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

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(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.)