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 (NS) should be the seat product, raise to the power, 1/6. The seat product is the assembly size, times the mean district magnitude. The SPM predictions for NS 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 NS=S1/6. Let’s call this “Equation 1.”

For Canada’s current assembly size (338), this means NS=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 NS=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=1.59S1/12 (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 (NS) 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 NS 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 NS. We can derive an expected connection of N’V to NS via basic algebra. We just substitute the value from one equation into the other (using Equations 1 and 2). If we have NS=S1/6 then it must be that S= NS6. So we can substitute:

N’V=1.59(NS6)1/12= 1.59√NS (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 22/3; 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 NS suggests that parties that win some share of seats (i.e., can contribute more or less to the value of NS) 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 NS 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 NS. 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 NS 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 NS (i.e., Equation 3). We have actual mean N’V=2.485 compared to expected N’V from actual NS 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 NS) 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 NS 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 NS (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 NS 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 NS 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-indued 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.

The Brexit Party

Just a quick add-on to my previous remarks on the UK 2019 election. Via @kiwiting on Twitter comes this example of a Brexit Party local leaflet.

Look closely and you might actually see the local candidate’s name! As I stress in the preceding post, I expect parties under FPTP (at least in parliamentary systems) to require a national presence in the party system in order normally to do well at the constituency level. That is a key insight of the Seat Product Model, and how it stands apart from “bottom-up” approaches that stress local district-level “coordination” as what drives a party system. But this is pretty extreme: the Brexit Party is not only a single-issue party, it is also a one-man band!

Even though this party at one point was polling above 20% (and won a plurality of the UK vote in the European Parliament elections), it was always hard for me to take the Brexit Party seriously. On the one hand, it certainly is a nationally focused party. On the other hand, the leader Nigel Farage made a decision not to contest any constituency, or to target even one seat somewhere that some candidate of the party might win. The process behind the SPM implies that voters respond to the “viability” of a smaller party, and tend to vote for it without too much regard for the viability of its candidate in their own district. But for that to work, it has to be viable–and preferably winning–somewhere. Not only did the Brexit Party not even try this, it pulled its candidates out of seats the Conservatives hold, while retaining candidates only in districts held by other parties. It is a bizarre strategy if the party was serious, and it is no wonder the party is on life support. Of course, they are going to get their one policy issue enacted (even if not as “hard” as they would like), precisely by not posing too big a risk to the incumbent government’s pursuit of a (manufactured) majority.

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 (NS), 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 (NV) 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 Nbased on the known NS=2.79, we would expect NV=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 NS. 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 NS, 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 NV 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.

Israel 2019b, compared to 2019a

Here, following up on the earlier discussion of post-election bargaining scenarios, I want to compare Israel’s two elections of 2019 on several statistical measures. The 2019b (September) results are not quite official yet, but are very unlikely to change other than in the smallest of voting detail.

The table below compares the votes for Netanyahu’s “Bibi bloc” of right-wing and Haredi parties, by various definitions, as well as the indicators of fragmentation: effective number of parties by seats and votes, total number of lists with seats, and the seats won by the largest list. For each measure, there is a comparison of change from April to September. The final three columns refer to output of the Seat Product Model (SPM) for the indicators of fragmentation–what is expected from the model (given an assembly size of 120 and district magnitude also of 120), and ratios of the actual indicators to the expectation.

Measure April Sept change SPM expected Ratio, April Ratio, Sept.
Bibi bloc (percent votes) 48.7 44.5 -4.2
… plus YB 52.7 51.5 -1.2
… plus Otzma 46.4
… plus YB & Otzma 52.7 53.3 0.6
Effective N, seats 5.24 5.67 0.43 4.93 1.06 1.15
Effective N, votes 6.33 6.11 -0.22 5.23 1.21 1.17
No. of lists with 1 or more seats 11 9 -2 11 1.00 0.82
Seats for largest list 35 33 -2 36 0.97 0.92

The scale of the defeat for the core Bibi bloc is clear. Already in April, these parties had less than 50% of the votes, at 48.7%, which is why they won only 60 seats under Israel’s proportional system. If we include Yisrael Beiteinu in the total Bibi bloc, we get 52.7% (which is why this larger definition of the bloc had 65 seats). As I have explained already–both before and after the most recent election–we should not count YB in the bloc, particularly since it was this party’s actions that precipitated the early elections of 2019–yes, both of them.

In the second election of 2019, this Bibi bloc fell to 44.5% of the vote, a drop of 4.2 percentage points. If we include YB, they do have a narrow majority of votes (51.2%), but we should not include them. However, we probably should include Otzma Yehudit, given that it was part of the Union of Right Wing Parties in April, and probably would have been invited to join a coalition had it cleared the threshold in the September election. But still this is short of a voting majority without YB, at 46.4% (which would mean a loss of 2.3 percentage points off the April showing of 48.7).

For a baseline, consider that the Bibi bloc had 48.4% in 2015, or 53.5% including YB (which was without doubt part of the bloc at that time–their staying out of the coalition initially in 2015 was a surprise). Note that, leaving out YB, they were already below majority voter support in 2015, but had managed 61 seats. The reason they gained ever so slightly in votes in April, yet got only 60 seats, was all the wasted votes for New Right (3.22%), which did not clear the threshold in the April, 2019, election.*

If we include both Otzma and YB in the 2019b election, it looks like a very small gain for the wider bloc. But we should not do this because some of YB’s increased votes probably came from Blue and White or other parties not in the right, due to YB’s promise not to return to a Likud-led government unless it was a “unity” government with Blue and White.

On the fragmentation indicators, the effective number of seat-winning parties went up, from 5.24 to 5.67, despite the drop of the total number of lists winning seats, from 11 to 9. The increase in the effective number is due to the smaller size of the largest party in the more recent election, 33 seats (Blue and White) vs. 35 (tie between Blue & White and Likud).

The effective number of vote-earning parties came down somewhat, from 6.33 to 6.11. None of these measures is much different than what we should expect under the SPM, although the raw number of represented lists this time is actually smaller than expected, while the effective number of seat winning parties was closer to the expectation in April than now.

We should expect the largest party, given this electoral system, to have 30.2% of the seats, which out of 120 works out to 36 (rounded down). The election pretty much nailed that in April, but this election saw a return to a smaller than expected plurality party.

So, strictly from the SPM, this was a slightly less “normal” election than 2019a, although not too far off. From the standpoint of the usual pattern with a “b” election (a second one within a year), it was, as I anticipated, a little unusual. Typically, the effective numbers go down and the size of the largest up. Israel went the opposite way between April and September, and thus government formation still will not be easy.


* We could go back and include Yachad (of which Otzma Yehudit was a part) in the 2015 count, which would bring it to 51.3%, but at the time I do not recall their being taken seriously as part of the bloc. Doing so, of course, increases the scale of the loss of voter support already as of the first election of 2019.

Israel is about to have a very unusual ‘b’ election

Israel is about to hold its second election of 2019, and it will be unusual, relative to other cases of a second election within a year elsewhere. While the number of lists winning seats is likely to go down, other indicators of fragmentation are likely to go up.

Using the National Level Party Systems Dataset (Struthers, Li, and Shugart, 2018), I performed calculations to find out how the standard indicators of party-system fragmentation change from a first election that fails to produce a “stable” government or any government at all, leading to a second election. I looked at all cases in the dataset in which two elections were held in the same Gregorian calendar year, plus all cases where an election is in the second half of a year and followed by another in the first half of the next year. The first table below gives the full list, including the first and second election in each sequence. In one case in the dataset (Greece, 1989-1990) the second election was followed by yet another within a year, indicated by a “3” in the final column. Note that a country’s data sequence begins in the early post-WWII era or when a country democratized and ends in 2016, so any cases outside that timeframe are not included.

country year date mo within_yr_seq
Denmark 1953 4/21/53 4 1
Denmark 1953 9/22/53 9 2
Denmark 1987 9/8/87 9 1
Denmark 1988 5/10/88 5 2
Greece 1989 6/18/89 6 1
Greece 1989 11/5/89 11 2
Greece 1990 4/8/90 4 3
Greece 2012 5/6/12 5 1
Greece 2012 6/17/12 6 2
Greece 2015 1/25/15 1 1
Greece 2015 9/20/15 9 2
Iceland 1959 6/28/59 6 1
Iceland 1959 10/25/59 10 2
Ireland 1982 2/18/82 2 1
Ireland 1982 11/24/82 11 2
Japan 1952 10/1/52 10 1
Japan 1953 4/19/53 4 2
Japan 1979 10/7/79 10 1
Japan 1980 6/22/80 6 2
Moldova 2009 4/5/09 4 1
Moldova 2009 7/29/09 7 2
Spain 2015 12/20/15 12 1
Spain 2016 6/26/16 6 2
Sri Lanka 1960 3/19/60 3 1
Sri Lanka 1960 7/20/60 7 2
St. Lucia 1987 4/6/87 4 1
St. Lucia 1987 4/30/87 4 2
Thailand 1992 3/22/92 3 1
Thailand 1992 9/13/92 9 2
Turkey 2015 6/7/15 6 1
Turkey 2015 11/1/15 11 2
UK 1974 2/28/74 2 1
UK 1974 10/10/74 10 2

The list contains 17 cases of an election within twelve months of the preceding one. Not a large sample; fortunately, this sort of thing does not happen very often. (There are 1,025 elections in the sample.)

If elites and/or voters “learn” from the experience of bargaining failure or lack of stability from the first election in such a sequence, we would expect the second to be less fragmented. We can test this by looking at mean differences between the second election and the first. The indicators I have are the number of parties (or lists, more precisely, counting an independent as a “list” of one) that win at least one seat (NS0), the effective number of seat-winning lists (NS), the effective number of vote-earning lists (NV), the seat share of the largest party (s1), and the vote share of the largest party (v1). The first three should go down if there’s an adaptation occurring, while the second two should go up (i.e., the largest party gets bigger).

Here is what we see from the results, reporting the mean differences:

NS0: –0.215

NS: –0.098

NV: –0.469

s1: +0.010

v1: +0.0035

In terms of raw direction, all are as expected. On the other hand, the number of lists winning seats hardly budges (recall that the first number is the actual number, not “effective”), and the effective number on seats changes much less than the one on votes. The implication is that fewer votes are wasted in the second election, as we would expect. On the other hand, the seat share of the largest party–the single most important quantity because it determines whether there is a single-party majority and if not, how far from majority it is–rises by a very small amount, on average. That is partly due to most of these systems being proportional, so large shifts should be unusual. The complete list of elections and their indicators is provided in an appendix below.

As far as statistical significance is concerned, only in NV and v1 is the difference significant (NV at p<0.03; v1 at p<0.10), when comparing these “second” elections to all others. (This is not meant to be a sophisticated test; I am not comparing to a country baseline as I really should.)

We might expect that the first election in such a sequence is anomalously fragmented, hence the need for a second election to calm things down once again. That is also supported, for NV and v1 again, but also, crucially, for s1.

Now, how might the Israeli second election of 2019 compare? We can use the polling average from Knesset Jeremy (using the poll of polls from three weeks before the actual election), and compare to the actual results of 2019a (the first election in the sequence) and the previous election (2015). Also included in the Seat Product Model expectation.

measure 2019b (poll avg) 2019a actual diff 2015 diff SPM expected
NS0 9 11 –2 10 1 11
NS 6.04 5.24 0.801 6.94 –1.70 4.93
NV ? 6.33 ? 7.71 –1.38 5.24
s1 0.258 0.292 -0.034 0.25 0.042 0.3
v1 ? 0.2646 ? 0.234 0.031 0.289

For the number of lists that look likely to clear the threshold, we have the direction expected: currently there are 9 likely to win seats, compared to 11 in April. In turn, the April figure was one seat-winning list higher than in 2015. However, in terms of both NS and s1, the case is anomalous. All indications are that the largest party will be smaller than it was in April, which also will drive up the effective number. Moreover, these measures in April were less fragmented than they had been in 2015; that is, the first election of the 2019 sequence was not unusually fragmented. Quite the contrary; I called it a “normal” election at the time for a reason.

So the Israeli sequence of two elections in 2019 is unusual indeed.


Appendix

Below are two tables. One has all the “second” elections, and changes in the various measures. The second has all “first” elections. In each case, the comparison is just to the immediately preceding election (not to all other elections), so we can see how much short-term fluctuations were affecting the process in each sequence.

Elections ocurring within one year of previous, compared to previous results
country year mo diff_Ns0 diff_Ns diff_Nv diff_s1 diff_v1
Denmark 1953 9 1 -0.2199998 -0.1000001 0.014 0.009
Denmark 1988 5 -1 0.0100002 0 0.005 0.005
Greece 1989 11 1 -0.0800002 -0.1700001 0 0
Greece 1990 4 5 0.05 0.0700002 0.005 0.017
Greece 2012 6 0 -1.07 -3.75 0.07 0.108
Greece 2015 9 1 0.1490002 -1.19 -0.014 -0.008
Iceland 1959 10 0 0.24 . 0 .
Ireland 1982 11 -1 -0.01 0.03 0 0
Japan 1953 4 . 0.8099999 0.8999999 -0.088 -0.091
Japan 1980 6 -8 -0.3999999 -0.24 0.074 0.033
Moldova 2009 7 1 0.8699999 0.27 0 -0.048
Spain 2016 6 -1 -0.3700004 -0.7999997 0.04 0.043
Sri Lanka 1960 7 . -1.22 -2.52 0.166 0.032
St. Lucia 1987 4 0 0 -0.1099999 0 0.007
Thailand 1992 9 0 -0.0999999 0.0999999 0 0.017
Turkey 2015 11 . -0.322 0.03 -0.126 -0.089
UK 1974 10 -1 -0.01 -0.02 0.028 0.021
Election that is the first in a series of two within a year, compared to preceding election
country year mo diff_Ns0 diff_Ns diff_Nv diff_s1 diff_v1
Denmark 1953 4 0 -0.1300001 -0.0900002 0.013 0.008
Denmark 1987 9 0 0.27 0.5799999 -0.009 -0.023
Greece 1989 6 1 0.26 0.1400001 -0.044 -0.006
Greece 2012 5 2 2.24 5.79 -0.173 -0.25
Greece 2015 1 0 -0.6700001 -0.77 0.067 0.066
Iceland 1959 6 0 -0.28 . 0.035 .
Ireland 1982 2 -2 -0.05 -0.1699998 -0.039 0.009
Japan 1952 10 . . . . .
Japan 1979 10 -1 0.1199999 -0.2199998 -0.002 0.027
Moldova 2009 4 1 0.1400001 0.1600001 -0.079 0.035
Spain 2015 12 -3 1.93 3.23 -0.18 -0.159
Sri Lanka 1960 3 . 1.456 2.26 -0.206 -0.043
St. Lucia 1987 4 -1 0.55 -0.0800002 -0.295 -0.049
Thailand 1992 3 . . . . .
Turkey 2015 6 . 0.4320002 0 0.002 0.005
UK 1974 2 2 0.1900001 0.6900001 -0.05 -0.077

 

Ukraine honeymoon election today

Ukrainians are voting today in an assembly election. It is a relatively extreme “honeymoon” election, as the new president, Volodomyr Zelensky, was just elected in March-April of this year (two rounds). There was already an assembly election scheduled for October of this year, which certainly would have qualified as a honeymoon election. But in his inauguration, Zelensky announced he would dissolve the Verkhovna Rada and call an election even earlier.

And why not? Based on much experience in presidential and semi-presidential systems, we know that there is a strong tendency for the party of a newly elected president to gain a large boost in votes the earlier it is held following the presidential election. This topic of the impact of election timing has been a theme of my research ever since my dissertation (1988), an early APSR article of mine (1995), and most recently in a whole chapter of Votes from Seats (2017).

At the time Zelensky was elected, various news commentary had the all-too-typical concern that the new president would be weak, because he is an “outsider” with no established political party. We got similar useless punditry when Emannuel Macron was elected in France in 2017. And we know how that turned out–his formed-on-the-fly party did slightly better than the 29% of votes I projected, based on an equation in Votes from Seats, prior to Macron’s own runoff win. (The electoral system helped turn that into a strong majority in the assembly.)

In May of this year, I projected that Zelensky’s Servant of the People party could get around 34.5% of the votes in an election held on 28 July. (One week earlier obviously does not change anything of substance.)

Early polling had him short of this (not even 25% just before the presidential first round), but predictably, SoP has been rising in the polls ever since Zelensky took office. The party almost certainly will beat this projection, and may even have an electoral majority. If short of 50% of votes, the party still looks likely to win a parliamentary majority, given the electoral system (discussed below).

A bigger boost than average (where the average across systems with nonconcurrent elections is what my projections are based on) is to be expected in a context like Ukraine, in which the party system is so weak. That is, poorly institutionalized party systems would tend to exaggerate the normal electoral cycle effect. The effect will be only further enhanced by low turnout, as opponents of the new president have little left in the way of viable political parties to rally behind. Thus a performance in the range of the mid-40s to over 50% of the vote would not be a surprise.

As for the electoral system and election itself, Ukraine is using again (for now, at least) its mixed-member majoritarian (MMM) system. It consists of 225 single-seat districts, decided by plurality, and 225 closed-list proportional representation seats, in a single nationwide district. The two components are in “parallel”, meaning seats won by any given party in districts and seats won from party lists are simply summed; there is no compensatory process (as with MMP). There is a 5% threshold on the list component; quite a few small opposition parties may waste votes below this bar. Due to parts of the country being under Russian occupation, only 199 single-seat contests will take place.

In some past MMM elections in Ukraine, a large share of the single-seat districts have been won by independents or minor parties, whereas the national parties (such as they are) have, obviously, dominated the nationwide list seats. It is probably quite likely that this rather extreme honeymoon election will result in most of the seats in both components being won by “Servants.”

On that theme, a tweet by Bermet Talant makes the following points (and also has some nice polling-place photos) based on conversations with voters in Kyiv:

• Ppl vote for leaders. Few know other candidates on party lists, even top5

• Servant of the People = Zelensky. Bscly, ppl vote for him again

• In single-member districts, ppl vote for a party too, not candidate

This is, of course, as expected. It is a completely new party. Many voters will be wanting to support the new president who created the party. The identity of candidates will not matter, either on party lists (where at least the top ones might be known in a more conventional party) or in the districts (where the vote is cast for a candidate). The single-seat districts themselves are referred to as the “twilight zone” of Ukrainian elections in a fascinating overview of the candidates and contests in the district component published in the Kyiv Post. These contests attract “shady candidates” many of whom are “largely unknown”. If a given election lacks a strong national focal point, it would tend to favor independents and local notables. In an election with an exceptionally strong focal point–as in a honeymoon election, more or less by definition–that will benefit whoever has the “Servant of the People” endorsement.

The timing of the election, and the likely dominance of an entirely new pro-Zelenskyy party, really is presidentialization at its very “finest”.

I am just going to quote myself, in the final paragraph of an earlier post about Macron’s honeymoon election, as it totally applies here, too: “All of the above should serve as a reminder of two things: (1) the purpose of the upcoming election is to ratify the new executive’s direction, not to be a second chance for an alternative vision; (2) the honeymoon electoral cycle matters.”

Expect the new Verkhovna Rada to be Servants of Zelenskyy.

Did Greece just have a normal election?

The Greek general election of July, 2019, may have been about as “normal” as they get. After the country’s period of crisis–economic and political–things seem to have settled down. The incumbent party, Syriza (“Radical Left”), which saw the country through the crisis got booted out, and the old conservative New Democracy got voted in.

Of course, around here when we refer to an election as “normal” it means it conforms to the Seat Product Model (SPM). Applying the SPM to an electoral system as complex as that of Greece is not straightforward. However, based on some calculations I did from breaking the system down to its component parts (an approach I always advocate in the face of complexity), it seems we have a result that conforms to a plausible interpretation of its “expectation”.

The basics of the electoral system are as follows: there are 300 seats, of which 50 are an automatic bonus to the party with a plurality of the vote, while the remainder are allocated as if there were one nationwide district. The “as if” is key here. In fact, there are 59 districts. In other words, the district magnitudes in which the election plays out for voters and candidates are quite small. There are 12 seats in a nationwide compensatory tier [EDIT: see below], so we have 288 basic-tier seats for a mean district magnitude of around 4. (I am not going to go into all the further details of this very complex system, as these will suffice for present purposes; Election Resources has a great detailed summary of the oft-changed Greek electoral system.)

To check my understanding that the system is as if nationwide PR for 250 seats, plus 50 for the plurality party, I offer the following table based on the official results. Note that there are two columns for percent of seats, one based on 250 and the other based on the full 300. For the largest party, ND, the “% seats out of 250” is based on 108 seats, because we are not including the 50 bonus seats in this column.

Party % votes seats % seats out of 250 % seats out of 300
Nea Dimokratia 39.9 158 43.2 52.7
Syriza 31.5 86 34.4 28.7
Kin.Al 8.1 22 8.8 7.3
KKE 5.3 15 6.0 5.0
Elliniki Lysi 3.7 10 4.0 3.6
Mera25 3.4 9 3.6 3.0
14 others 8.1 0 0.0 0.0

We can see that the seat percentages out of 250 are close to the vote percentages, as we would expect if the system acts as if it were nationwide PR (not counting the bonus). More to the point, we would expect all parties, even the smallest that win seats, to be over-represented somewhat, due to the nationwide threshold. That is indeed what we see. Over 8% of the votes were wasted on parties that failed to clear the threshold. The largest of these, Laikos Syndesmos, had 2.93%. The threshold is 3%. No other party had even 1.5%.

It is clear that the system has worked in this election exactly as intended. The largest party has a majority of seats, due to the bonus, but even the percentages out of 300 are close to proportionality–far more than they would be if Greece tried to “manufacture” majorities via FPTP or two-round majority instead of “bonus-adjusted PR”.

The effective number of seat-winning parties (NS) is 2.70. It would have been 3.13 based on the indicated parties’ percentages of seats out of 250. So the bonus provision has reduced NS by 13.7%. (The effective number of vote-earning parties, NV, is 3.68, calculated on all the separate parties’ actual vote shares.)

But what about the SPM? With 288 seats in districts and 12 nationwide, we technically have a basic-tier seat product of 288 x 4 (total seats in the basic tier, times the mean magnitude). However, this includes the 50 bonus seats, which are actually assigned to districts, but clearly not allocated according to the rules that the SPM works on: they are just cream on top, not a product of seat allocation rules in the basic tier and certainly not due to compensation. So, what percentage of seats, excluding the bonus, are allocated in districts? That would be 288/300=0.96, which out of 250 yields a “shadow” basic-tier size of 240 (96% of 250). So our adjusted basic-tier seat product is 240 x 4=960.

In a “simple” system (no compensatory tier as well as no bonus), we would expect, based on the Seat Product Model formula, that the effective number of seat-winning parties would be NS=9601/6=3.14. We would expect the size of the largest party to be s1=960–1/8=0.424. Note that these are already really close to the values we see in the table for the 250-seats, pre-bonus, allocation, which are 3.13 and 0.432. I mean, really, we could hardly get more “normal”.

[Added, 14 July: The following paragraph and calculations are based on a misunderstanding. However, they do not greatly affect the substantive conclusions, as best I can tell. The system is two-tier PR, of the “remainder-pooling” variety. However, the 12 seats referred to as a nationwide tier are not the full number of compensatory seats. With remainder-pooling systems it is not always straightforward to know the precise number of seats that were allocated above the level of the basic tier. Nonetheless, the definition here of the basic tier seems correct to me, even if I got the nationwide portion wrong. Thanks to comments by JD and Manuel for calling my attention to this.]

Nonetheless, there is a nationwide compensation tier, and if we take that into account through the “extended” SPM, we would multiply the above expected values by 2.50.04=1.037, according to the formula explained in Votes from Seats. (The 0.04 is the share of seats in the upper, compensatory, tier; 100–0.96). This is obviously a minor detail in this system, because the upper tier is so small (again, not yet counting the bonus seats). Anyway, with this we get expected values of NS=1.037 x 3.14=3.26. We do not have a formula for the largest seat winning party (s1) in two-tier PR, but one can be determined arithmetically to lead to the following adjustment: s1=0.973 x 0.424=0.413. (This is based on applying to the extended SPM for Nthe formula, s1=NS–3/4, as documented in Votes from Seats [and its online appendix] as well as Taagepera (2007).) I believe these are the “right” figures for what we should expect the outputs of this system to be, on average and without taking election-specific politics into account, given this is not a “simple” (single-tier PR) system even before the bonus seats are taken into account.

Out of 250 seats, 41.3% is 103. The ND actually won 108 pre-bonus seats. The 50 bonus seats then would get the party to an expected 153, which would be 51.0%. It actually got 52.7%.

So, as we deconstruct the electoral system into its relatively simpler components, we get an impact on the party system that is expected to result in a bare-majority party. As for NS, values are generally around s1–4/3, which with s1=0.51, would be 2.45, which is somewhat lower than the actually observed 2.70. But perhaps the actual relationship of s1 to NS should be something between a “typical” party system with a largest party on 51% of the seats (2.45) and the party system we expect from 250 seats with Greece’s pre-bonus two-tier PR system (3.26). The geometric average of these two figures would be 2.82. The actual election yielded NS=2.70, which is pretty close. OK, so maybe the similarly of this value of NS to our “expectation” came out via luck. But it sure looks like as normal a result as we could expect from this electoral system.

Of course, in 2015, when there were two elections, the country was in crisis and the outcome was rather more fragmented than this. I am not sure when the 50-seat bonus was implemented; it used to be 40. So I am reluctant to go back to the pre-crisis elections and see if outcomes were “normal” before, or if this 2019 result is just a one-off.

For the record, in September, 2015, the largest party had 48.3% and NS=3.24; in January, 2015, the figures were 49.7% and 3.09. These are hardly dramatic differences from the expectations I derived above (51.0% and 2.82), but they are more fragmented (particularly in terms of higher NS albeit only marginally in terms of a lower s1). So, all in all, maybe the Greek electoral system is not as complex as I think it is, and all its elections fall within the range of normal for such a system. But this 2019 election seems normaler than most.

Finally, Israel has a totally normal election

[Updated with final results]
Israel has seemingly defied the Seat Product Model in recent years, with a top seat-winning party smaller than expected, and a number of parties greater than expected, based on its electoral system. To be fair to the Seat Product Model (SPM)–and who would not want to be fair to the SPM?–in earlier years of the state, the largest party had been bigger than expected and the number of parties smaller. On average, over its 70+ years, the State of Israel is pretty close to a normal country, at least as far as the SPM is concerned. But, oh, those fluctuations! And it had been many years since it was not overly fragmented, even given an electoral system that invites fragmentation through use of a single nationwide district.

At last, 2019 produced a result over which we can all sigh with relief. Someone got the memo, and the election produce a totally compliant result!

Here are the seat totals and percentages for each of the parties that cleared the threshold.

Likud 35 29.17
B&W 35 29.17
Shas 8 6.67
UTJ 8 6.67
Hadash-Ta’al 6 5.00
Labor 6 5.00
URWP 5 4.17
Yisrael Beitenu 5 4.17
Kulanu 4 3.33
Meretz 4 3.33
Ra’am-Balad 4 3.33
120 100.00

The Seat Product Model gives us a baseline expectation from the “seat product”, which is defined as the mean district magnitude, times the assembly size. Then the seat product is raised to a given exponent, based on deductive logic as to what the outcome of interest should be expected to be, on average. In the case of the largest party, the exponent is –1/8. The largest party in the 2019 Israeli election, Likud, is one seat off the 30% (which would be 36, which actually was the number in the preliminary count), at 29.17%; the expectation is a share of 0.302=(120 x 120)^–1/8. So the ratio of actual to expected is 1.036. So just about right on target.

The SPM exponent for the number of parties winning at least one seat is 1/4, which yields an expectation of 10.95. The actual number was 11. For the effective number of seat-winning parties, the exponent is 1/6, for an expectation of 4.93. The actual value from the above seat shares works out to 5.24, which is 1.062 times the expectation.

All in all, totally normal!

So it will be fun to update the following graph for my forthcoming chapter in the Oxford Handbook of Israeli Politics and Society, and show the lines for observed values over time coming back to the expected values, which are marked by the horizontal solid line in each plot. The dashed line marks the mean for the entire period, through 2015. Vertical lines mark changes in electoral-system features other than the district magnitude and assembly size–specifically formula changes or threshold increases. (I have not yet run calculations for deviation from proportionally for 2019.)

So, how did this happen, quite apart from the strong pull of the SPM, given that everyone presumably had plenty of time to read the book, which was published in 2017?

My main answer is strategic voting, following upon strategic alliance formation. The forging of the Blue & White alliance in late February, gave the opposition at least a sense of momentum and opportunity to defeat Netanyahu and Likud. The alliance surely benefited a great deal from voters deserting other parties in the opposition in order to bolster B&W. At the same time, many voters on the right no doubt feared B&W just might win, and so defected to the strongest party in the bloc, Likud. Never mind that this sort of within-bloc strategic voting is not entirely rational–the government will be the set of parties that can reach 61 votes, whether or not that set includes the largest party overall. Voters may not understand that fully, or may expect that if one of the top two parties could be at least a few seats ahead of the other, it might be politically difficult for the second to form the government even if it was mathematically feasible.

Such strategic voting would explain why Labor did so poorly. It had been polling near ten seats, which would have been bad enough for the once grand party. But that it ended up on an embarrassing six is probably attributable to strategic defection to B&W. Similarly, Meretz’s very close scare, winning only 4 seats on 3.63% of the votes. The threshold is 3.25%.

Speaking of the threshold, one of the big stories of the election was the failure of New Right to clear it, ending up at 3.22%, despite having been at 6-8 seats in most polls throughout the campaign. That, too, may be due to strategic defection, to either Likud itself or back to the alliance that New Right leaders Naftali Bennet and Ayelet Shaked split from, Bayit Yehudi (running within the new Union of Right Wing Parties).

The result shows two relatively dominant parties, each at 29.2%, and then a smattering of small parties. The third largest seat total is shared by the two ultra-orthodox parties, Shas and UTJ, which have just 8 apiece (6.7%). Seven other parties have 4-6 seats each. This is a result that actually makes a lot of sense for an electoral system with such a high seat product, which allows sectarian interest (different flavors of religious politics, different tendencies within the Arab minority, different strands of left-Zionism, etc.) to win representation, while still featuring two parties around which potential coalitions could form. (Leave aside for now the trouble B&W would have had forming a government even had it been a couple of seats ahead of Likud; it was still a potential alternative pole of attraction.)

In the recent past, I have felt that the low threshold–formerly 2% and even lower farther back in time–was not the issue driving fragmentation. And, in fact, the increases in the threshold in 2003 and 2015 (with the last increase actually leading to a moderately high threshold, not a “low” one) did little to bring fragmentation down, as the graph above shows. The driver of fragmentation was the absence of a real “big” party–with even Likud struggling to break 25%–and a surplus of mid-sized parties, which I am defining as parties with around 10-20 seats apiece. Well, this time the party system really looks different, with a leading party almost exactly the expected size, a second party its equal, and then a bunch of little parties. That implies that a somewhat higher threshold–either 4% or 5%–could make a difference, after all. Now would be a good time to seize the day, and form a unity government to do just that. Of course, that is unlikely to happen for various reasons, some of which I mentioned in the previous post. And high thresholds can have perverse outcomes, leading to greater risk of some relevant segment of the electorate being left out.

Speaking, still, of thresholds, I should acknowledge something about the fit to the SPM. The SPM formulas used above do not take thresholds into account. Why not? Simple. Because the formulas work without taking them into account! However, had there been no threshold, the Israeli result would have been different, obviously. Even if we assume no change in party/alliance formation in the absence of a threshold (massive and unrealistic assumption), three more parties would have won seats: Zehut (2.7%) and Gesher (1.7%), in addition to New Right. So then we are up to 14 parties, and some corresponding increase in the effective number and decrease in size of the largest.

In Votes from Seats, we propose some “first approximation” predictive models based on thresholds instead of the seat product. Given a threshold of 3.25%, these predict a largest party of 42.5% (or a little less with a “second approximation” that I will leave aside here), and an effective number of parties of 3.13. As we can see, these do not do so well on the Israeli election of 2019. So the SPM has it, notwithstanding the complication of the threshold making the SPM fit better than it might otherwise for this election.

Finally, a totally normal election in Israel.

The datasets for Votes from Seats

I have neglected to publish a link that has been available since December–the article announcing the datasets used in Votes from Seats (national and district) was published in Research & Politics (open access).

Citation:

Cory L. Struthers, Yuhui Li, Matthew S. Shugart, “Introducing new multilevel datasets: Party systems at the district and national levels” (December 20, 2018). https://doi.org/10.1177/2053168018813508

The abstract:

For decades, datasets on national-level elections have contributed to knowledge on what shapes national party systems. More recently, datasets on elections at the district level have advanced research on subnational party competition. Yet, to our knowledge, no publicly accessible dataset with observations of the party system at both national and district levels exists, limiting the ease with which cross-level comparisons can be made. To fill this gap, we release two corresponding datasets, the National Level Party Systems dataset and the District Level Party Systems dataset, where the unit of analysis is the party system within either the national or district jurisdiction. More than 50 elections in the two datasets are overlapping, meaning they include observations for a single election at both the district and national levels. In addition to conventional measures such as the effective number of parties, we also include underutilized variables, such as the size of the largest party, list type, and the vote shares for presidential candidates in corresponding elections.

The datasets themselves can be accessed directly at Dataverse.

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.

Is the effective number of parties rising over time?

I was recently having a conversation with another political scientist who showed me a graph that suggests the effective number of vote-earning parties in established democracies has been increasing over time. I was skeptical that it was, relative to baseline. Of course, if we do not have a baseline, we do not really know what is causing any such possible increase. The baseline should be the Seat Product Model, which tells us what we should expect the effective number of parties to be, given the electoral system. When we do the baseline, the increase over time remains, but is not significant.

Here is a graph with no baseline. It is just the the effective number of vote-earning parties (NV) in Western Europe (most countries–see below for notes on coverage). The scatterplot marks elections by a three-letter abbreviation for each country. The x-axis is years since 1945, the earliest election year in the dataset. The graph’s y-axis is unlogged, but the plotted regression curve and 95% confidence intervals are based on a logged NV.

(The regression is a GLS with random effects by country. It would not be much different if OLS were used.)

There does seem to be an increase over time. The regression estimates NV averaging around 3.42 in 1945 and around 4.60 in 2011. The 95% confidence intervals on those estimates are 2.96 – 3.96 and 3.98 – 5.32, respectively. So, yes, the vote is getting more fragmented over time in Western Europe!

But hold on a moment. We should look at the fragmentation relative to baseline. As shown in Votes from Seats, the seat product (mean district magnitude times assembly size; in a two-tier system, also taking into account the size of the compensatory tier) explains around 60% of the variance in key party-system outcomes, including the effective number of parties. It would be useful to know if the Seat Product Model (SPM) is on its way to being unable to account for party-system fragmentation if current trends continue. It would be useful to know if recent fragmentation is part of that other 40% (i.e., the amount of variance in NV that the SPM can’t account for). That is, are we witnessing some inexorable fragmentation of party systems that is resulting from the breakdown of existing party alignments in the electorate, and which electoral systems have begun to lose their ability to constrain? Should Western European countries go so far as to reduce their proportionality, in order to contain fragmenting trends?

So the next data visualization asks the question from a different perspective. Is the ratio of observed fragmentation to the SPM prediction increasing over time? We can take any given election’s actual NV, divided by the SPM-predicted NV to arrive at a ratio, which is equal to 1.00 for any election in which the result exactly matches the predicted value. (In other words, if R2=100%, all elections would have a ratio of 1.00.)

Here it is, for NV, again with the estimates from a GLS regression and the 95% intervals. In the regression, the ratio is entered as its decimal log, but the graph uses the underlying values for ease of interpretation.

What we see is indeed an increase (note the slope of the dashed line). However, the reference line at 1.00 (the log of which is, of course, zero) is easily within the 95% confidence interval of the regression throughout the six and a half decades of the data series. The regression estimates a ratio of actual to SPM of 0.911 in 1945 and 1.054 in 2011. The 95% confidence intervals are 0.782 – 1.062 and 0.905 – 1.228, respectively.

In other words, the increase is not statistically significant. There may in fact be an increase, which is to say that something in that other 40% is driving, over time, the SPM to be less successful at predicting the fragmentation of the vote. However, it could just be “noise”; we really can’t say, statistically, because of 1.00 remaining well within the confidence interval.

If it continues on current pace, then 1.00 (or rather its log) will be outside the confidence interval on NV as soon as the year 2065. I will put it on my calendar to check how we are doing at that time.

Independent of the statistical significance, there could be something of interest going on. Note that the regression trend does not cross the 1.00 line till about 42 years into the time series (i.e., 1987). This suggests that, prior to that time, the average election in Western Europe saw the vote be less fragmented than it “should have been”, according to its electoral system. That could suggest that major party organizations were partially overriding the electoral-system effect (producing party systems on average around 90% as fragmented as expected) in the early post-war years. In more recent times, the weakening of party alignments could be making the electoral system expectation finally be realized, with some tendency to exceed in recent times. But we really can’t say, given that the main conclusion is the SPM is all right, and should be for a little while yet, even if the current trend continues (which, of course, it might not).

I also wanted to checked the parliamentary party systems, that is, the effective number of seat-winning parties (NS).

Here it is even more clear that the SPM is doing all right! It is only about now that the regression estimate has finally reached 1.00, but the rate of increase is more minor than with NV, and clearly of minimal significance.

The regression estimates a ratio of actual NS to SPM prediction of around 0.909 in 1945 and 0.991 in 2011. Confidence intervals are 0.773 – 1.068 and 0.843 – 1.164, respectively.

It is somewhat interesting that the trend in the ratio for NV is rising above 1.00 before the ratio for NS. Perhaps there’s an explanation of interest in there. The electoral system more directly constrains NS, after all, and voters perhaps are more willing to “waste” votes as party alignments decrease. But it could just be noise.

(If I do a graph like the first one, with NS with the baseline, there is an increase, but less significant than for NV.)

The conclusion is that there is indeed some truth to the notion that West European party systems are fragmenting. However, relative to the Seat Product Model, they are fragmenting at a slow and hardly significant pace. How can that be? Well, perhaps it is obvious, or perhaps it is not. But a country’s seat product tends to increase over time. Most countries included here have expanded their assemblies over time, and some have also increased district magnitudes and/or adopted upper (compensatory) tiers. So, the effective number of parties should increase to some degree over time, even if voters were just as moored to their party organizations and identities as they ever were!

_________

Appendix: some details.


On the last point above: Specifically, a GLS regression on expected NS says we should have seen on average NS=3.58 in 1945 but 3.67 in 2011. That is not much, but it means some increase is “baked in” even before we look at how actual voters behave. Some part of the increase is in the 60% rather than the 40%.


I dropped Belgium and Italy from the regressions, although they are included in the scatterplots for recent years. The reasons for dropping are that we could not obtain data for the share of seats allocated in upper (compensatory) tiers for the years when these countries used multi-tier PR systems; without that, we can’t calculate the extended version of the SPM (for 2-tier PR). In the later years in the Italy series, when we have such data, these are actually even more complex rules (involving a majoritarian component and alliance vote-pooling), and so the SPM really can’t predict them. In Belgium, the electoral system has been “simple” since 2003, but I think we can agree that there is no semblance of a national party system in that country.


France is also not included, partly due to the importance of the elected presidency (after 1965) and partly due to the two-round system for assembly (after 1958). We do show in Votes from Seats that the SPM works pretty well for France nonetheless. So I doubt its inclusion would have altered the results much. But I wanted to stick to the PR systems and FPTP, which the SPM is designed to handle.

Canadian provincial elections and the SPM: Those assemblies are too small

After posting my earlier overview of expectations from possible electoral reform in British Columbia, I was wondering how well the Seat Product Model has performed over time in Canadian provincial assembly elections. Spoiler alert: not nearly as well in the provinces as a whole as in BC, and better for votes than for seats. The latter is particularly puzzling; the model works by first estimating seats (which are more “mechanically” constrained by district magnitude and assembly size than are votes). That is why the book Rein Taagepera and I published in 2017 is called Votes from Seats. The key to the puzzle may be the serious under-sizing of Canadian provincial assemblies. As I will show in a table at the end of this entry, many provincial assemblies should be almost twice their current size, if we go by the cube root law.

I have a dataset originally constructed for my “to keep or change FPTP” project (published in Blais, ed., 2008). It has most provincial elections back to around 1960, although it stops around 2011. Maybe some day I will update it. For now, it will have to do.

The first graph shows the degree of correspondence between a given election’s observed effective number of vote-earning parties (NV) and the expectation from its seat product (i.e., for FPTP, the assembly size). The black diagonal is the equality line: perfect prediction would place an election on this line. The lighter diagonal is a regression line. Clearly, the mean Canadian provincial election exhibits NV higher than expected. On the other hand, the 95% confidence interval (dashed curves) includes the equality line other than very trivially near the middle of the x-axis range. Thus, in statistical terms, we are unable to reject the hypothesis that actual NV in Canadian provincial elections is, on average, as expected by the SPM. However, the regression-estimated line is systematically on the high side.

It might be noted that we never have NV expected to be 2.0, and in the largest provinces, the assemblies are large enough that we should expect NV>2.5. (“Large enough” here meaning independent of what they “should be”, by the cube root; this is referring only to actual assembly size.) So the classic “Duvergerian” outcome is really only expected in the one province with the smallest assembly. And such an outcome is more or less observed there, in PEI. Nonetheless, a bunch of elections are very much more fragmented than expected, with NV>3! And several are unexpectedly low; many of these are earlier elections in Quebec.

All individual elections are labelled; in a few cases the label generation did not work well (some elections in 2000s). The regression coefficient is significant, although the regression’s R2 is only 0.15. The basic conclusion is a marginally acceptable fit on average, but lots of scatter and some tendency for the average election to be more fragmented than expected.

Now, for the largest seat-winner in the assembly (s1). Here things get a little ugly.

This might be considered a rather poor fit. There is a systematic tendency for the largest party to be bigger than expected: note that the equality line is essentially never within the 95% confidence interval. When we expect, based on assembly size, the largest party to have 60% of the seats, it actually tends to have more like 68%. More importantly, the scatter is massive. In fact, the regression coefficient is insignificant here; please do not ask what the  R2 is!

The size of Canadian provincial parliaments is never so large that the leading party is expected to have only 50% of the seats (note the reference lines and where the equality line crosses the 50-50 point). Yet there is not a trivial number of elections with the largest party under 50%. More common, however, are the blowout wins, where the largest party has 80% or more of the seats. This has been a chronic feature of Canadian provincial politics, especially in a few provinces (notably Alberta, New Brunswick, and Prince Edward Island).

Why is Nv so much better predicted (even if not exceptionally well) than s1? It is hard to say. It is an unusual situation to have both NV and s1 trend higher than expected. After all, normally the more “significant” parties there are the smaller the largest party should tend to be. In the set of predictive equations, s1 (and the effective number of seat-winning parties, NS) are prior to NV, because the seat-based measures are more directly constrained. This is why the book is called Votes from Seats. In our diagram (p. 149) deriving the various quantities from the seat product, we show NV and s1 coming off separate branches from “Ns0” (the actual number of parties winning seats), which is expected to be (MS)0.25, where M is the magnitude and S the assembly size. Thus for FPTP, it is S0.25. A parliament with 81 seats is expected to feature three parties; the other formulas would predict that NS=2.08, s1=0.577, and NV=2.52. If the parliament had 256 seats, we would expect four parties, NS=2.52, s1=0.50, and NV=2.92.

Unfortunately, the dataset I am using does not (yet) have how many parties won seats–actual or effective number. Thus I can’t determine whether this is the point at which the connections get fuzzy in the Canadian provincial arena. Nonetheless, there should be a relationship between NV and s1. It can be calculated from the formulas displayed in Table 9.2 (also on p. 149). It would be:

s1=(NV1.5 -1)-0.5.

In this last graph, I plot this expectation with the solid dark line, and a regression on s1 and NV from Canadian provincial elections as the lighter line (with its 95% confidence intervals in dashed curves).

The pattern is obvious: there are many elections in Canadian provinces in which the leading party gets a majority or even 60% or 70% or more of the seats despite a very fragmented electorate. We should not expect a leading party with more than 50% of the seats when NV>2.92. And yet 17 elections (around 15% of the total) defy this logically derived expectation. Six have a party with 2/3 or more of the seats despite NV>2.92 (in order of increasing NV: BC91, AB04, NB91, QC70, AB67, BC72; in the last one, NV=3.37!).

I think the most likely explanation is Cube Root Law violations! Canadian provincial assemblies are much too small for their populations. So, the cause of the above patterns may be that voters in Canadian provinces vote as if their assemblies were the “right” size, but these votes are turned into seats in seriously undersized assemblies, which inflates the size of the largest party. (Yes, votes come from seats in terms of predictive models, but obviously in any given election it is the reverse!)

There is some support for this. I can calculate what s1 and NV would be expected to be, if the assemblies were the “right” size, which is to say the cube root of the number of voters (which is obviously smaller than the number of citizens, but this is what I have to work with). I will call these s1cr and NVcr. Then I can take ratios of actual s1 and actual NV to these “expectations”. The mean ratios are: NV/NVcr=0.994; s1/s1cr=1.19. If the assemblies were larger, the votes–already with a degree of fragmentation about as expected from more properly sized assemblies–would probably have stayed about the same. However, with these hypothetically larger assemblies, the largest party in parliament would be less inflated by the mechanics of the electoral system.

Canadian provinces would have a greatly reduced tendency to have lopsided majorities if only they would expand their assemblies up to the cube root of their active voting population. Of course, this assumes they stick to FPTP. The other thing they could do is switch to (moderately) proportional representation systems, like BC is currently considering. That would be seem to be a good idea regardless of whether they also correct their undersized assemblies.

Below is a table of suggested sizes compared to actual, for several provinces. (“Current” here is for the latest election actually in the dataset; some of these have been increased–somewhat–subsequently.)

Prov. Current S Increased S
Alberta 83 121
British Columbia 79 152
Manitoba 57 94
New Brunswick 55 91
Newfoundland 48 81
Nova Scotia 52 95
Ontario 107 208
Prince Edward Island 27 55
Quebec 125 197
Saskatchewan 58 95

_________

(By the way, the R2 on that s1 graph that I asked you not to ask about? If you must know, it is 0.03.)

 

What could we expect from electoral reform in BC?

This week is the beginning of the mail voting period for the referendum on whether to reform the electoral system for provincial assembly elections in British Columbia. The ballot asks two questions: (1) Do you want to keep the current FPTP system or “a proportional representation voting system”: (2) If BC adopts PR, which of three types of PR do you prefer?

The second question offers three choices, which voter may rank: Mixed-Member Proportional (MMP); Dual-Member Proportional (DMP); Rural-Urban Proportional (RUP).

I have reviewed before what these options entails, and will not repeat in detail here. Besides, the official BC Elections site explains them better than I could. What I want to try to get it here is how we might expect BC’s provincial party system to change, were any of these options adopted. To answer that question, I turn, of course, to the Seat Product Model, including the extended form for two-tier systems developed in Votes from Seats.

The punch line is that the various scenarios I ran on the options all suggest the effective number of parties in the legislative assembly would be, on average, somewhere in the 2.46 to 2.94 range, the effective number of vote-earning parties would tend to be in the 2.83 to 3.32 range, and the size of the largest party would be somewhere between 45% and 51% of the seats. Again, these are all on average. The ranges just provided do not mean elections would not produce a largest party smaller than 45% or larger than 51%. Actual elections will vary around whatever is the point prediction of the Seat Product Model for any given design that is adopted. And fine, yet important, details of whichever system is adopted (if FPTP is not retained) will remain to be fleshed out later.

The ranges I am giving are formula-predicted averages, given the inputs implied by the various scenarios. I explain more below how I arrived at these values. The key point is that all proposals on the ballot are quite moderate forms of PR, and thus the party system would not be expected to inflate dramatically. However, coalition governments, or minority governments with support from other parties, would become common; nonetheless, single-party majority governments would not likely disappear from the province’s future election outcomes. As we shall see, one of the proposals would make single-party governments reman as the default mean expectation.

Before going to the scenarios, it is important to see whether the real BC has been in “compliance” with the Seat Product Model (SPM). If it has tended to deviate from expectations under its actual FPTP system, we might expect it to continue to deviate under a new, proportional one.

Fortunately, deviations have been miniscule. For all elections since 1960, the actual effective number of vote-earning parties has averaged 1.117 times greater than predicted. That is really minor. More important is whether it captures the actual size of the largest party well. This, after all, is what determines whether a single-party majority government can form after any given election. For all elections since 1960, the average ratio of actual largest-party seat share to the SPM prediction is 1.068. So it is even closer. For an assembly the size of BC’s in recent years (mean 80.7 since 1991), the SPM predicts the largest party will have around 57.8% of seats. The mean in actual elections since 1991 has been 62.7%. That is a mean error on the order of 4 seats. So, the SPM captures something real about the current BC electoral system.

Going a little deeper, and looking only at the period starting in 1996, when something like the current party system became established (due to the emergence of the Liberals and the collapse of Social Credit), we find ratios of actual to predicted as follows: 1.07 for effective number of vote-earning parties; 1.07 for largest parliamentary party seat share; 0.905 for effective number of seat-winning parties. If we omit the highly unusual 2001 election, which had an effective number of parties in the assembly of only 1.05 and largest party with 96.2%, we get ratios of 0.98 for effective number of seat-winning party and 0.954 for largest party size. The 2017 election was the first one since some time before 1960 not to result in a majority party, and it is this balanced parliament that is responsible for the current electoral reform process.

As for the proposed new systems, all options call for the assembly to have between 87 seats (its current size) and 95 seats. So I used 91, the mean; such small changes will not matter much to the estimates.

The MMP proposal calls for 60% of seats to remain in single-seat districts (ridings) and the rest to be in the compensatory tier (which would be itself be regionally based; more on that later). So my scenarios involved a basic tier consisting of 55 seats and a resulting 36 seats for compensation. Those 36/91 seats mean a “tier ratio” of 0.395 (and I used the rounded 0.4). The formula for expected effective number of seat-winning parties (Ns) is:

Ns=2.5^t(MS)^0.167.

With t=.4, M=1 (in the basic tier) and S=91, this results in Ns=2.81. I will show the results for other outputs below.

For the DMP proposal, the calculations depend on how many districts we assume will continue to elect only one member of the legislative assembly (MLA). The proposal says “rural” districts will have just one, to avoid making them too large geographically, while all others will have two seats by combining existing adjacent districts (if the assembly size stays the same; as noted, the proposals all allow for a modest increase). In any case, the first seat in any district goes to the party with the plurality in the district, and the second is assigned based on province-wide proportionality. For my purposes, this is a two-tier PR system, in which the compensatory tier consists of a number of seats equivalent to the total number of districts that elect a second MLA to comprise this compensatory pool. Here is where the scenarios come in.

I did two scenarios, one with minimal districts classified as “rural” and one with more. The minimal scenario has 5 such districts–basically just the existing really large territorial ridings (see map). The other has 11 such districts, encompassing much more of the interior and north coast (including riding #72, which includes most of the northern part of Vancouver Island). I will demonstrate the effect with the minimal-rural scenario, because it turned out to the most substantial move to a more “permissive” (small-party-favoring) system of any that I looked at.

Of our 91 seats, we take out five for “rural” districts, leaving us with 86. These 86 seats are thus split into 43 “dual-member” districts. The same formula as above applies. (Votes from Seats develops it for two-tier PR, of which MMP is a subset.) The total number of basic-tier seats is 48 (the five rural seats plus the 43 DM seats). There are 43 compensation seats, which gives us a tier ratio, t=43/91=0.473. Ah ha! That is why this is the most permissive system of the group: more compensation seats! Anyway, the result is Ns=2.94.

If we do the 11-rural seat scenario, we are down to 80 seats in the DM portion of the system and thus 40+11 basic-tier seats. The tier ratio (40/91) drops to 0.44. The resulting prediction is Ns=2.61. This does not sound like much, and it really is not. But these results imply a difference for largest seat size between the first scenario (45%) and the second (49%) that makes a difference for how close the resulting system would be to making majority parties likely.

Finally, we have RUP. This one is a little complex to calculate because it is really two different systems for different parts of the province: MMP for “rural” areas and STV for the rest. I am going to go with my 11 seats from my second DMP scenario as my “rural” area. Moreover, I understand the spirit of this proposal to be one that avoids making the districts in rural areas larger than they currently are. Yet we need compensation seats for rural areas, and like the full MMP proposal, RUP says that that “No more than 40% of the total seats in an MMP region may be List PR seats”, so this region needs about 18 seats (the 11 districts, plus 7 list seats, allowing 11/18=0.61, thereby keeping the list seats just under 40%.) That leaves us with 91-18=73 seats for the STV districts. The proposal says these will have magnitudes in the 2-7 range. I will take the geometric mean and assume 3.7 seats per district, on average. This gives us a seat product for the STV area of 3.7*73=270.

In Votes from Seats, we show that at least for Ireland, STV has functioned just like any “simple” PR system, and thus the SPM works fine. We expect Ns=(MS)^.167=2.54. However, this is only part of the RUP system. We have to do the MMP part of the province separately. With just 11 basic-tier seats and a tier ratio of 0.39, this region is expected to have Ns=2.13. A weighted average (based on the STV region comprising 80.2% of all seats) yields Ns=2.46.

The key point from the above exercise is that RUP could result in single-party majority governments remaining the norm. Above I focused mainly on Ns expectations. However, all of the predictive formulas link together, such that if we know what we expect Ns to be, we can determine the likely seat-share of the largest party (s1) will be, as well as the effective number of vote-earning parties (Nv). While that means lots of assumptions built in, we already saw that the expectations work pretty well on the existing FPTP system.

Here are the results of the scenarios for all three output variables:

System Expected Ns Expected Nv Expected s1
MMP 2.81 3.19 0.46
DMP1 2.94 3.32 0.45
DMP2 2.61 3.01 0.49
RUP 2.46 2.87 0.51

“DMP1” refers to the minimal (5) seats considered “rural” and DMP2 to the one with 11 such seats. If we went with more such seats, a “DMP3” would have lower Ns and Nv and larger s1 than DMP2, and the same effect would be felt in RUP. I did a further scenario for RUP with the MMP region being 20 districts, and wound up with Ns=2.415, Nv=2.83, s1=.52; obviously these minor tweaks do not matter a lot, but it is clear which way the trend goes.And whether any given election is under or over s1=0.50 obviously makes a very large difference for how the province is governed for the following four years!

I would not really try to offer the above as a voter guide, because the differences across systems in the predicted outputs are not very large. However, if I wanted to maximize the chances that the leading party would need partners to govern the province, I’d probably be inclined to rank MMP first and RUP third. The latter proposal simply makes it harder to fit all the parameters together in a more than very marginally proportional system.

By the way, we might want to compare to the BC-STV proposal that was approved by 57% of voters in 2005 (but needed 60%, and came up for a second referendum in 2009, when no prevailed). That proposal could have been expected to yield averages of Ns=2.61, Nv=3.0, and s1=.49. By total coincidence, exactly the same as my DMP2 scenario.

Edit: The following paragraph is based on my own misreading of the proposal, which contrary to what I say here, calls for province-wide proportionality if MMP is chosen. What it says about Scotland still applies, so I will leave it here. [Then again, maybe not?]

A final note concerns the regional compensation in the MMP proposal vs. province-wide in DMP. In an on-line appendix to Votes from Seats, I explored whether regional compensation in the case of Scotland produces a less permissive system than if compensation were across all of Scotland. I concluded it made no difference to Ns or s1. (It did, however, result in lower proportionality.) Of course, if it made a difference, province-wide would have to be more favorable to small parties. Thus if this were a BC voter’s most important criterion, DMP might pull ahead of MMP. However, the benefit on this score of DMP is greater under a “low-rural” design. The benefit of DMP vanishes, relative to MMP, if the system adopted were to be one with a higher share of seats marked as rural. I certainly am unable to predict how the design details would play out, as this will be left up to Elections BC.

The bottom line is that all proposals are for very moderately proportional systems, with MMP likely the most permissive/proportional on offer.

Brazil’s open list is (a little bit of) a hybrid now

Brazil is a classic case of open-list proportional representation (OLPR): lists win seats in proportion to their collective votes in a district (state), but candidates within the list are ordered solely according to preference votes obtained as individuals. These rules can result in individual candidates elected with very small preference-vote totals.

For the most recent Brazilian election, a new provision has gone into force. There is now a threshold on preference votes that candidates must obtain to be elected. This means that, in a very technical sense, a hybrid element has been brought into the Brazilian system. However, the provision is not the usual hybrid seen in “preferential list” systems, whereby seats not filled on preference votes are filled instead according to a party’s (or coalition’s) pre-determined rank. That hybrid format is what is typically called a flexible list or a semi-open list. However, Brazilian lists remain unranked, except via the preference votes.

Rather, in Brazil, a list that has an insufficient number of candidates with above-threshold preference votes forfeits those seats to other lists in the district. The threshold is set at 10% of the electoral quota, which is a Hare quota (1/M, where M is district magnitude).

This provision changed the allocation of 8 seats. Given a Chamber of Deputies with over 500 seats, we should not exaggerate the significance of the change, although of course, some other parties might have adjusted either their nomination behavior or their “intra-party vote management” practices (defined below) to avoid being hit by the threshold.

The Chamber’s website has an article regarding the seat shifts, and a table with the details (in Portuguese). The PSL, which is the party of the likely next president, Jairo Bolsonaro, won 7 fewer seats than it would have without the threshold. All these seats were in São Paulo, which is the highest-magnitude district in Brazil (M=70). The threshold there is thus 0.143% of the votes cast in the state. The Novo list in Rio Grande do Sul (M=31) also lost 1 seat due to the intra-list threshold. (Novo, meaning “New”, is a small liberal party.)

In São Paulo, the seven PSL candidates who were not eligible to take seats the list otherwise would have won had vote totals ranging from 19,731 to 25,908. They were replaced by candidates on six different lists with preference votes ranging from 56,033 to 92,257, suggesting the replacements had, on average, about three times the votes of the forfeiting candidates. (The party that picked up two of these seats was the Democrats.) In Rio Grande do Sul, the seat Novo forfeited would have been won by a candidate with 11,003 votes, but was instead filled by a candidate the Brazilian Labor Party (PTB, not to be confused with the PT of Lula) who had 69,904 votes, a preference-vote total 6.35 times greater than that of the forfeiting candidate.

As is clear from the vote totals of those who lost under this provision and those who gained, if the intention was to prevent candidates with marginal personal followings from riding in on the “coattails” of strong list-pullers (whose popularity increases the votes of the collective list), then the reformers can declare “mission accomplished”.

I am personally quite excited by this provision, which I had missed when summarizing minor changes made to the electoral law in 2017, because I once wrote up a proposal for just such a hybrid. It is in some text that was going to be part of one of the chapters in Votes from Seats, but Rein Taagepera and I decided it was not directly germane to the book and left it out. The chapter it would have been part of compares OLPR to the single non-transferable vote (SNTV) with respect to vote shares of first and last winners, and regarding the extent to which parties do (or do not) manage their intra-party competition.

Managing intra-party competition refers to parties doing one or both of: (1) restricting the number of candidates nominated, or (2) intervening in the campaign in an effort to shift votes from non-viable candidates to viable ones.

Under SNTV, these intra-party competition-management practices are critical because the total number of seats a party (or set of cooperating parties) can elect is entirely dependent on how many individual candidates it has whose votes are in the top-M vote totals in the district. Under OLPR, parties have no incentive to do this, if their goal is simply to maximize list seats–a list under OLPR can never displace seats to another list due to having “too many” candidates or having the candidates’ vote totals be widely unequal. (Parties may have other reasons to care about which candidates win, and multiple parties running in alliance face an SNTV-like conundrum in that they are competing with one another inside the list to get their candidates into the top s, where s is the number of seats won by the list. But these are separate problems, and the latter is a problem covered in Votes from Seats).

The proposal I drafted was a hybrid of OLPR and SNTV (unlike flexible lists, which re a hybrid of OLPR and closed-list PR). A threshold would be set on preference votes, and if a list won more seats, via application of the inter-list allocation rule, than it had candidates over the threshold, it would forfeit these seats. Any such forfeited seats would go into an “SNTV pool” to be be won by the otherwise unelected candidates with the highest preference-vote totals, independent of which list they had run on. My intention in devising this proposal was to encourage parties to be more active in managing their intra-party competition–taking some aspects of SNTV as beneficial–in order to make victory by candidates with marginal personal popularity less likely. (I would have set the threshold a little higher than 10% of a Hare quota.)

The article on the Chamber website is not clear on the precise rule now used in Brazil for deciding on the replacement candidates. In any case, it certainly has a similar effect to my proposal. (From a comment by Manuel at the earlier thread, it seems the forfeited seats are assigned proportionally rather than SNTV-like.) I can’t claim credit, as there is no way any Brazilian official saw my unpublished proposal. But I am pleased that some such a provision has been adopted somewhere.

Thanks to Dr. Kristin Wylie (on Twitter) for calling my attention to this article.