The French thresholds for runoff participation

This week is the inter-round period in the French two-round assembly elections. The first round was on 12 June. The French way of electing members of the National Assembly is not top-two majority-runoff, like the country’s presidential elections (or most elections in California). Rather, it is majority-plurality. That is, it is possible to have more than two candidates in the second round in any given single-seat district, and when this happens, the winner is the one with the most votes, even if it is less than 50%+1.

In any system within the broader family of two-round systems, there need to be threshold provisions for both (1) determining whether a runoff is required, and (2) determining who is eligible to participate. Under typical majority runoff, the provisions are (1) 50%+1 in the first round, or else (2) there must be a second round in which only the top two may participate.1

France follows the same first provision–with a caveat that I will get to. A majority is required at the first round. If that does not occur, the rules are that any candidate with votes equivalent to at least 12.5% of the registered voters in the district may stand in the runoff. That is, an eighth of the electorate, not an eighth of the votes cast. This is an important distinction. There is a further twist on the runoff-participation rules: If there are not two candidates who clear the 12.5%-of-registered threshold, then the top two go to the runoff anyway.

Here are some examples. In the district of Paris no. 2, the Ensemble (pro-Macron) candidate had the first-round plurality, with 35.66%, followed by the candidate of Nupes (pro-Mélenchon) on 27.27%. The third candidate, from the Republicans (LR, traditional right) has 18.23%. So is it a three-way race? Non. This candidate’s votes are a mere 10.63% of the total registered voters. Hence it is a top-two runoff.

Then we have Paris no. 15. Here, the leading candidate (from Nupes) has 47.31% of the votes cast. Pretty close to a majority, but not good enough under criterion #1 (50%+1 or else runoff). However, no other candidate cleared 12.5% of the registered voters. So maybe a runoff is not required after all. Not so fast. There needs to be a runoff if the plurality candidate did not win a majority of votes cast. Thus the candidate with the second highest vote total (from a left party unaffiliated with Nupes) advance to a runoff despite having won only 9.44% of the registered electorate in the first round. (This candidate won 17.87% of votes cast, but for qualifying purposes, this is not even relevant.)

There is, as I mentioned, a caveat on the first criterion, that a majority in the first round obviates the need for a second. To be elected in the first round, the leading candidate’s vote total also must be greater than 25% of the registered electorate in the district. There is at one prominent case where this comes up in the current election–prominent because it involves a famous politician. The district is Pas-de-Calais-11. The leading candidate is Marine Le Pen (you’ve probably heard of her–leader of the National Rally (RN)). She won 53.96% of the votes. Good for her; she won a majority! However, her votes amount to just 22.52% of registered voters. As a result, she must face a second round against another Marine, Tondelier of Nupes, who qualifies despite a vote total that is only 9.79% of the registered electorate.

It would be strange to have a candidate win a majority in the first round and yet lose the runoff. It probably won’t happen. On the other hand, if the opposition to her could mobilize and vote for the Nupes Marine, it is theoretically possible. Looking at the rest of the field, we find an Ensemble candidate in third place with 5.15% of registered electorate and then no other candidate over 1.5%. It would be a tall order, needing a whole lot of abstainers to turn up. But the rules of the French two-round system create the opportunity.

I thank Giacomo Benedetto and Steven Verbank (both via Twitter) for the above examples and clarifications on rules.

Just for fun, I was clicking on districts somewhat randomly. (You can play the game, too, by going back a step or two in any of the links for the three examples above.) It seems that the low turnout might be responsible for quite a few districts being like the second example–a candidate qualifying for a runoff despite being below 12.5% of the electorate in the first round. I do not know how common this or a majority but not 25% of registered voters has been over the course of the Fifth Republic’s history (dating to 1958, with then exception of a list-PR system in 1986). It is also noteworthy that there are eight three-way runoffs this year (compared to just one in 2017). If anyone happens to have a list of them, please post in the comments.

Randomly, I will now mention a few cases that looked interesting to me. Yonne no. 1: The top three candidates (Nupes, RN, and LR) have vote percentages of 24.25, 23.92, and 22.61. Only the first one has more than 12.5% of registered, and the third narrowly missed it (having 11.50%). I have no idea how often a candidate might ever have won from third place in a three-way runoff in France. If there were to be such a case, this would seem to be a promising opportunity–a mainstream right candidate against the left and far right, who combine for less than a majority of votes cast. However, the LR candidate came up short of making it a three-way.

Another similar case is Moselle no. 5: Top three candidates from RN, LR and Ensemble, with vote percentages of 25.88%, 24.53%, and 23.48%. Tight three-way race. But third did not clear 12.5% of the registered electorate–in fact, the leader had only 10.35%–and so it can be only a two-way.

Another majority-but-runoff case in Bouches-du-Rhone no. 4: Nupes leader has 56.04% of votes cast, but only 21.36% of registered voters. So the second candidate, from Ensemble, who won won only 14.88% of votes (and 5.67% of the electorate) advances to a runoff.

I probably could do this all day. But I probably should not.

One more thing before I hit “publish.” This collection of candidate statements and photos for every candidate in every district is an incredible resource! It is interesting that many of them are pictured next to their party’s presidential candidate (did someone say presidentialization?), and I also like how some of them show a little photo of their replacement candidate. Every candidate is elected with a substitute, who takes the seat if the principal vacates it for any reason during the term. Such reasons include appointment to an executive post, as France does not allow simultaneous service in the assembly and executive.

  1. Or sometimes–including French presidential elections–the top two remaining if one of the initial top two has bowed out for whatever reason.

Ontario 2022

Ontario’s election on 2 June saw another Progressive Conservative seat majority on barely over 40% of the votes. The party, led by provincial Premier Doug Ford, barely increased its vote percentage from 2018, when it won 40.2%; this time the tally is about 40.8% (pending final count). Its vote total actually went down, because it was the lowest turnout in the province’s history. Yet it will have 83 of the 124 seats, whereas in the 2018 election it won 76.

For those keeping the stats, that would be a bare two-thirds majority (66.9%), and an advantage ratio (%seats/%votes) of 1.64. That is very much on the high side, even by the standards of FPTP with multiparty systems.

The main shifts in vote percentages were among the two largest opposition parties. The Liberals improved from 19.4% to 23.9%. The payoff in seats was minimal: the party won 8 seats this time, 7 last time. The NDP performed especially badly, going from 33.3% of the vote in 2018 to 23.7%. However, even though the NDP’s votes are marginally behind the Liberals’, the NDP will continue to have more seats–a lot more–with 31 (down from 40 at the last election). Yes, FPTP in multiparty systems!

Ontario objectively needs to shift to a proportional system. It is not as if the province has not had the opportunity to do that before.

Kosovo electoral system note

In light of our previous discussion about how Kosovo’s electoral system challenges our usual notion of what a “district” is, this note from Michael Gallagher‘s Election Indices is interesting.

I am not sure Michael has made the correct choice here–minority representation provisions are part of the electoral system, after all–but I am also not sure this is incorrect. The system really is challenging to classify and quantify. I note in particular his decision to count its assembly size–and therefore, its district magnitude, given there are no district divisions unless we count the ethnic reservation/guarantee as separate “districts”–as 100 before 2014 but as the full 120 since then. Here, for reference, are the indices he reports in the main part of the document:

The unusual nature of the system is what results in the effective number of seat-winning parties (NS) sometimes being higher than the effective number of vote-earning parties (NV), something that is otherwise rare, and certainly should not happen in a single-district nationwide proportional system. As I noted in the earlier discussion, in 2021 it was even the case that a single party list won a majority of votes, but did not win a majority of the full 120 seats. Because I assume all legislators are equal, and that a government needs a majority of the 120, and not just the 100, I think it is incorrect to treat assembly size as not including the 20 ethnic representatives. Gallagher’s data from 2014 do include them, and I think that should be the case for the earlier years as well.

The question of how to calculate the indices is indeed a vexing one. Gallagher very helpfully explains his choices and what would change if we use a different assumption about what “counts.” This allows the researcher using his valuable resource the ability easily to make his or her own decision. But this researcher still is not sure which decision to make with respect to this system!

I am not comfortable with the idea of counting these various ethnic guarantees as additional “districts” even though I see the case for it (which Henry made in a comment to the previous planting). That lack of comfort is not solely because these “districts” overlay the main one. That is, after all, the case of the Maori districts in New Zealand (each of which encompasses the territory of several general electorates). For that matter, it is also the case with any two-tier system. Rather, the conceptual difficulty is that a given party list may win seats in either component of the system–the general 100 or the set-aside for their ethnic group–if they qualify for additional seats beyond their ethnic group’s reservation/guarantee.

However we conceptualize the system, I believe all these parties should be taken into account in calculating the effective number of parties (votes and seats). The question of whether we count them for deviation from proportionality is less clear to me.

I think I need to count this as a non-simple system (by the criteria used on Votes from Seats), giving us a unique case of what could be called a single nationwide district PR system that is nonetheless complex. For countries whose electoral system has just a few ethnic set-asides (like Colombia or Croatia), I tend to ignore the reserved seats when thinking of whether they are “simple” districted or national-district systems. But when such seats are a sixth of the total, they are clearly a complicating feature, as the unusual outcomes reveal.

St Kitts and Nevis 2000–Crazy result

Given my sudden fascination with small assemblies, I was poking around in election results from St Kitts and Nevis, a Caribbean sovereign state with a population of just over 52,000. With 11 elected members, its assembly certainly counts as small. The 2000 election is really something. Look at the national result:

PartyCodeVotes% votesCandidatesSeats
St. Kitts and Nevis Labour PartySKNLP11,76253.85%88
People’s Action MovementPAM6,46829.61%80
Concerned Citizens MovementCCM1,9018.70%32
Nevis Reformation PartyNRP1,7107.83%31
Total Valid Votes21,841100%2211
Source: St Kitts and Nevis Election Center, Caribbean Elections.

The second largest party got no seats, while two parties with less than 10% each won a seat or two. This is a first-past-the-post system. The problem the PAM had was it came in second in all eight seats it contested, i.e., every district on the island of St. Christopher (none were close). The advantage the CCM and NRP had is they run only on the island of Nevis, which has three district. Here are the district results.

ConstituencyRegistered votersSKNALPPAMCCMNRPValid Votes
St Christopher #14,5191,7881,1492,937
St Christopher #25,6522,0111,5073,518
St Christopher #32,5961,2353771,612
St Christopher #42,4301,0137351,748
St Christopher #52,3288697691,638
St Christopher #62,5711,6131191,732
St Christopher #72,8741,4414791,920
St Christopher #84,3251,7921,3333,125
Nevis #92,9248087961,604
Nevis #101,517555184739
Nevis #112,4305387301,268
Total34,16611,7626,4681,9011,71021,841
Source: Same as for first table.

Note that there is some pretty serious malapportionment here, as well. Nevis constituencies have many fewer voters than St. Christopher constituencies. In fact, the three Nevis districts together have only about 1.2 times the population of the most populous St. Christopher district.

So what should we have according to the Seat Product Model? The seat product is 11 (magnitude of 1, times assembly size of 11), so the effective number of seat-winning parties should be 1.49. In this election it was actually 1.75. That’s actually not a terrible miss! But in most elections it has been considerably higher than that–as high as 3.90 in 2015. So just for fun, a quick look at that one:

PartyVotesvotes% votesCandidates
St. Kitts and Nevis Labour PartySKNLP11,89739.27%83
People’s Action MovementPAM8,45227.90%64
People’s Labour PartyPLP2,7238.99%21
Concerned Citizens MovementCCM3,95113.04%32
Nevis Reformation PartyNRP3,27610.81%31
Total Valid Votes30,299100%2211
(Last column is seats won, but the heading did not copy over.)

This time, the PAM benefitted greatly! It is in a clear second place in votes, yet won a plurality of seats. Not a majority, however. According to Wikipedia, there were alliances. But even at the alliance level, there was a plurality reversal: “The outgoing coalition (SKNLP and NRP) secured 50.08% of votes but got only 4 seats, the winning coalition (PAM, PLP and CCM) won 7 seats with only 49.92% of votes.” Oh, cool: Another case of pre-electoral alliances! The effective number of alliances was just 1.86.

And at the district level:

ConstituencyRegistered VotersSKNLPPAMPLPCCMNRPValid Votes
St. Christopher #15,0361,7271,7313,458
St. Christopher #24,7401,7581,6603,418
St. Christopher #33,2651,3481,0762,424
St. Christopher #43,1661,2161,2522,468
St. Christopher #53,1078841,2452,129
St. Christopher #62,8231,9692002,169
St. Christopher #73,1918671,6472,514
St. Christopher #85,7532,1282,3644,492
Nevis #96,1272,0331,7153,748
Nevis #101,3937543061,060
Nevis #113,5841,1641,2552,419
Total42,18511,8978,4522,7233,9513,27630,299

We might not expect regionalism in such a small country, with a small assembly. But the party preferences of the two islands obviously are genuinely different (and the PLP is “regional” in that it contested only two districts on St. Christopher); yet the parties aggregate into alliances for purposes of national politics.

The malapportionment is still noteworthy–look at the small population of Nevis 10. However, one of the other two districts is now the most populous in the country, quite unlike in 2000.

Final point: Its population may be small, but according to the cube root law St Kitts and Nevis should have an assembly more than three times what it actually has: 37. If they were proportional to registered voters, Nevis would be allotted nine of those 37 seats. It currently has 3 of the 11, so 27%, so quite close to its population share, unlike in 2000 when it was overrepresented. Making the seats allocated by island more easily fit population balance in itself would be a good argument for increasing assembly size, but an even better argument would be making anomalous results like the two elections shown here less likely–even if they insist on sticking with FPTP.

Kosovo 2021: A single-district electoral system that violates the rank-size principle

In the previous planting asking whether free-list PR violated my own definition of a “simple system, I mentioned the criterion of avoiding violation of the rank-size principle in allocation of seats to votes within district (see footnote 2). I later happened up a major example of violation of the rank-size principle: Kosovo in 2021!

It is a single district of 120 seats, but per Wikipedia:

The Assembly had [under the Constitutional Framework] 120 members elected for a three-year term: 100 members elected by proportional representation, and 20 members representing national minorities (10 Serbian, 4 Roma, Ashkali and Egyptian, 3 Bosniak, 2 Turkish and 1 Gorani). Under the new Constitution of 2008, the guaranteed seats for Serbs and other minorities remains the same, but in addition they may gain extra seats according to their share of the vote.

The result of this is that there are parties with as much as 2.5% of the votes but no seats (there is also a 5% legal threshold for non-ethnic parties), and parties with as little as 0.14% who have seats when somewhat larger ethnic parties do not. For instance, the United Roma Party of Kosovo has a seat with 1,208 votes while the Innovative Turkish Party, which I presume is an “ethnic” party, with 1,243 votes, has none. That would be because the two set-aside Turkish seats were won by the Turkish Democratic Party of Kosovo, which had almost 6,500 votes (0.75%).

Perhaps it is not a “single district” and we should think of each ethnic group’s set-aside representation as a distinct district in addition to the general constituency. But that is certainly not how I generally understand “district.” In any case, there is nothing “simple” about the provision or its impact on outcomes, particularly regarding the rank-size criterion.

In addition, these provisions result in the odd case of a party with a majority of the vote not getting a majority of the seats, which is certainly unusual for a proportional representation system!

Is free-list PR a “simple” electoral system?

This seems like a trick question. Of course, free-list has all sorts of complex features. In such a system, the typical rules are that any voter may vote for as many candidates as he or she wishes, even across different lists (panachage). A vote for any candidate on a list counts as a vote for that list for purposes of determining proportional seat allocation across lists, as well as for the candidate in competition among other candidates on that list.

However, this system handles votes and seats for lists just like any other list-PR system: It is designed to allocate seats to lists first, and only then to candidates. It thus is “simple” on the inter-party dimension, unlike SNTV or MNTV or STV (where candidate votes do not count towards aggregate party vote totals and seats are allocated based only on candidate votes).

My general definition of a “simple” electoral system is one that is a single-tier, single-round, party-vote system. The free-list could be said to violate that last part of the definition, in that “party vote” maybe should mean a single party vote per voter. My instinct is to keep free list in, because it remains “simple” in terms of how it processes the votes across lists. But I could be convinced otherwise, given that effectively every voter can vote for more than one list–a “dividual vote” in Gallagher’s terms.1

In Votes from Seats, Taagepera and I kept at least three free-list systems in our dataset: Honduras (since 2005), Luxembourg, and Switzerland. The issue came back to my mind because of my consideration of including some smaller countries and non-independent territories in a dataset for some further analysis of key questions. One of the smaller countries that could be added to the data is Liechtenstein, which I believe uses a free-list PR system. My gut says “yes, include” but now I wonder if we already violated our own criteria2 in having those free-list systems in the prior analysis. To be clear, none of our results would be changed if we had dropped them.3 It is just a matter of consistency of criteria.

Questions like this always nag comparative analysis, or science more generally. What things are part of the set being analyzed? It is not always clear-cut.

____

  1. Note that there is no question regarding standard open-list PR: Even if there are multiple candidate preference votes cast per voter, as in Peru, only a single list vote is registered per voter.
  2. In fact, on p. 31 of Votes from Seats, we say “Only categorical ballots and a single round of voting are simple, by our definition.” A free-list ballot is dividual and thus not categorical. However, the reason we give for limiting the coverage to categorical ballots is that “other ballot formats… may violate a basic criterion for simplicity in the translation of votes into seats: the rank-size principle” (emphasis in original). For example, the party with the most aggregate votes in a district may not have the most seats allocated in the district (or at least tied for most with the second-most voted party). This violation of the rank-size principle can occur with SNTV, STV, and MNTV, but as noted above it can’t occur in free-list PR (per my understanding, anyway). I note that in a later work, Party Personnel, my coauthors and I seem to adopt a stricter definition. On p. 53 of that book, we say that simple means “a voter votes once, and this vote counts for the entire party list of candidates.” Yet the conceptual point there is somewhat different, in that we are referring to “simple vote” not simple electoral system, and we remove open-list PR from the standard of simple vote because they permit differentiation of candidates within a list in the same district. But as for the vote counting for the entire list, free list still meets that part of the criterion. (A reminder that “voting system” is not a synonym for “electoral system”!)
  3. Although I did not think of this possible issue with free lists at the time, I definitely ran robustness-check regressions with Switzerland dropped. I did so mainly because of its multiparty alliance feature, which also is a complex feature for reasons discussed in the book (mainly with reference to Finland and Chile). Doing so did not affect the results, so we left the case in. There are not enough elections from the other free-list cases, nor are they observably different on our outcomes of interest, that they could affect results. (Switzerland is observably different–far more fragmented than expected for its seat product, and that seems to be mostly due to alliances, even above the impact of its ethnic fragmentation–see p. 269 of Votes from Seats. But the inclusion or exclusion fo the case is immaterial for the overall results.)

Small assemblies in non-independent territories

I am going to do a little crowd-sourcing here. What do people think is a reasonable way to define “autonomous enough” to include a territory in a set of small assemblies worthy of comparative analysis alongside independent nations with small assemblies?

That is, there are various countries with very small assemblies that are recognized as independent states, such as St. Kitts and Nevis (assembly of 11 seats) or Antigua & Barbuda (17). Like the two just mentioned, most of the small-assembly independent states that are also democracies with small assemblies are in one world region (Caribbean) and use one type of electoral system (FPTP).

Now suppose one wanted to branch out and include small territories that were either not in the Caribbean or used PR. Suppose further that one did not want to include obviously fully dependent territories that just happen to hold elections for an internal legislative council. Where would one draw the line?

For instance, are the Faroe Islands and Greenland “autonomous enough” to include? What about Aruba and Curaçao? These use PR systems, and the first two are not Caribbean. Or the Cook Islands, with is FPTP but non-Caribbean?

One would need a reasonable standard for autonomy. I sort of feel the places I just named might qualify, but I do not know why I feel that way. And I do not want the can of worms opened whereby I’d be asked–legitimately–why did you exclude Turks and Caicos (for example)? (Other than, well, I already had enough FPTP Caribbean cases.)

The smallest currently included independent country in my related datasets seems to be St. Kitts & Nevis (pop 54k). One of the territories I mentioned is much smaller than that (Cook Islands only 15k), but others are of the same order as St. Kitts (like Faroe Islands and Greenland, 53-55k). I probably have a floor somewhere on population–which might well exclude Cook Islands–but my current query is for a reasonable standard on what is sufficiently self-governing to be comparable to small independent states for purposes of analyzing their assemblies and electoral systems.

What do readers of this site think?

No, the parties:largest-size relationship is not different for two-tier PR

Might as well graph it.

(Click for larger version)

No reason here to doubt that the logical model, NS = s1–4/3, applies equally well to two-tier systems as it does to simple, single-tier systems. This was a question I raised in the earlier planting on the revision of the extended Seat Product Model (incorporating two-tier systems without an empirical constant).

Thus any deviations of regression output from the precise predictions of the models–as reported in that earlier post–are not caused by some systematic difference in this relationship for two-tier systems. Such deviations are just noise. For instance, the regression intercept on these 472 elections is significantly greater than zero. Yet a nonzero intercept is impossible. It can’t be that the effective number of parties is any different from 1.0 (the log of which is 0) if the largest party has 100% of the seats.* More to the point for the question I had, the regression shows no significant difference in slope (or intercept for that matter) between single-tier and two-tier systems. They behave the same in this sense, meaning that when the compensation tier increases the effective number of parties and reduces the seat share of the largest, it does so while preserving NS = s1–4/3, on average. And, by the way, for those who care about such things, the R2=0.899.

Bottom line: there is no statistically significant difference between single-tier and two-tier electoral systems in how the effective number of parties is related to the size of the largest.

* If I suppress the constant (while also eliminating the binary for “simple”) the coefficient is –1.341, or almost precisely the logically required –4/3. When run with the constant, it is –1.235, but the 95% confidence interval includes –1.333.

The Austrian Question: Or how I corrected some data I’ve been using on two-tier systems

In the previous planting, I presented a revised version of the extended Seat Product Model. I noted that in the process of attempting to improve on the logical model, I discovered some inconsistencies in the treatment of remainder pooling systems in the dataset used in Votes from Seats. Here I describe the problem and how I corrected it. The changes here may still require further refinement, but at least they make the treatment of the cases internally consistent.

This first began to bother me even before Votes from Seats had been published. Figure 17.2 in the book shows how well (or not) the extended seat product model accounts for the effective number of seat-winning parties (NS) over time in several two-tier PR systems (plus Japan, included despite not fitting the category for reasons explained in the book). It plots every election in the dataset for this set of countries, with the observed value of NS shown with the solid grey line in each country plot. The expectation from the extended Seat Product Model (Equation 15.2) is marked by the dashed line. This equation is:

NS = 2.5t(MSB)1/6,

where NS is the effective number of seat-winning parties (here, meaning the expected NS), M is the mean district magnitude of the basic tier, SB is the total number of seats in the basic tier, and t is the “tier ratio” defined as the share of the total number of assembly seats allocated in the compensatory tier.

For countries that changed from simple to complex, the plots also show the expectation in the era of the simple system with the solid dark line. The troublesome case here is Austria, but why? That is my “Austrian Question.” It led me down quite a rabbit hole, but I think I have it figured out, more or less.

Figure 17.2 in Votes from Seats. Expected and actual effective number of seat-winning parties (NS) over time in long-term democracies with two-tier electoral systems. (Click for larger version.)

It always seemed unlikely that the design of the Austrian electoral system was such that expected NS could have reached well over 6 in the latter part of the time series! But that is what our data showed, supposedly. When you get an absurd result, generally you should impeach the data, not the model.1

The problem turned out to be that for several remainder-pooling systems, including Austria, some seats were effectively counted twice in the derivation of the extended seat product. We drew most of our data from Bormann and Golder’s Democratic Electoral Systems Around the World. However, the manner in which we did so handled remainder-pooling systems poorly. With one important exception that I will note below, the error was not in the original data, but in our application of it.

In a remainder-pooling system there is no fixed upper tier. Most two-tier compensatory systems have a fixed number of seats which are allocated nationally (or regionally) to “correct” for distortions in votes-to-seats allocation produced by the lower district magnitude of the basic tier. An example would be the system of Denmark, with 139 basic-tier seats and 40 compensatory seats. Others have a fixed minimum, such as the MMP systems of Germany and New Zealand (where the upper tier can expand if needed due to “overhang” seats, but it has a fixed starting size). In a remainder-pooling system, on the other hand, the “upper tier” can be as large or as small as needed to generate compensation. In theory, all seats could be allocated in the basic tier, and it would end up no different from a simple system. Typically these systems work by stipulating that parties earn seats based on full quotas (usually Hare quota, sometimes Hagenbach-Bischoff) in the districts. Any seats not filled are then “pooled” in a supra-district tier where they are allocated based on pooled votes, rather than being filled within districts. The upper tier is thus whatever number of remainder seats there are from all of the districts, which can vary from election to election depending on how votes are distributed among the parties and across the basic-tier districts.

A particular challenge in the analysis of these systems is that all seats may be attached to districts, and national reporting agencies vary in whether they indicate that a given seat is actually based on supra-district allocation. Thus a district might have, say, five seats, and in the determination of quotas, two parties may have obtained a total of three seats through quotas in the district. The other two seats go to the remainder pool. Maybe, once all the national seat allocation is complete, one of those two remainder seats goes to the largest party (bringing it up to three) and one goes to a third party that was short of a quota in the initial allocation. The complication is that while all five seats are assigned to candidates who were on party lists in the district, only three were assigned based solely on votes cast in the district. The other two were assigned according to the compensation mechanism, drawing upon the pooled votes from across multiple districts. Where is the upper tier? It is sort of a phantom, and if we count the two seats in our hypothetical example as part of the upper tier, and we also count them as part of the district (basic tier), we have double counted them!

Here is where the Austrian case comes in. If we look at the 1990 election, we see an assembly size of 183, with 9 basic-tier districts, averaging 20.3 seats apiece. Bormann and Golder report that 25 seats were allocated as “upperseats”. In our Equation 15.2, the input parameters were MSB=20.3*(183–25)=3207 (rounding off) and t=(25/183)=0.137. That is, the basic-tier seat product is mean district magnitude multiplied by the size of the basic tier (which is total assembly size minus upper seats). But hold on! Those 25 upper-tier seats are taken out of the 20.3 seats per (average) district. Yet our original calculation takes them only out of the “S” part, but not out of the “M” part. They should not be counted in both tiers! Those 25 seats came from the 9 districts, so 25/9=2.8 remainder seats per district, on average. This gives us an adjusted basic-tier M=20.3–2.8=17.5. Now we have MSB=17.5*(183–25)=17.5*158=2765, and t=0.137. This changes the “expected” NS (based on Equation 15.2) from 4.35 to 4.25. Not a huge difference, but one that more accurately reflects how the system actually works.

Where things really went haywire was with the electoral reform that took place before the 1994 election. The Bormann and Golder dataset correctly notes that the number of basic-tier districts was increased to 43. With S=183 unchanged, this is a mean district magnitude of M=183/43=4.26, a figure which matches the description in Electoral System Change in Europe, maintained by Jean-Benoit Pilet and Alan Renwick. However, for some reason, the Bormann and Golder gives first-tier mean district magnitude for the post-1992 system as 17.2. The indicated values of “upper seats” range from 78 to 111 in the elections of 1994–2008. When we apply the same procedures of the preceding paragraph to elections in these years, we get a reduction in MSB from the 2669 we used in the book to a more accurate 196.7. That is quite a change! It comes from the reduction in district magnitude to 4.26, which in turn greatly pushes up the number of seats allocated in upper tiers.2 When we stop double counting the remainder seats, we actually have an adjusted basic-tier magnitude of less than 2, and an upper tier ratio, t=0.5. This changes that rather absurd “expected NS” depicted in Figure 17.2 as 6.3 for recent elections to a more reasonable 3.83. And, in fact actually observed NS in recent years has tended to be in the 3.4–4.2 range.

Here is the corrected version of the figure. (I left Japan off this one.) In addition to using the corrected data, as just explained, it also uses the revision of the extended Seat Product Model:

NS = (1–t)–2/3(MSB)1/6.

Version of Figure 17.2 in Votes from Seats using corrected data. (Click for larger version.)

Austria is no longer shown as system that should be “expected” to have an effective number of parties around six! It still has an observed NS in most years that is smaller than expected, but that’s another story. We are not the first to observe that Austria used to have an unusually consolidated party system for its electoral system.3 In fact, in recent years it seems that the revamped design of the system and the increasingly fragmented party system have finally come into closer agreement–provided we use the revised SPM (as explained in the previous planting) and the corrected electoral-system data, and not the inconsistent data we were using before.

And, here for the first time, is a graph of largest party seat share in these systems, compared to expectations. This seemed worth including because, as noted in the previous planting, the s1 model for two-tier works a little better than the one for NS. Moreover, it was on s1 that the revised logic was based.

Expected and actual largest party seat share (s1) over time in long-term democracies with two-tier electoral systems. (Click for larger version.)

Note that the data plots show a light horizontal line at s1=0.5, given the importance of that level of party seat share for so much of parliamentary politics.

Notes

1. Assuming the model is on solid grounds, which was very much not the case of the original version of application to two-tier PR. I hope it is now, with the revision!

2. Plural because the 9 provincial districts still exist but are now an intermediate compensation tier, and there is a single national final compensation tier. This additional complication should not affect our estimation of the system’s impact on party-system outputs. (It principally affects which candidates from which of a given party’s lists earn the various compensation seats.)

3. This is not unique to Austria. Several European party systems used to have effective number of parties smaller than expected for their electoral system. In recent decades, many have become more fragmented, although the fragmenting trend is not significant, when compared to the SPM baseline). The trend implies that, in many cases, their electoral systems are shaping their party systems more as expected now than in the early post-war decades. In the past, the full electoral system effect may have been tamped down by the stronger role of the major party organizations in society. This is a very big question that it far beyond the scope of my current tasks.

Further note

In order to attempt a further validation of the procedure, I calculated the number of quota seats expected in each district based on my district-level dataset, derived originally from CLEA. I can then sum this up across districts in a given election, and subtract the result from the total assembly size to arrive at an indicator of what the upper-tier size should have been in that election. When I do this, I usually come close to the value for “upperseats”in Bormann & Golder, although not always precisely. I do not know what explains the deviations, but in all but one election they are so small that I would not fret. For the two elections used as examples from Austria above, I get 24 remainder-pooled seats in 1990 (vs. B&G 25) and 111 in 2008 (identical to B&G). Ideally, we would be able to estimate what upper-tier seats should be, on average, for a given design of a remainder-pooling system. Then we could estimate the parameters needed for the extended SPM even if data sources do not separate out the seats allocated on district votes from those allocated via supra-district pooling. This would introduce some unknown error, given that the actual number of remainder-pooled seats can vary depending on election results, for constant institutions. For instance, for the current Austrian system, it has ranged from 81 to 111 between 1995 and 2008. Perhaps there is some mathematical relationship that connects this average (92) to fixed parameters of the electoral system, and that works across remainder-pooling electoral systems. If there is, it has not revealed itself to me yet.

Local government STV in New Zealand

Increasing numbers of local councils in New Zealand are switching to the single transferable vote (STV) system. An article by Tim Newman, Nelson Reporter (via Stuff), indicates that in “2022 Nelson will be one of 15 councils using the STV system, and one of four adopting it for the first time.”

The Nelson version of STV (which the article indicates is approved but still subject to an appeal process) will be somewhat more complex than I would think necessary.

Under the new model two general wards have been set up, Central and Stoke-Tāhunanui , with four councillors to be elected per ward. For each ward, the population per councillor will be approximately 6400.

Running parallel to the general wards will be the Whakatū Māori Ward, which covers the whole city and will only be eligible for those on the Māori Roll.

One councillor will be elected from this ward, which has a population per councillor of about 3300.

In addition to the wards, there will also three “at-large” councillors representing the whole city. The mayor will also be voted at large.

So if I am understanding this correctly, it will be doubly parallel. For electing the 12 council members there are both districts (wards) and a citywide component in addition to the Maori special district. And all by STV, except maybe the single Maori member (it is not clear if this is by STV (AV) or not). One would think they could simply use STV–either citywide or in districts–with a rule ensuring a minimal number of those elected are Maori. Or, slightly more complex than that, but less than what is now likely to be adopted, two sets of districts–general and Maori–but not three.

The current system seems to be MNTV, but the article is a little confusing on this point. It says:

In previous elections, voting in Nelson has been conducted “at large”, meaning that voters could vote for any of the 12 council candidates standing for election, along with mayoral candidates.

I am taking that to mean the voter had 12 votes and the top 12 were elected, but I wish it was clearer. The adoption of STV is a positive development, even if it has been done with more complex districting than seems necessary.

MMP as sub-category of two-tier PR–some basis for doubt

In yesterday’s review of the German election outcome, I used the extended Seat Product Model (SPM) formula for two-tier PR systems. I have done this many times, and Rein Taagepera and I (in our 2017 book, Votes from Seats) do explicitly include mixed-member proportional (MMP) in the category of two-tier PR systems.

However, there is one problem with that characterization. All other two-tier PR systems that I can think of entail a single vote, which is then used both for allocating seats in the basic tier and pooled across districts for national (or sometimes regional) compensation.

MMP, of course, usually entails two votes–a nominal (candidate) vote used only in the basic tier, and a second, party-list, vote used for determining overall proportionality. (In MMP, the basic tier is a “nominal tier” because the vote there is cast for a candidate, and the district winner earns the seat solely on votes cast for him or her by name.) This two-vote feature is a complex feature of MMP that is actually emphasized in my more recent coauthored book, Party Personnel Strategies, but which I may have tended to underplay in my comparative work on modeling the effects of electoral systems on party systems. Of course, by being two-tier, it is already a non-simple system, as Taagepera and I define that term. But we also say that two-tier PR, including MMP, is as simple as an electoral system can be and still be included in the complex category (see p. 263 and 299 of Votes from Seats).

Maybe that is not an accurate statement for two-vote MMP. Our definition of simple (pp. 31-36) concentrates on two features: (1) all seats allocated within districts, and (2) adherence to the rank-size principle, such that the largest party gets the first seat in a district, and remaining seats are allocated in a way that respects their relative sizes (i.e., by any of the common PR formulas). We further say that for simple PR, “the vote for candidate and for party is one act” (p. 35). This latter condition still holds for any two-tier list-PR system, because there is a list vote that applies both for allocating seats within a district, and also for the “complex” feature of the supra-district compensation mechanism. Obviously, however, MMP as used in Germany violates the principle that “the vote for candidate and for party is one act.” So maybe it is not “simple enough” to qualify as an almost-simple complex system. (Yes, that was a complex statement, but that’s kind of the point.)

If MMP were to tend to produce a party system more fragmented than expected from the extended SPM, it might be due to the “second” vote, i.e., the list vote. To test this, one could aggregate all the nominal votes and use them as the notional list votes in a simulated compensation. (This is how MMP in Germany worked in 1949, albeit with compensation only at state level. It is also how MMP now works in Lesotho.) The aggregation of basic-tier votes should work better from the standpoint of modeling the party system impact of the key features of a given MMP system–the size of the basic tier and the share of seats in the compensation tier.

The catch in all this is that, of course, till quite recently German MMP was under-fragmented, according to the SPM, despite using a separate list vote. Thus the issue did not arise. The New Zealand MMP system also has matched expectations well, after the first three post-reform elections were over-fragmented relative to model prediction. The graph below shows the relationship over time between the expectations of the SPM and the observed values of effective number of seat-winning parties (NS) in both Germany and New Zealand. For the latter country, it includes the pre-reform FPTP system. In the case of Germany, it plots NS alternately, with the CDU and CSU considered separately. As I noted in the previous discussion, I believe the “correct” procedure, for this purpose, is to count the “Union” as one party, but both are included here for the sake of transparency. In both panels, the dashed mostly horizontal line is the output of the extended SPM for the countries’ respective MMP systems1; it will change level only when the electoral system changes. (For New Zealand, the solid horizontal line is the expectation under the FPTP system in use before 1996.)

The German party system from 1953 through 2005 was clearly fitting quite poorly, due to how under-fragmented it was for the electoral system in use. The old CDU/CSU and SPD were just too strong and overwhelmed the considerable permissiveness of the electoral rules.2 So clearly the question I am raising here–whether the two-vote feature of MMP means it should not be modeled just like any (other) two-tier PR system–is moot for those years. However, perhaps it has become an issue in recent German elections, including 2021. The underlying feature of voter behavior pushing the actual NS to have risen to well above “expectation” would be the greater tendency of voters towards giving their two votes to different parties. At least that would be the cause in 2021, given that we saw in the previous post that the basic tier produced almost exactly the degree of fragmentation that the SPM says to expect. It is the compensation tier that pushed it above expectation, and the problem here (from a modeling perspective) is that the formula implicitly assumes the votes being used in the compensation mechanism are the same votes being cast and turned into seats in the basic (nominal) tier. But with two votes, they are not, and with more voters splitting tickets, the assumption becomes more and more untenable.

The previous planting on this matter emphasized that the SPM is actually performing well, even in this most recent, and quite fragmented, election. I am not trying to undermine that obviously crucial point! However, the marked rise in NS since 2009–excepting 2013 when the FDP failed to clear the threshold–may suggest that the model’s assumption that the two votes are pretty similar could be problematic.

Maybe two-vote MMP is more complex after all than its characterization as a two-tier PR system–the simplest form of complex electoral system–implies. In fact, maybe I should stop referring to MMP as a sub-category of two-tier PR. Yet for various reasons, it is a convenient way to conceptualize the system, and as yesterday’s discussion of the recent German election showed, it does work quite well nonetheless. It could be based on a flawed premise, however, and the more voters cast their nominal and list votes differently, the more that flaw becomes apparent.

A work in progress… in other words (fair warning), more such nerdy posts on this topic are likely coming.

Notes

1. The “expected NS” line for Germany takes the tier ratio to be 0.5, even though as I argued in the previous entry, we really should use the actual share of compensation seats in the final allocation. This would have only minimal impact in the elections before 2013; in 2021, it makes a difference in “expected” NS of 0.36.

2. Partly this is due to the 5% list-vote threshold, which is not a factor in the version of the SPM I am using. In Votes from Seats, we develop an alternate model based only on a legal threshold. For a 5% threshold, regardless of other features, it predicts NS=3.08. This would be somewhat better for much of the earlier period in Germany. In fact, from 1953 through 2002, mean observed NS=2.57. In the book we show that the SPM based only on mean district magnitude and assembly size–plus for two-tier PR, tier ratio–generally performs better than the threshold model even though the former ignores the impact of any legal threshold. This is not the place to get into why that might be, or why the threshold might have “worked” strongly to limit the party system in Germany for most of the postwar period, but the permissiveness of a large assembly and large compensation tier is having more impact in recent times. It is an interesting question, however! For New Zealand, either model actually works well for the simple reason that they just happen to arrive at almost identical predictions (3.08 vs. 3.00), and that for the entire MMP era so far, mean NS has been 3.14.

The Germany 2021 result and the electoral system

The German general election of 2021 has resulted in a situation in which neither major party can form a government without either the other, or more likely, a coalition that takes in both the liberal FDP and the Greens. With the largest party, the social-democratic SPD, under 30% of seats, it is an unusually fragmented result compared to most German elections. Naturally, this being Fruits & Votes, attention turns to how much more fragmented this outcome is than expected, given the electoral system. The answer may be a bit of a surprise: not all that much. I expected this outcome to be a significant miss for the Seat Product Model (SPM). But it is really not that far off.

For a two-tier PR system, of which Germany’s MMP can be thought of as a subtype, we need to use the extended version of the SPM developed in Votes from Seats.

NS = 2.5t(MSB)1/6,

where NS is the effective number of seat-winning parties (here, meaning the expected NS), M is the mean district magnitude of the basic tier, SB is the total number of seats in the basic tier, and t is the “tier ratio” defined as the share of the total number of assembly seats allocated in the compensatory tier. For Germany, basic-tier M=1 and SB=299. The tier ratio could be coded as 0.5, because the initial design of the system is that there are 299 list tier seats, allocated to bring the result in line with the overall party-list vote percentages of each party that clears the threshold. However, in Germany the electoral law provides that the list tier can be expanded further to the extent needed to reach overall proportionality. Thus t is not fixed; we should probably use the ratio that the final results are based on, as NS would necessarily be lower if only 299 list seats had been available. In the final result, the Bundestag will have 735 seats, meaning 436 list seats, which gives us a tier ratio of t=436/735=0.593. Plug all this into the formula, and you get:

NS = 2.50.5932991/6=1.72*2.59=4.45.

Now, what was the actual NS in the final result? We have to ask ourselves whether to count to two Christian “Union” parties, the CDU and the CSU, as one party or two. The answer really depends on the question being asked. They are separate parties, with distinct organization, and they bargain separately over portfolios and policy when they are negotiating a coalition with another party. However, for purposes of the SPM, I firmly believe that when two or more parties in a bloc do not compete against each other (or, alternatively, do so only within lists over which votes are pooled for seat-allocation1), they should be treated as one. The SPM does not “care” whether candidates of the bloc in question are branded as CSU (as they are in Bavaria) or as CDU (the rest of Germany). It simple estimates the effective number of “agents of the electorate” given the electoral rules. In terms of national politics, these are the same “agent”–they always enter government together or go into opposition together, and they jointly nominate a leader to be their Chancellor candidate.

Taking the CDU/CSU as a “party” for this purpose, we get actual NS =4.84 in the 2021 election. So, given an expectation of 4.45, the actual outcome is just over 8.75% higher than expected. That is nothing too extraordinary. For comparison purposes, we can just take the ratio of actual NS to expected NS. Here are some elections in the dataset used for Votes from Seats that are in the same range of over-fragmentation as Germany 2021:

      country   year   simple   Ns   exp_Ns   ratio 
     Barbados   1981        1    1.87   1.735597   1.077439  
       Norway   1965        1    3.51   3.255616   1.078137  
    Sri Lanka   1970        1    2.49   2.307612   1.079037  
Dominican Rep   1990        1    3.05   2.810847   1.085082  
     Trinidad   2002        1    1.98   1.824064   1.085488  
      Iceland   1963        0    3.33   3.060313   1.088124  
       Israel   1961        1    5.37   4.932424   1.088714  
     Trinidad   2001        1       2   1.824064   1.096452  
     Trinidad   2000        1       2   1.824064   1.096452  
      Iceland   1999        0    3.45   3.146183   1.096567  
      Denmark   1950        0    3.98   3.624933   1.097951  
     

(The table indicates as ‘simple’ those with a single tier; others are two-tier.)

The ratio variable has a mean of 1.021 in the full dataset and a standard deviation of 0.359. Its 75th percentile is 1.224 (and 25th is 0.745). So the German election of 2021 is actually very well explained by this method. The degree of fragmentation we saw in this election is not too surprising. It is about what should be expected with MMP consisting of 299 nominal-tier M=1 seats and a very generous and flexible compensation tier.

As an aside, if we used the initial tier size (299, so t=0.5) in the formula, we would get an “expected” NS=4.09. This would mean a ratio of 1.183, still short of the 75th percentile of the 584 elections included in the book’s main statistical test. Here is the company it would be keeping in that neighborhood:

            country   year   simple   Ns   exp_Ns   ratio 
            Germany   2009        0    4.83   4.121066   1.172027  
St. Kitts and Nevis   2000        1    1.75   1.491301   1.173472  
         Luxembourg   2009        1    3.63   3.077289    1.17961  
             Canada   2004        1    3.03   2.560218   1.183493  
            Denmark   1998        0    4.71   3.965222   1.187828  
          Venezuela   1963        0    4.32    3.63006   1.190063  
        Korea South   1988        0    3.55   2.981969   1.190488  
     Czech Republic   2010        1    4.51   3.767128   1.197199  
            Iceland   1991        0    3.77   3.146183   1.198277  

This would put the German 2021 election about as “over-fragmented” as the Canadian election of 2004. In other words, still not a big deal. If we count the two “Union” parties separately, obviously the degree of over-fragmentation goes up considerably. As I have said already, I think for this purpose counting them as one is the correct decision.2

As far as size of the largest seat-winning party is concerned, the SPD has 206 seats, for 28.03%. The SPM would predict, given expected NS=4.45, that the largest should have 32.6% (240 seats out of 735); that’s a ratio of 0.860 (which is a slightly bigger miss than the NS ratio of 1.088, the reciprocal of which would be 0.919). It is worth pausing on this for a bit. Polling before the election said the largest party might be only on a quarter of the votes. This was accurate, as the SPD won 25.7%. The advantage ratio (%seats/%votes) is 1.09, which is rather high for an electoral system that promises as near-perfect proportionality as Germany’s current system does, with its compensation for overhangs (cases in which a party has won more nominal-tier seats in a state than its list votes would have entitled it to). This bonus is a result of a rather high below-threshold vote. Not as high in 2013, of course, when two parties (FDP and AfD) narrowly missed the nationwide 5% threshold. But still considerably high, at 8.6% combined for all parties that failed to win a seat.

It is also worth asking whether the logic behind the extended SPM for two-tier systems holds for this German election. The formula says that the basic tier produces an initial allocation of seats consistent with the SPM for simple systems, and then inflates it based on the size of the compensation tier. So we can ask what the effective number of seat-winning parties is in the basic tier alone. It should be NS =(MSB)1/6= 2991/6= 2.59. In fact, the basic-tier NS in this election was 2.51 (as before, taking CSU/CSU as one party). The ratio of 0.969 is a pretty trivial miss. We should expect the largest party to have won 0.490 of these seats (about 146). Actually the Union parties, which together won the most single-seat districts, won 143 (0.478). Thus Germany’s MMP system, in the 2021 election, actually did produce a basic-tier (nominal-tier) party system pretty much just like it should, given 299 seats and M=1 plurality, and then augmented this through a large compensatory national tier. The actual inflator is a factor of 1.93=4.84/2.51, rather than the expected 1.72=2.50.593. Had it been 1.72 instead, the final effective number of seat-winning parties would have been 4.32, about “half a party” less than in reality, implying almost exactly one third of seats to the SPD instead of just 28%.

This surprised me (pleasantly, of course). When I saw that the Greens and AfD each had won 16 seats in the nominal tier, I thought that was too many! But in fact, it works out. Maybe sometimes even I think Duverger had a law, or something. But given 299 single-seat districts, this is pretty much in line with expectations.

The outcome is interesting in the many ways that it serves as a primer on details of the electoral system. Here I mean not only the substantial expansion of the Bundestag from 598 to 735 seats, due to the way the compensation mechanism works, but also the thresholds. One of the best known features of the German electoral system is the 5% nationwide threshold. But of course, the threshold is more complex than that. It is 5% of the national party-list vote or three single-seat wins, except if a party is an ethnic-minority party. All these provisions were on display. For instance, the Linke (Left) party fell below the 5.0% threshold, yet is represented at full proportionality. That is because it won three individual mandates, thus fulfilling the “or” clause of the threshold. There was a point on election night when it looked as if the Linke might hold only two single-seat districts. In that case, with less than 5% of the list votes nationwide, it would have held only those seats as its total. By winning three, it is entitled under the law to full proportional compensation, and as a result it was awarded 36 list seats. Then, for the first time in a very long time, an ethnic party has won a seat. The South Schleswig Voters’ Association (SSW), which had not contested federal elections in decades, ran in this one and was able to win a single (list) seat, because as a representative of the Danish and Frisian minorities, it is exempt from the usual threshold provisions, as long as its votes are sufficient to qualify it for a seat when the threshold is ignored. Its 0.1% of the national vote was good enough. The SSW has had some renewed success in state elections in Schleswig-Holstein recently, and now it has scored a seat in the federal parliament for the first time since 1949. In 1949, the MMP system was a bit different, in that the 5% threshold was determined state-by-state, rather than nationwide. If the threshold had been state-by-state in this election, one other party would have earned seats. The Free Voters won around 7.5% of party-list votes in Bavaria. However, they managed only 2.9% nationwide (and no district seat), so they are shut out.

Now attention turns to what the coalition will be. Two options are on the table: SPD+Greens+FDP (“traffic light”) or CDU/CSU+Greens+FDP (“Jamaica”). The possibility of a broad left coalition has been ruled out by the election results: SPD+Green+Linke is not a majority. It was never likely anyway; the SPD and Greens did not spend recent years convincing voters they were safe options near the center of German politics to team up with the far left. Nonetheless, had it been mathematically possible the SPD might have used it as leverage against the FDP. My guess is that the traffic light coalition will form. Despite some serious policy differences between the FDP and the other two, it would be a government made up of the winners of the election, as these three parties all gained votes compared to 2017. On the other hand, one led by the CDU/CSU would be led by a pretty big loser, even though it is mathematically possible and the Greens seem to have been positioning for it over the last several years.3 Following the election, the DW live blog has been reporting on comments by various prominent CDU and CSU politicians that could be interpreted as saying the bloc needs some time in opposition, after the disappointing result. I suspect this is the view that will prevail, and after a lot of intense and difficult bargaining, Germany will be led by a traffic light coalition for the first time.

____

Notes

1. Here I am thinking of cases like Chile, where alliance lists contain candidates of different parties, but for purposes of how the electoral system assigns seats between competing teams of candidates, we should count the alliances, not the component parties. The same condition applies in Brazil and Finland, only there it is essentially impossible to aggregate to a meaningful national alliance category because the combinations of parties are not always the same across districts. In Chile, and also in the FPTP case of India–as well as in the current case of Germany–there is no such problem, as the alliances are nationwide in scope and consistent across districts.

2. For the record, counting them separately yields NS=5.51 in this election, which would put the ratio just barely above the 75th percentile.

3.To be clear, they are much happier working with the SPD, but what I mean is that their positioning for the possibility of a coalition with the CDU/CSU should make finding common ground with the FDP easier than it otherwise would have been.

What electoral system should Canada have?

Once again, Canadians have voted as if they had a proportional representation (PR) electoral system, but obtained almost exactly the party system they should be expected to get, given the first-past-the-post (FPTP) system that they actually use.

If voters are voting as if they had PR already, why not just give them PR? Of course, it does not work that way, as the decision to adopt a new electoral system is rarely separable from party politics. Nonetheless, it is worth asking what electoral system the country should have, based on how voters are actually voting. They certainly are not playing the game as if it were FPTP. Even though it is.

To get at an answer to this question, we can start with the average value of the effective number of vote-earning parties over recent elections. (For those just tuning in or needing a refresher, the effective number of parties is a size-weighted count, where each party’s “weight” in the calculation is its own size–we square the vote (or seat) share of each party, sum up the squares, and take the reciprocal. If there were four equal size parties, the effective number would be 4.00. If there are four parties of varying sizes, the effective number will be smaller than four. For instance, if the four have percentages of 40%, 35%, 20%, and 5%, the effective number would be 3.08.) From the effective number, we can work backwards through the Seat Product Model (SPM) to determine what electoral system best fits the distribution of parties’ votes that Canadians have actually been providing. The SPM lets us estimate party system outputs based on a country’s mean district magnitude (number of seats elected per district (riding)) and assembly size. As noted above, Canada currently tends to have a distribution of seats among parties in the House of Commons consistent with what the SPM expects from a district magnitude of 1 and a House size of 338. The puzzle is that it does not have a distribution of votes consistent with the SPM. Instead, its distribution of votes across parties looks more like we would expect from a PR system. But what sort of PR system? That is the question the following calculations aim to answer.

Over the past eight elections, going back to 2000, the mean effective number of vote-earning parties (dubbed NV in systematic notation) has been 3.70. During this time, it has ranged from a low of 3.33 (2015 when Justin Trudeau won his first, and so far only, majority government) to a high of 3.87 (the second Conservative minority government of the period under leadership of Stephen Harper). In 2019 it was 3.79 and in 2021 it was very slightly higher (3.84, based on nearly complete results). Even the lowest value of this period is not very “two party” despite the use of FPTP, an electoral system allegedly favorable to two-party systems. (I say allegedly, because given FPTP with a House of 338 seats, we actually should expect NV=3.04, according to the SPM. In other words, a “two-party system” is not really what the current electoral system should deliver. Nonetheless, it would not be expected to be associated with as fragmented a voting outcome as Canadians typically deliver.)

How to get from actual voting output to the PR system Canadians act as if they already had

The SPM derives its expectation for NV via a phantom quantity called the number of “pertinent” vote-earning parties. This is posited in Shugart and Taagepera (2017), Votes from Seats, to be the number of parties winning at least one seat, plus one. It is theoretically expected, and empirically verifiable, that the effective number of seat-winning parties (NS) tends to equal the actual number of seat winning parties (NS0, with the 0 in the subscript indicating it is the unweighted, raw, count), raised to the exponent, 2/3. That is, NS=NS02/3. The same relationship logically would hold for votes, meaning NV=NV02/3, where NV0 is the aforementioned number of pertinent vote-earning parties. We can’t measure this directly, but we take it to be NV0=NS0+1, “strivers equal winners, plus one.” In Votes from Seats we show that this assumption works for estimating the impact of electoral systems on NV.

Thus we start with the recently observed mean NV=3.7. From that we can estimate what the number of pertinent parties would be: given NV=NV02/3, we must also have NV0=NV3/2. So NV0=3.73/2 = 7.12. This number by itself is not so interesting, but it makes all the remaining steps of answering our question possible.

Our expected number of seat-winning parties from a situation in which we know NV=3.7 works out to be 6.12 (which we might as well just round and call 6). We get that as follows. First, NS0=NV0-1: the number of pertinent vote-earning parties, minus one. We already estimated the pertinent vote-earning parties to be 7, so we have an estimated average of 6 parties winning at least one seat. This is realistic for current Canadian politics, as recently five parties have been winning seats (Liberal, Conservative, NDP, BQ, and since 2011, Greens). With PR, the PPC likely would win a few seats on current strength, and the Greens probably would continue to do so, assuming they either recover from their current doldrums (especially once PR were adopted) or that any legal threshold would not be applied nationally and thus even their 2.3% showing in the 2021 election would not lock them out of parliament. (In 2021, Greens still got 9.6% in PEI, 5.3% in BC and 5.2% in New Brunswick, for example (per Elections Canada).)

If we have an expected number of seat-winning parties, based actual mean NV, that is equal to six, what would be the seat product (MS) that would be expected? Once again, the seat product is the mean district magnitude (M), times the assembly size (S). Given M=1 (single-seat districts) and S=338, Canada’s current seat product is 338. Based on one of the formulas comprising the SPM, a seat product of 338 should be expected to result in an effective number of seat-winning parties (NS) of 2.64 and effective number of vote-earning parties (NV) of 3.04. It is working out pretty close to that for seats (average NS=2.8). Yet voters are voting more like they had a PR system given the average over recent elections of NV=3.7.

One of the formulas of the SPM, which like all of those referenced here, is empirically accurate on a worldwide sample of election results, predicts that NS0=(MS)1/4. Thus if we have an expected value of seat-winning parties of around 6, as expected from NV=3.7, we can simply raise it to the power, 4, to get what the seat product is expected to be: MS=64=1296. In other words, based on how Canadian voters are actually voting, it is as if their country had an electoral system whose seat product is around 1300, rather than the actual 338. For a comparative referent, this hypothetical PR system would be quite close to the model of PR used in Norway.1

Any electoral system’s mean district magnitude is M=(MS)/S,so taking a House of 338 seats,2 our hypothetical PR system has M=1300/338=3.85. That is, based on how Canadian voters are actually voting, it is as if their country had an electoral system whose mean district magnitude is around 3.85. Comparatively, this is quite close to the Irish PR system’s mean magnitude (but it should be noted that Ireland has a seat product of closer to 600, due to a much smaller assembly).

So there we have it. The mean district magnitude that would be most consistent with Canada’s current vote fragmentation would be just under 4, given the existing size of the House of Commons.

If Canada adopted a PR system with a seat product of 1300, its expected effective number of seat-winning parties (NS) would rise to 3.30, and its expected largest party would have, on average, 40.8% of the seats, or 138. (This is based on two other predictive formulas within the SPM: NS=(MS)1/6 and s1=(MS)–1/8, where s1 is the seat share of the largest party.)

A largest party with 138 seats (as an average expectation) would then require another party or parties with at least 32 seats to have a majority coalition, or a parliamentary majority supporting a minority government. The NDP would reach this easily under our hypothetical PR system, given it can win around 25 seats on under 18% of the votes under FPTP (and 44 seats on just under 20% as recently as 2015).

The Bloc Quebecois also would be available as a partner, presumably for a minority government, with which to develop budgets and other policy, thereby preventing the NDP from being able to hold the Liberal Party “hostage” to its demands. The BQ won 32 seats in 2019 and 33 in 2021. However, because it is a regionally concentrated party, we should entertain the possibility that it might do worse under PR than under FPTP, which rewards parties with concentrated votes. The only way to estimate how it would do might be to run the SPM within the province.

Estimating Quebec outcomes under PR

Quebec has 78 seats total, such that 33 seats is equivalent to 42% of the province’s seats. On Quebec’s current seat product (78) its largest party should win 45 seats (58%). So it is actually doing worse than expected under FPTP!

If the province had a mean district magnitude of 3.85, its seat product would be 300, for which the expected largest party size would be 49%, or 38 seats. In other words, when the BQ is the largest party in Quebec, it could do a little better on the very moderate form of PR being suggested here than it currently is doing under FPTP. (Suppose the model of PR had a mean magnitude of 9 instead, then we’d expect the largest provincial seat winner to have 44.1%, or 34 seats, or roughly what it has won in the last two elections. Only if the mean M is 16 or higher do we expect the largest party in Quebec—often the BQ—to have fewer than 32 of 78 seats. Obviously, in 2011 when the BQ fell all the way to 23.4% within the province, PR would have saved many of their seats when FPTP resulted in their having only 4 of 75 in that election. In 2015 they did even worse in votes—19.3%, third place—but much better in seats, with 10 of 78. Under the PR model being considered here, it is unlikely they would not have won at least 10 seats, which is 12.8%, on that provincial share of the vote.)

Do Canadians actually ‘want’ a still more proportional system than this?
It is possible we should use a higher NV as reflective of what Canadians would vote for if they really had a PR system. I have been using the actual mean NV of recent elections under FPTP, which has been around 3.7. But in the final CBC polling aggregate prior to the 2021 election, the implied NV was 4.12. It dropped by almost “half a party” from the final aggregate3 to the actual result either because some voters defected late from the NDP, Greens, and PPC, or because the polls simply overestimated the smaller parties. If we use 4.12 as our starting point, and run the above calculations, we’d end up with an estimated average of 7.4 parties winning at least one seat. Maybe this implies that the Maverick Party (western emulators of the BQ’s success as a regional party) might win a seat, and occasionally yet some other party. In any case, this would imply a seat product of 2939, for a mean M of 8.7. The largest party would be expected to have only 36.8% of the seats with such an electoral system, or about 125.

How to use this information when thinking about electoral reform

I would advise, as the way to think about this, that we start with what we’d like the parliamentary party system to look like. I am guessing most Canadians would think a largest party with only around 125 seats would be an overly drastic change, despite the fact that they are currently telling pollsters, in effect, that this is the party system they are voting for as of the weekend before the election!

The expected parliamentary party system from an average M around 4, yielding a largest party averaging just over 40% of the seats (around 138) is thus probably more palatable. Nonetheless, armed with the information in this post, drawn from the Seat Product Model, we could start with a desirable average share of the largest party, and work back to what seat product it implies: MS=s1–8, and then (assuming 338 seats in the House), derive the implied district magnitude from M=(MS)/S. Or one can start with how Canadians are actually voting, as I did above–or from how we think they would (or should) vote, using MS=[(NV3/2)–1]4, and followed by M=(MS)/S.

Whichever value of the seat product, MS, one arrives at based on the assumptions about the end state one is hoping to achieve, remember that we’d then expect the seat share of the largest party to be s1=(MS)–1/8. As we have seen here, that would tend to be around 40% if mean magnitude were just under 4. This implies a typical largest party of around 138 seats.4

But herein lies the rub. If you tell the Liberal Party we have this nifty new electoral system that will cut your seats by about 20 off your recent results, they probably will not jump at the offer. The parties that would benefit the most are the Conservatives (twice in a row having won more votes than the Liberals but fewer seats), NDP, and smaller parties, including apparently (based on above calculations) the BQ. But this isn’t a coalition likely to actually come together in favor of enacting PR. Thus FPTP is likely to stick around a while yet. But that’s no reason not to be thinking of what PR system would best suit Canadian voters, given that they have been voting for a while as if they already had a PR system.

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Notes

General note: At the time of writing, a few ridings remained uncalled. Thus the seat numbers mentioned above, based on who is leading these close ridings, could change slightly. Any such changes would not alter the overall conclusions.

1. More precisely, it would be almost identical in seat product to the Norwegian system from 1977 to 1985, after which point a small national compensation tier was added to make it more proportional.

2. I will assume electoral reform does not come with a change in the already almost perfect S for the population, based on the cube root law of assembly size, S=P1/3, where P is population, which for Canada is currently around 38 million. This suggests an “optimal” number of seats of about 336.

3. This is based on the Poll Tracker final aggregate having vote shares of 0.315, 0.310, 0.191, 0.070, 0.0680, 0.035 for the six main parties (and 0.011 for “other”).

4. I am deliberately not going into specific electoral system designs in this post. I am stopping at the seat product, implicitly assuming a simple (single-tier) districted PR system, meaning one with no regional or national compensation (“top up” seats). Arriving at a seat product to produce the desired party system should be the first step. Then one can get into the important finer details. If it is a two-tier system–including the possibility of mixed-member proportional (MMP)–one can generate its parameters by using the result of the calculations as the system’s “effective seat product,” and take it from there.

Canada 2021: Another good night for the Seat Product Model, and another case of anomalous FPTP

The 2021 Canadian federal election turned out almost the same as the 2019 election. Maybe voters just really do not want to entrust Justin Trudeau with another majority government, as he led from 2015 to 2019. The early election, called in an effort to turn the Liberal plurality into a Liberal majority, really changed almost nothing in the balance among parties.

The result in terms of the elected House of Commons is strikingly close to what we expect from the Seat Product Model (SPM). Just as it was in 2019. The predictive formulas of the SPM suggest that when your electoral system is FPTP and there are 338 total seats, the largest one should win 48.3% of the seats, or about 163. They further suggest that the effective number of seat-winning parties (NS) should be around 2.64. In the actual result–with five districts still to be called–the largest party, Liberal, has won or is leading in 159, or 47.0%., and NS=2.78. These results are hardly different from expected. They also are hardly different from 2019, when the Liberals won 157 seats; in that election we had NS=2.79.

While the parliamentary balance will be almost what the SPM expects, the voters continue to vote as if there were a proportional system in place. The largest party again has only around a third of the votes, and the effective number of vote-earning parties (NV) is around 3.8. For a FPTP system in a House the size of Canada’s, we should expect NV=3.04. Once again, the fragmentation of the vote continues to be considerably greater than expected.

Another bit of continuity from 2019 is the anomalous nature of FPTP in the current Canadian party votes distribution. For the second election in a row, the Conservative Party has won more votes than the Liberals, but will be second in seats. The votes margin between the two parties was about the same in the two elections, even though both parties declined a little bit in votes in 2021 compared to 2019. Moreover, as also has happened in 2019 (and several times before that), the third largest party in votes will have considerably fewer seats than the party with the fourth highest vote share nationwide. The NDP won 17.7% of the vote and 25 seats (7.4%), while the Bloc Quebecois, which runs only in Quebec, won 7.8% of vote and 33 seats (9.8%).

The Green Party and the People’s Party (PPC) more or less traded places in votes: Greens fell from 6.5% in 2019 to 2.3%, while the PPC increase from 1.6% to 5.0%. But the Greens’ seats fell only from 3 to 2, while the PPC remained at zero.

So, as in 2019, the 2021 election produced a good night for the Seat Product Model in terms of the all-important party balance in the elected House of Commons. However, once again, Canadians are not voting as if they still had FPTP. They are continuing to vote for smaller parties at a rate higher than expected–and not only in districts such parties might have a chance to win–and this is pushing down the vote share of the major parties and pushing up the overall fragmentation of the vote, relative to expectations for the very FPTP system the country actually uses.

It is worth adding that the virtual stasis at the national level masks some considerable swings at provincial level. Éric Grenier, at The Writ, has a table of swings in each province, and a discussion of what it might mean for the parties’ electoral coalitions. A particularly interesting point is that the Conservatives’ gains in Atlantic Canada and Quebec, balanced by vote loss in Alberta and other parts of the west, mirrors the old Progressive Conservative vs. Reform split. Current leader Erin O’Toole’s efforts to reposition the party towards the center may explain these regional swings.

In a follow up, I will explore what this tendency towards vote fragmentation implies for the sort of electoral system that would suit how Canadians actually are voting.

Below are the CBC screen shots of election results for 2021 and 2019. As of Thursday afternoon, there remain a few ridings uncalled.

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.