Note: This post was written some time ago (as you can see from the motivation!). Given the “shelter in place” order in San Francisco, I have more time indoors to dedicate to publishing these analyses. Full disclosure: I am close friends with Hamish & Kathryn Marshall — Hamish ran the Conservative campaign in October 2019.

A gaffe in Parliament

After the 2015 election, Prime Minister Justin Trudeau's Liberal Majority government established a Special Committee on Electoral Reform to study possible modifications to the existing "first past the post" (FPTP) electoral system.

This initiative was plagued with problems and the government eventually abandoned it. For a start, the Minister of Democratic Institutions proposed a committee structure allocating 60% of the voting seats to the Liberals, effectively giving the government the ability to steamroll any opposition. After objections from the opposition, the government eventually acceded by having the committee reflect the popular vote.

When the Committee's report was presented to Parliament in December of 2016, the Minister criticized the recommendations due to their reliance on the "Gallagher Index", which she called "an incomprehensible formula". She went so far as to hold up an oversize card with the formula in Parliament.

To be charitable, it is likely that the Minister intended to criticize the report's somewhat muddled emphasis on proportionality. Regardless, her comments came across as petty and tone-deaf: it was easy for critics to point out that a Minister from the party of "evidence-based policy"— who holds a Bachelor of Science no less — was claiming that a relatively simple formula was "incomprehensible". Several weeks after this incident, Minister Monsef was reassigned to another portfolio.

What is the Gallagher Index?

The Gallagher index is a simple metric to show how much the outcome of an election (e.g., the seats in the House of Commons) deviates from the popular vote proportions. That's it. In fact, it's even got a rather simple formula to describe it:

This is the formula for the Gallagher index (similar to the one Monsef held up in the House of Commons)

where Si the the percent of seats party i won and Vi is their national vote. The index is effectively measuring the "distance" between the popular vote and the outcome in the House of Commons.

Consider a simple example: suppose there is an election where the Church of the Militant Elvis party receives 10% of the vote and no seats, the Monster Raving Loony Party receives 40% of the popular vote but only 10% of the seats in Parliament and the Rhinoceros Party receives 60% of the popular vote and 90% of the seats in Parliament. Intuitively, such a result might seem "disproportionate".

It's reasonable to think that the Monster Raving Loonies were short-changed by 20 percentage points of support (30% minus 10%) and the Rhinoceros Party is over-represented by 30 percentage points (90% minus 60%). Further, the Church of the Militant Elvis saw none of their 10 percentage points of support represented in the HoC. Adding up all the absolute deviations between the vote and HoC proportions could give us a measure for how "disproportionate" an election outcome was. In fact, multiplying that sum by 1/2 would give you the "Loosemore-Hanby index".

Here is a quick example that shows how the Loosemore-Hanby and Gallagher indices would be calculated.

(As an aside, I am continually amazed how simple metrics are given such fancy names!)

The problem with looking at only the absolute deviations, as in the Loosemore-Hanby index, is that they treat small deviations the same as large ones. This is an issue since (often) small deviations in proportionality are not very important but larger differences can raise major concerns. The Gallagher index takes this into account by squaring these differences, which increases the relative importance of large deviations.

If you would like to understand more detail behind how proportionality indices work, please have a look at this longer discussion on measuring (dis)proportionality.

How does this relate to the Special Committee on Electoral Reform?

One of the recommendations from the Committee on Electoral Reform was that the new system achieve an adjusted Gallagher index of 5 (to be calculated across sub-regions of the country and then averaged into a national score). This recommendation is vague and poorly explained. For instance, the only justification I can find for this recommendation (page 69) appears to be an appeal to authority:

According to Professor Becker, a Gallagher Index of less than 5 is considered “excellent”

If you know of any more information on how the committee got "5" as the magic number (or if I just missed it), please contact me.

Since I didn't have all the regional electoral breakouts readily available, I've gone ahead and calculated the national Gallagher index at a National level for every Canadian Federal election since Confederation (1867-2015). Here are the results for just the post-WWII elections, the rest are further down the page:

The Gallagher index calculated for every Canadian Federal election since 1945. The chart is colour-coded by which party won the election and whether they achieved a majority or minority.

Quick reactions to the data:

Our current system typically does not come close to 5 on the national Gallagher index: only a single election since 1945 has done so
Liberal and Conservative governments have about the same average national Gallagher index: since 1945, Conservative governments average 11.7 and Liberal governments 11.6
Majorities have higher on the Gallagher index than minorities: Perhaps unsurprisingly, Majority governments are less proportionate. Since 1945, majority Parliaments have an average score of 13.4 while minority Parliaments average 8.4. This should be obvious since most majorities are largely a result of the convexity in our first pas the post system - few parties ever achieve more than 50% of the vote yet majorities are quite common. (I've got another post with a chart that clearly shows this)

Here is the Gallagher index calculated for all elections since 1867. It is apparent that the index increases (on average) after the 1920's with the emergence of more political parties (e.g., the CCF, Reform, the BQ, etc.). Six elections after 1920 h… — Here is the Gallagher index calculated for all elections since 1867. It is apparent that the index increases (on average) after the 1920's with the emergence of more political parties (e.g., the CCF, Reform, the BQ, etc.). Six elections after 1920 have indices over 15, while all elections before that were below.

A Gallagher target is not a recommendation

The Gallagher index is an emergent property of a complex system. It's not like we can turn a dial to reduce the Gallagher index while also keeping other aspects of our system constant. Recommending that the new system achieve a certain Gallagher score tells us little about how we ought to achieve this outcome.

To use a crude analogy, it is like recommending someone lose weight. Well, yes, many people need to lose weight - but how? Will you hit the gym, take up running, eat less, get liposuction? Stating a desired outcome provides little help in designing a future electoral system.

In fact, focusing on a single, highly simplified metric to design a complex institution is missing the point. Minister Monsef was correct to criticize the vague recommendation from the committee that Canada adopt "proportional electoral system that achieves a Gallagher Index score of 5 or less". But Minister Monsef got it backwards: it's not that the Gallagher index is too complex, it's far too simple.

Unfortunately, the report was full of "motherhood and apple pie" recommendations about desirable end states (e.g., increasing voter turnout, electing more women) but remained silent on concrete proposals to reform the electoral system. A Gallagher target is not a reform.