Gallagher index

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The Gallagher index measures an electoral system's relative disproportionality between votes received and seats 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] The Gallagher index is a statistical analysis methodology utilised within political science, notably the branch of psephology.

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 (${\displaystyle V_{i}}$) and percent of seats (${\displaystyle S_{i}}$) for each of the political parties (${\displaystyle i=1,\ldots ,n}$).[4]

${\displaystyle \mathrm {LSq} ={\sqrt {{\frac {1}{2}}\sum _{i=1}^{n}(V_{i}-S_{i})^{2}}}}$  [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. Compared to the Loosemore–Hanby index, the Gallagher index is more sensitive to large discrepancies.[6] Other indices measuring the proportionality between seat share and party vote share are the Loosemore–Hanby index, Rae index, and the Sainte-Laguë Index.

History

The first publication of the use of least squares in measuring the dis-proportionality of election outcomes was by Michael Gallagher in 1991: In Michael Gallagher's Electoral Systems web site he offers a PDF download under "Values of Indices" [7] in which he writes: "These [election] indices were originally outlined in Markku Laakso and Rein Taagepera, ‘ “Effective” number of parties: a measure with application to west Europe’, Comparative Political Studies 12:1 (1979), pp. 3–27 (effective number of parties), and Michael Gallagher, ‘Proportionality, disproportionality and electoral systems’, Electoral Studies 10:1 (1991), pp. 33–51 (least squares index)."

The Gallagher index gained considerable attention in Canada in December 2016 in the context of efforts to reform Canada's electoral system.[8][9] 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."[10][11]

Examples of calculating disproportionality

In the 2015 Canadian federal election, the Gallagher index was 12.02, where 0 would be a perfectly proportional election outcome.[12]

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

Australia

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

EU

The 7 political groups of the European Parliament instead of the 203 political parties[13] allow a concise calculation of disproportionality between votes and seats. The Gallagher index for the European Parliament is 7.87.

Gallagher Index for the 2019 European Parliament election[14]
Party Votes (%) Seats (%) Difference Difference
squared
EPP 20.80% 24.23% 3.43 11.7649
S&D 17.88% 20.51% 2.63 6.9169
RE 12.01% 14.38% 2.37 5.6169
G/EFA 10.04% 9.85% -0.19 0.0361
ID 10.59% 9.72% -0.87 0.7569
ECR 7.17% 8.26% 1.09 1.1881
GUE/NGL 5.16% 5.46% 0.3 0.09
NI 6.52% 7.59% 1.07 1.1449
Wasted vote 9.82% 0.00% -9.82 96.4324
Total of differences squared 123.9471
Total / 2 61.9736
Square root of (Total / 2): Gallagher Index result 7.87

Sweden

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

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

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

In Michael Gallagher's Electoral Systems web site he offers a PDF download under "Values of Indices." [15] Those Gallagher indices for individual countries are listed below. Only the last available index for each country is shown.

Notes

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. ^ "Election Indices" (PDF).
8. ^ Cash, Colby (December 2, 2016). "Colby Cosh: Did Maryam Monsef actually read the whole electoral reform report?". National Post. Retrieved 10 December 2016.
9. ^ Wherry, Aaron (December 1, 2016). "Minister 'disappointed' as electoral reform committee recommends referendum on proportional representation". CBC News. Retrieved 10 December 2016.
10. ^ 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.
11. ^ "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.
12. ^ 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...
13. ^ "European Parliament: Facts and Figures" (PDF). Retrieved 13 February 2023.
14. ^ "2019 European election results – Comparative tool". European Parliament. Retrieved 5 July 2019.
15. ^ "Election Indices" (PDF).