Why cross multiplications are not enough
(And what must be done instead)

by Maurice Y. Michaud (he/him)

To have this simulator of past elections is the reason why PoliCan exists today.

After studying the different electoral systems used in democracies elsewhere in the world, the members of commissions tasked with considering electoral reform in their jurisdiction have tended to take the results of their previous general election to simulate the results in their proposed proportional system. And while they reported having tried adjusting some settings, like the regular to proportional seat ratio or the thresholds that parties should reach to get compensatory or list seats, they would only publish simulations based on the settings upon which they settled. These were reasonable decisions, as gathering and calculating alternate outcomes takes a lot of time, not to mention that it would be overwhelming for anyone having to review all the information and recommendations that the commissioners gathered.

But that was before a fool like me took it upon himself to gather all the numbers of every election ever held in Canada and enter them into a single database. The PoliCan MMP simulator allows you to take any of the 434 partisan general elections held in Canada and estimate how many seats each party might have won if the voting been done with a mixed-member pro­por­tional (or MMP) electoral system.

It would be too simplistic to look at the percentage of votes each party has received in a general election and assume that, in a proportional voting system, they would inevitably have obtained the same percentage of seats. That’s because a well-designed system needs safeguards, or thresholds, to prevent a transitory or mar­gin­al party from obtaining a seat simply because a few thousand people in the entire jurisdiction may have voted for it. If most of the votes for such a party are concentrated in a single riding, they could lead to that party’s candidate winning the seat, and so be it. But if the remainder of that party’s votes are distributed thinly over several or all the other ridings, then its performance in the jurisdiction overall should rightly be deemed insignificant.

Simple cross multiplications of each party’s result and seats available fail to address fundamental concerns with regards to an electoral system, the most obvious being geographical representativity. Objectively it is true that, in 2021, the federal Conservatives won a plurality of votes and 33.7% of the popular vote nationwide. However, when looking at the results for the 104 seats in the four western provinces only, their popular vote jumped to 44.5%, but dropped to 28.9% for the remaining seats in the six other provinces. In a 338-seat Parliament where one seat was equal to 0.30% of the seats, a cross-multiplication would only indicate that the Conservatives should have had 114 seats, without any indication of WHERE those seats should have been. By the same token, a cross-multiplication would also have suggested that the now defunct Free Party of Canada, with only 47,252 votes throughout Canada and 0.4% of the popular vote, could have been entitled to one seat (really?!) but again, with no indication of WHERE. Geographical representativity is such an important concern that it overrides pure proportionality in an MMP system.

In 1792 in the newly formed United States, it was being proposed that the number of seats for northern states in the House of Representatives should be increased. Thomas Jefferson — yes, later the third president — invented a formula, used until 1842, that instead proposed a proportional ap­por­tion­ment of seats based strictly on the population of each state. In 1878, a Belgian mathematician, Victor D’Hondt , independently “reinvented” the formula for the assignment of seats in a legislature, substituting the population of a state with the number of votes obtained by a political party. It is perhaps because his version was designed specifically for assigning seats after an election that the same formula is more widly attributed to him today than to Jefferson.
 

The D’Hondt method

A seat is attributed in the D’Hondt method to the party obtaining the highest quotient (Q) in this equation:

where V is the number of votes and s is the number of seats. But since every party begins with no seat, this means every party’s initial divider (or denominator) can be simplified as 1:

So if in a two-party race, Party A had obtained 4 million votes and Party B had obtained 3 million votes, the first round would go like this:

Party A would obviously get that first seat. Before going into the next round, Party A’s divider would be increased by 1, but Party B’s wouldn’t change. Therefore, the next round would look like this:

Thus Party B would get the second seat, and this process would be repeated until all the seats are allocated.
 

The Sainte-Laguë method

The most common alternative to the D’Hondt method is the Sainte-Laguë method , named after André Sainte-Laguë and used in Germany and New Zealand, among other countries. It is said to give a more equal seats-to-votes ratio for different sized parties compared to D’Hondt, and not to favour larger parties as much as D’Hondt. The quotient begins as:

where V is the number of votes and s is the number of seats. Since every party begins with no seat, this means every party’s initial divider (or denominator) can be simplified as 1:

If in a two-party race, Party A had obtained 4 million votes and Party B had obtained 3 million votes, the first round would go like this:

Party A would get that first seat. Before going into the next round, like in D’Hondt, Party A’s divider would be increased but by 2 — or (2×1 seat)+1 — and Party B’s wouldn’t change. Therefore, the next round would look like this:

Thus Party B would get the second seat, and this process would be repeated until all the seats are allocated.

In Norway, the initial round begins with V ÷ 0.7 and with V ÷ 0.6 in Sweden. So the first round in Norway would be like this:

But then, when a party wins its first seat, it passes to 3 and subsequent increments are +2 (so 0.7, 3, 5, 7, ...). As a result, the next three rounds would go like this:

In short, the divider is always an odd number, except for the first round in Sweden.

You will notice that, if you choose a country’s settings instead of customizing the ratio and the threshold, and that country happens to be using the Sainte-Laguë method, the PoliCan MMP simulator will adapt its calculations accordingly.
 

The formula in majoritarian systems

Quite frankly, there is no formula in majoritarian systems. There are only summations, and the focus is on the number of seats rather than the number of votes. The votes are summed up separately within each riding, and all a candidate needs to win the seat in a riding is one vote more than the candidate finishing second. The sum of the votes in all the ridings becomes a “for information only” data point, as the only sum that matters is that of the seats won, by party.

This is obviously an oversimplified example, but it illustrates perfectly how majoritarian electoral systems are absolutely determined to yield majority governments. Consider that in Ontario in 1879, 2 votes more for the Liberals gave these results.

Let’s look at how the D’Hondt method would have affected a more recent FPTP election had it been in MMP mode.

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