Will Macron lose his assembly majority?

French election season is upon us. In four rounds of elections over the next three months France will choose their President and National Assembly. The presidency is elected by two-round majority (10 and 24 April), followed closely by the assembly using two-round majority-plurality (12 and 19 June). Predictably, the news media are already starting to suggest that President Emmanuel Macron, while likely to be reelected, might be at risk of losing his assembly majority (e.g., The Economist). Will he?

What is almost as predictable as the media expressing this outcome as a real possibility is that presidents–just elected or reelected–see their parties do really well in honeymoon assembly elections. You can’t get much more honeymoon-ish than the French cycle. The assembly election occurs with approximately 1/60 of the time between presidential elections having elapsed. It just so happens that we have a formula for this.

Rp=1.20–0.725E,

where Rp is the “presidential vote ratio”– vote share of the president’s party in the assembly election, divided by the president’s own vote share (in the first round, if two-round system)–and E is the elapsed time (the number of months into the presidential inter-electoral period in which the assembly election takes place, divided by the total months comprising that period).

In 2017, there were actually news reports suggesting that because Macron at the time he was elected did not yet have a true political party, he would face cohabitation. That would mean an opposition majority, which under French institutions would also mean a premier (head of cabinet) from parties opposed to the president. This was, even at the time, obviously hogwash.

The formula suggested that, once we knew Macron’s first-round vote percentage, we could estimate his (proto-) party’s first-round assembly vote percentage–assuming he would go on to win his own runoff (which was never seriously in doubt). Given that Macron had won 24% of the vote in his own first round, that implied 29% of the vote for the party in the first round for assembly.

What did his party, branded by then La République En Marche!, get? The answer would be… 28.2%. Not too bad for a political science formula. Not too surprising, either. It does not sound impressive as a vote percentage, but when you have the plurality of the vote in a multiparty field with a two-round majority-plurality electoral system, it can be pretty helpful in terms of seats won. Even more when you are a center party, and your opponents are split between left, right, and farther right (and we should not leave out farther left, too). After the second round, LREM ended up with about 54% of the seats. When combined with a pre-election ally, Democratic Movement, the seat total was over 60% (the two parties had combined for about a third of the first-round votes and got 49% of second-round votes).

The Economist article I linked to in the first paragraph was published in the March 5 edition. I want to check how plausible its claim was, using the Economist’s own election forecast model. As of a few days before March 5, that model was basing its forecast on aggregated polls that averaged about 27% of expected first-round vote for Macron himself. In other words, a few percentage points higher than he ended up winning in the first round in 2017. The model also gave Macron at the time an 88% chance of winning the presidency. Thus on the basis of information available at the time–including the Shugart-Taagepera formula for expected presidential-party vote share–we should conclude that LREM would win about 32% of the vote in the first-round assembly election. Assuming this would be the plurality share–a very safe assumption–that would again imply a strong chance of a single-party majority of seats. Not a loss of the majority, or even the need to forge a post-electoral coalition.

Now, since that article was published, Macron has been enjoying quite a surge in the polls. As of today, the forecast model at The Economist has his odds of winning the presidency above 95%. His polling aggregate as of March 12 is up to 31% (Marine Le Pen, his runoff opponent in 2017, is a distant second with 18%). From this we could estimate the first-round assembly vote share is up to 38%.

I will caution that the formula is not a logical model. It is empirical. There is good logical basis behind the general idea of honeymoon surge (and midterm decline, for countries with such cycles). But the specific parameters of the formula do not have a logical basis. At least yet. The graph of the relationship that is shown in Chapter 12 of Votes from Seats (and also included in the 2017 “predictive” post on France) shows a couple honeymoon elections in various countries that have defied the expected surge. However, only one has an elapsed time of less than 0.1 (the specific example of a relatively early honeymoon decline was Chile 1965, in an election held at 0.083 of the presidential inter-election period.1)

So I can’t predict what LREM will get in June. But it would be a surprise if it was worse than around a third of the vote, even if Macron’s own polling surge does not hold. Given the fragmentation of the party system–which looks even higher now than it was in 2017–and the majoritarian nature of the electoral system, anything short of a majority of seats for Macron would be a surprise at this point.

The notion that voters will come out and vote to “check” a just-elected president that they maybe were not all that enthusiastic about is a hard notion for the news media (not only The Economist) to shake. But there just is not much evidence that politics in presidential and semi-presidential systems works like that.2

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1. This election saw the Christian Democratic Party of newly elected President Eduardo Frei win a very strong plurality, 43.6%, but Frei himself had won 56%. The problem–for the formula–is that there were only two serious candidates and three total in the presidential election, whereas the PR-elected legislature featured many parties, including allies of the president running separately. The formula implicitly assumes that all parties contest both elections. This is one of the reasons I can’t call it a logical model, because such conditions have not been incorporated, and perhaps can’t be without making it too complicated to be useful. It is pretty useful as it is, even with its oversimplification and lack of true logical basis!

(By the way, in the next Chilean assembly election, held with 75% of the term elapsed, the party’s vote percentage fell to 31%. The formula suggests 37%, but given that we already know the party did worse than “expected” at the honeymoon, we should just use the expected drop from what it actually had. That would “predict” about 25% of the vote at the late-term election. So they did better than expected, actually.)

2. On this point, let me shout out a just-published article by some recent UC Davis Ph.D.s Carlos Algara, Isaac Hale, and Cory L. Struthers on the Georgia (US) Senate runoffs. Even I was skeptical that honeymoon logic could apply to those elections. And in fact it did not turn out as a Dem surge, but there was clearly no evidence of “checking the president” behavior by voters.

Costa Rica 2022: Continued high fragmentation

Costa Rica recently (6 Feb.) held its presidential and national assembly elections. In the case of the presidency, it was the first round; a runoff will be needed (3 April), as no candidate came close to the 40% required for a first-round victory. The result shows a continuation of the impressive degree of fragmentation that has occurred in recent elections, following a prolonged period of dominance by two major parties.

I will focus first on the assembly election. The largest party in the new assembly will be the National Liberation Party (PLN), one of those formerly major two parties, but in this election it won only 24.5% of the votes for assembly party lists and 18 of the 57 seats, or 31.6%. That is a one seat gain from what it had in the outgoing assembly, elected in 2018, when it was also the largest party. No other party broke 15%. Six parties have won at least one seat, and a large number of parties obtained vote shares of around 2% or less but no seats.

In terms of effective numbers, for votes this works out to 8.3. Yes, eight point three! That is up there with the world’s highest observed values. In seats, the effective number is 5.02, which is also high but less remarkably so in world comparative terms. For comparison, the 99th percentile of effective number of vote-earning parties from over a thousand elections in the dataset I use is 8.6. On the other hand, Costa Rica’s value for seats in this election is just above the 75th percentile (which is 4.77). Another way of stating this is that Costa Rica is experiencing an unusually large gap between effective numbers of parties by votes and seats. This is not the first time, as the values in 2018 were, respectively, 7.79 and 4.78.

The precise reasons for why the votes are fragmenting so much would require someone versed in Costa Rican politics, which I certainly am not. However, it is obvious that the electoral system is struggling to accommodate the voting fragmentation that is being fed into it, and at at the same time, voters are no longer coordinating their votes around what the electoral system can sustain. That leads to a lot of wasted votes.

This is a new phenomenon for Costa Rica. Over the entire period of the current electoral system, which has been in place since 1962 (the year the current assembly size and the current mean district magnitude (8.14) went into effect), the mean effective number of vote-earning parties has been 3.67, and the mean effective number of seat-winning parties has been 2.97. The mean largest party vote share has been 0.413. The mean seat share for the largest party has been 0.453. So the recent two elections (and to some notable degree those since 2006) have been quite a break with the old “textbook” Costa Rican party system.

A point I wish to emphasize is that the old party system was what we should expect of an electoral system like Costa Rica’s. It is a proportional representation (PR) system, but one with a modest seat product. Its seat product (mean district magnitude times assembly size) is only 464, or a little higher than that of the USA (435). So it should be expected to have a party system with two major parties, one of which averages close to a majority of seats, plus some smaller parties–as indeed the USA should have! And that is what Costa Rica had. The expected outcomes of this system, from the seat product model, would be a mean effective number of seat winning parties of 2.78 (barely below the observed fifty-year mean of 2.97). For votes we should expect 3.17 (not far below the long term observed mean, 3.67). For largest party seat share, we expect 0.464 (nearly matching the observed mean of 0.453); for vote share, 0.421 (actual mean 0.413).

In other words, the longterm party system of Costa Rica is basically what we should expect to see, given the modest value of its seat product. We do not need to invoke a presidential electoral rule that allegedly supports a two-party system, as some scholars have done in the past (hey, including me!). In fact, it is not even clear that the presidential electoral system–40% or runoff–should support two-candidate competition. In some past works I classified it as close enough to plurality, which some folks allege supports two-party systems. Of course, it does. Except when it does not. And the runoff provision makes that “except when it does not” even more accurate a description of the systemic effect. Sure, if 40% in within reach for a leading contender, others may have incentive to coordinate and try to beat the leader to 40% When the PLN was politically dominant, that was exactly what the game was. But when expectations are that no one will get to 40%, all bets are off, because to a significant degree political forces can coordinate between rounds, rather than before the first one.

In Votes from Seats (2017), Taagepera and I showed that we can actually predict presidential vote fragmentation from the assembly seat product better than we can predict it from either the rule used to elect the president or the actual number of competitors in the presidential election. And Costa Rica was, until recently, a great demonstration of that effect, with (as noted) an assembly party system that was a near perfect fit for the assembly electoral system’s seat product. The presidential party system followed right along, as expected, with a mean effective number of presidential candidates of 2.5 since 1962. The predictive model Taagepera and I propose in our 2017 book suggests that with Costa Rica’s seat product, the effective number of presidential candidates should average 2.49–so there was basically perfect prediction of Costa Rican presidential competitiveness. However, something clearly has upset the old equilibrium.

In this election, the effective number of presidential candidates was 6.15! For comparison, this is almost the 99th percentile of over 200 presidential elections from around the world in the dataset (6.25). [Update: see my own first comment below.] The leading candidate, José María Figueres had only 27.3%. His opponent in the upcoming runoff, Rodrigo Chaves Robles of Social Democratic Program, won 16.7%, and three other candidates had between 12% and 14.8%. The party of outgoing President Carlos Alvarado, Citizens Action, collapsed, with its candidate getting only 0.66% of the presidential vote (and 2.2% of the assembly vote, and no seat–in 2018, despite winning the presidency it had won only 10 seats, good for third place; further, presidents are not eligible for immediate reelection in Costa Rica).

The level of fragmentation of the presidential vote in 2022 is an increase over 2018, when the effective number of presidential candidates was 5.51, and the leading candidate (who lost the runoff) had just under 25%. It is the third election in a row in which no candidate broke 31%. (In 2010, the leading candidate who was from the PLN, won without a runoff, getting just under 47%.)

While on average, the seat product model leads us to expect presidential systems to have assembly party systems similar to what their seat product predicts, and a mean presidential competition also predictable from the seat product, individual elections can upset this. That is, short term presidential politics–who is entering competition and who is seen as a viable presidential candidate–can shock the assembly party system, due to a “coattail” effect. So we generally get longterm predictability from the assembly electoral system’s seat product, but short term disruptions from “presidentialization” of competition. This is now Costa Rica’s third consecutive election with effective number of seat-winning parties over 4.5. That seems unsustainable, based on the electoral system. But at some point maybe a short-term shock settles down and becomes the new normal. I guess we will have to wait till at least 2026 to see if the seat product reasserts itself, or if fragmentation really is the norm. And not just any fragmentation, but an exceptionally high level by world standards, particularly in the votes for both assembly and president.

Portugal 2022–unexpected majority, but not that rare (for Portugal)

The majority of seats obtained by the Socialist Party (PS) in the recent general election in Portugal was seen as a surprise. Polling generally had not shown a majority as within reach and indeed showed a likely close result. However, Portugal has had relatively frequent parliamentary majorities over the years, despite its proportional representation (PR) system. How unusual was the 2022 outcome?

From 1976 to 2019, the mean seat share for the largest party in Portugal has been 0.478–not a majority, but pretty close. In this election the PS obtained 117 of 230 seats, which is 0.509. (The total includes the four seats for Portuguese abroad.) This is the fifth absolute majority won in 16 Portuguese assembly elections since 1976. Thus in terms of Portugal’s electoral history, the result was not so unusual. How unusual is it relative to what is expected from Portugal’s PR system?

Portugal’s electoral system has a seat product of around 2400. This is a modest seat product by standards of proportional representation, stemming from a moderate assembly size, S (currently 230; 250 before 1991), and a middle-range district magnitude, M (currently 10.5 on average), yielding a seat product, MS=2415. For such a seat product, the expected largest party seat share is 0.378, derived from the formula expecting this share to be (MS)–1/8. Thus Portugal’s actual largest party seat share has averaged 1.26 times the seat product model prediction.1 This indicates that while Portugal’s electoral system is not expected to produce a high degree of fragmentation (38% of the seats is a decent sized largest party2), actual Portuguese politics supports a more de-fragmented party system–at least so far–than what its electoral system could sustain.

As for votes, the associated formula of the seat product model implies we should expect the largest to have 35.4% of the votes, but the average has been 41.8% instead. In this election the PS won 41.7%. So, whatever people expected, it was a pretty ordinary voting result by the standard of Portuguese electoral history. There was a somewhat higher boost for the largest party, however, than the norm. The average advantage ratio (%seats/%votes) has been 1.14; in this election it was 1.22. I would guess that this larger seat bonus for the largest party comes in significant part from the main rival for national power, the Social Democrats (PSD, actually a center-right party) losing votes to a farther right-wing/nationalist party, CHEGA. The latter party was the big gainer in votes and seats in the election, as it had only one seat from 2019 but won 12 in this election. However, it had a very low advantage ratio, with its 5.31% of seats coming on 7.15% of votes, for a ratio of 0.74. Its votes thus did not translate efficiently into seats, which may have helped the PS harvest more seats than normally would be the case for a party with just over 41% of the votes given Portuguese electoral laws.3

Notes

  1. The mean actual largest party seat share in a sample of 634 simple electoral systems is only 1.048 times the model prediction; for PR systems the model is even better, with a ratio of 1.033. So a ratio of 1.26 indicates a strong degree of politics being needed in addition to institutions to explain an outcome. Less than a quarter of PR elections have ratios that high or higher.
  2. The mean largest party seat share for the sample of 280 PR elections in parliamentary (or semi-presidential) democracies that I am working with happens to be 38.2%.
  3. Relative punishment of smaller parties is an inherent feature of the system’s moderate seat product. For instance, in this election the significantly smaller Liberal Initiative won 3.5% of seats on just under 5% of votes. The wasted votes by smaller parties have to go somewhere; given that Portugal uses the D’Hondt formula, the result will tend to be generally more favorable to the largest party than it would be with other PR formulas, for a given seat product. (This is not unusual; more than two thirds of all simple PR systems use D’Hondt.) Still, for a party in its range of vote percentage, CHEGA’s advantage ratio is quite low. For instance, in 2019, the Left Bloc and Unitary Democratic Coalition, with 9.5% and 6.3% of votes, respectively, had advantage ratios of 0.86 and 0.82. So CHEGA must have had an unusually inefficient geographic spread for a party of its approximate size. Indeed, skimming the table the Wikipedia page offers for district-level results, it is easy to spot districts where CHEGA received above its nationwide vote share yet won no seats. As a final note on CHEGA, I will add that its single seat in 2019 was won in Lisbon, where the district magnitude is 48, on 2% of the vote.

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.

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.

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  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.)

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.

The Extended Seat Product Model: Getting rid of that annoying “2.5”

[Update, late April, 2022: I have continued to refine this method, and the specific values mentioned below no longer hold (due a revision of the estimation procedure outlined below), although the basic framework remains the same. In fact, the revision is based on what is described as “a further extension” towards the end of this post. This also means that the datasets linked at the end of the linked post are not accurate. I will upload corrected ones at some point.]

The extended version of the Seat Product Model (SPM), devised to be applicable to two-tier PR systems as well as simple electoral systems, states:

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. In the case of a simple (single-tier) system, this reduces to the basic SPM: NS =(MS)1/6, given that for simple systems, by definition, t=0 and SB=S, the total size of the elected assembly.

Ever since this formula first appeared in my 2016 Electoral Studies article with Huey Li (and later as Equation 15.2 in Shugart and Taagepera, 2017,  Votes from Seats) I have been bothered by that “2.5.” The SPM for simple systems is a logical model, meaning its parameters are derived without recourse to the data. That is, the SPM is not an empirical regression fit, but a deductive model of how the effective number of seat-winning parties (and other electoral-system outputs) should be connected to two key inputs of the electoral system, if certain starting assumptions hold. When we turn to statistical analysis, if the logic is on the right track, we will be able to confirm both the final model’s prediction and the various steps that go into it. For simple systems, such confirmation was already done in Taagepera’s 2007 book, Predicting Party Sizes; Li and Shugart (2016) and Shugart and Taagepera (2017) tested the model and its logical antecedents on a much larger dataset and then engaged in the process of extending the model and its regression test in various ways, including to cover more complex systems. Yet the derivation of the “2.5” was not grounded in logic, but in an empirical average effect, as explained in a convoluted footnote on p. 263 of Votes from Seats (and in an online appendix to the Li-Shugart piece).

If one is committed to logical models, one should aim to rid oneself of empirically determined constants of this sort (although, to be fair, such constants do exist in some otherwise logical formulas in physics and other sciences). Well, a recent Eureka! moment led me to the discovery of a logical basis, which results in a somewhat revised formula. This revised version of the extended Seat Product Model is:

NS = (1–t)–2/3(MSB)1/6.

The variables included are the same, but the “2.5” is gone! This revision produces results that are almost identical to the original version, but stand on a firmer logical foundation, as I shall elaborate below.

Consider a few examples for hypothetical electoral systems.

MSBt1-t(1–t)2/32.5tNS (rev.) NS (Eq. 15.2)
100.5.51.591.583.423.40
100.25.751.211.262.612.71
250.3.71.271.324.684.85
250.4.61.411.443.533.62
250.6.41.841.734.624.35
2500.3.71.271.324.684.85
2500.15.851.111.154.114.23

It may not work especially well with very high MSB, or with t>>.5. But neither does equation 15.2 (the original version); in fact, in the book we say it is valid only for t≤0.5. While not ideal from a modelling perspective, it is not too important in the real world of electoral systems: cases we would recognize as two-tier PR rarely have an upper compensation tier consisting of much more than 60% of total S; relatedly, SB much greater than around 300 is not likely to be very common. My examples of MSB =2,500 are motivated by the notion of SB=300 and a decently proportional basic-tier M=8.3.

Testing on our dataset via OLS works out well, for both versions of the formula. Our largest-sample regression test of Equation 15.2, in Table 15.1 of Votes from Seats, regression 3, yields:

            log NS = –0.066 + 0.166log MSB + 0.399t .

Logically, we expect a constant of zero and a coefficient of 0.167 on the log of MSB; the coefficient on t is expected to be 0.398=log2.5 (but as noted, the latter is not logically based but rather expected only from knowledge of relationships in the data for two-tier systems). In other words, it works to almost point predictions for what we expected before running the regression! Now, let’s consider the revised formula. Using the same data as in the test of Equation 15.2 in the book, OLS yields:

            log NS = –0.059 + 0.165log MSB – 0.654 log(1–t) .

Again we expect a constant at zero and 0.167 on log MSB . Per the revised logic presented here, the coefficient on log(1–t) should be –0.667. This result is not too bad!1

OK, how did I get to this point? Glad you asked. It was staring me in the face all along, but I could not see it.

I started the logical (re-)modeling with seat share of the largest party, s1, as it was easier to conceptualize how it would work. First of all, we know that for simple systems we have s1= (MS)1/8; this is another of the logical models comprising the SPM and it is confirmed statistically. So this must also be the starting point for the extension to two-tier systems (although none of my published works to date reports any such extended model for s1). Knowing nothing else about the components of a two-tier system, we have a range of possible impact of the upper-tier compensation on the basic-tier largest party size (s1B). It can have no effect, in which case it is 1*s1B. In other words, in this minimal-effect scenario the party with the largest share of seats can emerge with the same share of overall seats after compensation as it already had from basic-tier allocation. At the maximum impact, all compensation seats go to parties other than the largest, in which case the effect is (1–t)*s1B. A fundamental law of compensation systems is that s1 ≤ s1B. (and NS ≥ NSB); by definition, they can’t enhance the position of the largest party relative to its basic-tier performance.2

Let’s see from some hypothetical examples. Suppose there are 100 seats, 50 of which are in the basic tier. The largest party gets 20 of those 50 seats, for s1B = 0.4. If compensation also nets it 20 of the 50 compensation seats, it emerges with 40 of 100 seats, for s1=0.4 = 1*s1B. If, on the other hand, it gets none of the upper-tier seats, it ends up with 20 of 100 seats, for s1=0.2 = (1–t)*s1B. For a smaller t example… Suppose there are 100 seats, 80 of which are in the basic tier, and the largest gets 32 seats, so again s1B = 0.4. If compensation nets it 8 of the 20 compensation seats (t=0.2), it emerges with 40 of 100 seats, for s1=0.4 = 1*s1B. If, on the other hand, it gets none of the upper-tier seats, it ends up with 32 of 100 seats, for s1=0.32 = (1–0.2)*s1B = 0.8*0.4=0.32.

In the absence of other information, we can assume the upper tier effect is the geometric average of these logical extremes (i.e, the square root of the product of 1 and 1–t), so:

            s1= (1–t)1/2(MSB)1/8,

and then because of the established relationship of NS = s1–4/3, which was also posited and confirmed by Taagepera (2007) and further confirmed by Shugart and Taagepera (2017), we must also have:

            NS = (1–t)2/3(MSB)1/6.

Testing of the s1 formula on the original data used for testing Equation 15.2 is less impressive than what was reported above for NS, but statistically still works. The coefficient on log(1–t) is actually 0.344 instead of 0.5, but its 95% confidence interval is 0.098–0.591. It is possible that the better fit to the expectation of NS than that of s1 is telling us that these systems have a different relationship of NS to s1, which I could imagine being so. This remains to be explored further. In the meantime, however, an issue with the data used in the original tests has come to light. This might seem like bad news, but in fact it is not.

The data we used in the article and book contain some inconsistencies for a few two-tier systems, specifically those that use “remainder pooling” for the compensation mechanism. The good news is that when these inconsistencies are corrected, the models remain robust! In fact, with the corrections, the s1 model turns out much better than with the original data. Given that s1 is the quantity on which the logic of the revised equation was based, it is good to know that when testing with the correct data, it is s1 that fits revised expectations best! On the other hand, the NS model ends up being a little more off.3 Again, this must be due to the compensation mechanism of at least some of these systems affecting the relationship of s1 to NS in some way. This is not terribly surprising. The fact that–by definition–only under-represented parties can obtain compensation seats could alter this relationship by boosting some parties and not others. However, this remains to be explored.

A further extension of the extended SPM would be to allow the exponent on (1–t) to vary with the size of the basic tier. Logically, the first term of the right-hand side of the equation should be closer to (1–t)0=1 if the basic tier already delivers a high degree of proportionality, and closer to (1–t)1=1–t when the upper tier has to “work” harder to correct deviations arising from basic-tier allocation. In fact, this is clearly the case, as two real-world examples will show. In South Africa, where the basic tier consists of 200 seats and a mean district magnitude of 22.2, there can’t possibly be much disproportionality to correct. Indeed, the largest party–the hegemonic ANC– had 69% of the basic tier seats in 2009. Once the compensation tier (with t=0.5) went to work, the ANC emerged with 65.9%. This is much less change from basic tier to final overall s1 than expected from the equation. (Never mind that this observed s1 is “too high” for such a proportional system in the first place! I am simply focusing on what the compensation tier does with what it has to work with.) The ratio of overall s1 to the basic-tier s1B in this case is 0.956, which is approximately (1–t)0.066, or very close to the minimum impact possible. On the other hand, there is Albania 2001. The largest party emerged from the basic tier (100 seats, all M=1)4 with 69% of the seats–just like in the South Africa example, but in this case that was significant overrepresentation. Once the upper tier (with t=0.258) got to work, this was cut down to 52.1%. The ratio of overall s1 to the basic-tier s1B here is 0.755, which is approximately (1–t)0.95, or very close to the maximum impact possible given the size of the upper tier relative to the total assembly.

These two examples show that the actual exponent on (1–t) really can vary over the theoretical range (0–1); the 0.5 proposed in the formula above is just an average (“in the absence of any other information”). Ideally, we would incorporate the expected s1 or NS from the basic tier into the derivation of the exponent for the impact of the upper tier. Doing so would allow the formula to recognize that how much impact the upper tier has depends on two things: (1) how large it is, relative to the total assembly (as explained by 1–t), and (2) how much distortion exists in the basic tier to be corrected (as represented by the basic-tier seat product, MSB).

However, incorporating this “other information” is not so straightforward. At least I have not found a way to do it. Nonetheless, the two examples provide further validation of the logic of the connection of the impact through 1–t. This, coupled with regression validation of the posited average effect in the dataset, as reported above, suggests that there really is a theoretical basis to the impact of upper-tier compensation on the basic-tier’s seat product, and that it rests on firmer logical grounds than the “2.5” in the originally proposed formula.

This a step forward for the scientific understanding of two-tier proportional representation!

In the next installment of the series, I will explain what went wrong with the original data on certain two-tier systems and how correcting it improves model fit (as it should!).

______

Notes.

1. The reported results here ignore the coefficients on the log of the effective number of ethnic groups and the latter’s interaction with the the log of the seat product. These are of no theoretical interest and are, in any case, statistically insignificant. (As explained at length in both Li & Shugart and Shugart & Taagepera, the interaction of district magnitude and ethnic fragmentation posited in widely cited earlier works almost completely vanishes once the electoral-system effect is specified properly–via the seat product and not simply magnitude.)

2. Perhaps in bizarre circumstances they can; but leave these aside.

3. This is what we get with the corrected data, First, for seat share of the largest party:

  log s1 = 0.047 – 0.126log MSB + 0.433 log(1–t) .

(Recall from above that we expect a constant of zero, a coefficient of –0.125 on log MSB and 0.5 on log(1–t).)

For effective number of seat-winning parties:

  log NS = –0.111 + 0.186log MSB – 0.792 log(1–t).

Both of those coefficients are somewhat removed from the logical expectations (0.167 and –0.667, respectively). However, the expectations are easily within the 95% confidence intervals. The constant term, expected to be zero, is part of the problem. While insignificant, its value of –0.111 could affect the others. Logically, it must be zero (if MSB=1 and t=0, there is an anchor point at which NS =1; anything else is absurd). If we suppress the constant, we get:

  log NS = 0.152log MSB – 0.713 log(1–t).

These are acceptably close (and statistically indistinguishable from expected values, but then so were those in the version with constant). Nonetheless, as noted above, the deviation of this result from the near-precise fit of most tests of the SPM probably tells us something about the relationship between s1 and NS in these two-tier systems. Just what remains to be seen.

4. In other words, it was an MMP system, conceived as a subtype of two-tier PR.

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.

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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.

_______

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.

Small national parties in Canada in the 2021 election and the connection of district voting to national outcomes

One of the notable trends in polling leading up to the Canadian election of 20 September is the increasing vote share of the Peoples Party of Canada (PPC). At the same time, polls have captured a steady decline of the Green Party as the campaign reaches its end. These two small parties’ trends in national support appear to be happening in all regions of the country, albeit to different degrees (see the graphs at the previous link). That is, while these parties have different levels of support regionally, their trends are not principally regional. Rather, all regions seem to be moving together. This will be a key theme of this post–that politics is fundamentally national, notwithstanding real difference in regional strengths1 and the use of an electoral system in which all seat winning is very local (in each of 338 single-seat districts or “ridings”).

The PPC is a “populist” party of the right. It seems that the Conservatives’ attempt to position themselves closer to the median voter during this campaign has provoked some backlash on the party’s right flank, with increasing numbers of these voters telling pollsters they will vote PPC.

At The Writ, Éric Grenier offers a look into what the polls say about the type of voter turning to the PPC, and whether they might cost the Conservatives seats. The PPC vote share ranges widely across pollsters but in the CBC Poll Tracker (also maintained by Grenier) it currently averages 6.7%. This would be quite a strikingly high figure for a party that is not favored to win even one seat and probably very unlikely to win more than one.2 The Poll Tracker shows a stronger surge in the Prairies region than elsewhere (3.6% on 14 Aug. just before the election was called to 10.9% when I checked on 19 Sept.) and Alberta (4.6% to 9.0% now), but it is being picked up in polling in all regions (for example, from 2.2% to 4.4% in Quebec and 2.9% to 6.1% in Atlantic Canada).

What I wish I knew: Is a voter more likely to vote PPC if he or she perceives that the party is likely to win at least one seat? This question is central to the “all politics is national” model developed in Shugart & Taagepera (2017) Votes from Seats, in chapter 10. We do not mean “all” to be taken literally. Of course, regional and local political factors matter. We mean that one can model the average district’s effective number of parties based on the national electoral system. More to the point, we argue that the way to think of how party systems form under FPTP (or any simple districted system) is not to think in terms of local “coordination” that then somehow gets projected up to a national party system, but rather that the national electoral system shapes the national party system, which then sets the baseline competition in the district contests.

If the PPC or Greens are perceived as likely to have a voice in parliament–and perhaps especially if the parliament is unlikely to have a majority party– voters who like what a small party proposes may be more inclined to support it, even though few voters live in a district where it has any chance of winning locally. Below I will show two graphs, each based on a mathematical model, showing a relationship of local votes to national seats. The first is based on the total available seats–the assembly size–while the second will be based on the seat outcome, specifically the nationwide effective number of seat-winning parties. The formula derived in the book for the connection to assembly size states the following for FPTP systems (every district with magnitude, M=1, and plurality rule):

NV=1.59S1/12,

where NV is the mean district-level effective number of vote-earning parties and S is the assembly size. Please see the book for derivation and justification. It may seem utterly nuts, but yes, the mean district’s votes distribution in FPTP systems can be predicted when we know only how many districts there are (i.e., the total number of seats). In the book (Fig. 10.2 on p. 156) we show that this sparse model accurately tracks the trend in the data across a wide range of FPTP countries, particularly if they are parliamentary. Here is what that figure looks like:

Of course, individual election averages (shown by diamonds) vary around the trend (the line, representing the above equation), and individual districts (the smear of heavily “jittered” gray dots) have a wide variation within any given election. But there is indeed a pattern whereby larger assemblies tend to be associated more fragmented district voting than is the case when assembly size is smaller. At S=338, Canada has a relatively large assembly (which happens to be almost precisely the size it “should be,” per the cube root law of assembly size).

The model for NV under FPTP is premised on the notion that voters are less attuned to the likely outcome in their own district than they are to the national scene. There is thus a systematic relationship between the national electoral system and the average district’s votes distribution.

Moreover, by combining the known relationship between the national electoral system and the national party system, we can see there should be a direct connection of the district vote distribution to the national distribution of seats. The Seat Product Model (SPM) states that:

NS=(MS)1/6,

where NS is the nationwide effective number of seat-winning parties. For FPTP, this reduces to NS=S1/6, because M=1. In terms of a FPTP system, this basically just means that because there are more districts overall, there is room for more parties, because local variation in strengths is, all else equal, likelier to allow a small party to have a local plurality in one of 400 seats than in one of 100. So, more seats available in the assembly (and thus more districts), more parties winning seats. The model, shown above, connecting district-level votes (NV) to the assembly size (S) then suggests that the more such seat-winning opportunities the assembly affords for small parties, the more local voters are likely to give their vote for such parties, pushing NV up. The process probably works something like this: Voters are aware that some small parties might win one or more seats somewhere, providing these parties a voice in parliament, and hence are likelier to support small parties to some degree regardless of their local viability. It is national viability that counts. “All politics is national.” The posited connection would be more convincing if it could be made with election-specific seat outcomes rather than with the total number of available seats. We can do that! Given the SPM for the national seat distribution (summarized in NS) based on assembly size, and the model for district-level votes distribution (NV), also based on assembly size, we can connect NV to NS algebraically:

NV=1.59NS1/2.

(Note that this comes about because if NS=S1/6, then S=NS6, giving us NV=1.59(NS6)1/12, in which we multiply the exponents in the final term of the equation to get the exponent, 1/2, which is also the square root. A full discussion and test of this formula may be found in my forthcoming chapter with Cory Struthers in an volume in honor of Richard Johnston being edited by Amanda Bittner, Scott Matthews, and Stuart Soroka. Johnston’s tour de force, The Canadian Party System likewise emphasizes that voters think more in terms of national politic than their local contest.)

Here is how this graph looks:

This again shows elections with diamonds and individual districts in small gray dots. The diagonal line is the preceding equation. It most definitely fits well. Note that it even fits India if we base the nationwide party system on the alliances (shown by squares), as we should, given that they and not the many parties are the nationwide actors, whereas each alliance is represented by a given component party in each district. (The graph also shows India if we use individual parties in the calculation of NS, which is useful because it makes clear just how well India, in the era of competing alliances, follows the S model–the one in the first graph. It obviously would not fit the NS model if we did not use the alliances, but again, it is the alliances that it should track with if the model is correct in its grounding district-level vote outcomes in the national balance of seats among the national political forces–parties elsewhere, including Canada, but alliances in India.)4

By implication, this connection of district-level NV to national NS may arise because voters have some estimate of how the national parliament is going to look when they decide whether or not to support a party other than one of the two leading national parties. For instance, a voter wavering between the NDP and the Liberals might be more likely to support the NDP if she estimates that there will be no majority, thereby allowing a smaller party like the NDP to be more influential than if one of the big parties has a majority on its own.

A vote for a much smaller party, like the PPC, might be simply expressive–“sending a message” to the Conservatives that they are not sufficiently right wing or populist. For a purely expressive voter, the national seat outcome may be irrelevant. Such a voter simply wants to register a protest. There still might be a connection to expected national votes: If such a voter thinks the PPC can get 7% he might be likelier to vote for it than if it’s only 3%.3 If, however, the connection runs through thinking about the national parliament, and whether one’s party will have voice there, it should help the party win votes around the country if its potential voters perceive that it will win one or more seats–in other words, that it is viable somewhere. I hope there is some polling data that I can find some day that allows us to get at this question, as it would connect the aggregate outcome demonstrated here with individual-level voter behavior. As the Canadian 2021 campaign has developed, it would be an especially good test of the model’s underlying individual-voter premise, given the surge of a small national party that is probably not likely to have a voice in the House of Commons. (But maybe its voters believe it will! They might even turn out to be correct.)

I do not, however, currently know if any polling or voter surveys exist to get at these questions. Such a survey ideally would ask the respondent how many seats they believe the various parties will get in the election. This would allow a rough construction of voter-expected effective number of seat-winning parties even though no voter actually has to know what that concept means or how to calculate it for the premise of the model to work. Minimally, as noted, it would at least be useful to know if voters choosing a small party think that party will indeed get one or more seats.

I have so far focused on the PPC in the Canadian 2021 election, as a possible example of a wider phenomena connecting local voting to the (expected) national seat outcome. A similar logic on the left side of politics should apply for the Green Party. Does its perceived viability for seats in parliament affect the tendency of voters to vote for it outside the specific districts where it is locally viable? The very big wrinkle this time around for the Greens, however, is that the party is struggling mightily, with an ongoing conflict between its leader and much of the rest of the party. It is currently projected to win no more than two seats, and perhaps none. It might be expected to retain the former leader’s seat in British Columbia, but even that may be in jeopardy with the national party in such disarray.

It is even questionable whether the Green Party still meets the criteria of a “national” party this time around; I do not (yet) have a really precise working definition of how many districts the party must be present in to qualify as “national.” The Green Party has not fielded a candidate in about a quarter of the ridings nationwide. Grenier has reviewed the 86 Green-less constituencies and whether their absence could affect outcomes among the contesting parties. Obviously the connection between expected seat winning nationally and obtaining votes in contests around the country is broken in any district in which there is no candidate running for the party. No candidate, no possibility of the local voters augmenting the party’s aggregate vote total. In any case, the party has dropped in national polls from 5.4% on 14 August to 3.2% now.

Further emphasizing now the Greens may not be a “national” party in this election is the campaign behavior of the leader. The CBC recently noted that the leader, Annamie Paul, is not exactly campaigning like the leader of a national party:

Asked why she hasn’t campaigned in more ridings, Paul acknowledged Friday that some candidates may want her to steer clear. She has campaigned outside of her home riding of Toronto Centre twice so far — once in a neighbouring riding and then Monday, in P.E.I.

Candidates distancing themselves from the leader is not normally a good sign for a party, particularly in a parliamentary system. “All politics is national,” after all. As explained in Votes from Seats (ch. 10), the impact of national politics on local voting is likely enhanced by parties bringing resources into districts to “show the flag” even where they are not likely to win a seat. (The PPC leader is certainly doing this.) If your leader remains mostly ensconced in her own district, the party is not deploying what is normally one of its best resources–the leader making the case for her party.

Nonetheless, it still might matter for the party’s ability to get votes, even in ridings it surely will not win, whether its potential voters believe it is viable for seat-winning somewhere. The good news for the party–and there is little of that–is that the province where it currently holds two seats, BC, is one of those where its polling has declined least: 7.0% on 14 August to 6.3% now. So, politics is still at least a bit more regional for the Greens than for other “national” parties, perhaps.

In conclusion, the district-level extension of the Seat Product Model states that in FPTP systems, district-level effective number of vote-earning parties can be predicted from the national electoral system–specifically, the assembly size. By further extension (in the aforementioned chapter I am working on with Struthers for the volume honoring Johnston), it should also be tied to the nationwide effective number of seat-winning parties, and to voter perceptions in the campaign as to how parties are doing at the national level. The result would be that voters are more likely to vote for even a small party under FPTP to the extent that they expect it to have a voice in parliament, and to the extent that the parliament may not have a majority party. The Canadian 2021 election, with a surging small party (the PPC) and another one declining (the Greens) offers an excellent case study of the phenomenon that is behind these models.

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Notes:

1. Obviously, things are different for an explicitly regional party (one that does not present candidates outside its region) like the Bloc Quebecois, which I will leave aside for this current discussion.

2. Perhaps it has some chance of winning the leader’s riding of Beauce (in Quebec), but as Grenier notes in a post the day before the election:

There’s nothing about Bernier’s Beauce riding that makes it particularly open to a party that has been courting the anti-vaxxer, anti-vaccine mandates and anti-lockdowns crowd. It’s hard to know where in the country that crowd would be big enough to elect a PPC MP.

He does also note that one poll, by EKOS, has put the party second in Alberta, albeit with only 20% of the vote. Maybe they could get a local surge somewhere and pick up a seat there.

3. Indeed, it might seem that we could make a similar algebraic connection. The Seat Product Model expects national effective number of vote-earning parties to be NV=[(MS)1/4 +1]2/3. This is confirmed in Votes from Seats. However, this can’t easily be expressed in terms of just S (even for FPTP, where the term for M drops out) and therefore is complicated to connect to the NV formula. In any case, the theoretical argument works better from seats–that voters key on the expected outcome of the election, which is a distribution of seats in parliament and whether one or another party has a majority or not. These outcomes are summarized in the effective number of seat-winning parties.

4. This graph is a version of the one that will be shown in the previouysly mentioned Shugart & Struthers chapter.

Israel government update and the likelihood of a 2021b election

It has been some time since I did an update on the election and government-formation process in Israel, 2021 (or, as I called it, 2021a, giving away my expectation that a 2021b was likely). The election was on 23 March, and as all readers likely know, it was the fourth election since an early call of elections was legislated at the end of 2018.

Since the March election, the government-formation process has been playing out in its usual manner. President Reuven Rivlin received recommendations from party leaders about who should be tasked to form a government. As expected, no candidate had recommendations from parties totaling 61 or more seats, but incumbent PM Benjamin Netanyahu (Likud) had more than opposition leader Yair Lapid (Yesh Atid), so he got the first nod. As everyone pretty much understood would happen, Netanyahu failed to cobble together a government. Arguably he did not even try very hard, “negotiating” mainly through press statements trying to shame leaders of small right-wing parties to rejoin his bloc. So, again as expected, Lapid received the mandate to try. And he most certainly has been trying hard. But as I write this he has one week remaining before his time expires.* If Lapid’s mandate expires with no government to present to the Knesset, there is a period in which any Knesset member can be nominated to be the PM via 61 signatures from members of the Knesset. However, with two blocs (using the term loosely) having both failed to win 61 seats, such a path to a government is highly unlikely to work.

The attempt to strike an agreement with Yamina, whose head Naftali Bennett would have gone first as PM, with Lapid taking over after a year (based on the same Basic Law amendments that the aborted Netanyahu–Gantz rotation was to follow), seemed close to fruition as the second week of May began. It would have been a strange government, given Bennett’s party won only 7 seats to Lapid’s 17, and because it would span nearly the entire Israeli political spectrum, including one Arab party (most likely as an outside supporter to a minority government, not as a full cabinet partner). Then once Hamas decided to escalate ongoing tensions in Jerusalem (including over things such as those I was writing about a decade ago) by firing their terrorist rockets directly at the capital city on Jerusalem Day, the ensuing war led Bennett to get cold feet and abandon a plan that apparently was all but final. On the other hand, he apparently also never quite ruled out returning to the plan. For instance, he never said in front of cameras that the deal was off, and there was a letter on 20 May from major activists in Yamina calling for the party to avoid another election and back an anti-Bibi government. Just today Bennett has supposedly told Likud he will return to talking with Lapid about forming a government if Netanyahu can’t form one (which he can’t).

So the “change” government remains a possibility even now (given the cessation of hostilities after 11 days) and may remain so right up until Lapid’s mandate expires. Frankly, it was always uphill to to form this proposed government, and would be a challenge for it to last if it did form. Yet it is the only current option, aside from another election later this year. Bennett has claimed numerous times that he will do everything he can to prevent another election. He has claimed a lot of things, so no one really can claim to know what he will do. (This is sometimes a good negotiating tactic, although it seems to have failed badly for Bennett, and in any case it is a terrible trait in a governing partner.) Although it is easy to mock Bennett for his flip-flops, we should acknowledge that he is in a genuinely difficult place. He has spent the last several years carving out a niche for his party to the right of Likud on security matters, so he can’t appear too eager to form a government with left-wing parties and reliant on Arab support. Thus even if he has intended all along to back such a government–and who knows–he and his no. 2, Ayalet Shaked, would need to make a good show of “leaving no stone unturned to form a nationalist government” before signing up to a deal with Lapid and Labor, Meretz, and Ra’am.

The bottom line is that the election produced a genuine stalemate. Even if Yamina sides with Netanyahu, that is not a majority without Ra’am, the Islamist party that broke off from the Joint List and has a pragmatic leader, Mansour Abbas, who seeks to be relevant in Israeli politics (unlike the Joint List itself). Such a government would also need the Religious Zionist list, which has said repeatedly it opposes any cooperation with Ra’am. The parties we are talking about here for a potential right-wing government are Likud (30 seats), the Haredi parties–Shas (9) and UTJ (7)–Yamina (7), plus Religious Zionist (6). These reach only 59 seats, hence the need for Ra’am (4) to back it; and, yes, Ra’am is certainly a right wing party within Arab Israeli politics, particularly on matters of social/religious policy. There is also New Hope (6), the party formed by Gideon Sa’ar and other Likud defectors. Obviously, if they joined, it would obviate the need to have the backing of Ra’am. However, Sa’ar has said over and over that he will not back Netanyahu. The entire reason his party formed was to offer an option for Likud without Bibi. While one should never rule anything out, and reports occasionally circulate that he is talking with Bibi, he looks like he just might mean it when he says no.

The “change” government would be Lapid (17), Blue and White (8), Labor (7), Yisrael Beiteinu (7), Meretz (6) New Hope (6), plus 6** from Yamina. Together, that “bloc” of left and right parties would have 58 seats, hence the inability to form a government without backing of Ra’am (who remains “brave” in evidently being willing to do a deal despite the violence of recent weeks). If Yamina is really out of this group, then that leaves it on only 51 seats, ten seats short. Yes, the two Arab lists just happen to combine for 10 seats, but it is highly unlikely that the Joint List is going to be part of such a government. And it is just as unlikely that the either or both Haredi parties are going to defect from the Bibi bloc to lend Lapid a hand.

I concluded my preview of the last election by saying, ” I don’t see a government being formed from this mess… the safe call is continuing deadlock and a 2021b election being necessary.” While that almost proved too pessimistic as of early May, and maybe yet will be shown to be the wrong call, it still could end up that way.

Finally, because this is Fruits and Votes, I want to highlight just how crazy the fragmentation was in the 2021(a) election. Throughout the three elections of 2019-20 the party system had reached a period of being almost exactly as fragmented as expected for its electoral system, as emphasized in my chapter in the Oxford Handbook of Israeli Politics and Society. In my post-election blog post, I even called the 2019a election “a totally normal election” based on the effective number of seat-winning parties being just over five and the largest party having 29% of the seats. These are almost precisely what we expect from the Seat Product Model (SPM) for such a high seat product (120-seat assembly elected in a single district). The indicators stayed in that general range for the next two elections. But check out the disruption of that trend in 2021! This graph is an updated version of the plots in the Handbook chapter (also a version of this was shown in the just-linked earlier post following 2019a).

The plots, for four party-system indicators, show lines for observed values over time with the expected values from the SPM marked by the horizontal solid line in each plot. The dashed line marks the mean for the entire period, through 2021a. Vertical lines mark changes in electoral-system features other than the district magnitude and assembly size–specifically formula changes or threshold increases.

Look at those spikes in the plots of the top row! The number of seat-winning lists (not parties, per se, given that many lists actually are alliances of two or more parties) jumped to 13, and the effective number to 8.52, almost as high as in 1999 (8.69). In 1999, a key reason for the spike was the directly elected PM, which freed voters to vote sincerely rather than for their preferred PM party in Knesset elections. In 2021, it is a product of the breakup of Blue and White (which happened as soon as the “unity” government was formed), the breakaway New Hope, the split of the Labor-Meretz list that contested the 2020 election, and Ra’am splitting from the (Dis)Joint List.

In the bottom row at left we see the corresponding collapse in the size of the largest party, although not quite to the depths reached a few times previously. In the lower right, we see a new record for lowest deviation from proportionality, thanks to no parties just missing the threshold (as happened in 2019a spectacularly and to a lesser degree in the subsequent election).

If there is a 2021b, will the fragmentation again be this high? The number of seat-winning lists could very well turn downward again as some parties re-enter pre-election pacts. On the other hand, as long as the Bibi-or-no cleavage continues to cross-cut all the others, it is entirely possible that fragmentation will remain “unnaturally” high. Barring Bennett and Lapid getting back together in the next week, we will find out later this year. And if that happens, then in the meantime, Bibi would continue benefitting from the stalemate.

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* By coincidence, Rivlin’s successor as president will be elected by the Knesset the same day Lapid’s current mandate to form a government expires.

** Yamina won 7 seats but one of the party’s MKs has said he will not support the government that was being negotiated with Lapid. Today he said his position has not changed.