Gallagher index

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Comparison of Gallagher indices of various countries since 1945.

The Gallagher index is a statistical analysis methodology utilised within political science (notably the branch of psephology) which "measures an electoral system’s relative disproportionality between votes received and seats allotted in a legislature."[1][2] As such, it measures the difference between the percentage of votes each party gets and the percentage of seats each party gets in the resulting legislature, and it also measures this disproportionality from all parties collectively in any one given election. That collective disproportionality from the election is given a precise score, which can then be used in comparing various levels of proportionality among various elections from various electoral systems.[3]

Michael Gallagher, who created the index, referred to it as a "least squares index", inspired by the sum of squares of residuals used in the method of least squares. The index is therefore commonly abbreviated as "LSq" even though the measured allocation is not necessarily a least squares fit. The Gallagher index is computed by taking the square root of half the sum of the squares of the difference between percent of votes () and percent of seats () for each of the political parties ().[4]

  [5]

The division by 2 gives an index whose values range between 0 and 100. The larger the differences between the percentage of the votes and the percentage of seats summed over all parties, the larger the Gallagher index. The larger the index value the larger the disproportionality and vice versa. Michael Gallagher included "other" parties as a whole category, and Arend Lijphart modified it, excluding those parties. Unlike the Loosemore–Hanby index, the Gallagher index is less sensitive to small discrepancies.[6]

Application in Canada[edit]

The Gallagher index gained considerable attention in Canada in December 2016 in the context of efforts to reform Canada's electoral system.[7][8] The Special Committee on Electoral Reform (a Parliamentary Committee) recommended "that the Government should, as it develops a new electoral system, use the Gallagher index in order to minimize the level of distortion between the popular will of the electorate and the resultant seat allocations in Parliament." The committee recommended that "the government should seek to design a system that achieves a Gallagher score of 5 or less."[9][10] In the 2015 Canadian federal election, the Gallagher index was 12.02, where 0 would be a perfectly proportional election outcome.[11]

Gallagher Index for the 2015 Canadian federal election
Party Votes (%) Seats (%) Difference Difference
squared
Liberal 39.47% 54.44% 14.97 224.1009
Conservatives 31.89% 29.29% -2.6 6.76
New Democratic 19.71% 13.02% -6.69 44.7561
Bloc Québecois 4.66% 2.96% -1.7 2.89
Green 3.45% 0.29% -3.16 9.9856
Other 0.82% 0.00% -0.82 0.6724
Total of differences squared 289.165
Total / 2 144.5825
Square root of (Total / 2): Gallagher Index result 12.02
The Gallagher Index ranges from 0 to 100. Low indexes (close to 0)
are very proportionate, high indexes (20 or greater) are very disproportionate.


Examples of calculating disproportionality[edit]

New Zealand[edit]

This table uses the New Zealand 2005 election result.[12] Note that since New Zealand uses the MMP voting system, voters have two votes. This list uses the party vote, which determines the proportionality of the House; the electorate vote determines the local member.

Party Votes (%) Seats (%) Difference Difference
squared
Labour 41.10 41.32 -0.22 0.0484
National 39.10 39.67 -0.57 0.3249
NZ First 5.72 5.79 -0.07 0.0049
Green 5.30 4.96 0.34 0.1156
Māori Party 2.12 3.30 -1.18 1.3924
United Future 2.67 2.48 0.19 0.0361
ACT 1.51 1.65 -0.14 0.0196
Progressive 1.16 0.82 0.34 0.1156
Destiny 0.62 0 0.62 0.3844
Legalise Cannabis 0.25 0 0.25 0.0625
Christian Heritage 0.12 0 0.12 0.0144
Alliance 0.07 0 0.07 0.0049
Family Rights 0.05 0 0.05 0.0025
Democrats 0.05 0 0.05 0.0025
Libertarianz 0.04 0 0.04 0.0016
Direct Democracy 0.03 0 0.04 0.0016
99 MP 0.03 0 0.03 0.0009
One NZ 0.02 0 0.02 0.0004
Republicans 0.02 0 0.02 0.0004
Total of squares of differences 2.5336
Total / 2 1.2668
Square root of (total / 2) 1.13

Thus the disproportionality of the 2005 New Zealand election is 1.13, which is very low by international standards.[13]

Note that the Māori Party has the highest difference, which is significantly above the others. This is due to New Zealand's system of reserved seats for Māori. The Māori seats are allocated by votes on a separate electoral roll, and while any party can contest these seats, they have historically been won by either the Māori Party, the Labour Party, or New Zealand First.

Australia[edit]

This table uses for example the 2012 Queensland state election, one of the largest landslides in Australian electoral history. Though Australia and New Zealand have somewhat similar political histories, Australia uses preferential voting in Single-member districts for Commonwealth House of Representative and most state and territory Legislative Assembly elections, which tends to result in far less proportionality compared to New Zealand's MMP system (or other proportional electoral systems), especially for larger minor parties, such as The Greens or, historically, the Australian Democrats. The 2012 Queensland election had an extremely high Gallagher Index, at 31.16, due to the massive landslide in seats for the victorious LNP. The LNP gained 88% of the seats with less than 50% of the vote. Most recent Australian state and federal elections however score between 10 and 12.

Gallagher Index for the 2012 Queensland state election
Party Votes (%) Seats (%) Difference Difference
squared
Liberal National 49.65% 87.64% 37.99 1443.2401
Labor 26.66% 7.87% -18.79 353.0641
Katter 11.53% 2.25% -9.28 86.1184
Greens 7.53% 0.00% -7.53 56.7009
Other 1.47% 0.00% -1.47 2.1609
Independent 3.16% 2.25% -0.91 0.8281
Total of differences squared 1942.1125
Total / 2 971.0563
Square root of (Total / 2): Gallagher Index result 31.16


For comparison, the 2010 Australian Federal Election was more typical of Australian state and federal politics. The overall score was 11.34, with maily between Labor and the Greens.

Gallagher Index for the 2010 Australian federal election
Party Votes (%) Seats (%) Difference Difference
squared
Labor 37.99% 48.00% 10.01 100.2001
Liberal 30.46% 29.33% -1.13 1.2769
Greens 11.76% 0.67% -11.09 122.9881
Liberal National (QLD) 9.12% 14.00% 4.88 23.8144
National 3.43% 4.00% 0.57 0.3249
Family First 2.25% 0.00% -2.25 5.0625
National (WA) 0.34% 0.67% 0.33 0.1089
Country Liberal (NT) 0.31% 0.67% 0.36 0.1296
Other 1.82% 0.00% -1.82 3.3124
Independent 2.52% 2.67% 0.15 0.0225
Total of differences squared 257.2403
Total / 2 128.6202
Square root of (Total / 2): Gallagher Index result 11.34


Sweden[edit]

The disproportionality of the 2018 Swedish general election was 1.8 according to the Gallagher index, which is extremely low by international standards (resulting in almost perfectly proportional seat allocations), due to Sweden's use of the modified Sainte-Laguë method in elections to the Riksdag.

Republic of Ireland[edit]

The disproportionality of the 2020 Irish general election was 1.96 according to the Gallagher index. The Republic of Ireland uses the single transferable vote (STV) system with Droop quota in elections to the Dáil Éireann.

United States[edit]

This table uses the aggregate results of the 2016 elections to the United States House of Representatives. These 435 single-seat elections are winner-take-all, which would tend to create disproportionate results, but this is moderated by the extremely high share of votes obtained by the two major parties—more than 97%, likely in part caused by fears of wasted votes and vote splitting. The Gallagher index ignores the effect of the primaries on the proportionality.

Gallagher Index for the 2016 United States House of Representatives elections
Party Votes (%) Seats (%) Difference Difference
squared
Republican Party 49.11% 55.40% 6.29 39.5641
Democratic Party 48.02% 44.60% -3.42 11.6964
Libertarian Party 1.29% 0.00% -1.29 1.6641
Independents and minor parties 1.18% 0.00% -1.18 1.3924
Green Party 0.39% 0.00% -0.39 0.1521
Total of differences squared 54.4691
Total / 2 27.2346
Square root of (Total / 2): Gallagher Index result 5.22


Countries[edit]

A list of Gallagher indices from 2015[14] is shown below, only the latest index is shown for each country. Countries considered Dictatorships measured by Democracy-Dictatorship Index are excluded.

Country Year Gallagher Index
 Albania 2013 5.58
 Andorra 2011 17.2
 Antigua and Barbuda 2014 24.94
 Argentina 2013 4.82
 Armenia 2012 6.73
 Australia 2013 9.54
 Austria 2013 3.31
 Bahamas 2012 24.3
 Barbados 2013 1.88
 Belgium 2014 4.6
 Belize 2012 3.74
 Benin 1999 15.56
 Bermuda 2012 1.28
 Bhutan 2013 29.57
 Bolivia 2009 3.76
 Brazil 2010 2.5
 Bulgaria 2013 10.88
 Canada 2011 12.42
 Cape Verde 2011 1.95
 Chile 2013 8.04
 Colombia 2002 3.98
 Costa Rica 2014 6.34
 Croatia 2011 12.31
 Cyprus 2011 1.69
 Czech Republic 2013 6.12
 Denmark 2011 0.73
 Dominican Republic 2002 4.99
 Ecuador 1998 4.6
 El Salvador 2012 3.36
 Estonia 2011 5.09
 Finland 2011 2.95
 France 2012 17.66
 Georgia 2012 2.98
 Germany 2013 7.83
 Ghana 2012 5.76
 Greece 2012 9.96
 Grenada 2013 40.95
 Guatemala 2011 9.33
 Guinea-Bissau 2014 9.71
 Honduras 2013 5.6
 Hungary 2014 17.8
 Iceland 2013 6.23
 India 2014 17.53
 Indonesia 2014 2.79
 Ireland 2011 8.69
 Israel 2013 3.09
 Italy 2013 17.34
 Jamaica 2011 13.33
 Japan 2012 19.96
 Kenya 2013 8.26
 Latvia 2011 2.76
 Liberia 2011 11.14
 Liechtenstein 2013 1.17
 Lithuania 2012 9.08
 Luxembourg 2013 5.2
 Malawi 2014 7.43
 Malta 2013 1.75
 Mauritius 2010 12.96
 Mexico 2012 6.87
 Moldova 2010 3.65
 Monaco 2013 30.27
 Mongolia 2000 33.25
 Montenegro 2012 2.28
 Nepal 2013 6.09
 Netherlands 2012 0.99
 New Zealand 2011 2.38
 Nicaragua 2011 6.41
 Niger 2011 2.83
 North Macedonia 2014 4.45
 Norway 2013 2.56
 Panama 2014 7.87
 Papua New Guinea 1997 23.99
 Paraguay 2013 11.51
 Peru 2001 8.87
 Philippines 1998 25.16
 Poland 2011 5.95
 Portugal 2011 5.68
 Romania 2012 6.2
 Saint Kitts and Nevis 2010 12.04
 Saint Lucia 2011 12.83
 Saint Vincent and the Grenadines 2010 2.12
 San Marino 2008 3.81
 Sao Tome and Principe 2010 6.29
 Senegal 2012 19.8
 Serbia 2012 6.53
 Sierra Leone 2012 6.09
 Slovakia 2012 9.77
 Slovenia 2014 6.57
 South Africa 2014 0.37
 South Korea 2012 7.15
 Spain 2011 6.93
 Sri Lanka 2010 4.33
 Suriname 2010 8.94
 Sweden 2010 1.25
 Switzerland 2011 3.76
 Taiwan 2012 9.07
 Thailand 2011 4.92
 Timor-Leste 2007 4.48
 Trinidad and Tobago 2010 10.55
 Turkey 2011 7.4
 Ukraine 2007 3.59
 United Kingdom 2010 15.1
 United States 2012 4.79
 Uruguay 2009 1.1
 Zambia 2011 3.39

Alternative indices[edit]

Other indices measuring the proportionality between seat share and party vote share are the Loosemore–Hanby index, Rae index and the Sainte-Laguë Index.

Notes[edit]

  1. ^ Special Committee on Electoral Reform (a Canadian Parliamentary Committee) (December 1, 2016). Report 3: Strengthening Democracy in Canada : Principles, Process and Public Engagement for Electoral Reform (Report). Parliament of Canada. p. 69 (or p. 83 in PDF search). Retrieved December 26, 2016. One tool that has been developed to measure an electoral system's relative disproportionality between votes received and seats allotted in a legislature is the Gallagher Index, which was developed by Michael Gallagher (who appeared before the Committee).
  2. ^ O'Malley, Kady (December 1, 2016). "Read the full electoral reform committee report, plus Liberal and NDP/Green opinions". Ottawa Citizen. Ottawa Citizen. Retrieved December 26, 2016.
  3. ^ This is discussed in simple English at "Gallagher Index Made Easy". 2016-12-31.
  4. ^ Gallagher 1991, pp. 33–51.
  5. ^ Gallagher 1991, p. 40.
  6. ^ Gallagher 1991, p. 41.
  7. ^ Cash, Colby (December 2, 2016). "Colby Cosh: Did Maryam Monsef actually read the whole electoral reform report?". National Post. Retrieved 10 December 2016.
  8. ^ Wherry, Aaron (December 1, 2016). "Minister 'disappointed' as electoral reform committee recommends referendum on proportional representation". CBC News. Retrieved 10 December 2016.
  9. ^ O'Malley, Kady (December 1, 2016). "Read the full electoral reform committee report, plus Liberal and NDP/Green opinions". Ottawa Citizen. Ottawa Citizen. Retrieved 10 December 2016.
  10. ^ "Is Canada Fair?". Measuring Unfairness — Calculating Canada's Gallagher Index. (This website includes the Gallagher Index in adjustable table format. It initially shows the data for Canada's 2015 federal election, but some variables in some table cells are adjustable by the visitor to the website, and then the rest of the table is automatically adjusted to reflect this visitor's new input.). Retrieved 10 December 2016.
  11. ^ The rules for federal elections in Canada require that certain provinces always get a certain quantity of seats – on a province by province basis. If so, then Byron Weber Becker proposed that the Gallagher index for Canada ought to ALSO reflect that. In other words, the Gallagher data should be collected on a province by province basis; and the Gallagher score should be calculated on a province by province basis. Only after that is done, can we then add up all of those provincial scores and then average them out to get the true national "composite Gallagher index" score. If we do that, then the illustrated table calculation of 12 for Canada is incorrect. It should instead show a "composite Gallagher index" of 17.1. Byron Weber Becker developed this "composite" index. See citation here: Special Committee on Electoral Reform (a Canadian Parliamentary Committee) (December 1, 2016). Report 3: Strengthening Democracy in Canada : Principles, Process and Public Engagement for Electoral Reform (Report). Parliament of Canada. p. 69 (or p. 83 in PDF search). Retrieved December 26, 2016. ...Professor Becker developed the “Gallagher Index Composite” for the Committee’s study...
  12. ^ "Official Count Results – Electorate Status". Chief Electoral Office- New Zealand. 1 October 2005. Retrieved 9 August 2013.
  13. ^ Gallagher, Michael (2013). "Election indices" (PDF). Trinity College, Dublin. Retrieved 9 August 2013.
  14. ^ "Disproportionality data, A .csv file of Gallagher Disproportionality Data, v2, Data Updated: 20 March 2015".

References[edit]

External links[edit]