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.

3 thoughts on “Why 1.59√Ns?

  1. If one treats the 46 members of the House Freedom Caucus as a distinct part that is in coalition with the 166 other members of the GOP, in the HOuse of Reps, the effective number of parties in the US House is 2.52 (treating the 2 independents as Democrats). This predicts an effective vote-earning party of 2.52 as well. This may account for the tensions within the GOP, as there are effectively two parties competing in the primaries and it also accounts for why they’re relying on Trumpism to supposedly help them win in 2022 and to mitigate “somewhat” their tensions.

    But I doubt Trump will keep the momentum going. He wants to waste his energy attacking McConnell and settling scores and his legal issues, the truth behind 1/6, his poor spending of donations, the lack of love from Russia and the pandemic will hurt him.

  2. I actually forgot that I had explained that equation before. Well, it is time to revise the chapter, so it was a good exercise to re-familiarize myself with the the logic!

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