What electoral system should Canada have?

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_S0, with the 0 in the subscript indicating it is the unweighted, raw, count), raised to the exponent, 2/3. That is, N_S=N_S0^2/3. The same relationship logically would hold for votes, meaning N_V=N_V0^2/3, where N_V0 is the aforementioned number of pertinent vote-earning parties. We can’t measure this directly, but we take it to be N_V0=N_S0+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_V0^2/3, we must also have N_V0=N_V^3/2. So N_V0=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_S0=N_V0-1: 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_S0=(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=6⁴=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 s₁=(MS)^–1/8, where s₁ 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 aggregate³ 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=s₁^–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 s₁=(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.

Canada 2021: Another good night for the Seat Product Model, and another case of anomalous FPTP

The 2021 Canadian federal election turned out almost the same as the 2019 election. Maybe voters just really do not want to entrust Justin Trudeau with another majority government, as he led from 2015 to 2019. The early election, called in an effort to turn the Liberal plurality into a Liberal majority, really changed almost nothing in the balance among parties.

The result in terms of the elected House of Commons is strikingly close to what we expect from the Seat Product Model (SPM). Just as it was in 2019. The predictive formulas of the SPM suggest that when your electoral system is FPTP and there are 338 total seats, the largest one should win 48.3% of the seats, or about 163. They further suggest that the effective number of seat-winning parties (N_S) of around 2.64. In the actual result–with five districts still to be called–the largest party, the Liberals, has won or is leading in 159, or 47.0%., and N_S=2.78. These results are hardly different from expected. They also are hardly different from 2019, when the Liberals won 157 seats; in that election we had N_S=2.79.

While the parliamentary balance will be almost what the SPM expects, the voters continue to vote as if there were a proportional system in place. The largest party again has only around a third of the votes, and the effective number of vote-earning parties (N_V) is around 3.8. For a FPTP system in a House the size of Canada’s, we should expect N_V=3.04. Once again, the fragmentation of the vote continues to be considerably greater than expected.

Another bit of continuity from 2019 is the anomalous nature of FPTP in the current Canadian party votes distribution. For the second election in a row, the Conservative Party has won more votes than the Liberals, but will be second in seats. The votes margin between the two parties was about the same in the two elections, even though both parties declined a little bit in votes in 2021 compared to 2019. Moreover, as also has happened in 2019 (and several times before that), the third largest party in votes will have considerably fewer seats than the party with the fourth highest vote share nationwide. The NDP won 17.7% of the vote and 25 seats (7.4%), while the Bloc Quebecois, which runs only in Quebec, won 7.8% of vote and 33 seats (9.8%).

The Green Party and the People’s Party (PPC) more or less traded places in votes: Greens fell from 6.5% in 2019 to 2.3%, while the PPC increase from 1.6% to 5.0%. But the Greens’ seats fell only from 3 to 2, while the PPC remained at zero.

So, as in 2019, the 2021 election produced a good night for the Seat Product Model in terms of the all-important party balance in the elected House of Commons. However, once again, Canadians are not voting as if they still had FPTP. They are continuing to vote for smaller parties at a rate higher than expected–and not only in districts such parties might have a chance to win–and this is pushing down the vote share of the major parties and pushing up the overall fragmentation of the vote, relative to expectations for the very FPTP system the country actually uses.

It is worth adding that the virtual stasis at the national level masks some considerable swings at provincial level. Éric Grenier, at The Writ, has a table of swings in each province, and a discussion of what it might mean for the parties’ electoral coalitions. A particularly interesting point is that the Conservatives’ gains in Atlantic Canada and Quebec, balanced by vote loss in Alberta and other parts of the west, mirrors the old Progressive Conservative vs. Reform split. Current leader Erin O’Toole’s efforts to reposition the party towards the center may explain these regional swings.

In a follow up, I will explore what this tendency towards vote fragmentation implies for the sort of electoral system that would suit how Canadians actually are voting.

Below are the CBC screen shots of election results for 2021 and 2019. As of Thursday afternoon, there remain a few ridings uncalled. I may update the view for 2021 once they have all been called.

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Canada 2021

So election day is here already in the Canadian federal general election of 2021. The election was called in mid August, but otherwise would not have been due till 2023.

The final CBC Poll Tracker has the nationwide votes really close, at 31.5% to 31.0%, the Liberals being barely in front. The NDP is on 19.1%. For comparison, in 2019, these parties’ vote percentages were 33.1, 34.4, and 15.9, respectively. Note that the Conservatives led in the votes, but the Liberals led in seats (157 to the Conservative’s 121 and NDP’s 24). The Poll Tracker for the other parties has the following vote percentages (with last election’s results in parentheses) has the PPC on 7.0 (1.6), Bloc Quebecois on 6.8 (7.7), and Greens 3.5 (6.5).

The Poll Tracker’s seat projections currently have Liberals at 155, Conservatives 119, NDP 32, BQ 31, Green 1, PPC 0. The “likely” range for the Liberals extends to 168, which would be two seats short of the majority that PM Justin Trudeau was seeking by calling this election. If they have a really good result and there is some poling error or last-minute changes of minds (for those who have not already voted early), they might yet make it. On the other hand, the likely range for the party extends as low as 121 in the projection, while that for the Conservatives extends from 105 to 143. It would not be a surprise to see the NDP’s actual vote and seat numbers drop from the projection–their final “likely” range is 24 to 48 (indicating they also have some significant potential upside). They have been declining a little bit in projections in recent days, and they came short of the final projection in 2019.

So, unless there is a surprise, the results will not be fundamentally different from the last time. That would be good news for the Seat Product Model (SPM), as the projected outcome is an effective number of seat-winning parties (N_S) around 2.84. For an assembly the size of Canada’s, with M=1, the expected result is N_S=2.64. In 2019, the actual result was 2.79, a small excess over the model expectation. Additionally, the SPM expects the largest party to have 48.3% of the seats (163), and the projected outcome of this election is 45.9%, also a small deviation from the expectation, albeit a potentially consequential one politically. On the effective number of vote-earning parties, the current poll tracker projection works out to about 4.1! That is far above expectation. The SPM would expect 3.22; as was already the case in 2019 and indeed earlier, but even more so now, Canadian voters are not playing along with the FPTP game anymore, even if the translation of their votes into seats is still giving them the parliamentary party system expected for FPTP, given their assembly size.

News flash: Canada still needs a new electoral system! Only with some kind of PR will they get the parliamentary party system closer to the one they vote for, instead of the the SPM says they “should” have.

As results come in, or as you have any questions or thoughts about this election, here is the “open planting hole.”

Please be advised that I will not be monitoring it after about my local sundown, as the holiday of Sukkot starts tonight. But the virtual orchard is always open.

Why 1.59√Ns?

In the previous planting, I showed that there is a systematic relationship under FPTP parliamentary systems of the mean district-level effective number of vote-earning parties (N‘_V) to the nationwide effective number of seat-winning parties (N_S). Specifically,

N‘_V =1.59√N_S .

But why? I noticed this about a year after the publication of Votes from Seats (2017) while working on a paper for a conference in October, 2018, honoring the career of Richard Johnston, to which I was most honored to have been invited. The paper will be a chapter in the conference volume (currently in revision), coauthored with Cory Struthers.

In VfrS Rein Taagepera and I derived N‘_V =1.59S^1/12. And as explained in yesterday’s planting, it is simply a matter of algebraic transformation to get from expressing of N‘_V in terms of assembly size (S) to its expression in terms of N_S. But perhaps the discovery of this connection points the way towards a logic underlying how the nationwide party system gets reflected in the average district under FPTP. In the paper draft, we have an explanation that I will quote below. It is on to something, I am sure, but it remains imperfect; perhaps readers of this post can help improve it. But first a little set-up is needed.

To state clearly the question posed in the title above, why would the average district-level effective number of vote-winning parties in a FPTP system tend be equal to the square root of the nationwide effective number of seat-winning parties, multiplied by 1.59?

We can deal with the 1.59 first. It is simply 2^2/3, which should be the effective number of vote-earning party in an “isolated” district; that is, one that is not “embedded” in a national electoral system consisting of other seats elected in other districts (this idea of embedded districts is the key theme of Chapter 10 of VfrS). The underlying equation for N‘_V, applicable to any simple districted electoral system, starts with the premise that there is a number of “pertinent” parties that can be expressed as the (observed or expected) actual (i.e., not ‘effective’) number of seat-winning parties, plus one. That is, the number of parties winning at least one seat in the district, augmented by one close loser. For M=1 (as under FPTP), we obviously have one seat winning party, and then one additional close loser, for two “pertinent” parties. Thus with M=1 it is the same as the “M+1 rule” previously noted by Reed and Cox, but Taagepera and I (in Ch. 7 of our 2017 book) replace it with an “N+1″ rule, and find it works to help understand the effective number of vote-earning parties both nationwide and at district level. Raising this number of pertinent vote-earning parties to an exponent (explained in the book) gets one to N_V (nationwide) or N‘_V (district-level). When M=1, the number of pertinent parties is by definition two, and for reasons shown by Taagepera in his 2007 book, the effective number of seat-winning parties tends to be the actual number of seat-winning parties, raised to the exponent, 2/3. The same relationship between actual and effective should work for votes, where we need the “pertinent” number only because “actual number of parties winning at least one vote” is a useless concept. Hence the first component of the equation, 2^2/3=1.5874.

As for the second component of the equation, S^1/12, it is also an algebraic transformation of the formula for the exponent on the quantity defined as the number of seat-winning parties, plus one. At the district level, if M>1, the exponent is itself mathematically complex, but the principle is it takes into account the impact of extra-district politics on any given district, via the assembly size. The total size of the assembly has a bigger impact the smaller the district is, relative to the entire assembly. Of course, if M=1, that maximizes the impact of national politics for any given S –meaning the impact of politics playing out in other districts on the district of interest. And the larger S is, given all districts of M=1, the more such extra-district impact our district of interest experiences. With all districts being M=1, the exponent reduces to the simple 1/12 on assembly size (shown in Shugart and Taagepera, 2017: 170). Then, as explained yesterday we can express N‘_V in terms of N_S via the Seat Product Model. It should be possible to verify N‘_V =1.59√N_S empirically; indeed, we find it works empirically. I showed a plot as the second figure in yesterday’s post, but here is another view that does not add in the Indian national alliances as I did in yesterday’s. This one shows only Canada, Britain, and several smaller FPTP parliamentary systems. The Canadian election mean values are shown as open squares, and several of them are labelled. (As with the previous post’s graphs, the individual districts are also shown as the small light gray dots).

It is striking how well the Canadian elections, especially those with the highest nationwide effective number of seat-winning parties (e.g., 1962, 2006, and 2008) conform to the model, indicated with the diagonal line. But can we derive an explanation for why it works? Following is an extended quotation from the draft paper (complete with footnotes from the original) that attempts to answer that question:

Equation 4 [in the paper, i.e. N‘_V =1.59√N_S ] captures the relationship between the two levels as follows: If an additional party wins representation in the national parliament, thus increasing nationwide N_S to some degree, then this new party has some probabilistic chance of inflating the district-level voting outcome as well. It may not inflate district-level voting fragmentation everywhere (so the exponent on N_S is not 1), but it will not inflate it only in the few districts it wins (which would make the exponent near 0 for the average district in the whole country). A party with no seats obviously contributes nothing to N_S, but as a party wins more seats, it contributes more.^[1] According to Equation 4, as a party emerges as capable of winning more seats, it tends also to obtain more votes in the average district.

As Johnston and Cutler (2009: 94) put it, voters’ “judgements of a party’s viability may hinge on its ability to win seats.” Our logical model quantitively captures precisely this notion of “viability” of parties as players on the national scene through its square root of N_S component. Most of the time, viability requires winning seats. For a new party, this might mean the expectation that it will win seats in the current election. Thus our idea is that the more voters see a given party as viable (likely to win representation somewhere), the more they are likely to vote for it.^[2] This increased tendency to vote for viable national parties is predicated on voters being more tuned in to the national contest than they are concerned over the outcome in their own district, which might even be a “sideshow” (Johnston and Cutler 2009: 94). Thus the approach starts with the national party system, and projects downward, rather than the conventional approach of starting with district-level coordination and projecting upward.

[Paragraph on the origin of 2^2/3 =1.5874 skipped, given I already explained it above as stemming from the number of pertinent parties when M=1.³]

Thus the two terms of the right-hand side of Equation 4 express a district component (two locally pertinent parties) and a nationwide one (how many seat-winning parties are there effectively in the parliament being elected?) Note, again, that only the latter component can vary (with the size of the assembly, per Equation 2, or with a given election’s national politics), while the district component is always the same because there is always just one seat to be fought over. Consider some hypothetical cases as illustration. Suppose there are exactly two evenly balanced parties in parliament (N_S =2.00), these contribute 1.41=√2 to a district’s N’_V, while the district’s essential tendency towards two pertinent parties contributes 1.59=2^2/3. Multiply the two together and get 1.59*1.41=2.25. That extra “0.25” thus implies some voting for either local politicians (perhaps independents) not affiliated with the two national seat-winning parties or for national parties that are expected to win few or no seats.^[4] On the other hand, suppose the nationwide N_S is close to three, such as the 3.03 observed in Canada in 2004. The formula suggests the national seat-winning outcome contributes √3.03=1.74 at the district level; multiply this by our usual 1.59, for a predicted value of N’_V =2.77. […] this is almost precisely what the actual average value of N’_V was in 2004.^[5]

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^[1] The formula for the index, the effective number, squares each party’s seat share. Thus larger parties contribute more to the final calculation.

^[2] Likely the key effect is earlier in the sequence of events in which voters decide the party is viable. For instance, parties themselves decide they want to be “national” and so they recruit candidates, raise funds, have leaders visit, etc., even for districts where they may not win. Breaking out these steps is beyond the scope of this paper, but would be essential for a more detailed understanding of the process captured by our logic. 

^[3] Because the actual number of vote-earning parties (or independent candidates) is a useless quantity, inasmuch as it may include tiny vanity parties that are of no political consequence.

^[4] A party having one or two seats in a large parliament makes little difference to N_S. However, having just one seat may make some voters perceive the party a somehow “viable” in the national policy debate—for instance the Green parties of Canada and the UK.

^[5] The actual average was 2.71.

Small national parties in Canada in the 2021 election and the connection of district voting to national outcomes

One of the notable trends in polling leading up to the Canadian election of 20 September is the increasing vote share of the Peoples Party of Canada (PPC). At the same time, polls have captured a steady decline of the Green Party as the campaign reaches its end. These two small parties’ trends in national support appear to be happening in all regions of the country, albeit to different degrees (see the graphs at the previous link). That is, while these parties have different levels of support regionally, their trends are not principally regional. Rather, all regions seem to be moving together. This will be a key theme of this post–that politics is fundamentally national, notwithstanding real difference in regional strengths¹ and the use of an electoral system in which all seat winning is very local (in each of 338 single-seat districts or “ridings”).

The PPC is a “populist” party of the right. It seems that the Conservatives’ attempt to position themselves closer to the median voter during this campaign has provoked some backlash on the party’s right flank, with increasing numbers of these voters telling pollsters they will vote PPC.

At The Writ, Éric Grenier offers a look into what the polls say about the type of voter turning to the PPC, and whether they might cost the Conservatives seats. The PPC vote share ranges widely across pollsters but in the CBC Poll Tracker (also maintained by Grenier) it currently averages 6.7%. This would be quite a strikingly high figure for a party that is not favored to win even one seat and probably very unlikely to win more than one.² The Poll Tracker shows a stronger surge in the Prairies region than elsewhere (3.6% on 14 Aug. just before the election was called to 10.9% when I checked on 19 Sept.) and Alberta (4.6% to 9.0% now), but it is being picked up in polling in all regions (for example, from 2.2% to 4.4% in Quebec and 2.9% to 6.1% in Atlantic Canada).

What I wish I knew: Is a voter more likely to vote PPC if he or she perceives that the party is likely to win at least one seat? This question is central to the “all politics is national” model developed in Shugart & Taagepera (2017) Votes from Seats, in chapter 10. We do not mean “all” to be taken literally. Of course, regional and local political factors matter. We mean that one can model the average district’s effective number of parties based on the national electoral system. More to the point, we argue that the way to think of how party systems form under FPTP (or any simple districted system) is not to think in terms of local “coordination” that then somehow gets projected up to a national party system, but rather that the national electoral system shapes the national party system, which then sets the baseline competition in the district contests.

If the PPC or Greens are perceived as likely to have a voice in parliament–and perhaps especially if the parliament is unlikely to have a majority party– voters who like what a small party proposes may be more inclined to support it, even though few voters live in a district where it has any chance of winning locally. Below I will show two graphs, each based on a mathematical model, showing a relationship of local votes to national seats. The first is based on the total available seats–the assembly size–while the second will be based on the seat outcome, specifically the nationwide effective number of seat-winning parties. The formula derived in the book for the connection to assembly size states the following for FPTP systems (every district with magnitude, M=1, and plurality rule):

N‘_V=1.59S^1/12,

where N‘_V is the mean district-level effective number of vote-earning parties and S is the assembly size. Please see the book for derivation and justification. It may seem utterly nuts, but yes, the mean district’s votes distribution in FPTP systems can be predicted when we know only how many districts there are (i.e., the total number of seats). In the book (Fig. 10.2 on p. 156) we show that this sparse model accurately tracks the trend in the data across a wide range of FPTP countries, particularly if they are parliamentary. Here is what that figure looks like:

Of course, individual election averages (shown by diamonds) vary around the trend (the line, representing the above equation), and individual districts (the smear of heavily “jittered” gray dots) have a wide variation within any given election. But there is indeed a pattern whereby larger assemblies tend to be associated more fragmented district voting than is the case when assembly size is smaller. At S=338, Canada has a relatively large assembly (which happens to be almost precisely the size it “should be,” per the cube root law of assembly size).

The model for N‘_V under FPTP is premised on the notion that voters are less attuned to the likely outcome in their own district than they are to the national scene. There is thus a systematic relationship between the national electoral system and the average district’s votes distribution.

Moreover, by combining the known relationship between the national electoral system and the national party system, we can see there should be a direct connection of the district vote distribution to the national distribution of seats. The Seat Product Model (SPM) states that:

N_S=(MS)^1/6,

where N_S is the nationwide effective number of seat-winning parties. For FPTP, this reduces to N_S=S^1/6, because M=1. In terms of a FPTP system, this basically just means that because there are more districts overall, there is room for more parties, because local variation in strengths is, all else equal, likelier to allow a small party to have a local plurality in one of 400 seats than in one of 100. So, more seats available in the assembly (and thus more districts), more parties winning seats. The model, shown above, connecting district-level votes (N‘_V) to the assembly size (S) then suggests that the more such seat-winning opportunities the assembly affords for small parties, the more local voters are likely to give their vote for such parties, pushing N‘_V up. The process probably works something like this: Voters are aware that some small parties might win one or more seats somewhere, providing these parties a voice in parliament, and hence are likelier to support small parties to some degree regardless of their local viability. It is national viability that counts. “All politics is national.” The posited connection would be more convincing if it could be made with election-specific seat outcomes rather than with the total number of available seats. We can do that! Given the SPM for the national seat distribution (summarized in N_S) based on assembly size, and the model for district-level votes distribution (N‘_V), also based on assembly size, we can connect N‘_V to N_S algebraically:

N‘_V=1.59N_S^1/2.

(Note that this comes about because if N_S=S^1/6, then S=N_S⁶, giving us N‘_V=1.59(N_S⁶)^1/12, in which we multiply the exponents in the final term of the equation to get the exponent, 1/2, which is also the square root. A full discussion and test of this formula may be found in my forthcoming chapter with Cory Struthers in an volume in honor of Richard Johnston being edited by Amanda Bittner, Scott Matthews, and Stuart Soroka. Johnston’s tour de force, The Canadian Party System likewise emphasizes that voters think more in terms of national politic than their local contest.)

Here is how this graph looks:

This again shows elections with diamonds and individual districts in small gray dots. The diagonal line is the preceding equation. It most definitely fits well. Note that it even fits India if we base the nationwide party system on the alliances (shown by squares), as we should, given that they and not the many parties are the nationwide actors, whereas each alliance is represented by a given component party in each district. (The graph also shows India if we use individual parties in the calculation of N_S, which is useful because it makes clear just how well India, in the era of competing alliances, follows the S model–the one in the first graph. It obviously would not fit the N_S model if we did not use the alliances, but again, it is the alliances that it should track with if the model is correct in its grounding district-level vote outcomes in the national balance of seats among the national political forces–parties elsewhere, including Canada, but alliances in India.)⁴

By implication, this connection of district-level N‘_V to national N_S may arise because voters have some estimate of how the national parliament is going to look when they decide whether or not to support a party other than one of the two leading national parties. For instance, a voter wavering between the NDP and the Liberals might be more likely to support the NDP if she estimates that there will be no majority, thereby allowing a smaller party like the NDP to be more influential than if one of the big parties has a majority on its own.

A vote for a much smaller party, like the PPC, might be simply expressive–“sending a message” to the Conservatives that they are not sufficiently right wing or populist. For a purely expressive voter, the national seat outcome may be irrelevant. Such a voter simply wants to register a protest. There still might be a connection to expected national votes: If such a voter thinks the PPC can get 7% he might be likelier to vote for it than if it’s only 3%.³ If, however, the connection runs through thinking about the national parliament, and whether one’s party will have voice there, it should help the party win votes around the country if its potential voters perceive that it will win one or more seats–in other words, that it is viable somewhere. I hope there is some polling data that I can find some day that allows us to get at this question, as it would connect the aggregate outcome demonstrated here with individual-level voter behavior. As the Canadian 2021 campaign has developed, it would be an especially good test of the model’s underlying individual-voter premise, given the surge of a small national party that is probably not likely to have a voice in the House of Commons. (But maybe its voters believe it will! They might even turn out to be correct.)

I do not, however, currently know if any polling or voter surveys exist to get at these questions. Such a survey ideally would ask the respondent how many seats they believe the various parties will get in the election. This would allow a rough construction of voter-expected effective number of seat-winning parties even though no voter actually has to know what that concept means or how to calculate it for the premise of the model to work. Minimally, as noted, it would at least be useful to know if voters choosing a small party think that party will indeed get one or more seats.

I have so far focused on the PPC in the Canadian 2021 election, as a possible example of a wider phenomena connecting local voting to the (expected) national seat outcome. A similar logic on the left side of politics should apply for the Green Party. Does its perceived viability for seats in parliament affect the tendency of voters to vote for it outside the specific districts where it is locally viable? The very big wrinkle this time around for the Greens, however, is that the party is struggling mightily, with an ongoing conflict between its leader and much of the rest of the party. It is currently projected to win no more than two seats, and perhaps none. It might be expected to retain the former leader’s seat in British Columbia, but even that may be in jeopardy with the national party in such disarray.

It is even questionable whether the Green Party still meets the criteria of a “national” party this time around; I do not (yet) have a really precise working definition of how many districts the party must be present in to qualify as “national.” The Green Party has not fielded a candidate in about a quarter of the ridings nationwide. Grenier has reviewed the 86 Green-less constituencies and whether their absence could affect outcomes among the contesting parties. Obviously the connection between expected seat winning nationally and obtaining votes in contests around the country is broken in any district in which there is no candidate running for the party. No candidate, no possibility of the local voters augmenting the party’s aggregate vote total. In any case, the party has dropped in national polls from 5.4% on 14 August to 3.2% now.

Further emphasizing now the Greens may not be a “national” party in this election is the campaign behavior of the leader. The CBC recently noted that the leader, Annamie Paul, is not exactly campaigning like the leader of a national party:

Asked why she hasn’t campaigned in more ridings, Paul acknowledged Friday that some candidates may want her to steer clear. She has campaigned outside of her home riding of Toronto Centre twice so far — once in a neighbouring riding and then Monday, in P.E.I.

Candidates distancing themselves from the leader is not normally a good sign for a party, particularly in a parliamentary system. “All politics is national,” after all. As explained in Votes from Seats (ch. 10), the impact of national politics on local voting is likely enhanced by parties bringing resources into districts to “show the flag” even where they are not likely to win a seat. (The PPC leader is certainly doing this.) If your leader remains mostly ensconced in her own district, the party is not deploying what is normally one of its best resources–the leader making the case for her party.

Nonetheless, it still might matter for the party’s ability to get votes, even in ridings it surely will not win, whether its potential voters believe it is viable for seat-winning somewhere. The good news for the party–and there is little of that–is that the province where it currently holds two seats, BC, is one of those where its polling has declined least: 7.0% on 14 August to 6.3% now. So, politics is still at least a bit more regional for the Greens than for other “national” parties, perhaps.

In conclusion, the district-level extension of the Seat Product Model states that in FPTP systems, district-level effective number of vote-earning parties can be predicted from the national electoral system–specifically, the assembly size. By further extension (in the aforementioned chapter I am working on with Struthers for the volume honoring Johnston), it should also be tied to the nationwide effective number of seat-winning parties, and to voter perceptions in the campaign as to how parties are doing at the national level. The result would be that voters are more likely to vote for even a small party under FPTP to the extent that they expect it to have a voice in parliament, and to the extent that the parliament may not have a majority party. The Canadian 2021 election, with a surging small party (the PPC) and another one declining (the Greens) offers an excellent case study of the phenomenon that is behind these models.

___________

Notes:

1. Obviously, things are different for an explicitly regional party (one that does not present candidates outside its region) like the Bloc Quebecois, which I will leave aside for this current discussion.

2. Perhaps it has some chance of winning the leader’s riding of Beauce (in Quebec), but as Grenier notes in a post the day before the election:

There’s nothing about Bernier’s Beauce riding that makes it particularly open to a party that has been courting the anti-vaxxer, anti-vaccine mandates and anti-lockdowns crowd. It’s hard to know where in the country that crowd would be big enough to elect a PPC MP.

He does also note that one poll, by EKOS, has put the party second in Alberta, albeit with only 20% of the vote. Maybe they could get a local surge somewhere and pick up a seat there.

3. Indeed, it might seem that we could make a similar algebraic connection. The Seat Product Model expects national effective number of vote-earning parties to be N_V=[(MS)^1/4 +1]^2/3. This is confirmed in Votes from Seats. However, this can’t easily be expressed in terms of just S (even for FPTP, where the term for M drops out) and therefore is complicated to connect to the N‘_V formula. In any case, the theoretical argument works better from seats–that voters key on the expected outcome of the election, which is a distribution of seats in parliament and whether one or another party has a majority or not. These outcomes are summarized in the effective number of seat-winning parties.

4. This graph is a version of the one that will be shown in the previouysly mentioned Shugart & Struthers chapter.

Are soft NDP voters switching to save Liberals?

In my earlier preview of September elections, I noted that the surge in polls for the Conservatives in the Canadian election might lead to the NDP losing votes. It is possible this is happening now.

In recent days, the Liberals have returned to a narrow lead in votes and strengthened their existing seat lead, according to the CBC poll tracker. At the same time, there is a notable dip in NDP votes and seats.

We need to be careful about inferring individual change from aggregate trends. But the most likely cause of what the poll tracker and its seat estimator are picking up is softer NDP voters worried about a Conservative plurality.

Note that this would be strategic voting, but not based on district-level expected outcomes (“coordination”), rather on national-level expectations. “All politics is national”, as Taagepera and I put it in the title of our chapter (10) on predicting district patterns from the Seat Product Model in Votes from Seats.

Here are screen shots from the poll tracker on the morning of 12 Sept.

Votes:

Estimated seats:

The challenge for voters who prefer the NDP over Liberals but are motivated more by stopping the Conservatives is that some of them may be in districts where the Conservatives wouldn’t have won anyway. But some of these voters may help the Liberals win a majority of seats—the poll tracker shows this outcome back within its 95% confidence interval. Yet the sort of voter I am describing wound surely prefer a Liberal minority with a strong NDP third-party caucus.

Getting just the right amount of strategic voting is hard when seats are determined one-by-one, but voters key mostly in national expectations. Yet this is exactly the best available information, which voters tend to employ in choosing voting strategy, according not only to the Seat Product Model, but also Richard Johnston’s The Canadian Party System.

It appears the Liberals have regained their stalled momentum and thus Justin Trudeau just might get what he was seeking after all. On the other hand, a short-term trend need not continue, and at the moment the most likely result still seems to be a Liberal plurality of seats.

Elections in September, 2021–campaigns matter

It won’t be quite like September, 2005, back when the virtual orchard was just a sapling, but somewhat like that September sixteen years ago featured several interesting elections, this month also looks great for election watchers.

In September, 2005, we had elections in three major examples of mixed-member systems: Japan, New Zealand, and Germany. (As I look back, I see I wrote several times about Afghanistan’s election that month; I am guessing there will no longer be a need for new Afghanistan elections plantings. I also see lost of posts about a hurricane disaster in New Orleans. Some things do recur, though fortunately in this case not on as horrific a scale, though bad enough.)

In September, 2021, we have the California recall. I don’t have anything at the moment to say about that beyond what I’ve already said. We have Germany, again, with its general election on 26 Sept. And we have Canada, on 20 Sept. There are also various other elections this month, but these are the two I will focus on here.

The German election for this September has been known about for a long time, as it is occurring on schedule (unlike the one in 2005). The Canadian one, on the other hand, is a snap election as one would not have been due till 2023. Both of these elections are going to become case studies in how campaigns really can matter.

For months it has seemed certain that in Germany, the ruling Christian Democratic/Christian Social “Union” bloc (CDU/CSU) would again come out on top, with the Greens in second place and a likely new coalition partner. Then a funny thing happened: the Social Democratic Party (SPD), for years seemingly finished as a major party, started to surge and seemingly now has pulled ahead of the CDU/CSU. The range of coalition possibilities is suddenly rather large, with some novel possibilities in the cards.

The election is also notable compared to past German elections in that the incumbent Chancellor (prime minister), Angela Merkel (CDU) is not a candidate to remain head of government. In fact, it seems likely that the campaign has caused voters to reckon with the with the less than inspiring leadership of the party’s chosen successor to Merkel (Armin Laschet), to feel less than sure they are ready to make the Greens potentially the largest or even second largest party, and to have turned to the SPD and its new leader, Olaf Scholz as the potential safe pair of hands to lead the government. These three parties seem certain to be the big three, but with the largest still below 25% in polls, it will probably take three parties to forge a majority coalition (taking the Union as one for purposes of government-formation, even though it is actually two parties). Unless all three govern together, or the current no-so-grand coalition clears a majority fo seats and continues in power, only with the SPD on top, it is going to take some combination involving the Free Democrats (FDP) to have a majority coalition. The post-election bargaining will be interesting (FDP with either Greens or SPD is not “natural”), and thus the election will really matter for which combines are possible and how much bargaining power each party has.

In Canada, the Liberal Party of incumbent Prime Minister Justin Trudeau looked safe to be not only the largest party, but also to win a majority of parliamentary seats. In fact, given that the election timing was the government’s choice, that is precisely why it is happening–to convert a minority Liberal government into a majority Liberal government. Then Trudeau called the election and a funny thing happened: his party started sinking in the polls and the Conservatives appear to have caught up. In votes, that is.

As in 2019, the Conservatives could lead in votes and still come second in seats. The Conservatives probably need to win the votes by more than a couple percentage points to have a reasonably good chance at a plurality of seats, due to their inefficient vote distribution across the country. A majority Conservative government probably requires that party to keep adding support at a rapid pace, and that may be happening. Yet if that continues, some voters would probably dessert the New Democratic Party (NDP), currently running at around 20%, in favor of the Liberals. Given the FPTP electoral system, of course, such NDP desertion for the Liberals would not be guaranteed to help the latter at the aggregate seat-winning level, although it probably would do so. (Canada’s Green Party, meanwhile, is in a shambles, largely because its black, female, Jewish leader was not sufficiently anti-Israel for others in the party.) One thing seems safe to say at this point: A Liberal majority has become rather unlikely.

That’s the funny thing about campaigns. Sometimes they actually matter.

Quebec electoral-reform referendum on hold

Quebec was to have a referendum on a proposed new mixed-member electoral system concurrent with the next provincial election in 2022. However, that plan is now “on hold” as the bill will not be passed in time.

The Canadian parliamentary vote on using the term, genocide

On 22 February, the House of Commons of Canada voted to label persecution of the Uighur people by the Chinese authorities a genocide. I am not interested for purposes of this blog post in whether that is the right label or not (that’s way beyond my competence or the focus of this blog). I am interested in the unusual nature of the vote.

It was unanimous among those voting, 266-0. However, the government did not take part in the vote. The governing Liberal Party currently has 154 of the House’s 338 seats. Thus as a minority government (see 2019 election result), the possibility of a measure passing over its abstention (or outright objection) is always a possibility even if the party itself votes with the government. In this case, obviously, some Liberals voted for the measure, but most were absent. Only two MPs were present but formally registered an abstention, including the Minister of Foreign Affairs, who declared he was abstaining “on behalf of the Government of Canada.”

I am not sure how frequently votes pass in this manner, especially on sensitive diplomatic matters, either in Canada or in other parliamentary systems. I am also not sure what the practical (as opposed to symbolic) meaning of such a vote is when the government is not on board with it.

Canada and UK 2019: District level fragmentation

With two of the big Westminster parliamentary democracies having had general elections in 2019, we have a good opportunity to assess the state of district-level competition in FPTP electoral systems.

(Caution: Deep nerd’s dive here!)

Before we turn to the district level, a short overview of what is expected at the national level is in order.

As noted previously, Canada’s election produced a nationwide seat balance that was extremely close to what we expect from the Seat Product Model (SPM), yet the nationwide votes were exceedingly fragmented (and, anomalously, the largest seat-winning party was second in votes). The UK election, on the other hand, was significantly less fragmented in the parliamentary outcome than we expect from the SPM, even if it was in key respects a “typical” FPTP outcome in terms of manufacturing a majority for a party with less than a majority of the vote.

In general, over decades, Canada tends to conform well to the SPM expectation for the shape of its parliamentary party system, whereas the UK is a more challenging case from the SPM’s perspective.

The SPM states that the effective number of seat-winning parties (N_S) should be the seat product, raised to the power, 1/6. The seat product is the assembly size, times the mean district magnitude. The SPM predictions for N_S explain around 60% of the variance in actual outcomes for elections around the world under a wide variety of electoral systems. SPM predictions for other output quantities also explain in the neighborhood of 60%. So the SPM is both successful at explaining the real world of seat and vote fragmentation, and leaves plenty of room for country-specific or election-specific “other factors” (i.e., the other 40%). The SPM is based on deductive logic, starting from the minimum and maximum possible outcomes for a given number of seats at stake (in a district or an assembly). The logic is spelled out in Votes from Seats.

In the case of a FPTP system, the SPM makes the bold claim that we can understand the shape of a party system by knowing only the assembly size. That is because with district magnitude, M=1, the seat product is fully described by the country’s total number of seats, S, which is also the number of districts in which the voting is carried out. Thus we expect N_S=S^1/6. Let’s call this “Equation 1.”

For Canada’s current assembly size (338), this means N_S=2.64, as an average expectation. Actual elections have tended to come pretty close–again, on average. Of course, individual elections might vary in one direction or the other. (The assembly size was also formerly smaller, but in recent times, not by enough to concern ourselves too much for purposes of this analysis.) For the UK, the corresponding expectation would be 2.94 based on a seat product of 650.

The actual Canadian election of 2019 resulted in N_S=2.79; for the UK it was 2.39. Thus for Canada, we have a result very close to the expectation (ratio of actual to expected is 1.0578). For the UK, the actual result was quite short (ratio of 0.8913). As I said, the UK is a challenging, even aberrant, case– at least at the national level.

What about the district level? A national outcome is obviously somehow an aggregation of all those separate district-level outcomes. The SPM, however, sees it differently. It says that the districts are just arenas in which the nationwide election plays out. That is, we have a logical grounding that says, given a national electoral system with some seat product, we know what the nationwide party system should look like. From that we can further deduce what the average district should look like, given that each district is “embedded” in the very same national electoral system. (The logic behind this is spelled out in Votes from Seats, Chapter 10).

The crazy claim of the SPM, district-level extension, is that under FPTP, assembly size alone shapes the effective number of votes-earning parties in the average district (N’_V, where the prime mark reminds us that we are talking about the district-level quantity rather than the nationwide one). (Note that for FPTP, it must be the case that N’_S=1, always and in every district).

The formula for expected N’_V under FPTP is: N’_V=1.59S^1/12 (Equation 2). It has a strictly logical basis, but I am not going to take the space to spell it out here; I will come back to that “1.59” below, however. It is verified empirically on a wide set of elections, including those from large-assembly FPTP cases like Canada, India, and the UK. So what I want to do now is see how the elections of 2019 in Canada and UK compare to this expectation. (Some day I will do this for India’s 2019 election, too.)

If the effective number of seat-winning parties at the national level (N_S) is off, relative to the SPM, then it should be expected that the average district-level effective number of vote-earning parties (N’_V) would be off as well. They are, after all, derived from the same underlying factor–the number of single-seat districts, i.e., the assembly size (S). We already know that N_S was close to expectation in Canada, but well off in the UK in 2019. So how about the districts? In addition to checking this against the expectation from S alone, we can also check one other way: from actual national N_S. We can derive an expected connection of N’_V to N_S via basic algebra. We just substitute the value from one equation into the other (using Equations 1 and 2). If we have N_S=S^1/6 then it must be that S= N_S⁶. So we can substitute:

N’_V=1.59(N_S⁶)^1/12= 1.59√N_S (Equation 3).

In a forthcoming book chapter, Cory L. Struthers and I show that this works not only algebraically, but also empirically. We also suggest a logical foundation to it, which would require further analysis before we would know if it is really on target. The short version suggested by the equation is that the voting in any given district tends to be some function of (1) the basic tendency of M=1 to yield two-candidate competition (yes, Duverger!) in isolation and (2) the extra-district viability of competing parties due to the district’s not being isolated, but rather embedded in the national system. The 1.59, which we already saw in Equation 2, is just 2^2/3; it is the expected N’_V if there were exactly two vote-earning parties, because it is already established–by Taagepera (2007)–that the effective number tends to be the actual number, raised to the power, two thirds. And the square root of N_S suggests that parties that win some share of seats (i.e., can contribute more or less to the value of N_S) tend to attract votes even though they may have no chance of winning in any given district. By having some tendency to attract votes based on their overall parliamentary representation, they contribute to N’_V because voters tend to vote based on the national (expected, given it is the same election) outcome rather than what is going on in their district (about which they may have poor information or simply not actually care about). If the parliamentary party system were fully replicated in each district, the exponent on N_S would be 1. If it were not replicated at all, the exponent would be zero. On average, and in absence of any other information, it can be expected to be 0.5, i.e., the square root.

How does this hold up in the two elections we are looking at in 2019? Spoiler alert: quite well in the UK, and quite badly in Canada. Here are graphs, which are kernel density plots (basically, smoothed histograms). These plots show how actual districts in each election were distributed across the range of observed values of N’_V, which in both elections ranged from around 1.35 to just short of 4.5. The curve peaks near the median, and I have marked the arithmetic mean with a thin gray line. The line of most interest, given the question of how the actual parliamentary outcome played out in each district is the long-dash line–the expected value of N’_V based on actual N_S. This corresponds to Equation 3. I also show the expectation based solely on assembly size (light dashed line); we already have no reason to expect this to be close in the UK, but maybe it would be in Canada, given that the actual nationwide N_S was close to the SPM expectation, based on S (Equation 2).

Here is the UK, then Canada, 2019.

What we see here is interesting (OK, to me) and also a little unexpected. It is the UK in which the actual mean N’_V is almost the same as the expectation from nationwide N_S (i.e., Equation 3). We have actual mean N’_V=2.485 compared to expected N’_V from actual N_S of 2.45; the ratio of actual to expected is 1.014. We can hardly ask for better than that! So, the nationwide party system (as measured by N_S) itself may be well off the SPM expectation, but the vote fragmentation of the average district (N’_V) closely tracks the logic that seems to stand behind Equation 3. Voters in the UK 2019 election tended to vote in the average district as if parties’ national viability mattered in their choice.

In Canada, on the other hand, even though national N_S was very close to SPM expectation, the actual average district’s N’_V (2.97) was really nowhere near either the expectation solely from S (the light dashed line, at 2.58) or the expectation from the actual N_S (2.66). The average district was just so much more fragmented than it “should be” by either definition of how things ought to be! (The ratio of actual to that expected from Equation 3 is 1.116; the Equation 3 expectation is almost exactly the 25th percentile of the distribution.)

The Canadian outcome looks as if the exponent on actual N_S in Equation 3 were around 0.64 instead of 0.5. Why? Who knows, but one implication is that the NDP (the third national party) performed far better in votes than the party’s contribution to N_S implies that it should have. Such an overvaluing of a party’s “viability” would result if voters expected the party to do much better in terms of seats than it did. This is probably a good description of what happened, given that pre-election seat extrapolations implied the NDP would win many more seats than it did (and the Liberals fewer). The NDP also underperformed its polling aggregate in votes (while Liberals over-performed), but it held on to many more voters than it “should have” given its final seat-winning ability would imply. That is, the actual result in votes suggests a failure to update fully as the parties’ seat prospects shifted downward at the very end of the campaign. In fact, if we compare the final CBC poll tracker and seat projections to the ultimate result, we find that their actual votes dropped by 13.6% but their seats dropped by 31.7% (percent change, not percentage points!). In other words, this was just an unusually difficult context for voters to calibrate the expectations that Equation 3 implies they tend to make. (I am assuming the polls were “correct” at the time they were produced; however, if we assume they were wrong and the voters believed them anyway, I think the implications would be the same.)

It should be understood that the divergence from expectation is not caused by certain provinces, like Quebec, having a different party system due to a regional party, as some conventional expectations might point towards. While Quebec’s size is sufficient to exert a significant impact on the overall mean, it is not capable of shifting it from an expected 2.6 or 2.7 towards an observed 3.0! In fact, if we drop the Quebec observations, we still have a mean N’_V=2.876 for the rest of Canada. The high fragmentation of the average district in the 2019 Canadian election is thus due to a Canada-wide phenomenon of voters voting for smaller parties at a greater rate than their actual viability would suggest they “should”. In other words, voters seem to have acted as if Trudeau’s promise that 2015 would be the last election under FPTP had actually come true! It did not, and the electoral system did its SPM-induced duty as it should, even if the voters were not playing along.

On the other hand, in the UK, voters played along just as they should. Their behavior produced a district-level mean vote fragmentation that logically fits the actual nationwide seat balance resulting from how their votes translated into seats under FPTP. There’s some solace in that, I suppose.

Canada 2019: Results and a good night for the Seat Product Model

Add Canada 2019 to the set of plurality reversals. As anticipated before the election, the two largest parties each ended up with around one third of the vote. This is the lowest vote percentage for a governing party in Canada ever, I believe. The seats are somewhat less close than the CBC’s Poll Tracker estimated they would be. Instead of 133 seats to 123, the seats split 157 to 121. The Liberals are indeed that largest seat-winner, despite trailing the Conservatives in votes percentage, 34.4 – 33.1.

The NDP was either overestimated by polls or, more likely, suffered some late strategic defection. Instead of the near 19% of the vote in the final Poll Tracker, the party ended up with only 15.9%. More importantly, its seats stand at only 24, well below where estimates late in the campaign had them (per the CBC Poll Tracker).

As excepted the BQ had a good night, with 32 seats. The Greens picked up one new seat to augment the two they already held. The new seat is Fredricton, New Brunswick, whereas the other two are both on Vancouver Island.

In what I will call the two best pieces of news form the night (other than there being no single-party majority), the People’s Party crashed and burned, winning only 1.6% and seeing its leader lose his seat. That and the fact that Jody Wilson-Raybould, the former Attorney General who was kicked out of the Liberal caucus, retained her seat, Vancouver-Granville, as an independent.

 

Anomalous FPTP

I will certainly use this result often as a demonstration of how the first-past-the-post (FPTP) system can produce strange results.

Not only the plurality reversal for the top two, but the differential treatment of the next three parties, show anomalies of the sort that are inherent to FPTP. The BQ is only somewhat larger in votes than the Green Party, but will have more than ten times the number of seats. Under FPTP, it is good to have efficient regional distribution of support, and getting all your votes in one province, where you perform exceptionally well, is really efficient. The Greens, on the other hand, gained in almost all provinces, but it was good enough to add only one seat.

The NDP’s situation is one of a quite strong third party, but also inefficient regional distribution: 7.1% of the seats on 16% of the votes is a punishing result, but nothing at all unexpected, given the electoral system.

For that matter, the plurality reversal is itself a signal of the problem of inefficient vote distribution. The Conservative Party mostly gained votes where they could not help the party win seats, whereas the Liberals were much more successful winning close contests.

In his victory speech, PM Justin Trudeau was bold enough to use the M-word (mandate), but this most certainly is not one. For the moment, he can be pretty happy he broke that promise on 2015 being the last FPTP election. His party remains in position to form the government, and has a substantial seat bonus. The advantage ratio (%seats/%seats) is 1.40. (How does that compare with past elections? Click to see.)

Canada would be well served by at least some degree of proportionality. In fact, so would the Conservatives, given their tendency to run up margins where they are already strong. (Note that they are only barely over-represented in seats, with 35.8%.) However, this result is unlikely to advance the cause of reform, as the Liberals’ position–46% of the seats and a 36-seat (more than ten percentage point) edge over the runner-up–looks quite solid.

The other reason the country could really use electoral reform is the map. There is no Liberal red to be seen from central Ontario westward, except around Vancouver (and two northern territories). The party lost some of its ministers’ reelection bids in Alberta and Saskatchewan. With even a minimally proportional system, the situation of a governing party without members of its caucus in nearly every province would not happen.

While a PR system would be beneficial, the country is stuck with FPTP at least for now. So how did this result compare to what we should expect from the electoral system actually in use?

 

The Seat Product Model and the outcome

The Seat Product Model (SPM) performed better than the CBC Poll Tracker’s seat estimator. For an assembly of 338 and districts with magnitude of 1, we should expect the largest party to have, on average, 48.3% of the seats, which would be 163 seats. So the actual result (46.4%) misses the expectation by 6 seats, or 1.78 percentage points (compared to the a 20-plus, or 6 percentage point, miss by the Poll Tracker).

Of course, the SPM has one advantage in its favor: it does not “know” that the seat-winning party would have under 33.3% of the vote, whereas the Poll Tracker must work with this expectation (and, as it turned out, reality). In fact, when a party wins 48.3% of the seats, the formulas of SPM (collected in Table 9.2 of Votes from Seats) expect it to have won 43.3% of the votes. (Theoretically, we do not expect the SPM to perform as well with votes as with the seats that are at its core; but in Votes from Seats, we show that, on average, it performs about equally as well with both.) The Liberals underperformed this expectation by more than ten percentage points! The voters genuinely voted for something their electoral system could not deliver, even if the system indeed delivered what should be expected solely on institutional grounds.

In terms of the effective number of seat-winning parties (N_S), the actual result was 2.79. This is slightly higher than the SPM expectation, which is 2.64. The miss is minor, with a result only 1.057 times expectation.

On the other hand, the effective number of vote-earning parties (N_V) was 3.79. The SPM expects 3.04. Let me pause and emphasize that point. Because Canada uses FPTP in a 338-seat assembly, we should expect the votes to resemble a “three-party system” and not the two-party system that all the conventional “Duvergerian” wisdom claims. If we calculated expected N_V based on the known N_S=2.79, we would expect N_V=3.17. However, neither the SPM nor Duverger’s “law” expects that the largest party nationwide should have only around a third of the votes. That is the really remarkable thing about this outcome.

 

The district level

At the district level, there were numerous non-Duvergerian outcomes, as would be expected with the known distribution of nationwide votes among parties. According to an extension of the SPM (in a forthcoming book chapter), we should expect the effective number of vote-earning parties at the average district (N’_V) to be 1.59 times the square root of the nationwide N_S. That would be 2.66. It will be a while before I am able to calculate what it actually was, but it would not surprise me if it was a fair bit higher than that. But, again, let me pause and say that a Duvergerian two-party competition at the district level is NOT to be expected, given both the nationwide electoral system and the actual aggregate seat outcome. (If we went off expected nationwide N_S, instead of the known outcome, the district-level mean still would be predicted to be 2.58; see Chapter 10 of Votes from Seats.) Canadian elections of the past several decades have tended to conform closely to this expectation for district-level N’_V.

The country does not tend to have two-party contests at district level, nor should it (when we have the Seat Product Model to guide our expectations). In other words, voters do not tend to vote in order to “coordinate” their district outcome around the two most viable candidates. They tend to vote more towards their expectation (or desire) about what the nationwide parliamentary outcome will be. This is so even in Quebec where, in this election, many Francophone voters returned to the regional party, the Bloc Québécois. Quebec has numerous district contests that feature three or four viable parties.

So if your image of Canada’s party system is that in Quebec districts it is BQ vs. Liberal, with other parties barely registering, while elsewhere it is Liberal vs. Conservative, except where it is one of those vs. NDP, it is well past time to update. Canada does not have nationwide multiparty politics because it has separate regional two-party systems (as many folks, even political scientists, seem to believe). Canada has district-level multipartism because it has nationwide multipartism. (See Richard Johnston’s outstanding book for a rich “analytic history” that supports this point.) And this may be even more true in the one province in which there is (again) a strong regional party. Consider the aggregate provincial outcome in terms of vote percentages in Quebec: Liberal 34.2% (slightly higher than nationwide), BQ 32.5%, Conservative 16.0%, NDP 10.7%, Green 4.5%. This gives a provincial-level N_V of 3.82, a bit higher than nationwide.

I will offer a few striking examples of multiparty contests at district level, just to illustrate the point. The new Green Party MP from Fredericton, Jenica Atwin, won 32.8% of the vote. The Conservative had 31.1%, the Liberal 27.3%, and the NDP 6.0%. There may indeed have been strategic voting happening here, with some NDP voters–the party had 9.9% in 2015–switching to Atwin to stop the Conservative (and perhaps some who don’t like the Greens boosting the Liberal). But the outcome here is N’_V=3.53!

The change from 2015 in Fredericton is really striking, as the Liberal candidate was an incumbent who had won 49.3% in 2015 (against 28.4% for the Conservative, meaning this party gained a little here in 2019). Clearly many Liberals defected from their party to the Green following that party’s success, including a local win, in the recent provincial election. In doing so they only narrowly avoided the serious “coordination failure” that would have been a Conservative win.

Another Green MP, the reelected Paul Manly in Nanaimo-Ladysmith, won 34.5%. This was actually a pretty clear victory despite being barely over a third of the vote; Manly had been elected in a by-election this past May with 37.3%. The runner-up Conservative had only 25.9% in the general election contest, the NDP 23.7%, Liberal 13.6%. N’_V=3.83!

Wilson-Raybould’s win in Vancouver-Granville as an independent was also with under a third of the vote. She had 32.3%, beating the Liberal’s candidate (26.6%) and the Conservatives’ (22.1%). The NDP candidate had 13.1%. The Greens, who tried to recruit Wilson-Raybould to be their candidate, put up their own against her, who got 5.0%. It should be noted that the NDP candidate in this riding last time won 26.9%, so it would appear there was ample strategic voting here in Wilson-Raybould’s favor. (She won 43.9% as the Liberal candidate in 2015.) The Green voters, on the other hand, did not seem to warm to their near-candidate; the party’s actual candidate did better in this district in 2019 than in 2015 (when the party got 3.1%).

One of my favorite cases is Sherbrooke, in Quebec. The winner was Liberal Elisabeth Briere with 29.3%, edging out an NDP incumbent who won 28.3% in this election. He had won the seat with 37.3% in 2015. Close behind in this year’s contest was the BQ candidate who had 25.8%. Following behind them was a Conservative (10.7%), and Green (4.5%). N’_V=4.06!! The Liberals won this by basically standing still in vote share, having lost this district by a wide margin in 2015 when their candidate had 29.8%.

A few interesting tidbits from candidate backgrounds. Bernier’s defeat in his own riding of Beauce was at the hands of a dairy farmer, Richard Lehoux. The Conservatives recruited him because of Bernier’s opposition to supply management policies in the dairy sector. (Info found in the CBC’s Live Blog.) Lehoux won only 38.6% of the vote, but it was sufficient to beat Bernier rather badly, as the latter (elected as a Conservative in 2015 and previously) had just 28.4%.

There were several mayors recruited to run, including a case in Quebec where the Conservatives hoped the candidate’s local popularity would overcome the party leader’s unpopularity. (The specific case was Trois-Rivières; the Conservative finished a close third in a riding the BQ candidate won with 28.5%.) There was also an Olympic medal-winning kayaker, Adam van Koeverden, whom the Liberals recruited in Milton (in Toronto, Ontario) to run against the Conservative Deputy Leader, Lisa Raitt. He defeated her–easily, winning 51.4% to her 36.5%. Presumably his celebrity (and perhaps his local roots, which he made a point to emphasize in an interview after his victory was confirmed) helped him win despite a nationwide swing against the Liberals and in favor of the Conservatives. (She had won 54.4% in 2015.) In other words, while I may emphasize that district politics under FPTP in a parliamentary system is mostly national politics, there is still plenty of room for local and personal factors to matter.

 

What it means for the near term

As to the shape of the government to result, it should be a reasonably stable minority government, although it may not last full term. It can form legislative majorities with either the BQ or the NDP, and thus need not be tied to either one in a coalition. And the NDP certainly is not strong enough to demand a coalition (even if it wanted to try). Nor is it likely strong enough to demand action on electoral reform, even if an election in which two thirds of the voters voted against the governing party, and various other aspects of the outcome can be seen as anomalous, suggests that reform is needed more than ever.

Canada 2019

It is the final Friday before Shemini Atzeret, also known as Election Day in Canada this year, And what an interesting campaign it has been! The polls have moved quite a lot, especially recently. The New Democrats (NDP) seem to be enjoying a surge. Not on anything like the scale of 2011, but still something notable, as it was not long ago that there was talk about the Greens possibly passing them for third place. The Greens have slipped somewhat, as has been the case in past campaigns. No longer do they look likely to win as many as four seats; two (which is their current number) looks most likely.

The striking thing is that the poll aggregate at CBC (compiled by Éric Grenier) shows both major parties–incumbent seat-majority Liberal and opposition Conservative) barely above 30% of the vote (31.7-30.8 at my latest check). From 1949 to present, the largest party has never had a vote percentage below 36.3% (in 2006). So if there is not a late surge of strategic voting, this will be quite a record-breaker.

Projecting seats under FPTP is always a challenge. The CBC Poll Tracker currently has the Liberals significantly favored, despite being marginally behind in votes, 133 seats to 123 (but with wide confidence bands on both).  That would be 39.3% of the seats for the largest party, which would also break the record (from 1949 on) set in 2006 (40.3%, or 124 in what was then a smaller parliament).

Despite being both a plurality reversal and a record low vote percentage (and an extremely close vote margin), the advantage ratio (%seats/%votes) of 1.278 for the largest seat-winner would be just about average. Over 22 elections, the mean advantage ratio has been 1.2897. (Note: I am calculating this as the share of the largest seat-winner over its vote share, not over the share of the largest vote-winner, when those diverge.) For those who know Canadian electoral history, I will note that advantage ratios of around 1.2-1.3 have occurred in 1965, 1968, and 2008 (among others). Thus even if the specific vote totals may be very unusual, the workings of FPTP, given the actual votes, is fairly “typical” for Canada.

As for the other parties, I mentioned the NDP surge. But just as noteworthy is the surge of the Bloc Quebecois, which may turn out (again) to be the single most important factor in preventing a majority of seats. The BQ is currently polling just under 7% nationwide, while the Greens are just over 8%.

Of course, the BQ and Green fortunes will diverge in seats. It is very helpful for votes-seats conversion to be a regional party under FPTP, and not useful to be relatively more dispersed. So the BQ is currently estimated to get 38 seats, about the same as a much larger national party, the NDP (41) and vastly more than the also larger–in votes–Greens (2).

Regarding those surges I mentioned. The BQ was, according to the polling aggregate on only about 20% in Quebec as recently as one month ago. Now it is up to almost 30%, and just behind the Liberals’ percentage in the province (31%, having been 37% a month ago). The Conservatives have really crashed in Quebec, down from 22% a month ago to just under 16% now. The latter puts them not too far ahead of the NDP, who are now on about 14% in the province.

Nationally, the NDP was at only about 13% a month ago, but is approaching 19%. A rising vote share tends to lift the seat share–even for a national third party under FPTP. While a month ago, the Poll Tracker had the party at only 15 seats, its 41 projected now represents an increase by a factor of 2.7 when its votes have increased only 1.27 (19/15). The party would still be significantly under-represented by the electoral system, but it has reached a point where it gains a lot of seats by a small increase in votes (assuming it holds and that Grenier’s swing assumptions are reasonable, etc.).

The NDP has also pulled narrowly ahead of the Liberals in the polling aggregate in British Columbia, although still well behind the leading Conservatives.

As for the Greens, their slide has been quite abrupt. They were over 10% as recently as the first of October and were projected to win 4 seats as recently as 16 Oct.

The sixth party in the picture, the far-right Peoples Party of Canada, looks likely to win only the seat of its leader, Maxime Bernier. The riding (district) is Beauce, in Quebec, in which Bernier has held as a Conservative since 2006 until defecting from that party in 2018. (I see the Rhinoceros Party has found a candidate with the same name to put up against him.) For months, the PPC has been at either zero or one seat in the projection.

As for who will form a government, the Liberals seem best placed, even if the result is as short of majority as the Poll Tracker projects. It is possible that they will be weak enough to have to form a coalition with the NDP, even though probably the Liberals would prefer a minority government. On current numbers, Liberal+NDP would be a very bare majority. The coalition or a minority government might need working arrangements of some sort with the Greens and/or BQ as well.

It is much harder to see how the Conservatives can form government, even if they end up edging out the Liberals for a seat plurality. Conservative leader Scheer has already begun the spin just in case, claiming this week (incorrectly) that the party with the most seats gets the first shot at forming a government.

NDP leader Jagmeet Singh has said he would try to form a coalition with the Liberal Party if the Conservatives have the most seats. And PM Justin Trudeau would have the legal right to attempt to work out such a deal and meet parliament to try to retain office. Presumably, Singh (and the Green leader, Elizabeth May) would attempt to extract a concession that 2019 be the last election under FPTP.

Quebec to have electoral reform referendum

Per CTV News Montreal, the CAQ governing party in Quebec promises a referendum on a specific electoral-reform proposal to be held concurrent with the next general election in 2022.

The CAQ government on Wednesday introduced its electoral reform law, but backed away from its 2018 campaign promise to have it in place in time for the next general election.

The system is a form of mixed-member proportional representation (MMP), but a complex one. And not very proportional.

The number of seats in the National Assembly would remain 125. The new electoral law would divide Quebec into 80 larger electoral divisions that will mainly be the same as federal ridings, and 17 wider regions:

• 80 MNAs would be elected according to the current system

• 45 MNAs on a list of candidates would be divided among the parties, according to the percentage of votes obtained in the 17 regions

Thirty six percent of seats for the list tier is certainly on the small side, and the proportionality would be reduced further by that list tier itself being districted. Note that the mean number of seats per compensation region is only 2.65.

Based on further detail that I learned from Manuel on Twitter (and that I trust he will not mind my sharing here), we can see yet more ways that this proposal is designed to limit proportionality.

…it’s a very constrained implementation of PR, limited by a provincial-level 10% (yes, ten percent) threshold; and districted MMP in seventeen regions, with a new variation of the D’Hondt rule that skews seat distributions in favor of the larger parties.

Regarding the seat allocation method for the compensation seats (which will limit how compensatory it actually will be):

In Scotland and Wales the modified D’Hondt divisors are N+1,N+2,N+3, and so on, where N = number of single-member seats. In the Quebec proposal, they are N/2 + 1, N/2 + 2, N/2 + 3, N/2 being *half* the number of single-member seats, rounded up, and resulting in lower divisors.

One additional detail: the bill provides for separate allocations of single-member and PR list seats among regions – according to the number of registered voters – which guarantee all but one region a minimum of two seats. This would cost Montreal – a PLQ bastion – three seats.

I would still consider this MMP*, as there is a compensation mechanism. I am on record as considering even the Jenkins Commission proposal in the UK to be MMP, albeit with lots of caveats given it was also designed to be about as weak on the P as could be.

Regarding the election of 2018 when the CAQ came into power–surprisingly, with an absolute majority of seats–and the electoral-reform promises made at the time, see this earlier planting.

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• * UPDATE: I am now not sure about this; I need more time to think it through, and that will have to wait till some time in 5780! In the meantime, see this Twitter thread. I think the issue hinges on whether the “modification” to D’Hondt is actually more like Imperiali. While Taagepera and I list Imperiali divisors in our book as being part of the family of PR allocation formulas, we both now believe that it should not be. We were prompted to this view by an email exchange earlier in 2019 with Steven Verbanck (regular F&V commenter). Anyway, to be continued…

Newfoundland and Labrador 2019

Now, this is a close election! Newfoundland and Labrador held its provincial election on 16 May. The result was the Liberal Party winning exactly half the seats–apparently. One of their losses is by five votes. Not five percentage points. Five votes. The apparent winner is the NDP candidate. There is likely to be a recount.

If the result holds, the NDP has won three seats in the 40-seat assembly, up from two in the previous election. The Liberals had 31 seats last time, so this is some electoral rebuke, even if it ends up back with a majority. The Progressive Conservative Party, which won 15 seats, has not conceded defeat, and the leader has said he will be calling on the NDP and two independents (who are ex-Liberals) to consider a possible bid for power if the Liberals do not win their 21st seat.

The Liberal vote total percentage was 43.9, PCs 42.6, NDP 6.3.

PEI 2019: Provincial assembly election and MMP referendum

Prince Edward Island election day was today. Results appear to be pointing towards a lead in votes and seats for the Progressive Conservatives. Thus the expectation (at least according to some polling) that the Greens might form the government looks unlikely now. The Green Party appears to be in second place, although closer (in votes) to the third place Liberals than to first place. The Liberals are the outgoing governing party, with a seat majority.

It looks like it will be a minority situation, which I think will be a first for PEI.

Voters were also voting in a referendum on whether to replace FPTP with a mixed-member proportional (MMP) system, with the voter brochure showing it as an open-list variant. That looks like a close call at this point, but most likely it has been defeated. To pass, the MMP proposal requires not only a majority of votes, but also a win in 17 of the 27 districts. At the moment, CBC is reporting it will win no more than 15 (and may not have a provincewide majority anyway).

PEI has been here before, having voted in a non-binding ballot (conducted by phone or online) for MMP in 2016, and having defeated an earlier MMP referendum by a big majority in 2005.

Today’s referendum had an odd ballot format: the NO option appears above the YES option. I am not sure I have ever seen that before.

(There is one district that did not vote today in the assembly election; a candidate, of the Greens, died in a canoe accident last Friday, so there will be a by-election at a future date.)