Why 1.59√Ns?

In the previous planting, I showed that there is a systematic relationship under FPTP parliamentary systems of the mean district-level effective number of vote-earning parties (NV) to the nationwide effective number of seat-winning parties (NS). Specifically,

NV =1.59√NS .

But why? I noticed this about a year after the publication of Votes from Seats (2017) while working on a paper for a conference in October, 2018, honoring the career of Richard Johnston, to which I was most honored to have been invited. The paper will be a chapter in the conference volume (currently in revision), coauthored with Cory Struthers.

In VfrS Rein Taagepera and I derived NV =1.59S1/12. And as explained in yesterday’s planting, it is simply a matter of algebraic transformation to get from expressing of NV in terms of assembly size (S) to its expression in terms of NS. But perhaps the discovery of this connection points the way towards a logic underlying how the nationwide party system gets reflected in the average district under FPTP. In the paper draft, we have an explanation that I will quote below. It is on to something, I am sure, but it remains imperfect; perhaps readers of this post can help improve it. But first a little set-up is needed.

To state clearly the question posed in the title above, why would the average district-level effective number of vote-winning parties in a FPTP system tend be equal to the square root of the nationwide effective number of seat-winning parties, multiplied by 1.59?

We can deal with the 1.59 first. It is simply 22/3, which should be the effective number of vote-earning party in an “isolated” district; that is, one that is not “embedded” in a national electoral system consisting of other seats elected in other districts (this idea of embedded districts is the key theme of Chapter 10 of VfrS). The underlying equation for NV, applicable to any simple districted electoral system, starts with the premise that there is a number of “pertinent” parties that can be expressed as the (observed or expected) actual (i.e., not ‘effective’) number of seat-winning parties, plus one. That is, the number of parties winning at least one seat in the district, augmented by one close loser. For M=1 (as under FPTP), we obviously have one seat winning party, and then one additional close loser, for two “pertinent” parties. Thus with M=1 it is the same as the “M+1 rule” previously noted by Reed and Cox, but Taagepera and I (in Ch. 7 of our 2017 book) replace it with an “N+1″ rule, and find it works to help understand the effective number of vote-earning parties both nationwide and at district level. Raising this number of pertinent vote-earning parties to an exponent (explained in the book) gets one to NV (nationwide) or NV (district-level). When M=1, the number of pertinent parties is by definition two, and for reasons shown by Taagepera in his 2007 book, the effective number of seat-winning parties tends to be the actual number of seat-winning parties, raised to the exponent, 2/3. The same relationship between actual and effective should work for votes, where we need the “pertinent” number only because “actual number of parties winning at least one vote” is a useless concept. Hence the first component of the equation, 22/3=1.5874.

As for the second component of the equation, S1/12, it is also an algebraic transformation of the formula for the exponent on the quantity defined as the number of seat-winning parties, plus one. At the district level, if M>1, the exponent is itself mathematically complex, but the principle is it takes into account the impact of extra-district politics on any given district, via the assembly size. The total size of the assembly has a bigger impact the smaller the district is, relative to the entire assembly. Of course, if M=1, that maximizes the impact of national politics for any given S –meaning the impact of politics playing out in other districts on the district of interest. And the larger S is, given all districts of M=1, the more such extra-district impact our district of interest experiences. With all districts being M=1, the exponent reduces to the simple 1/12 on assembly size (shown in Shugart and Taagepera, 2017: 170). Then, as explained yesterday we can express NV in terms of NS via the Seat Product Model. It should be possible to verify NV =1.59√NS empirically; indeed, we find it works empirically. I showed a plot as the second figure in yesterday’s post, but here is another view that does not add in the Indian national alliances as I did in yesterday’s. This one shows only Canada, Britain, and several smaller FPTP parliamentary systems. The Canadian election mean values are shown as open squares, and several of them are labelled. (As with the previous post’s graphs, the individual districts are also shown as the small light gray dots).

It is striking how well the Canadian elections, especially those with the highest nationwide effective number of seat-winning parties (e.g., 1962, 2006, and 2008) conform to the model, indicated with the diagonal line. But can we derive an explanation for why it works? Following is an extended quotation from the draft paper (complete with footnotes from the original) that attempts to answer that question:

Equation 4 [in the paper, i.e. NV =1.59√NS ] captures the relationship between the two levels as follows: If an additional party wins representation in the national parliament, thus increasing nationwide NS to some degree, then this new party has some probabilistic chance of inflating the district-level voting outcome as well. It may not inflate district-level voting fragmentation everywhere (so the exponent on NS is not 1), but it will not inflate it only in the few districts it wins (which would make the exponent near 0 for the average district in the whole country). A party with no seats obviously contributes nothing to NS, but as a party wins more seats, it contributes more.[1] According to Equation 4, as a party emerges as capable of winning more seats, it tends also to obtain more votes in the average district.

As Johnston and Cutler (2009: 94) put it, voters’ “judgements of a party’s viability may hinge on its ability to win seats.” Our logical model quantitively captures precisely this notion of “viability” of parties as players on the national scene through its square root of NS component. Most of the time, viability requires winning seats. For a new party, this might mean the expectation that it will win seats in the current election. Thus our idea is that the more voters see a given party as viable (likely to win representation somewhere), the more they are likely to vote for it.[2] This increased tendency to vote for viable national parties is predicated on voters being more tuned in to the national contest than they are concerned over the outcome in their own district, which might even be a “sideshow” (Johnston and Cutler 2009: 94). Thus the approach starts with the national party system, and projects downward, rather than the conventional approach of starting with district-level coordination and projecting upward.

[Paragraph on the origin of 22/3 =1.5874 skipped, given I already explained it above as stemming from the number of pertinent parties when M=1.3]

Thus the two terms of the right-hand side of Equation 4 express a district component (two locally pertinent parties) and a nationwide one (how many seat-winning parties are there effectively in the parliament being elected?) Note, again, that only the latter component can vary (with the size of the assembly, per Equation 2, or with a given election’s national politics), while the district component is always the same because there is always just one seat to be fought over. Consider some hypothetical cases as illustration. Suppose there are exactly two evenly balanced parties in parliament (NS =2.00), these contribute 1.41=√2 to a district’s N’V, while the district’s essential tendency towards two pertinent parties contributes 1.59=22/3. Multiply the two together and get 1.59*1.41=2.25. That extra “0.25” thus implies some voting for either local politicians (perhaps independents) not affiliated with the two national seat-winning parties or for national parties that are expected to win few or no seats.[4] On the other hand, suppose the nationwide NS is close to three, such as the 3.03 observed in Canada in 2004. The formula suggests the national seat-winning outcome contributes √3.03=1.74 at the district level; multiply this by our usual 1.59, for a predicted value of N’V =2.77. […] this is almost precisely what the actual average value of N’V was in 2004.[5]


[1] The formula for the index, the effective number, squares each party’s seat share. Thus larger parties contribute more to the final calculation.

[2] Likely the key effect is earlier in the sequence of events in which voters decide the party is viable. For instance, parties themselves decide they want to be “national” and so they recruit candidates, raise funds, have leaders visit, etc., even for districts where they may not win. Breaking out these steps is beyond the scope of this paper, but would be essential for a more detailed understanding of the process captured by our logic. 

[3] Because the actual number of vote-earning parties (or independent candidates) is a useless quantity, inasmuch as it may include tiny vanity parties that are of no political consequence.

[4] A party having one or two seats in a large parliament makes little difference to NS. However, having just one seat may make some voters perceive the party a somehow “viable” in the national policy debate—for instance the Green parties of Canada and the UK.

[5] The actual average was 2.71.

Canada 2019

It is the final Friday before Shemini Atzeret, also known as Election Day in Canada this year, And what an interesting campaign it has been! The polls have moved quite a lot, especially recently. The New Democrats (NDP) seem to be enjoying a surge. Not on anything like the scale of 2011, but still something notable, as it was not long ago that there was talk about the Greens possibly passing them for third place. The Greens have slipped somewhat, as has been the case in past campaigns. No longer do they look likely to win as many as four seats; two (which is their current number) looks most likely.

The striking thing is that the poll aggregate at CBC (compiled by Éric Grenier) shows both major parties–incumbent seat-majority Liberal and opposition Conservative) barely above 30% of the vote (31.7-30.8 at my latest check). From 1949 to present, the largest party has never had a vote percentage below 36.3% (in 2006). So if there is not a late surge of strategic voting, this will be quite a record-breaker.

Projecting seats under FPTP is always a challenge. The CBC Poll Tracker currently has the Liberals significantly favored, despite being marginally behind in votes, 133 seats to 123 (but with wide confidence bands on both).  That would be 39.3% of the seats for the largest party, which would also break the record (from 1949 on) set in 2006 (40.3%, or 124 in what was then a smaller parliament).

Despite being both a plurality reversal and a record low vote percentage (and an extremely close vote margin), the advantage ratio (%seats/%votes) of 1.278 for the largest seat-winner would be just about average. Over 22 elections, the mean advantage ratio has been 1.2897. (Note: I am calculating this as the share of the largest seat-winner over its vote share, not over the share of the largest vote-winner, when those diverge.) For those who know Canadian electoral history, I will note that advantage ratios of around 1.2-1.3 have occurred in 1965, 1968, and 2008 (among others). Thus even if the specific vote totals may be very unusual, the workings of FPTP, given the actual votes, is fairly “typical” for Canada.

As for the other parties, I mentioned the NDP surge. But just as noteworthy is the surge of the Bloc Quebecois, which may turn out (again) to be the single most important factor in preventing a majority of seats. The BQ is currently polling just under 7% nationwide, while the Greens are just over 8%.

Of course, the BQ and Green fortunes will diverge in seats. It is very helpful for votes-seats conversion to be a regional party under FPTP, and not useful to be relatively more dispersed. So the BQ is currently estimated to get 38 seats, about the same as a much larger national party, the NDP (41) and vastly more than the also larger–in votes–Greens (2).

Regarding those surges I mentioned. The BQ was, according to the polling aggregate on only about 20% in Quebec as recently as one month ago. Now it is up to almost 30%, and just behind the Liberals’ percentage in the province (31%, having been 37% a month ago). The Conservatives have really crashed in Quebec, down from 22% a month ago to just under 16% now. The latter puts them not too far ahead of the NDP, who are now on about 14% in the province.

Nationally, the NDP was at only about 13% a month ago, but is approaching 19%. A rising vote share tends to lift the seat share–even for a national third party under FPTP. While a month ago, the Poll Tracker had the party at only 15 seats, its 41 projected now represents an increase by a factor of 2.7 when its votes have increased only 1.27 (19/15). The party would still be significantly under-represented by the electoral system, but it has reached a point where it gains a lot of seats by a small increase in votes (assuming it holds and that Grenier’s swing assumptions are reasonable, etc.).

The NDP has also pulled narrowly ahead of the Liberals in the polling aggregate in British Columbia, although still well behind the leading Conservatives.

As for the Greens, their slide has been quite abrupt. They were over 10% as recently as the first of October and were projected to win 4 seats as recently as 16 Oct.

The sixth party in the picture, the far-right Peoples Party of Canada, looks likely to win only the seat of its leader, Maxime Bernier. The riding (district) is Beauce, in Quebec, in which Bernier has held as a Conservative since 2006 until defecting from that party in 2018. (I see the Rhinoceros Party has found a candidate with the same name to put up against him.) For months, the PPC has been at either zero or one seat in the projection.

As for who will form a government, the Liberals seem best placed, even if the result is as short of majority as the Poll Tracker projects. It is possible that they will be weak enough to have to form a coalition with the NDP, even though probably the Liberals would prefer a minority government. On current numbers, Liberal+NDP would be a very bare majority. The coalition or a minority government might need working arrangements of some sort with the Greens and/or BQ as well.

It is much harder to see how the Conservatives can form government, even if they end up edging out the Liberals for a seat plurality. Conservative leader Scheer has already begun the spin just in case, claiming this week (incorrectly) that the party with the most seats gets the first shot at forming a government.

NDP leader Jagmeet Singh has said he would try to form a coalition with the Liberal Party if the Conservatives have the most seats. And PM Justin Trudeau would have the legal right to attempt to work out such a deal and meet parliament to try to retain office. Presumably, Singh (and the Green leader, Elizabeth May) would attempt to extract a concession that 2019 be the last election under FPTP.

Canadian Senate being debated in Supreme Court

Via CBC:

Prime Minister Stephen Harper’s government has asked the Supreme Court of Canada to advise whether it can proceed unilaterally to impose term limits on senators and create a process for electing them.

The government contends that some such reforms can be imposed by the central government, citing the imposition of a retirement age for senators in 1965. However, the government’s question also considers the question of possible abolition of the senate. Here the question is whether unanimous consent of the provinces would be required, or whether the “750 formula” must be adhered to. The latter means seven provinces, accounting for half the national population.