Once again, Canadians have voted as if they had a proportional representation (PR) electoral system, but obtained almost exactly the party system they should be expected to get, given the first-past-the-post (FPTP) system that they actually use.

If voters are voting as if they had PR already, why not just give them PR? Of course, it does not work that way, as the decision to adopt a new electoral system is rarely separable from party politics. Nonetheless, it is worth asking what electoral system the country should have, based on how voters are actually voting. They certainly are not playing the game as if it were FPTP. Even though it is.

To get at an answer to this question, we can start with the average value of the effective number of vote-earning parties over recent elections. (For those just tuning in or needing a refresher, the *effective number of parties* is a size-weighted count, where each party’s “weight” in the calculation is its own size–we square the vote (or seat) share of each party, sum up the squares, and take the reciprocal. If there were four equal size parties, the effective number would be 4.00. If there are four parties of varying sizes, the effective number will be smaller than four. For instance, if the four have percentages of 40%, 35%, 20%, and 5%, the effective number would be 3.08.) From the effective number, we can work backwards through the Seat Product Model (SPM) to determine what electoral system best fits the distribution of parties’ votes that Canadians have actually been providing. The SPM lets us estimate party system outputs based on a country’s mean district magnitude (number of seats elected per district (riding)) and assembly size. As noted above, Canada currently tends to have a distribution of seats among parties in the House of Commons consistent with what the SPM expects from a district magnitude of 1 and a House size of 338. The puzzle is that it does not have a distribution of votes consistent with the SPM. Instead, its distribution of votes across parties looks more like we would expect from a PR system. But what sort of PR system? That is the question the following calculations aim to answer.

Over the past eight elections, going back to 2000, the mean effective number of vote-earning parties (dubbed *N*_{V} in systematic notation) has been 3.70. During this time, it has ranged from a low of 3.33 (2015 when Justin Trudeau won his first, and so far only, majority government) to a high of 3.87 (the second Conservative minority government of the period under leadership of Stephen Harper). In 2019 it was 3.79 and in 2021 it was very slightly higher (3.84, based on nearly complete results). Even the lowest value of this period is not very “two party” despite the use of FPTP, an electoral system allegedly favorable to two-party systems. (I say allegedly, because given FPTP with a House of 338 seats, we actually should expect *N*_{V}=3.04, according to the SPM. In other words, a “two-party system” is not really what the current electoral system should deliver. Nonetheless, it would not be expected to be associated with as fragmented a voting outcome as Canadians typically deliver.)

**How to get from actual voting output to the PR system Canadians act as if they already had**

The SPM derives its expectation for *N*_{V} via a phantom quantity called the number of “pertinent” vote-earning parties. This is posited in Shugart and Taagepera (2017), *Votes from Seats*, to be the number of parties winning at least one seat, plus one. It is theoretically expected, and empirically verifiable, that the effective number of seat-winning parties (*N*_{S}) tends to equal the actual number of seat winning parties (*N*_{S}_{0}, with the 0 in the subscript indicating it is the unweighted, raw, count), raised to the exponent, 2/3. That is, *N*_{S}=*N*_{S}_{0}^{2/3}. The same relationship logically would hold for votes, meaning *N*_{V}=*N*_{V}_{0}^{2/3}, where *N*_{V}_{0} is the aforementioned number of pertinent vote-earning parties. We can’t measure this directly, but we take it to be *N*_{V}_{0}=*N*_{S}_{0}+1, “strivers equal winners, plus one.” In *Votes from Seats* we show that this assumption works for estimating the impact of electoral systems on *N*_{V}.

Thus we start with the recently observed mean *N*_{V}=3.7. From that we can estimate what the number of pertinent parties would be: given *N*_{V}=*N*_{V}_{0}^{2/3}, we must also have *N*_{V}_{0}=*N*_{V}^{3/2}. So *N*_{V}_{0}=3.7^{3/2} = 7.12. This number by itself is not so interesting, but it makes all the remaining steps of answering our question possible.

Our expected number of *seat*-winning parties from a situation in which we know *N*_{V}=3.7 works out to be 6.12 (which we might as well just round and call 6). We get that as follows. First, *N*_{S}_{0}=*N*_{V}_{0}-1: the number of pertinent vote-earning parties, minus one. We already estimated the pertinent vote-earning parties to be 7, so we have an estimated average of **6 parties winning at least one seat**. This is realistic for current Canadian politics, as recently five parties have been winning seats (Liberal, Conservative, NDP, BQ, and since 2011, Greens). With PR, the PPC likely would win a few seats on current strength, and the Greens probably would continue to do so, assuming they either recover from their current doldrums (especially once PR were adopted) or that any legal threshold would not be applied nationally and thus even their 2.3% showing in the 2021 election would not lock them out of parliament. (In 2021, Greens still got 9.6% in PEI, 5.3% in BC and 5.2% in New Brunswick, for example (per Elections Canada).)

If we have an expected number of seat-winning parties, based actual mean *N*_{V}, that is equal to six, what would be the seat product (*MS*) that would be expected? Once again, the seat product is the mean district magnitude (*M*), times the assembly size (*S*). Given *M*=1 (single-seat districts) and *S*=338, Canada’s current seat product is 338. Based on one of the formulas comprising the SPM, a seat product of 338 should be expected to result in an effective number of seat-winning parties (*N*_{S}) of 2.64 and effective number of vote-earning parties (*N*_{V}) of 3.04. It is working out pretty close to that for seats (average *N*_{S}=2.8). Yet voters are voting more like they had a PR system given the average over recent elections of *N*_{V}=3.7.

One of the formulas of the SPM, which like all of those referenced here, is empirically accurate on a worldwide sample of election results, predicts that *N*_{S}_{0}=(MS)^{1/4}. Thus if we have an expected value of seat-winning parties of around 6, as expected from *N*_{V}=3.7, we can simply raise it to the power, 4, to get what the seat product is expected to be: *MS*=6^{4}=1296. In other words, based on how Canadian voters are actually voting, it is *as if their country had an electoral system whose seat product is around 1300*, rather than the actual 338. For a comparative referent, this hypothetical PR system would be quite close to the model of PR used in Norway.^{1}

Any electoral system’s mean district magnitude is *M*=(*MS*)/*S*,so taking a House of 338 seats,^{2} our hypothetical PR system has *M*=1300/338=3.85. That is, *based on how Canadian voters are actually voting, it is as if their country had an electoral system whose mean district magnitude is around 3.85*. Comparatively, this is quite close to the Irish PR system’s mean magnitude (but it should be noted that Ireland has a seat product of closer to 600, due to a much smaller assembly).

So there we have it. The mean district magnitude that would be most consistent with Canada’s current vote fragmentation would be just under 4, given the existing size of the House of Commons.

If Canada adopted a PR system with a seat product of 1300, its expected effective number of seat-winning parties (*N*_{S}) would rise to 3.30, and its expected largest party would have, on average, 40.8% of the seats, or 138. (This is based on two other predictive formulas within the SPM: *N*_{S}=(*MS*)^{1/6} and *s*_{1}=(*MS*)^{–1/8}, where *s*_{1} is the seat share of the largest party.)

A largest party with 138 seats (as an average expectation) would then require another party or parties with at least 32 seats to have a majority coalition, or a parliamentary majority supporting a minority government. The NDP would reach this easily under our hypothetical PR system, given it can win around 25 seats on under 18% of the votes under FPTP (and 44 seats on just under 20% as recently as 2015).

The Bloc Quebecois also would be available as a partner, presumably for a minority government, with which to develop budgets and other policy, thereby preventing the NDP from being able to hold the Liberal Party “hostage” to its demands. The BQ won 32 seats in 2019 and 33 in 2021. However, because it is a regionally concentrated party, we should entertain the possibility that it might do *worse* under PR than under FPTP, which rewards parties with concentrated votes. The only way to estimate how it would do might be to run the SPM within the province.

**Estimating Quebec outcomes under PR**

Quebec has 78 seats total, such that 33 seats is equivalent to 42% of the province’s seats. On Quebec’s current seat product (78) its largest party should win 45 seats (58%). So it is actually doing worse than expected under FPTP!

If the province had a mean district magnitude of 3.85, its seat product would be 300, for which the expected largest party size would be 49%, or 38 seats. In other words, when the BQ is the largest party in Quebec, it could do a little better on the very moderate form of PR being suggested here than it currently is doing under FPTP. (Suppose the model of PR had a mean magnitude of 9 instead, then we’d expect the largest provincial seat winner to have 44.1%, or 34 seats, or roughly what it has won in the last two elections. Only if the mean *M* is 16 or higher do we expect the largest party in Quebec—often the BQ—to have fewer than 32 of 78 seats. Obviously, in 2011 when the BQ fell all the way to 23.4% within the province, PR would have saved many of their seats when FPTP resulted in their having only 4 of 75 in that election. In 2015 they did even worse in votes—19.3%, third place—but much better in seats, with 10 of 78. Under the PR model being considered here, it is unlikely they would not have won at least 10 seats, which is 12.8%, on that provincial share of the vote.)

**Do Canadians actually ‘want’ a still more proportional system than this?**

It is possible we should use a higher *N*_{V} as reflective of what Canadians *would vote for* if they really had a PR system. I have been using the actual mean *N*_{V} of recent elections *under FPTP*, which has been around 3.7. But in the final CBC polling aggregate prior to the 2021 election, the implied *N*_{V} was 4.12. It dropped by almost “half a party” from the final aggregate^{3} to the actual result either because some voters defected late from the NDP, Greens, and PPC, or because the polls simply overestimated the smaller parties. If we use 4.12 as our starting point, and run the above calculations, we’d end up with an estimated average of 7.4 parties winning at least one seat. Maybe this implies that the Maverick Party (western emulators of the BQ’s success as a regional party) might win a seat, and occasionally yet some other party. In any case, this would imply a seat product of 2939, for a mean *M* of 8.7. The largest party would be expected to have only 36.8% of the seats with such an electoral system, or about 125.

**How to use this information when thinking about electoral reform**

I would advise, as the way to think about this, that we start with *what we’d like the parliamentary party system to look like*. I am guessing most Canadians would think a largest party with only around 125 seats would be an overly drastic change, despite the fact that they are currently telling pollsters, in effect, that this is the party system they are voting for as of the weekend before the election!

The expected parliamentary party system from an average *M* around 4, yielding a largest party averaging just over 40% of the seats (around 138) is thus probably more palatable. Nonetheless, armed with the information in this post, drawn from the Seat Product Model, we could start with a desirable average share of the largest party, and work back to what seat product it implies: *MS*=*s*_{1}^{–8}, and then (assuming 338 seats in the House), derive the implied district magnitude from *M*=(*MS*)/*S*. Or one can start with how Canadians are actually voting, as I did above–or from how we think they would (or should) vote, using *MS*=[(*N*_{V}^{3/2})–1]^{4}, and followed by *M*=(*MS*)/*S*.

Whichever value of the seat product, *MS*, one arrives at based on the assumptions about the end state one is hoping to achieve, remember that we’d then expect the seat share of the largest party to be *s*_{1}=(*MS*)^{–1/8}. As we have seen here, that would tend to be around 40% if mean magnitude were just under 4. This implies a typical largest party of around 138 seats.^{4}

But herein lies the rub. If you tell the Liberal Party we have this nifty new electoral system that will cut your seats by about 20 off your recent results, they probably will not jump at the offer. The parties that would benefit the most are the Conservatives (twice in a row having won more votes than the Liberals but fewer seats), NDP, and smaller parties, including apparently (based on above calculations) the BQ. But this isn’t a coalition likely to actually come together in favor of enacting PR. Thus FPTP is likely to stick around a while yet. But that’s no reason not to be thinking of what PR system would best suit Canadian voters, given that they have been voting for a while as if they already had a PR system.

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**Notes**

General note: At the time of writing, a few ridings remained uncalled. Thus the seat numbers mentioned above, based on who is leading these close ridings, could change slightly. Any such changes would not alter the overall conclusions.

1. More precisely, it would be almost identical in seat product to the Norwegian system from 1977 to 1985, after which point a small national compensation tier was added to make it more proportional.

2. I will assume electoral reform does not come with a change in the already almost perfect *S* for the population, based on the cube root law of assembly size, *S*=*P*^{1/3}, where *P* is population, which for Canada is currently around 38 million. This suggests an “optimal” number of seats of about 336.

3. This is based on the Poll Tracker final aggregate having vote shares of 0.315, 0.310, 0.191, 0.070, 0.0680, 0.035 for the six main parties (and 0.011 for “other”).

4. I am deliberately not going into specific electoral system designs in this post. I am stopping at the seat product, implicitly assuming a simple (single-tier) districted PR system, meaning one with no regional or national compensation (“top up” seats). *Arriving at a seat product to produce the desired party system should be the first step*. Then one can get into the important finer details. If it is a two-tier system–including the possibility of mixed-member proportional (MMP)–one can generate its parameters by using the result of the calculations as the system’s “*effective* seat product,” and take it from there.