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.)


15 thoughts on “Canadian provincial elections and the SPM: Those assemblies are too small

  1. Regarding the final table: I believe some of the figures in the ‘S’ column are out of date, or simply erroneous. The NL legislature was reduced a couple of years ago to just 40(!) seats, whilst NB had an election just last month involving only 49 ridings. Alberta and BC look slightly on the low side, whereas Ontario re-expanded its Provincial Parliament earlier this year to a slightly less egregious 124 seats.

    All the statistical findings before that are of course brilliant as usual.

    • There was a disclaimer about the assembly sizes in the post: some are not current, because the dataset is not up to date. Thus, for some provinces the figure given may be from as late as 2011, while for others it may be from as early as 1999 (I think).

      For the statistics, 40 vs. 49 would not matter at all (given logarithmic scale). But 49 vs. 80 would matter.

      • I did scroll back up once I saw the numbers but I must’ve missed the disclaimer, sorry.

        I don’t think any assembly in Canada has changed in size as radically as that (from 80 seats down to 49) in the past two decades, so you’re in the clear there!

  2. In British Columbia the cube root rule bumps up against Architecture. In 2014 the Liberal government put forward Bill 2, instructions to the Electoral Boundaries Commission that included a cap of 85-87 members. During second reading, the opposition speaker L. Krog stated “Now, the kicker is that we’re also going to limit it to 85 seats. The opposition supports that. This place is a little crowded as it is.”

    In fact, Bill 2 included a nifty bit of malapportionment in the government’s favour. The opposition had to swallow it or else propose (1)increasing the area of the already-large northern districts or (2)increasing the number of Members, proposals that would generate strong public push-back.

    • Regarding the architecture, look at the UK House of Commons! That chamber is significantly oversized by cube root, but also quite oversized by the benches’ capacity.

      I am also reminded of a DW TV news segment I saw years ago of workers literally adding seats, due to the overhangs. I am not sure how they handled the current assembly size, given that the new rules can greatly expand the number of members.

      (I guess we should not say “seats” but just “members”; some, especially in the UK, may have to stand!)

      In light of what Dave says, it is interesting that the BC proposals for electoral reform all permit the assembly size to rise to as much as 95. Still well short of the cube root, but a few small steps in the right direction.

      I am glad you raised this issue, because it actually is likely that it comes up in other jurisdictions when size is discussed. I was not aware that it had specifically come in BC. This is interesting. Thank you.

      • It is a shame that publics, in general, object to assembly-size increase. It is the public being represented, and more representation should always be better, at least up to some ceiling that must be rather high. This improved representation has to be balanced off against other considerations (in the cube-root law of assembly size, the one other factor is intra-chamber communication). But most of those other considerations would allow for assemblies quite a bit larger than is currently the case in many countries, as well as Canadian provinces.

        And, yet, while I am not aware of any systematic polling on the idea, I suspect the idea of having more legislators is nowhere popular.

        Many readers may know that the opposition to electoral reform in New Zealand in the 1990s used the phrase, “MMP means more MPs.” Fortunately, it did not doom the proposal’s considerable improvement in overall representation and functionality of the two-tier model.

      • Encouraging populist sentiment against has been a common tactic for a long time: in 1903, the people of New South Wales voted to reduce the size of the lower house from the 125 that it had pre-Federation (i.e. as a colony in its own right) to just 90.

        Despite the huge increase in population in the interim – not to mention the expansion of the franchise – it currently stands at 93 seats, only 3 more than in 1904.

      • I’m a big fan of the House of Commons chamber’s under-provision of seating. More countries should adopt that.

  3. Although Canadian sub-national assemblies are too small according to the cube root rule, they are too small in a very systematic way. Using current values for assembly sizes, I find assemblies of the 13 provinces and territories coming in at 0.55 of the cube root value, with a standard deviation 0.05. Other former British colonies also have undersized assemblies in states/territories: Australia 0.48 (SD 0.08); India 0.45 (0.12); United States excluding NH 0.65 (0.27).

    The cube root rule has been explained in terms of legislators’ conflicting needs when communicating with each other versus communicating with constituents. Does the balance change when the issues are secondary roads rather than foreign policy? Do peer comparisons matter more for Canadian provinces than for US states?

    • … And complain that “I had to wait two weeks for an appointment with my local MP/ legislator, and they’d only give me 15 minutes’ face time”.
      Funny how it’s the libertarians of New Hampshire who have realised that the perceived link between “big, bloated bureaucracy” and “more legislators” is a cognitive fallacy.

      • It seems funny that New Hampshire has the biggest state legislature and yet it is not a very populated state. One of the largest legislative assemblies in the English speaking world, the House is huge, but the Senate is tiny, what is the reason for the size? It seems odd that no one has complained that it is too big.

        Has anyone did any studies on comparing the sizes of bicameral legislative bodies, lower house sizes vs upper houses? What is the biggest ratio between the two chambers, and the smallest ratios? Are there any upper houses bigger than the lower or any that are of the same size?

      • The House of Lords is bigger than the House of Commons.

        That may be the only upper house larger than the lower.

  4. On sizes of assemblies in bicameral and federal systems, in A Different Democracy (Yale, 2014), my coauthors and I observe that the mean ratio of second chamber to first chamber size is 0.34 for federal systems and 0.53 for unitary. The difference is statistically significant. This is only for our 31 democracies. Leaving the UK out still leaves the difference significant. The UK is the only one of the countries covered in the book and the only country I know of in which the second chamber is the larger one.

    I don’t think there is any systematic tendency of federal systems to deviate more from he cube root law than unitary systems do. I think I looked into that once and did not see an effect. However, in Votes from Seats (Cambridge, 2017) we report that presidential systems (i.e., the first chambers thereof) tend to be systematically undersized. We do not offer an explanation.

    For state legislatures in the US, I defer to JD. He looked at this in a paper a couple of years ago. My recollection is that many states are close to the cube root if we take the combined size of the two chambers.

    Clearly there is much more that could be done with these questions. Maybe some lifetime…

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