REVISION (Jan 22 at 1030 Pacific): My estimates based on a closer national result should also assume it would be closer in Quebec. I have changed them accordingly below, with the new or altered text in italics. (Jan 22 at 1507 Pacific: adding links at the bottom of this post to other projections.)

How big will the Conservative victory plurality be in Monday’s Canadian election? That is the only remaining question, barring some dramatic reversal of current fortunes of the parties in voter opinion. A reversal is not out of the question, as the undecided vote is still substantial and the parties seem to be trending closer to one another in recent polls, but it looks like the Conservatives will win the most votes and seats. The question is, will they have a majority of seats in parliament, or will the next Canadian government be a second consecutive minority cabinet (only of a different party)?

An average of four recent polls suggests the Conservatives lead with between 35.5 and 38% of the vote, with the Liberals at 26-29%. This is a significant swing from 2004, when the Liberals led with 36.7% of the vote, and the Conservatives had 29.6%. It is also quite a turnaround from polls as recently as mid-December, which implied a votes distribution not greatly different from the 2004 result.

But in Canada, there is no direct relationship between votes and seats at the national level, because it is a first-past-the-post (FPTP) system complicated by multiparty competition and significant regional variations in the four largest parties’ votes.

Ipsos-Reid published a seats forecast today: Conservatives, 143-147, Liberals 59-63, New Democrats (NDP) 39-43, Bloc Quebecois (BQ) 59-63. A majority in the House of Commons is 155 seats, so this projection says Canada will continue to have minority government as it has had since the 2004 election, only this time with Conservatives replacing Liberals. (For a discusion of an Ipsos-Reid projection from a few days earlier, see Election Canada 2006.)

However, it is worth noting that the Ipsos Reid poll gives the Conservatives the greatest lead of any of the five major polls published on January 19. For the record, these polls show the parties’ likely votes percentages as follows (always listed as Cons-Lib-NDP-BQ-Green):

Leger, 38-29-17-11-4
Ekos, 37.4-27.3-20.8-10.1-3.9
Ipsos Reid, 38-26-19-12-5
Strategic Counsel, 37-28-16-12-7
SES, 35.5-29-18.8-11.1-5.6

Also see Charles Franklin’s Canada poll tracker at Political Arithmetik. The trends suggest that the Conservative momentum may have peaked in the last week as it became likely that its level of support would be high enough to give the party a parliamentary majority. Whether this trend will continue–which would imply the Conservatives falling back to 35% or below–or stabilize will determine the size of the plurality the party has in the Commons. A majority remains within reach, but would require a reversal of the last several days’ trends. The party pretty much has to run the table to have a majority.

I have run several simulations based on various polls and other assumptions. Before going further, I want to make a very important disclaimer. What follows are not predictions. They are estimates from a mathematical model that has several assumptions behind it. If any of the assumptions are wrong, the estimate will be wrong. Therefore, I would discourage any Canadian from basing his or her voting on these scenarios or anyone else from taking the following too seriously. I am going to put this out there, explain my assumptions, give a few alternatives, and compare how the model has performed in several past Canadian elections.

If all you care about is the seat estimates themselves, scroll down to the bottom of this post (after clicking on the “more” that appears shortly after the graph, below). However, it makes little sense to show the estimates without explaining the methods first, and indicating the assumptions that underly them.

The estimator model is based on something called the seat-vote equation. The simplest form of this equation is the well known cube law of plurality elections, which states that the ratio of seats for the two largest parties tends to be the cube of the ratio of the votes of those parties. In mathematical notation:

s_k/s_l = (v_k/v_l)³,

where s is a party’s seat share and v the vote share and the subscripts refer to parties k and l.

While the cube law is well established, it has also been known for decades that the exponent, 3, is not quite right, at least not for all situations. In 1989, Rein Taagepera and I published Seats and Votes, in which we offer a generalization of the seat-vote equation. The form is the same:

s_k/s_l = (v_k/v_l)ⁿ (later to be referred to as equation 1)

where the exponent, n, is derived as follows:

n=logV/logE,

where V is the number of voters and E is the number of districts.

For Canada, around 13.5 million voters and 308 ridings (districts) results in an exponent of 2.865 instead of 3. In other words, for the size of Canada’s voting population and the size of its chamber, the expectation is that the two largest parties will be somewhat closer in seat shares than what the standard cube law would predict. (If the two largest parties are at .38 and .28, as Ipsos Reid says, then a cube relationship would predict a seat ratio of 2.499, while the 2.865 exponent would suggest a seat ratio of 2.399.)

How does this model perform? The graph below shows elections over the past 40 years in Canada, each of the Canadian provinces, the U.K., and New Zealand (before its change in electoral system). The horizontal axis shows the actual seat ratio of the two largest parties that resulted from the election, while the vertical axis shows the expected ratio, based on equation 1, with n defined according to the number of votes in the respective election and the size of the assembly. Diamonds indicate elections that produced a majority party; crosses indicate minority situations (such as Canada currently).

(Click on the image for a much larger version; I apologize that even on the large version some squinting is necessary.)

It is clear that most elections fall fairly close to the solid diagonal line that indicates a 1:1 agreement between the estimated and the actual seat ratios. There are some dramatic outliers–especially on the right side of the graph. More on them later, in a future post. For now, what we care about is Canada, at the federal level.

It is noteworthy that of the points that are left of and above the diagonal, many of the elections are Canadian federal elections. In other words, the Canadian electoral system results in a closer ratio of seats for the two largest parties than what would be expected. To put it less clumsily, the Canadian electoral system is relatively more proportional than most FPTP systems. (Note: I said relatively, as in relative to the expected degree of disproportionality inherent in FPTP.)

The seat-vote equation does not take into account regional dispersion of votes by parties. Or, rather, it blindly assumes that the degree of such dispersion is more or less the same across countries and individual elections under FPTP. Of course, Canada is famous for having sharp regional divisions, and in this regard it is hardly surprising that the election in which the Bloc Quebecois emerged for the first time (1993) would be the single greatest outlier in the upper left (where the seat ratio is much less than expected). The 1997 election is also an outlier. The 2000 election is above the 1:1 line, though closer to it. (The 2004 election, like other minority situations, is not labeled, but it is one of the small cluster of crosses near the “UK92″ label, thus again indicating a lower seat ratio that expected.)

The seat-vote equation forms the basis for a more complex model that allows one to estimate individual seat totals, based on a known or expected distribution of votes for individual parties (instead of just the ratios for the top two). It is this model that I will use, on several different plausible distributions of the parties’ votes, and then see how well it post-dicts previous elections in Canada.

The equation is the following:

S_k = V_kⁿ / [V_kⁿ + (N-1)^(1-n)(1-V_k)ⁿ] (hereinafter referred to as equation 2),

where all the variables have already been defined in connection with equation 1, above, except for upper-case N, which is the effective number of parties. N is an index, now widely used in the study of party systems, of the degree of fractionalization of a party system. N equals 2 if there are two parties at 50-50 and 3.0 if three parties each have exactly one third of the votes, and so on. Canada’s N in recent years has been in the 3.7-4.1 range.

The regionalized nature of Canada’s party system really gives the model problems, which is why–as we saw in the graph above–many Canadian elections have seat ratios that are smaller than would be expected. In fact, equation 2 yields an average error of +31 seats in Canadian elections since 1980. The error is smaller in the two elections in which the Conservatives were in the lead (+17 in 1988 and an almost spot-on -1 in 1984). Its worst error was in 1997, when it said the leading party, the Liberals, should have won 205 seats, but they actually won a bare majority of 155 (in what was then a 295-seat Commons). The error in 1993 was +29 and in 2004, +47. That is, for 2004, the model says the Liberals should have had a relatively comfortable majority instead of just a plurality. The systematically less disproportional nature of Canada’s FPTP, compared to most other such systems, is what produced the current minority government. (By contrast, the U.K. in 2005 should have had a minority government, but the Labour party was significantly more over-represented than it “should” have been. UK05 is not labeled, but it is one of the two points right above the label for QC94, which in turn is above the word “Ratio” in the lower legend.)

But why should we accept for Canada a model that has such a high error? Even if 2006 turns out to be more like 1984 and 1988 on account of it sharing with those earlier years a Conservative plurality, we can do better.

It is apparent that Quebec causes the model problems. That is not surprising in the BQ era, when there is a party that wins around 10% of the national vote, but all of that in one province. In both 2000 and 2004 (I did not look at 1997 and 1993), equation 2 is accurate for the largest party to within 3 seats for all of Canada outside Quebec. So I will estimate 2006 without Quebec and then add in a separate estimate for that province.

I will use equation 2 on Quebec alone (its exponent is 3.49), and the Ipsos-Reid poll of voting intentions in Quebec, taken Jan. 13-15. Below are their vote estimates and then my seat estimates from equation 2:

Con:25% (9)
Lib: 13% (1)
BQ: 48% (62)
NDP: 10% (–)

I did not estimate the NDP because I know what the equation does not know: There is no way the NDP will win any riding in Quebec. So, that leaves 3 of Quebec’s ridings unfilled. I will “fix” that later.

I also ran equation 2 on the 2004 and 2000 elections in Quebec. The equation does very well for the BQ, predicting 54 in 2004 (when it actually won 58) and 37 in 2000 (actually 38). It has a harder time discriminating among trailing parties that are so dependent on their degree of local concentration to win any seats.

To cope with the problem of prediction of the smaller parties–which in Quebec, just happen to be the two biggest national parties–I also looked at each riding individually to see which would swing from the party that won it in 2004 to another party under the following assumptions. First, that the Ipsos-Reid poll will prove a reliable predictor of the Quebec voting breakdown. Second, that the swing from 2004 to 2006 will be uniform across the province. I make no claims as to how reliable either of these assumptions will prove to be.

My riding-by-riding analysis agrees with equation 2 that the BQ should win 62 seats. That would be an increase of 8 seats from 2004, even though its votes are expected to remain almost the same. The reason is that the collapse of the Liberals in the province will throw many seats to the BQ, but the Conservatives’ gains will not be enough to take many of the seats currently held by the Liberals. In fact, my riding-by-riding analysis suggests the Conservatives would win at most 6 seats, or only half what equation 2 predicts, while the Liberals would win 7. It is these latter numbers that I will add to the equation 2 estimates for Canada as a whole.

So, now to Canada as a whole. The average of the five January 19 polls shown above has the parties’ vote shares as follows:

Con: 37.2
Lib: 27.9
NDP: 18.3
BQ: 11.2
Grn: 5.1

The sum of equation 2 estimates (based on the 233 non-Quebec seats) and the above Quebec estimates would translate this poll average as follows:

Conservative: 143
Liberal: 85
NDP: 19
BQ: 62

(The astute reader will note that this actually sums to 309, so one party should have one less, but I can’t determine which party. I would never claim these are accurate to within 1 seat anyway!)

I should note that it is possible that all my estimates understate the NDP seats, as that party, too, is somewhat concentrated, though far less than the BQ. The model has problems with concentrated parties, as I have already noted. Most of any additional NDP gains would probably be at Liberal expense, relative to my model estimates. Having said that, the Ipsos-Reid estimate–about double mine–strikes me as well out of line.

So, if the Conservatives actually win 12 seats in Quebec, then the total estimate for the party rises to 149. That is six seats short of a majority, or close enough that I can’t be sure that there will not be a Tory majority government, but I think 150 or so might be the outer limit.

What about the poll above that is most favorable to the Conservatives–the Ipsos-Reid? It would yield estimates like the following:

Conservative: 152
Liberal: 71
NDP: 23
BQ: 62

This still puts the Tories short of a majority, though if the equation-2 estimate for Quebec is used, they would be at 158 and over the top. However, my riding-by-riding analysis was already quite liberal–so to speak–in giving swing ridings to the Tories, so this really is the way outer limit of what the Conservatives can get, barring a new boomlet in their support in the last days or a really favorable district-level swing in Quebec (i.e. swings in specific districts that are well beyond the provincewide swing).

I also want to run with the poll from January 19 that is least favorable to the Conservatives (SES). In this application, I also use an estimate within Quebec that is closer than the estimates used above. In this scenario, the Conservatives win only 3 seats in Quebec, based on the January 5 Ekos poll that I discussed here previously.

Conservative: 127
Liberal: 103
NDP: 22
BQ: 60

(This adds to 312, but again, it would be arbitrary to reduce any parties’ estimates.)

OK, everyone sit back (well, if you are Canadian, get out and vote and then sit back) and see how closely the real voting result conforms to any of these polls, and then how closely the seat allocation conforms to the estimate that is based on a votes distribution closest to what Canadian voters actually produce.

If I were a betting man (and I am not) I would guess the result will be closer to the last one that I presented (Conservatives at around 127) than to any of the others. But that’s just because I expect the race to narrow at the very end. And if it narrows as much as my hunches (and that is all they are) lead me to believe, to something like Conservative 35% and Liberal 30%, it would actually be something more like this:

Conservative: 117
Liberal: 110
NDP: 19
BQ: 60

As you can see, a convergence of the two parties’ votes nationally to a margin of five percentage points or less would put the parties in the range in which a reversed plurality–one party with the most votes, the other with the most seats–becomes possible. Given the tendency of the Canadian electoral system to give the largest party in votes–especially when that is the Liberals–fewer seats than the seat-vote equation would predict, a reversed plurality is more likely if the Liberals have the most votes than if the Conservatives do. But in a very close election, it could happen either way.

But, remember, that the two parties’ votes might converge to a five-point margin or less is only a hunch, not a prediction!!