI and UI are bound between 0 and 1 (or 0 and 100%), and complement each other. A value of

UI equals zero (FI = 100) indicates ideal fair elections, and a value of UI equals 100 (FI = 0) indicates

wholly unfair elections. Obviously the lower the value of UI the more proportional the election system

is. Note however that UI is not linear so a UI value of 50 does not indicate that the election fairness is

halfway.

UI/FI calculations are only applied to parties that receive votes in an election. If for any reason

a party received no votes but still was awarded seats, the calculation is performed excluding these

seats, since UI/FI examine the relationship between vote and seat shares and not appointed seats. The

fact is that any appointed seats are not determined by an election.

In true fairness any party which gains seats without any votes should render the election entirely

unfair! Surprisingly most disproportionality indices include these parties in the calculation and in some

examples indices such as LHI and GhI may give such elections high proportionality scores, especially

if the number of seats given to the zero-vote-receiving party is not high.

Table 1: Application of UI (Vi and Si are reported as percentages)

Party Vi Si Ai Si Ai

A 50 60 1.20 72.0

B 38 25 0.66 16.4

C 10 15 1.50 22.5

D 2 0 0.00 0.0

VR = 98 FI = 88.3 LHI = 15.0

F = 1.132 UI = 11.7 GhI = 12.2

The application of FI and UI is illustrated in Table 1, which considers a hypothetical 100 seat

elected body with four competing parties. Vi and Si indicate the percentages of the votes and seats

obtained by each parties. Applying the methodology explained above, disproportionality results (in

percent) are shown in Table 1.


The UI score is 11.7, which indicates that the election of the hypothetical example is fairly

proportional. Both LHI and GhI agree with this assessment with scores of 15 and 12.2 respectively.

3.1 Assessment of UI against Taapegara and Grofman (2003) criteria

UI is examined against the 11 criteria proposed by Taapegara and Grofman (2003) which are

considered desirable for any disproportionality index. A twelfth criterion to use Vi and Si

symmetrically is not evaluated as it is stated by Taapegara and Grofman (2003) themselves that it is a

superfluous requirement for disproportionality indices. Table 2 shows the criteria and the evaluation of

UI against them.

Table 2: Evaluation of UI against Taapegara and Grofman (2003) criteria

No. Criterion Satisfied Notes

1 Completeness (makes use of all Vi and Si data for all parties) Yes

2 Uniformity for all parties (e.g. no special role to largest two parties) Yes

3 Range 0 and 1 (or 0 and 100 percent) Yes

4 Lower limit = 0 Yes

5 Higher limit = 1 (or 100 percent) Yes Example Al

6 Satisfies Dalton's principle of transfers Yes Example A2

7 Does not include P, the number of parties Yes

8 Insensitive to lumping of residuals Yes Example A3

9 Simple to compute Yes

10 Insensitive to shift from fractional to percent shares Yes

11 The input data consist only of vote and/or seat shares (i.e. does not depend on

total number of votes or seats)

Yes

The main conclusion from Table 2 is that UI satisfies all criteria set by Taapegara and Grofman

(2003). This was not achieved by all other indices reviewed by them including LHI and GhI.

To elaborate, the scoring method used by Taapegara and Grofman (2003) is applied to UI. The

application excluded Vi-Si symmetry criterion, and corrected the implementation of insensitivity to

lumping criterion (see Appendix A). UI scores 11 (out of 11) compared to LHI of 10.5 (correcting for

insensitivity of lumping) and GhI 8.5 if the advantage ratio in satisfying Dalton's transfer principle is

considered. If the difference (in number of seats) is used in satisfying Dalton's transfer principle then

the scores for UI, LHI and GhI are 10, 10.5 (correcting for insensitivity of lumping) and 9.5

respectively.

Three examples are illustrated in Appendix A discussing UI's satisfaction to criteria 5, 6 and 8

in Table 2. Satisfaction to other criteria is self explanatory.

4. Comparison of FL LHI and GhI applied to British General Elections

The UK Independent Commission on Voting System (1998) considered full proportionality is achieved

as is normally practicable if LHI value was below 8. Kestelman (2005) characterised an election

system full proportional if LHI < 10, semi (broad) proportional if LHI ranged between 10 and 15, and

non proportional if LHI > 15. GhI is highly correlated to LHI (for 44 UK general elections 1832-2005,

r2 = 0.964) Kestelman (2005), and its value is always lower than LHI Taapegara and Grofman (2003).

To compare the results of UI with established disproportionality indices LHI and GhI, analysis

of the British general elections from 1979 to 2010 was performed. Table 3 shows the results of the

three indices. Note that throughout the analysis UI, FI, LHI and GhI values are reported in percentages.

As shown in Table 3, the three indices UI, LHI and GhI are comparable and characterise the

UK general election as disproportional. This is expected as the British electoral system is an FPTP

system which inherently creates "unfairness" in the system (Reynolds et al., 2005).

To understand the reasons behind the relatively high disproportionality in the UK general

elections, the values of the party advantage ratio Ai for the UK’s three major parties (Labour,

Conservative and Liberal Democrats) are analysed, since these values are the main component in UI's

calculation.


Table 3: Disproportionality analysis of the British general election 1979-2010

UK Election UI LHI GhI

1979 15.59 15.34 11.58

1983 24.08 24.16 20.58

1987 20.35 20.85 17.75

1992 17.84 17.97 13.55

1997 21.87 21.25 16.56

2002 22.23 22.14 17.82

2005 24.07 20.67 16.74

2010 25.59 22.82 15.11

Figure 1 shows Ai values for the UK three major parties. The figure shows that the election

system consistently favours the Labour party on the expense of the Liberal Democrats. The

Conservative party was penalised between 1997 and 2005 when it suffered three consecutive electoral

defeats, however it still fairs considerably better than the Liberal Democrats. The significant increase

in party advantage values for the Liberal Democrats since 1997 is attributed to the recent public

awareness and the use of "strategic" voting by the voters to keep the Conservatives out. This may

change following the coalition government of the Conservative and Liberal Democrats, which may

drive voters away from the Liberal Democrats in future elections.

Figure 1: Advantage ratio for the three major parties in the UK, 1979-2010

Data Sources: Electoral Commission (2011), Kimber (2011), Rallings and Thrasher (2009)

Figures 2 to 4 show detailed Ai values for the three major parties in England, Scotland and

Wales. Northern Ireland is not included as the Labour and the Liberal Democrats do not contest

elections there, and the Conservative party organisation there is minimal. It can be seen clearly that the

system favours Labour in all three countries, with a significant bias in Scotland and Wales. The

Conservative party is generally favoured in England but penalised heavily in Scotland and Wales. The

Liberal Democrats were historically penalised in all countries, but their results appear to be balancing

in Scotland in recent elections.

From the analysis of the Ai values it is clear that the imbalance in the party advantage towards

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the third major party is significantly high and is the main cause for high disproportionality.

Figure 2: Advantage ratio for the Conservative Party in the UK, England, Scotland and

Wales, 1979-2010


Figure 3: Advantage ratio for the Labour Party in the UK, England, Scotland and Wales,

1979-2010


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United Kingdom England Scotland Wales

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Figure 4: Advantage ratio for the Liberal Democrats in the UK, England, Scotland and Wales,

1979-2010


5. Application of UI to various elections around the world

The UI is further applied to the results of elections in ten countries since 1979. The countries are

divided into three groups according to the election system:

1. Plurality/Majority systems: Canada, Botswana, and Zimbabwe.

2. Proportional systems: Netherlands, Israel, South Africa, and Turkey

3. Mixed systems: Germany, New Zealand, and Lesotho.

The three plurality/majority system countries (Canada, Botswana, and Zimbabwe) use FPTP.

All proportional system countries considered use LPR system with natural threshold in South Africa

and the Netherlands, 1.5% in Israel and 10% in Turkey. Germany and New Zealand use an MMP

system with a threshold of 5%, but with backdoor policy of 3 and 1 direct mandates respectively.

Lesotho uses natural threshold.

Figures 5 to 7 shows the Ul results for the three groups compared to the results of the United

Kingdom elections 1979-2010. The key results observed from these figures show that generally the UK

election system scores similar results to other FPTP systems analysed. Note the peculiar results for

Zimbabwe. The highly controversial 2005 and 2008 elections show surprisingly low disproportionality

not observed usually in FPTP systems, which may indicate government manipulation to enhance its

image. The results of the 1980 and 1985 elections show very high disproportionality with UI scores of

88.4 and 79.2 respectively. These scores indicate clearly the unfairness in these two elections where

20% of the seats were reserved to the white minority and the vote shares to elect these 20 seats were

0.5% and 1.2% in 1980 and 1985 respectively. LHI and GhI fail to highlight this and for the same two

elections LHI scores are 20.9 and 18.9 respectively, and GhI scores are 16.4 and 14.0 respectively.

Compared to LPR systems, the UK elections show significantly higher disproportionality when

compared to the results of South Africa, Netherlands and Israel. The latter three countries' results

illustrate that the lower the threshold is, the lower is the disproportionality. However the results for

Turkey prove that a high threshold in LPR system renders the system unfair. In 2002 elections UI score

is 71.2 which indicates very high disproportionality. This reflects clearly the facts that 45.8% of the

voters have no representatives, and that AK, the party with only 34.3% of the vote, won almost two-

thirds (66%) seats majority and was on the verge of being able to amend the constitution. LHI and GhI

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scores are 45.8 and 27.1 respectively. These scores do not reflect enough the severity of the unfairness,

which UI highlights more clearly.

Figure 5: UI values for the UK and selected FPTP countries

Data Sources: Electoral Commission (2011), Kimber (2011), Rallings and Thrasher (2009), Elections

Canada (2011), Nunley (2011)

Figure 6: UI values for the UK and selected List PR countries

Data Sources: Electoral Commission (2011), Kimber (2011), Rallings and Thrasher (2009), Statistics

Netherlands (2011), Independent Electoral Commission (2011), Knesset (2011), The Supreme Election

Board (2011), State Institute of Statistics (2011)

The results in Figure 7 signify the effects MMP can bring to lower disproportionality. Bearing

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in mind that New Zealand moved from FPTP to MMP in 1996 and Lesotho in 2002, the difference in

results is obvious, and the two countries now significantly outperform the United Kingdom Note that

before moving to MMP the elections in Lesotho were highly disproportional. For example in 1998

elections the LCD captured 79 out of 80 seats with 60.7% of the vote, leaving the BNP with 1 seat

despite 24.5% of the vote and 14.8% wasted votes. UI score for this election was 47.0 while LHI and

GhI scores were 38.0 and 32.5 respectively. Unlike UI scores, LHI and GhI scores do not differ by a

big margin from the scores observed in 2002 elections in Turkey. The proximity of the latter scores

highlights the insensitivity of LHI and GhI compared to UI. As a party with absolute majority of the

votes gaining absolute majority of the seats is much fairer than a party with only a third of the votes

winning two-thirds of the seats. UI can reflect this difference, whereas the other two indices cannot.

Figure 7: UI values for the UK and selected MMP countries

Data Sources: Electoral Commission (2011), Kimber (2011), Rallings and Thrasher (2009), Federal

Returning Officer (2011), Elections New Zealand (2011), Nunley (2011)

The analysis above illustrates the versatility of the UI and its ability to translate election results

better than LHI and GhI.

Based on the analysis in this section and Section 4 it is suggested that UI value of 15% is taken

as the characteristic of proportional elections.

6. Conclusions

The Unfairness Index (UI) is proposed as a generalised versatile easy-to-use disproportionality index.

UI satisfies 11 out of 11 criteria proposed by Taapegara and Grofman (2003) which are considered

desirable for any disproportionality index, and thus it scores higher than the scores of Loosemane-

Hanby Index (LHI) and Galleger Index (GhI).

UI is bound between 0 and 1 (0 and 100%), with 0 indicates no disproportionality. UI value of

15% is taken as the characteristic of proportional elections.

Applying UI to elections in the UK and ten other countries for the period 1979-2010 established

that UI not only agrees with LHI and GhI in its assessment regarding election disproportionality, but it

also interprets better the unfairness in an election compared to LHI and GhI. Examples are given to

demonstrate this.

Not surprisingly, UI showed that the current FPTP system used in the UK general election

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distorts significantly the voters' choice and is inherently less fair than either MMP or LPR systems.

References

[1] Elections Canada, 2011. http://www.elections.ca

[2] Elections New Zealand, 2011. http://www.elections.org.nz

[3] Electoral Commission, 2011. http://www.electoralcommission.org.uk

[4] Federal Returning Officer, 2011. http://www.bundeswahlleiter.de/wahlen/e/indexe.htm

[5] Gallagher, M., 1991. “Proportionality, Disproportionality and Electoral Systems”, Electoral

Studies, 10(1), pp. 33-51.

[6] Independent Commission on Voting System, 1998. “The Report of Independent Commission

on Voting System”, Volume 1, Cm 4090-I, The Stationery Office, London

[7] Independent Electoral Commission, 2011. http://www.elections.org.za

[8] Kestelman, P., 2005. “Apportionment and Proportionality: A Measured View”, Voting Matters,

20. PP. 12-22.

[9] Karpov, A., 2007. “Measurement of Disproportionality in PR Systems”, Centre for

Advances Studies Research Notes, CAS_RN_2007/09,

http://www.hse.ru/data/836/292/1234/CAS_RN_2007_9.pdf

[10] Kimber, R., 2011. “Political Science Resources”, http://www.politicsresources.net

[11] Knesset, 2011. Knesset Election Results

http://www.knesset.govil/description/eng/eng_mimshal_res.htm

[12] Koppel, M. and Diskin, A., 2009. “Measuring Disproportionality, Volatility and

Malapportionment: Axiomatization and Solutions”, Social Choice and Welfare, 33, PP. 281-

286.

[13] Lijphart, A., 1994. “Electoral Systems and Party Systems”, New York and Oxford, Oxford

University Press.

[14] Loosemore, J. and Hanby, V., 1971. “The Theoretical Limits of Maximum Distortion: Some

Analytical Expressions for Electoral Systems”, British Journal of Political Science, 1, pp. 467-

477.

[15] Monroe, B. L., 1994. “Disproportionality and Malapportionment: Measuring Electoral

Inequality”, Electoral Studies, 19, pp. 132-149.

[16] Nunley, A., 2011. “African Elections Database”, http://africanelections.tripod.com

[17] Pennisi, A., 1998. “Disproportionality Indexes and Robustness of Proportional References

Allocation Methods”, Electoral Studies, 17, pp. 3-19.

[18] Rallings, C. and Thrasher, M., 2009. “British Electoral Facts”, Total Politics, Biteback

Publishing Ltd., London, UK.

[19] Reynolds, A., Reilly B., Ellis A., Cheibub, J.A., Cox, K., Lisheng. D., Elklit, J., Gallagher, M.,

Hicken, A., Huneeus, C., Huskey, E., Larserud, S., Patidar, V., Roberts, N.S., Vengroff, R., and

Weldon, J.A., 2005. “Electoral System Design: The New International IDEA Handbook”,

International Institute for Democracy and Electoral Assistance. Stockholm, Sweden.

[20] State Institute of Statistics, 2011.

http://www.tbmm.gov.tr/develop/owa/genel_secimler.genel_secimler

[21] Statistics Netherlands, 2011. http://statline.cbs.nl

[22] Taagepera, R. and Grofman, B., 2003. “Mapping the Indices of Seats-Votes Disproportionality

and Inter-Election Volatility”, Party Politics, 9, pp. 659-677.

[23] The Supreme Election Board, 2011. http://www.ysk.gov.tr


Appendix A

Note that party vote shares (Vi) and seat shares (Si) are reported in percentages in all examples below.

UI, FI, LHI and GhI are reported in percentages as well.

Al. Satisfaction of Criteria 5 in Table 2: Highest limit =1 (or 100%)

Example Al:

Consider a 100 seat elected body with four competing parties. The election results and UI calculations

are shown in Table Al.

Table A1: Example A1


A 50 0 0 0

B 40 0 0 0

C 9.999 0 0 0

D 0.001 100 100000 10000000

VR = 0.001 FI = 1.0e-8 LHI = 99.999 ≅ 100

F = 1.0e10 UI ≅ 100 GhI = 84.3

Note that the smaller the share of the votes for party D the closer is the result to 100%.

However as explained in Section 3, UI calculation is not applicable to parties who get seats without a

single vote as these seats are not elected but appointed seats.

A2. Satisfaction of Criteria 6 in Table 2: Satisfaction of Dalton's principle of transfers

Dalton's principle of transfer states that when a seat is transferred from a richer to a poorer component,

the disproportionality index should decrease. Equally if the transfer is in the opposite direction the

index should increase. In this analysis party advantage ratios are considered prevailing when

determining which party is richer even though some may argue that the number of seats is more

important politically (but not mathematically).

Example A2:

Consider a 100 seat elected body with three competing parties. The election results and UI calculations

are shown in Table A2 for 3 cases. In all cases the share of the vote for parties A, B and C remains

unchanged. However in Case II party A loses a seat to party C, the opposite occurs in Case III, where

party A gains a seat at the expense of Party C.


Case I


A 50 60 1.2 72

B 40 25 0.625 15.625

C 10 15 1.5 22.5

VR = 100 FI = 90.8 LHI = 15.0

F = 1.101 UI = 9.2 GhI = 13.2

Case II


A 50 59 1.18 69.62

B 40 25 0.625 15.625

C 10 16 1.6 25.6

VR = 100 FI = 90.2 LHI = 15.0

F = 1.108 UI = 9.8 GhI = 13.1


Case III


A 50 61 1.22 74.42

B 40 25 0.625 15.625

C 10 14 1.4 19.6

VR = 100 FI = 91.2 LHI = 15.0

F = 1.096 UI = 8.8 GhI = 13.5

As shown from Table A2, UI satisfies the Dalton principle of transfer (based on the richest

party is the one with highest party advantage ratio) as transferring a seat from the richer party C (AC =

1.5) to the poorer party A (AA = 1.2) resulted in a decrease in UI's value as shown in Case C. A change

in the other direction resulted in increased UI's value. As noticed both GhI and LHI do not satisfy the

criteria. Note however that Taapegara and Grofman (2003) considered GhI to satisfy the criteria as

they considered the party with the largest number of the seats to be the richest, even though they

admitted there is an ambiguity and even are quoted saying for the same example considered here "yet

the mind baulks at adding seats to a party that is already 50% overpaid [i.e. Party C] and calling it a

reduction in disproportionality". According to their argument UI will not satisfy the criteria. However,

mathematically using party advantages is sounder even though it may not have the same political

significance, and as a result it is considered here that UI not GhI satisfies the criteria.

A3. Satisfaction of Criteria 8 in Table 2: Insensitive to lumping of residuals

Example A3:

Consider a 100 seat elected body. The election results and UI calculations are shown in Table A3 for 5

cases. In all cases the vote and seat shares for parties A, B and C remain unchanged. In Case I parties D

and E each get 5% of the votes and the seats. In Case II party D wins 9% of the vote but no seats,

whereas 10 parties E to N each wins 0.1% of the vote and 1% of the seats. While keeping Parties A, B

and C separate, the results of all other parties are lumped as one forth party in Case III, or as two

parties in Cases IV and V, but with different lumping scenarios.

As expected Cases I and III give identical results as the lumped parties D and E each has an

ideal party advantage of 1, and thus combining them together will have no effect. In this comparison,

UI is not sensitive to lumping, which is also applies to LHI and GhI.

Comparing Cases II and IV gives identical results for UI and LHI, but surprisingly GhI which is

claimed by Taagepera and Grofman (2003) to be insensitive to lumping increases, which is clearly

wrong considering that the results have not changed.

In comparing Cases II and V, UI and LHI indicate that Case V is more proportional than Case

II. GhI indicates the opposite. The change in values of all indices is relatively small indicating mild

sensitivity exists. Results in Case V imply no wasted votes while in Case II 9% of the votes were

wasted. Therefore GhI conclusion of less proportional elections in Case IV seems wrong.

Comparing Cases II and III shows that UI and LHI are sensitive to lumping while GhI is not.

Taken on face value this may be concluded. However it is wrong to lump parties that gained no seats to

parties that gained seats to determine sensitivity.

Based on the above analysis it is concluded that to assess sensitivity to lumping "wasted" votes

should not be lumped with votes for parties that gained seats as this will distort the results. Two

lumped values must be used: one to lump all other parties that gained seats, and the other for wasted

votes. Taking this into consideration UI and even LHI are insensitive to lumping unlike GhI. This

contradicts the conclusions of Taapegara and Grofman (2003). As a result in comparing which criteria

each index satisfies, LHI and UI are awarded scores for insensitivity to lumping. However, GhI keeps

the scores as well as the change in its values does not vary significantly.



Case I


A 40 50 1.25 62.5

B 30 30 1 30

C 20 10 0.5 5

D 5 5 1 5

E 5 5 1 5

VR = 100 FI = 93.0 LHI = 10.0

F = 1.075 UI = 7.0 GhI = 10.0

Case II


A 40 50 1.25 62.5

B 30 30 1 30

C 20 10 0.5 5

D 9 0 0 0

E 0.1 1 10 10

F 0.1 1 10 10

G 0.1 1 10 10

H 0.1 1 10 10

I 0.1 1 10 10

J 0.1 1 10 10

K 0.1 1 10 10

L 0.1 1 10 10

M 0.1 1 10 10

N 0.1 1 10 10

VR = 91 FI = 46.1 LHI = 19.0

F = 2.170 UI = 53.9 GhI = 12.0

Case III


A 40 50 1.25 62.5

B 30 30 1 30

C 20 10 0.5 5

D 10 10 1 10

VR = 100 FI = 93.0 LHI = 10.0

F = 1.075 UI = 7.0 GhI = 10.0

Case IV


A 40 50 1.25 62.5

B 30 30 1 30

C 20 10 0.5 5

D 9.1 1 0.11 0.11

E 0.9 9 10 90

VR = 100 FI = 53.3 LHI = 18.1

F = 1.876 UI = 46.7 GhI = 12.9

Case V


A 40 50 1.25 62.5

B 30 30 1 30

C 20 10 0.5 5

D 9 0 0 0

E 1 10 10 100

VR = 91 FI = 46.1 LHI = 19.0

F = 2.170 UI = 53.9 GhI = 13.5

Documents

Measuring Election Fairness: A New Approach