Psephology: The Gallagher Index and what it means

I have touched on psephological concepts like the magnification effect in electoral analyses in the past. This year, I will run a series of phenomena and concepts within electoral analysis itself, once every month

1. The Gallagher Index

British elections, at a local and national level, are notoriously unfair in terms of who gets represented on the council and in Parliament, since smaller parties are squeezed out whilst larger parties can control the council or Parliament by themselves despite not receiving close to half of the votes cast. This is due to the disproportionality of first past the post electoral politics, which can produce even more disproportional results elsewhere, most notably in Canada.

Meanwhile, countries using list-PR only elections, which includes most European countries, and mixed member proportional representation, produce much fairer electoral outcomes, although this depends on any threshold applied to list seats along with electoral laws determining distribution of seats to parties and the method used (d'Hondt vs. Sainte-League, for example).

How proportional, and therefore fair,a particular nation's election is can be determined by what is known as the Gallagher Index, named after Michael Gallagher, a politics professor at Trinity College Dublin, Ireland's oldest and most prestigious university.

The Gallagher Index can be calculated like this: first calculate the differences between the percentage of votes won by a political party or group and the percentage of seats it won in the election in question. Then square each of the differences and add them all together to show how fair or unfair the election was to dominant and smaller parties. Halve that total and then obtain the square root of half the total of differences squared This gives the Gallagher Index for that particular election. The lower the Gallagher Index, the more proportional and fair the election was.

As a good example, let us compare the most recent Canadian general election of 2015:

Party:                                    % of votes:  Seats: % of seats:  Difference: Difference squared:
Liberal:                                        39.47      184        54.44       14.97          224.10
Conservative                              31.89         99        29.29        -2.60           6.76
New Democratic Party               19.71         44        13.02        -6.69           44.76
Bloc Quebecois                          4.66          10           2.96        -1.70             2.89
Green                                          3.45           1            0.29        -3.16            9.99
Others                                        0.82            0             0.00        -0.82           0.67
Total                                         100.00         338         100.00     0.00            289.17
Half of total:  144.08
Square root (of half of total): 12.02


And the Dutch general election of 2017:

Party:                                     % of votes:  Seats: % of seats: Difference: Difference squared:
VVD                                         21.29          33        22.00          0.71              0.50
PVV                                         13.06          20        13.33          0.27             0.07
CDA                                         12.38         19         12.66          0.28             0.08
D66                                          12.22         19          12.66          0.44            0.17
GL                                            9.13           14           9.33           0.53            0.27
SP                                            9.09            14          9.33           0.57             0.32
PvdA                                        5.70               9          6.00          0.30             0.09
CU                                           3.39               5          3.33           -0.06            0.01
PvdD                                        3.19              5           3.33           0.14           0.02
50PLUS                                    3.11              5          3.33           0.22           0.05
SGP                                         2.08                3         2.00          -0.08           0.01
DENK                                       2.06                3         2.00          -0.06           0.01
FvD                                           1.78                2        1.33           -0.45          0.20
Others                                        1.52               0         0.00         -1.52          2.31
Total: 4.11
Half of total: 2.05
Square root (of half of total): 1.43

The Dutch electoral system has no official threshold, making the outcome very fair and proportional. By contrast, the Canadian election produced a very unfair result, as British parliamentary elections do, although so can proportional representation if the election threshold is set too high, as it is in Turkey (the 10% threshold heavily benefits the ruling AKP).

The Gallagher Index can help develop fair electoral systems for countries transitioning from FPTP, including those that need to adopt proportional representation. So, how can the Gallagher Index be used to make elections fairer, apart from pointing out how much fairer proportional representation is than a majoritarian system like FPTP?

1. Do not set the election threshold too high. Stability is important but so is ensuring groups with significant support obtain fair representation. The Danish threshold of 2% is relatively fair and does not in practice cause significant issues for governance.

2. Ensure that each district/constitiuency in a PR election, in countries too large and populous for just one national constituency (e.g. the UK) has as many seats as is practicable and respectful of community ties.

Ironically, it is Ireland where Professor Gallagher lectures whose electoral system cannot be accurately measured by the Gallagher Index, because the Gallagher Index cannot accurately account for how transfers can distort proportionality of results in a Single Transferable Vote (STV) system, or an Alternative Vote (AV) system.