Could Nigeria’s presidential election require a runoff?

On 25 February, Nigeria holds its elections for president, house, and senate. The country uses single-seat plurality elections, but with an important qualification in the case of the federal presidency. The winner of that contest must have not only a nationwide plurality of votes, but also meet a “distribution” requirement: earn at least a quarter of the vote in at least two thirds of the federal entities (consisting of 36 states and a capital territory).

If the first-round plurality candidate fails to meet the distribution qualification, then there is a runoff. The two candidates in a runoff would be the plurality candidate and the other candidate who wins the most states with a majority of the vote. Note that if there are three or more serious candidates, with regional bases, the runoff opponent might not be the one with the second highest votes in the first round.

Nigeria this year has precisely the situation that could lead to a runoff and possibly not with the top-two candidates: There are at least three candidates who might be considered “serious” and all exhibit Nigeria’s typical tendency towards regional concentration of support. See Africa Elects. Nigeria has not previously had a runoff.

Polling is apparently sparse, but shows Peter Obi of the Labour Party (LP) in the lead–some recent polls show him far ahead. I am not sure how reliable Nigerian polls are, let alone whether they effectively capture state-by-state trends. The next two candidates after Obi appear close in polling: Bola Tinubu (All Progressive Congress, APC) and Atiku Abubakar (Peoples Democratic Party, PDP). A fourth candidate, Rabiu Kwankwaso of the New Nigeria Peoples Party (NNPP) also has regionally concentrated support. Given the distribution requirement, even a candidate with little chance of winning might deprive a more nationally popular candidate of 25% of the vote in some states or of a majority in enough states to affect runoff qualifications should there be a second round.

The APC is the incumbent president’s party, and the PDP has provided the winning ticket in various past elections (including its presidential candidate this year as vice president in the past). Obi was the PDP’s vice presidential candidate in the last national election, in 2019, with Abubakar at the top of the ticket.

As far as congress is concerned, at the 2019 election the APC won 202 seats and the PDP 126 in the House of Representatives, which has 360 seats. In the 109-seat Senate, the APC won 63 and the PDP 44. It seems likely that this year’s results will be more fragmented, but given the use of single-seat plurality in both houses, obviously it depends entirely on the parties’ relative regional concentration in the voting.

Nigeria uses a pure presidential system.

Belize to correct its egregious malapportionment

The Supreme Court of Belize has brokered a consent order that will lead to a redistricting to correct the country’s extremely high malapportionment.

Co-Counsel Michelle Trapp stated that [Sean P.] Trende’s report, among other things, found at the time it was written, only five of Belize’s 31 electoral divisions met the international standard deviation of registered voters, generally between five to 15 percent with allocation made for sparsely populated areas.

Trende’s report itself indicates some districts have around five times the voters as some others.

The country has had some lopsided results in the past, including 2020 when a party with almost 60% of the votes won 26 of the 31 seats. Of course, FPTP in a very small assembly will tend to do that, but presumably it has been exacerbated by the malapportionment. (By the cube root law, Belize would be expected to have around 75 seats in total given its population.) Several other elections have been similarly lopsided.

Farther back, it even had a majority reversal. In FPTP systems, plurality reversals happen from time to time–defined as when the party with the plurality of votes does not win the plurality of seats–but majority reversals are rather unusual. Yet in 1993 the People’s United Party won 52.1% of the vote but only 13 of the 29 seats parliament then consisted of. The UDP-NABR alliance won the other 16 seats, despite just 48.7% of the vote.

Looking at the 2020 election at district level, I see one constituency, Ft. George, that had only around 1,500 votes cast (and the PUP nabbed with just 983) while Stann Creek West has over 7,600 votes cast and several others have more than 6,000. There is another around the voter base of Ft. George, as well as several in the mid two thousands or around three thousand. So it is indeed severely malapportioned.

The 1993 majority reversal did not take place under malapportionment this severe, but it was substantial. At that time Stann Creek West was among the relatively lower-population districts (just over 2,000 votes cast) while a few other districts had over 3,000. There were others with only around 1,500. So the problem is not new, but has got worse. And there seem to have been some really significant internal population shifts in the past thirty years. Fortunately, this will be corrected. They still should increase the size of their assembly, too–or adopt some sort of PR. But at least they will improve their small-assembly FPTP.

Finally, now that I’ve poked around Wikipedia pages on Belize constituencies a bit, I want to go see the Belize City Swing Bridge, one end of which is in Ft. George constituency, and which requires four operators to manually crank it.

Ontario 2022

Ontario’s election on 2 June saw another Progressive Conservative seat majority on barely over 40% of the votes. The party, led by provincial Premier Doug Ford, barely increased its vote percentage from 2018, when it won 40.2%; this time the tally is about 40.8% (pending final count). Its vote total actually went down, because it was the lowest turnout in the province’s history. Yet it will have 83 of the 124 seats, whereas in the 2018 election it won 76.

For those keeping the stats, that would be a bare two-thirds majority (66.9%), and an advantage ratio (%seats/%votes) of 1.64. That is very much on the high side, even by the standards of FPTP with multiparty systems.

The main shifts in vote percentages were among the two largest opposition parties. The Liberals improved from 19.4% to 23.9%. The payoff in seats was minimal: the party won 8 seats this time, 7 last time. The NDP performed especially badly, going from 33.3% of the vote in 2018 to 23.7%. However, even though the NDP’s votes are marginally behind the Liberals’, the NDP will continue to have more seats–a lot more–with 31 (down from 40 at the last election). Yes, FPTP in multiparty systems!

Ontario objectively needs to shift to a proportional system. It is not as if the province has not had the opportunity to do that before.

St Kitts and Nevis 2000–Crazy result

Given my sudden fascination with small assemblies, I was poking around in election results from St Kitts and Nevis, a Caribbean sovereign state with a population of just over 52,000. With 11 elected members, its assembly certainly counts as small. The 2000 election is really something. Look at the national result:

Party	Code	Votes	% votes	Candidates	Seats
St. Kitts and Nevis Labour Party	SKNLP	11,762	53.85%	8	8
People’s Action Movement	PAM	6,468	29.61%	8	0
Concerned Citizens Movement	CCM	1,901	8.70%	3	2
Nevis Reformation Party	NRP	1,710	7.83%	3	1
Total Valid Votes		21,841	100%	22	11

Source: St Kitts and Nevis Election Center, Caribbean Elections.

The second largest party got no seats, while two parties with less than 10% each won a seat or two. This is a first-past-the-post system. The problem the PAM had was it came in second in all eight seats it contested, i.e., every district on the island of St. Christopher (none were close). The advantage the CCM and NRP had is they run only on the island of Nevis, which has three district. Here are the district results.

Constituency	Registered voters	SKNALP	PAM	CCM	NRP	Valid Votes
St Christopher #1	4,519	1,788	1,149	–	–	2,937
St Christopher #2	5,652	2,011	1,507	–	–	3,518
St Christopher #3	2,596	1,235	377	–	–	1,612
St Christopher #4	2,430	1,013	735	–	–	1,748
St Christopher #5	2,328	869	769	–	–	1,638
St Christopher #6	2,571	1,613	119	–	–	1,732
St Christopher #7	2,874	1,441	479	–	–	1,920
St Christopher #8	4,325	1,792	1,333	–	–	3,125
Nevis #9	2,924	–	–	808	796	1,604
Nevis #10	1,517	–	–	555	184	739
Nevis #11	2,430	–	–	538	730	1,268
Total	34,166	11,762	6,468	1,901	1,710	21,841

Source: Same as for first table.

Note that there is some pretty serious malapportionment here, as well. Nevis constituencies have many fewer voters than St. Christopher constituencies. In fact, the three Nevis districts together have only about 1.2 times the population of the most populous St. Christopher district.

So what should we have according to the Seat Product Model? The seat product is 11 (magnitude of 1, times assembly size of 11), so the effective number of seat-winning parties should be 1.49. In this election it was actually 1.75. That’s actually not a terrible miss! But in most elections it has been considerably higher than that–as high as 3.90 in 2015. So just for fun, a quick look at that one:

	Party	Votes	votes	% votes	Candidates
	St. Kitts and Nevis Labour Party	SKNLP	11,897	39.27%	8	3
	People’s Action Movement	PAM	8,452	27.90%	6	4
	People’s Labour Party	PLP	2,723	8.99%	2	1
	Concerned Citizens Movement	CCM	3,951	13.04%	3	2
	Nevis Reformation Party	NRP	3,276	10.81%	3	1
	Total Valid Votes		30,299	100%	22	11

(Last column is seats won, but the heading did not copy over.)

This time, the PAM benefitted greatly! It is in a clear second place in votes, yet won a plurality of seats. Not a majority, however. According to Wikipedia, there were alliances. But even at the alliance level, there was a plurality reversal: “The outgoing coalition (SKNLP and NRP) secured 50.08% of votes but got only 4 seats, the winning coalition (PAM, PLP and CCM) won 7 seats with only 49.92% of votes.” Oh, cool: Another case of pre-electoral alliances! The effective number of alliances was just 1.86.

And at the district level:

Constituency	Registered Voters	SKNLP	PAM	PLP	CCM	NRP	Valid Votes
St. Christopher #1	5,036	1,727	1,731	–	–	–	3,458
St. Christopher #2	4,740	1,758	1,660	–	–	–	3,418
St. Christopher #3	3,265	1,348	–	1,076	–	–	2,424
St. Christopher #4	3,166	1,216	1,252	–	–	–	2,468
St. Christopher #5	3,107	884	1,245	–	–	–	2,129
St. Christopher #6	2,823	1,969	200	–	–	–	2,169
St. Christopher #7	3,191	867	–	1,647	–	–	2,514
St. Christopher #8	5,753	2,128	2,364	–	–	–	4,492
Nevis #9	6,127	–	–	–	2,033	1,715	3,748
Nevis #10	1,393	–	–	–	754	306	1,060
Nevis #11	3,584	–	–	–	1,164	1,255	2,419
Total	42,185	11,897	8,452	2,723	3,951	3,276	30,299

We might not expect regionalism in such a small country, with a small assembly. But the party preferences of the two islands obviously are genuinely different (and the PLP is “regional” in that it contested only two districts on St. Christopher); yet the parties aggregate into alliances for purposes of national politics.

The malapportionment is still noteworthy–look at the small population of Nevis 10. However, one of the other two districts is now the most populous in the country, quite unlike in 2000.

Final point: Its population may be small, but according to the cube root law St Kitts and Nevis should have an assembly more than three times what it actually has: 37. If they were proportional to registered voters, Nevis would be allotted nine of those 37 seats. It currently has 3 of the 11, so 27%, so quite close to its population share, unlike in 2000 when it was overrepresented. Making the seats allocated by island more easily fit population balance in itself would be a good argument for increasing assembly size, but an even better argument would be making anomalous results like the two elections shown here less likely–even if they insist on sticking with FPTP.

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.

_______

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) should be around 2.64. In the actual result–with five districts still to be called–the largest party, Liberal, 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.

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]

---

^[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.

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.

Votes, seats, and exit polls: UK 2019 edition

Two political scientists, Pippa Norris and Patrick Dunleavy, have accused the BBC and others of “systemic media bias” on the recent UK election night for not emphasizing the voting outcome and instead focusing on the seats. Their claims appear at the LSE blog. Of course, I am very much inclined to agree that votes and seats both matter–I’ve (co-) written two books that have both words, votes and seats, in their titles, after all! Thus I largely agree with Norris and Dunleavy’s bigger point that media coverage in majoritarian electoral systems tends to exaggerate the notion that a party that wins the seat outcome has a “mandate”. As I said in my own election post-mortem, the “mandate” claim is a stretch, at best, and very much depends on how the electoral system manufactures majorities–not only for Conservatives overall but also for the SNP among Westminster constituencies within Scotland.

Nonetheless, the claims in the LSE blog piece are somewhat hard to swallow. The main argument is that at 10:00 p.m., when polls closed, only the seats were mentioned. The votes did not come till 5:00 a.m., they claim. Anthony B. Masters has already shown that is not actually true, in a really excellent rebuttal. I won’t repeat Masters many points regarding misleading evidence that the LSE blog authors present to make their case.

The deeper issue here is that the exit poll is bound to be more accurate for seats–the initial projection almost nailed the result for the UK as a whole–than for votes. The voting estimates are subject to more error, because of uncertainty about turnout. Moreover, seats are the currency of power. Votes are relevant as a “currency of legitimacy” (as Jonathan Hopkin put it on Twitter), which is important for the subsequent narratives and intraparty soul-searching for the losers. That is, however, very much the kind of stuff that can only happen once the full results are known (not that it stops the media talking heads from engaging in speculation all night long). Basically, it is just very odd to slam as “biased” the media for reporting what was proven to be an actually accurate projection of the one thing the poll was designed to do and that matters most on election night–who won the most seats, was it a majority, and if so, how big?

Besides, as Masters notes in his rebuttal, it is not even true that votes were not being reported all night long. They simply are subject to more revisions as the picture gets clearer because, as noted above, the vote estimate is subject to more error.

Finally, I’d note that it could be much worse. In US elections, the topic of votes hardly comes up in the media, particularly for congressional elections. Even if you stayed up till 5:00 a.m. on election night (not that I ever have), you would not hear what percentage of the House votes each party had.

Reminder from UK 2019 result: Electoral systems matter

Keep this in mind about the UK result. The Conservatives won less than 44% of the vote. Polling has consistently shown that if there were another referendum on Brexit, a majority would vote for Remain. But the Conservatives won 56% of the seats, so Johnson is banging on about his great “mandate” to “get Brexit done”.

You see, electoral systems matter.

Even if you add in the Brexit Party votes (which got no seats), the combined votes cast for parties still advocating outright for leaving the EU do not reach a majority. In fact, it barely breaks 45%.

Meanwhile, the SNP has won 81% of the Scottish seats, with 45% of the votes cast in Scotland. And their leader, Nicola Sturgeon, is going on and on about the mandate for Scotland to decide on independence. It’s a fishy claim.

Which party gained the most in votes, relative to the last general election? That would be the Liberal Democrats. But the party suffered a net loss of one seat (and its leader was defeated).

The first-past-the-post (FPTP) electoral system makes a country seem more divided than it is, and often leads to policy outcomes a majority of voters actually oppose.

FPTP certainly is not very representative. But it can produce a decisive government, and Boris Johnson now looks like he could take his place among the significant Prime Ministers in the country’s recent history.

At least this result means my old lectures about British majoritarianism do not to be heavily caveated as they’ve been for the past several years.

UK election 2019

The UK general election is almost here. At this point, it seems quite unlikely that the result will be anything other than a good old fashioned FPTP manufactured majority. Boris Johnson and his Conservatives will win a majority of seats, barring a surprise, despite under 45% of the votes, and will be able to pass their Brexit deal.

If one looks at the polling aggregate graph by the Economist, one might be tempted to conclude it was also a good old fashioned “Duvergerian” pattern at work. As recently as early October, before the election was legislated, the Conservatives were leading on about 33% of the votes, and three other parties ranged from 12% to 25%. Go back further, to June, and all for were in the 18–25% range (with Labour then on top, and the Brexit Party ahead of the Conservatives). Since the latter part of October, and especially since the campaign formally got underway, Conservatives and Labour have both taken off, at the expense of the LibDem and Brexit parties. Notably, the gap between the top two has been quite steady, at 8-10 percentage points. Unlike 2017, there is no evidence at all that Labour is closing the gap. Labour simply are hoovering up the non-Tory (and Remain or second-referendum) votes at the same time as Leave voters have realized there’s no point in voting for a single-issue Brexit Party when the Tories have a pretty “hard” Brexit deal already to go, if only they win a majority of seats.

So, on the one hand, a far more “normal” election for a FPTP-parliamentary system than seemed possible during the long parliamentary deadlock of the past year or more. Just like Duverger’s “law” predicts, right? Desertion of the third and fourth parties for the top two.

Only sort of. Let’s take the current polling estimates for the parties (and not forgetting to include the current 5% “other”, which I will treat as one party, given most of it is one party–the Scottish National Party). It results in an effective number of vote-earning parties of 3.05. That’s a little high for a supposedly classic two-party system! It is, however, lower than seen at any election from 1997 through 2015. In 2017, however, it was 2.89, which was the lowest since 1979. The top two would be combining for 78% of the votes, which is a little higher than most elections from 1974 (February, in a two-election year) through 2001. Even in 2017, hailed by many at the time as the return to two-party politics–albeit dubiously–had a combined top-two of just 82.4%. (It looks like a high figure only compared to 2005-2015, when it ranged from 65.1% to 67.6%.)

Of course, it is the seats that really matter. Seat projections based on election polls under FPTP are never easy. There are various ones out there, but I will go with YouGov‘s.* It has the Conservatives with a projected 359 seats, which is 55.2%, with Labour on 211 (32.5%). Taking all the parties (and here breaking the “Northern Ireland” bloc down a bit, as we know it will consist of more than one such party), we get an effective number of seat-winning parties around 2.4. That is even lower than 2015, driven mainly by the presence of an expected single-party majority.

[*Note: just after I posted this, YouGov posted an update of their projections. I am not going to revise the numbers here. The differences are small, though potentially politically significant. See my first comment below this post.]

The problem with the standard Duvergerian claims about FPTP is that they ignore assembly size: In a larger assembly, we should expect more parties, other things (like district magnitude and formula) equal. While we could argue over how much the expected results of the 2019 election correspond to the so-called law, I’d rather not. What is of interest to me is that the UK case continues its long-term defiance of the Seat Product Model (SPM), and that’s something that I can’t take lying down.

While the conventional wisdom would see 2017 and 2019 as some sort of return to normalcy, it’s actually a challenging case for me. From the SPM (which explains over 60% of the variation in party-system outcomes worldwide, including FPTP systems), we should expect:

Effective number of seat-winning parties: 2.95.

Seat share of the largest party: 0.445.

Effective number of vote-earning parties: 3.33.

The seat outcomes actually never have come very close to the expectations. As for votes, the 1987 election got it right, but was a terrible performer in terms of seats (effective N=2.17!). Taking all the indicators together, the 2010 election is about the closest to what should be “normal” for a FPTP system with such a large assembly: effective N on votes 3.72, seats 2.57, and largest seat share of 0.47. So why was that not finally the start of the kind of party system the country “should” have? I guess we need to blame Nick Clegg. Or David Cameron. (I’d rather blame the latter; he was the one, after all, who thought a Brexit referendum was a good enough idea to go ahead with it.) More to the point, voters’ reaction to Clegg and the LibDems entering a coalition and–gasp–making policy compromises. After which, voters reverted to supporting the big two in greater shares than they are supposed to. In other words, contingency and path dependency overcome the SPM in this case. I hate to admit it, but it’s the best I’ve got!

Speaking of the LibDems, they should have had an opportunity here. Labour has the most unpopular opposition leader in decades. (Deservedly so, but I digress.) And the best hope for stopping Brexit would be tactical voting to increase their chances to win seats where Labour is not best positioned to defeat a Tory. Yet, despite lots of constituency-level tactical voting advice being offered in this campaign, there’s little evidence the message is getting though.

There is tactical voting happening, but as Rob Johns points out in a short video, it is happening based on the national outcome and not on district level. Under the Duvergerian conventional wisdom, voters are alleged to think of their constituency, and vote tactically (strategically) to effect the local outcome. Yet in real life, only a relatively small minority of voters behave that way. That voters use a strategy based on who is best placed to defeat a party they do not like on the national level, instead of at the constituency level, is a point made forcefully by Richard Johnston in his book, The Canadian Party System. It is also the underlying logic of the SPM itself.

So from the standpoint of the SPM, what is surprising is not that there isn’t more tactical voting at the constituency level. It is that there does not remain (so to speak) a strong enough third party, such as the Liberal Democrats, to appear viable nationally so that voters would be willing to vote for its district candidates. Quite apart from the legacy of the coalition that I referred to above, the case for the LibDems as a viable counterweight probably was not helped by a tactical decision it made in this campaign. Its leader, Jo Swinson, declared that a LibDem government would revoke the Article 50 notification and cancel Brexit. Put aside the ridiculous idea that there would have been a LibDem government. If one had resulted from this election, it would have been on far less than 50% of the votes. So you have a government resting on a minority promising to go back on the majority voice of the 2016 referendum without even bothering with a second referendum. That seemed at the time like a dumb position for the party to take. Only recently has Swinson offered the message of what the LibDems could accomplish in a no-majority parliament. But it’s too late. There almost certainly won’t be such a parliament.

The UK really needs a national third party (and fourth…). Contrary to the Duvergerian conventional wisdom, the electoral system actually could sustain it; we would expect the party system to look more like Canada’s (which conforms to the SPM very well, both over time and, in terms of seats, in 2019). Given the larger assembly, the British party system should be even less two-party dominated than Canada’s actually is. It is by now rather apparent that the LibDems are not the third party the system needs to realize its full potential. Will one emerge? Alas, not soon enough to stop a hard Brexit from being implemented by a manufactured majority (for a leader who is pretty unpopular himself) while Labour gobbles up most of the opposition, but falls well short.

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.

Is AV just FPTP on steroids?

In debates over electoral systems in Canada, one often hears, from otherwise pro-reform people, that a shift to the alternative vote would be worse than the status quo. It is easy to understand why this view might be held. The alternative vote (AV), also known as instant runoff (IRV), keeps the single-seat districts of a system like Canada’s current first-past-the-post (FPTP) system, but replaces the plurality election rule in each district with a ranked-ballot and a counting procedure aimed at producing a majority winner. (Plurality winners are still possible if, unlike in Australia, ranking all candidates is not mandatory. The point is that pluralities of first or sole-preference votes are not sufficient.)

Of course, the claim that AV would be FPTP on steroids implies that, were Canada to switch to AV, the current tendency towards inflated majorities for a party favored by less than half the voters would be even more intensified. This is plausible, inasmuch as AV should favor a center-positioned party. A noteworthy feature of the Canadian party system is the dominance, most of the time, by a centrist party. This is unusual in comparison with most other FPTP systems, notably the UK (I highly recommend Richard Johnston’s fascinating book on the topic). The party in question, the Liberal Party, would pick up many second preferences, mainly from the leftist New Democratic Party (NDP) and so, according to the “steroids” thesis, it would thus win many more seats than it does now. It might even become a “permanent majority”, able to win a parliamentary majority even if it is second in (first-preference) votes to the Conservatives (who thus win the majority or at least plurality of seats under FPTP). The “steroids” claim further implies that the NDP would win many fewer seats, and thus Canada would end up with more of a two-party system rather than the multiparty system it has under FPTP.

There is a strong plausibility to this claim. We can look to the UK, where AV was considered in a referendum. Simulations at the time showed that the Liberal Democrats would stand to benefit rather nicely from a change to AV. While the LibDems are a third party, heavily punished by the FPTP electoral system even when they have had 20% or so of the votes, what they have in common with the Canadian Liberals is their centrist placement. Thus, perhaps we have an iron law of AV: the centrist party gains in seats, whether or not it is already one of the two largest parties. An important caveat applies here: with the LibDems having fallen in support since their coalition with the Conservatives (2010-15), the assumptions they would gain from AV probably no longer apply.

On the other hand, we have the case of the Australian House of Representatives, which is elected by AV. There, a two-party system is even stronger in national politics than in the FPTP case of the UK, and far more so than in Canada. (When I say “two party” I am counting the Coalition as a party because it mostly operates as such in parliament and its distinct component parties seldom compete against one another in districts.)

It is not as if Australia has never had a center-positioned party. The Australian Democrats, for example, reached as high as 11.3% of the first-preference votes in 1990, but managed exactly zero seats (in what was then a 148-seat chamber). Thus being centrist is insufficient to gain from AV.

Nonetheless, the combination of centrism and largeness does imply that Canada’s Liberals would be richly rewarded by a change to AV. Or at least it seems that Justin Trudeau thought so. His campaign promised 2015 would be the last election under FPTP. While he did not say what would replace it, he’s previously said he likes a “ranked ballot” and he pulled the plug on an electoral-reform process when it was veering dangerously towards proportional representation.

Still, there are reasons to be somewhat skeptical, at least of the generalization of the Australian two-party experience. The reasons for my caution against the “steroids” view are two-fold: (1) the overlooked role of assembly size; (2) the ability of parties and voters to adapt.

Assembly size is the most important predictor of the size of the largest party, disproportionality, and the effective number of seat-winning parties in countries that use single-seat districts. (It is likely relatively less important when there are two rounds of voting, as in France, but still likely the most important factor.) This is a key conclusion of Votes from Seats. It is thus important not to overlook the fact that Australia has an assembly size considerably smaller than Canada’s. In the book, Taagepera and I show that Australia’s effective number of seat-winning parties and size of largest parliamentary party are almost what we would expect from its assembly size, even if FPTP were used. (See also this earlier post and its comment thread; how close it is to expectation depends on how we count what a “party” is.) The data are calculated over the 1949-2011 period, and the effective number of parties has been just 1.10 times the expectation from the Seat Product Model (which is based only on assembly size when single-seat districts are used). Similarly, the average largest party has been 93% of the expected size (averaging 50.5%  of seats when we would expect 54.2%).

Thus we do not need to invoke the alleged steroids aspect of AV to understand the dominance of two parties in Australia. But this does not mean it would not make a difference in Canada. Consider that the current effective number of parties and size of the largest party in that country, averaged over a similar period, are also just about what we should expect. The multipartism, including periodic minority governments, that characterize Canada are not surprising, when you use the Seat Product Model (SPM). They are surprising only if you think district magnitude is all that matters, and that FPTP is FPTP. But it isn’t! An electoral system using the FPTP electoral rule with an assembly of more than 300 seats is a different, and more multiparty-favoring, electoral system than one with 150 seats. Replace “FPTP” in that sentence with “AV” and it is surely still true.

But what about the centrist party, the Canadian Liberals? Surely AV would work differently in this context, and the Liberals would be a much more advantaged party. Right? Maybe. If so, then it would mean that the SPM would be overridden, at least partially, in Canada, and the largest party would be bigger than expected, for the assembly size, while the effective number of parties would be lower than expected. Of course, that’s possible! The SPM is devised for “simple” systems. AV is not simple, as we define that term. Maybe the SPM is just “lucky” that the one country to have used AV for a long time has the expected party system; or it is lucky that country has the “correct” assembly size to sustain two-party dominance. (Australia is the Lucky Country, after all, so if the SPM is going to get lucky somewhere, it might as well be Australia.)

This is where that other factor comes in. While no one has a crystal ball, I am going to go with the next best thing. I am going to say that the SPM is reliable enough that we can predict that, were Canada to have AV, it would have an effective number of parties around 2.6 and a largest party with around 48% of seats. In other words, just about where it has been for quite some time (adjusting for the House size having been a bit smaller in the past than it is now). Note these are averages, over many elections. Any one election might deviate–in either direction. I won’t claim that a first election using AV would not be really good for the Liberals! I am doubting that would be a new equilibrium. (Similarly, back in 2016 I said my inclination would not be to predict the effective number of parties to go down under AV.)

Parties and voters have a way of adapting to rules. Yes the Liberals are centrist, and yes the Conservatives are mostly alone on the right of the spectrum (albeit not quite as much now, heading into 2019, as in recent years). But that need not be an immutable fact of Canadian politics. Under AV, the Liberals might move leftward to attract NDP second preferences, the NDP center-ward to attract Liberal and even Conservative second preferences, the Conservatives also towards the center. It would be a different game! The Greens and other parties might be more viable in some districts than is currently the case, but also potentially less viable in others where they might win a plurality, but struggle to get lower ranked preferences. The point is, it could be fluid, and there is no reason to believe scenarios that have the largest party increasing in size (and being almost always the Liberals), and correspondingly the effective number of parties falling. With 338 or so districts, likely there would remain room for several parties, and periodic minority governments (and alternations between leading parties), just as the SPM predicts for a country with that assembly size and single-seat districts.

As I have noted before, it is the UK that is the surprising case. Its largest party tends to be far too large for that huge assembly (currently 650 seats), and its effective number of seat-winning parties is “too low”. Maybe it needs AV to realize its full potential, given that the simulations there showed the third party benefitting (at least when it was larger than it’s been in the two most recent elections).

Bottom line: I do not buy the “FPTP on steroids” characterization of AV. I can understand were it comes from, given the presence in Canada of a large centrist party. I just do not believe Liberal dominance would become entrenched. The large assembly and the diversity of the country’s politics (including its federal structure) both work against that.

I agree with electoral reformers that PR would be better for Canada than AV. I also happen to think it would be better for the Liberals! But would AV be worse than FPTP? Likely, it would not be as different as the “steroids” claim implies.

New Brunswick 2018

The Canadian province of New Brunswick held its provincial assembly election on 24 September. The result is an assembly with no majority of seats.

The incumbent Liberal government, which won a majority in 2014 but had fallen to minority in the interim, came in second place in seats but first in votes in the 2018 election. The main opposition, Progressive Conservatives (PC), won just one seat more, and are short of a majority.

The Liberals have 21 seats on 37.8% of the vote, the PCs 22 seats on 31.9%, while a previously underrepresented party, People’s Alliance (PANB) and the Green Party each won three seats. The PANB won 12.6% of the vote and the Greens 11.9%. The NDP won 5% of the votes, but no seats.

The district-level results are interesting. The People’s Alliance leader, Kris Austin, won a clear majority (54.6% in Fredricton-Grand Lake, with the runner up being a PC incumbent with only 27.7%; in 2014, Austin had lost to the PC candidate 28.8%-28.5%!). In another riding, Fredricton-York, the PANB candidate defeated another PC incumbent, 33.7%-30.9%. The third PANB winner was in Miramichi and won 57.0% to 35.0% over a Liberal. I counted six other seats in which a PANB candidate came in second, although only one of these was really close (Southwest Miramichi-Bay Du Vin, where a PC has 35.4% over the PANB on 35.0%). The three districts the PANB won and the one where they are very narrowly behind, are all contiguous. It is clearly a regional party; it ran in 30 of the 49 ridings.

As for the Green winners, leader David Coon, who was their first elected MLA (2014) retained his seat easily, 56.3%-20.1% over a Liberal. In Kent North their candidate won 45.9%-37.4% over a Liberal. In Memramcook-Tantramar, Megan “Landslide” Mitton won by 11 votes (38.3%-38.2%) over a Liberal.  It seems there are two districts in which a Green came in second, but neither was close; in both cases the Liberal winner had a majority. The Green wins are not contiguous districts; the leader represents a seat in Fredricton and the other two are geographically large coastal districts. (See results and map at CBC; these are, of course, not necessarily final at this point, and there is even one Liberal lead of just 10 votes over a PC.)

It is not clear what the government result will be. I’ve been listening to CBC on the post-election discussions, and it seems the Greens have rejected a possible coalition with the Liberals; given that the results revised above suggest the Liberals are the Greens’ main opponent at district level, this reluctance has some (FPTP-based) logic to it. The Conservatives have said they will vote down a Liberal throne speech (not surprisingly). The PCs have declared all of their members are unwilling to stand for Speaker, and the Liberals also do not want any of their own to take the post. Without a Speaker, no other business can be transacted. So, for now at least, we have a stand off. (Update: The Liberal leader and current Premier Brian Gallant has said his party will put forth a candidate for Speaker today.)

It is worth noting that New Brunswick has quite a record of unusual election outcomes, and electoral-reform proposals. Just click “N.B.” at the bottom of this post to see previous entries on this recent history. Of particular interest is the time the Liberals took power thanks to a plurality reversal and promptly called off the previous (Conservative) government’s planned referendum on adopting MMP. Maybe it is time to dust off those proposals. The voters of the province seem unwilling to play the old FPTP game the way “the law” says to play it.