For the second time in 2023, I have made a foray into podcast land. I was interviewed by Ben Raue for his podcast, The Tally Room: “From seats in to votes,” in which we discuss what the Effective Number of Parties is and how the Seat Product of an electoral system shapes party systems. We discuss Australia extensively, but also Canada and various other systems. Even St. Kitts & Nevis gets a reference.

# Category Archives: Votes from Seats

# Italy 2022

Italy votes in general elections today. The Brothers of Italy is expected to be the largest party, in a pre-electoral alliance with the League and Forza Italia that may end up with a substantial majority of seats in both houses.

The electoral system is similar to that used in 2018 in that it is mixed-member majoritarian despite having just over 60% of seats elected in the party-list proportional component of the system. In an important sense, however, *this year’s version is even more majoritarian*–the size of both chambers has been reduced substantially. Other things equal–as they are–a smaller assembly is less proportional (or “permissive” to small parties). And when you combine a relatively majoritarian system with a smaller assembly, you get a more majoritarian system overall. The new Chamber of Deputies, at 400 seats, is closer to the cube root law expectation for a country the size of Italy, but nonetheless the impact would be to favor more substantially than before the largest party or pre-electoral alliance, relative to the 2018 system which had a Chamber size of 630. The size of the Senate has been reduced correspondingly from 315 to 200 seats.

How is the system mixed-member *majoritarian* (MMM) and not mixed-member proportional (MMP)? This question has been asked before. The answer is straightforward: the seats a party wins in the list component are simply added on to those that it wins in the nominal component (single-seat districts decided by plurality). There is no compensation mechanism, not even a partial one like in the 1994–2001 version Italy used.^{1} There is a single vote, but whether voters can split their votes between nominal and list components has no bearing on the classification, which depends entirely on whether the list seats are allocated so as to compensate for deviations from proportionality arising from the district results (as under MMP) or not (as with “parallel” allocation under MMM).

The results from 2018, aggregated by pre-election alliances that coordinate nominations in the single-seat districts, certainly made this clear. The center-right alliance combined for 37% of the votes. This alliance won 42% of the seats, which is not terribly disproportional. However, we have to remember that more than three fifths of the seats are elected by PR. The nature of the system can be seen by looking at the detailed breakdown. The alliance won 111 nominal seats (out of 232, for 47.8%). Thus they were over-represented in this component of the system, as expected from single-seat plurality. If the list component were compensatory, as under MMP, the share of list seats won by this alliance should have been lower than its share of the vote. Yet it won 39.1% of them (111 of 386). It should have ended up with somewhere around 233 seats were these seats compensatory, but instead won 265 (including 3 seats for Italians overseas).

If we take the largest opposition force, the dynamic is even clearer. This was Five Star, which ran on its own, not as a part of any pre-electoral alliance. It won 32.7% of the vote, and 93 of the 232 nominal seats. That is 40%, so it is also slightly overrepresented in this component. To this it added 133 list seats, which is 34.5%, ending up with 227 seats total (including 1 abroad), or 36.0%. That the system was MMM becomes clearer still if we consider the second largest opposition alliance, the center-left. It had 22.9% of the vote, and won 28 nominal seats. This is only 12.1% of these seats–sever underrepresentation, as expected for a third party under single-seat plurality. Its list seat total was 88, which is 22.8% of the list component. Yes, 22.8%, so it got near-perfect proportional representation. However, it got this proportional result *only in the list seats* themselves. Overall, due to the punishment in the nominal seats, it was underrepresented, ending up with 122 seats (including 6 from Italians abroad), which is 19.4%. It was not severely underrepresented in the final result because–again–the list component is so large. However, were the system MMP they should have had approximately 110 list seats instead of just 88, in order to make their overall seats proportional to list votes. And, as already covered, the other alliances and parties would have had their list seats cut somewhat due to a compensation mechanism, if it were MMP. Thus the system is MMM, albeit with a large list component. I should also add that when I say “list votes” I mean votes aggregated from the nominal contests, given there is only a single fused ballot and not separate list and nominal votes (as there are in the MMM systems of Japan and Lithuania, or in the MMP systems of Germany and New Zealand).

Because polling for today’s election shows the Brothers of Italy in the lead and the combined center-right alliance clearing 40% of the vote while the second place center-left alliance looks to be under 30%, the system likely would provide a substantially larger boost to the center-right this time around than last, even if the rules were unchanged. However, *assembly size is a core defining characteristic of an electoral system.* If the rules for how seats are allocated are unchanged, and the balance in an MMM system between nominal and list seats is also unchanged, the key variable in how majoritarian it will be overall is assembly size. As already noted, both houses are half as large in the 2022 system as they were in 2018. This change promises a further boost to the winning alliance. There are only 147 single-seat contests in the Chamber of Deputies this time (around as many as in the Australian House of Representatives) and only 74 in the Italian Senate (about as many as in Liberia’s first chamber), it will be even more “work” for the list-PR component allocation to offset, despite its size relative to the nominal, given it is non-compensatory.

In terms of **effective seat product**, my estimations have it at 920 in the 2018 election. The goal behind the effective seat product is to allow us a rough approximation of what simple electoral system a given complex system is most similar to, in terms of its impact on the party system. Simple, single-tier systems with seat products in the 900–1000 ballpark include Luxembourg (900) and Greenland (961). The former has an assembly about ten percent the size of Italy’s in 2018, yet in terms of impact of the party system, the design of Italy’s system made it more like the simple PR system for the 60-seat assembly of Luxembourg than like other assemblies with 600+ seats and PR allocation (e.g., Germany’s effective seat product is currently around 1800 and Italy’s under its old PR system prior to the early 1990s was around 9800). As for Greenland, they get an effective seat product of 961 from an assembly of only 31 seats by allocating in a single territory-wide district. In other words, while Italy 2018 was a system of MMM, the large assembly and large share of seats allocated in the list component make the Chamber system of 2018 *similar to a small-assembly PR system*. But what about 2022?

The calculation of the effective seat product for the new Chamber of Deputies system would be around 650. In other words, roughly the same effect on a party system as Britain’s FPTP system, despite the election of over three fifths of deputies in a PR component. This is a fairly substantial reduction. It is based on the “as if” calculation of (1) an MMP system with same parameters as Italy’s new system, which would be an effective seat product of around 2860, and (2) a FPTP system of the actual size of Italy’s nominal component (147). For MMM, we take the geometric average of these two values, which is (rounded) about 650. This is very slightly less restrictive than the MMM system that was in use from 1994 to 2011 (for which the effective seat product could be said to have been around 660). Applying the same procedure to the Senate electoral system of 2022 would yield an effective seat product of around 370, implying roughly the same impact on the party system as the FPTP system of the Canadian House of Commons has.

In conclusion, Italy now has the most restrictive and thus plurality-favoring electoral system it has had in the post-WWII era.^{2} Despite still having a fragmented multiparty system in which parties enter pre-electoral alliances, it has an electoral system that is more like FPTP in the UK (in the case of the Chamber) or Canada (in the case of Italy’s Senate) than like a PR or MMP system. If the largest alliance clears 40% of the votes, as expected, it should obtain a substantial bonus in seats, due to the relatively majoritarian design of the system.

____

Notes

- That system was also MMM. It was often mis-classified in various sources as MMP. The misunderstanding was somewhat more justifiable than for the current one, because of the partial compensation mechanism, which was based on adjusting party-list votes according to nominal seat performance (rather than allocating list seats with regard to nominal seats won as is done under MMP). Even with the partial-compensation mechanism, that former system also should be classified as MMM.
- All of Italy’s post-war electoral systems have been complex in one way or another. Above I mentioned that the system in use as of the early 1990s had an effective seat product around 9800. That was a remainder-pooling PR system and
*Italy has not used a PR system since then*. The mixed-member system put in place in 1994 had an effective seat product around 660. The bonus-adjusted system from 2006 through 2013 comes out to around 1325 (but this is a more challenging system to estimate because of its unusual features). In all cases, these numbers refer only to the Chamber. Also, the calculation of effective seat product for the 1994–2001 system does not take the partial compensation mechanism into account. Perhaps it should, which would increase the effective seat product of that former system to some (small) degree. However, it is not clear how one would carry out such an adjustment, given the unusual nature of the mechanism. I do not think it is necessary or worthwhile to attempt.

# The effects (or their lack) of fused presidential–assembly ballots

A question that has arisen* is whether fused ballots–a single vote electing president and assembly, i.e., with no opportunity for ticket-splitting–suppress the number of parties, particularly when the president is elected by plurality and assembly by PR.

A challenge in addressing this question is that fused ballots are rather rare. Moreover, they may be adopted/abolished by ruling parties/coalitions based on expectations of advantage. In other words, the direction of causality between party-system outputs and rules is more ambiguous than usual. With such caveats reiterated, here is what I find.

This is for pure presidential systems, only because I am not aware of cases of semi-presidential systems that fuse presidential and assembly votes. (In parliamentary systems, the option does not arise, or in a sense the vote is always fused. I did not include the brief case in Israel of separate and direct election of an executive who was still responsible to the parliamentary majority.)

My outcome of interest is the ratio of expected effective number of seat-winning parties (*N _{S}*) or seat share of the largest party (

*s*

_{1}) to the expectation, given the seat product of the assembly (first chamber) electoral system.

For *N _{S}*, the ratio in non-fused cases is 1.13, for fused it is 0.927. This looks like good news for the hypothesis that fused ballots restrict party systems more than the separate vote does. However, the difference is not close to significant (p=0.12).

For *s*_{1}, the ratio in non-fused is 1.012, and in fused it is 1.047. Obviously that’s not significant. (Also, the seat product model is pretty good–even for presidential systems!)

Note that for *N _{S}*, the mean assembly party system in a presidential democracy tends to be

**more fragmented**than expected from its electoral system. Probably not what most people expect. Perhaps this is driven by the unusually fragmented case of Brazil. If I take it out, the ratios in non-fused are 1.083 for Ns and 1.031 for

*s*

_{1}. So not much impact.

Perhaps one should drop Uruguay from the set of fused cases. Not because ballots are not clearly fused, but because the electoral system is so different. Before 1999, parties could present multiple presidential candidates (and pool votes at party level for determining which party would win), and since then the fused ballot is only for the first round of a two-round presidential election. However, if we do this, we have only four cases left, so it is kind of meaningless. For the record, we would then have about a p=0.1 signifiant result in the expected direction. But I would put no stock in a result comparing four elections (in two countries) in one group to over 150 in the other group!

This is the list of cases with fused ballots that I am using. If I missed some, please let me know. (Angola, the case that prompted me to investigate this, is not in the dataset, nor are other countries that are not generally classified as democratic.)

```
country year
Dominican Rep 1978
Honduras 1993
Honduras 1997
Honduras 2001
Uruguay 1989
Uruguay 1994
Uruguay 1999
Uruguay 2004
Uruguay 2009
Uruguay 2014
Uruguay 2019
```

To this list could be added Bolivia. However, I did not include it because elections for president were not direct before 2005 (congress chose from top three if the popular vote did not yield a majority) and since 1997 the fusion has been only between the presidential vote and the party list vote of an MMP system.

(* A version of this text was originally posted as a comment in a thread on Angola, but it seemed to warrant a place in the center row of the virtual orchard.)

# Angola 2022: What (effective) seat product and impact on the outcome?

Earlier this week, in trying to understand the Angolan electoral system, I was unsure whether the allocation of the national list seats was compensatory, or in parallel to the provincial district results. In the comments, Miguel was kind enough to quote the relevant sections of the electoral law, confirming that allocation is parallel.

The results show the ruling MPLA won 51% of the vote and the main opposition UNITA 44%. I will take these as given, and not speculate on whether they are the “real” vote totals or a product of “electoral alchemy.” Rather, I am interested in whether the translation of these votes into seats suggests the MPLA chose a system that would benefit it considerably, or not.

The MPLA has won 124 of the 220 seats. That is 56.3% of the seats, for an advantage ratio (%seats/%votes) of 1.10. How does this compare with an “average” electoral system? I checked my dataset, restricting it to “simple” systems, even though Angola’s is not simple, and to those that are not FPTP or other M=1. The average across 377 such elections is… 1.12.

In other words, if the MPLA was trying to give itself a considerable seat advantage from this electoral system design, it kind of failed.

There is certainly one aspect of the electoral system design that looks like “rigging” via the rules: The provincial tier is highly malapportioned. The 18 provinces vary widely in population, yet each elects five members. See the images with preliminary vote totals in another comment from Miguel or see the CNE site, which also includes seats now. Given the use of D’Hondt at this level and the ample margins in rural provinces, the MPLA won 4-1 in several districts (and 5-0 in one)^{1} and 3-2 in all others aside from the three where UNITA was ahead. (UNITA won 4-1 in Cabinda.)

What undermines the MPLA’s own advantage considerably is the nationwide list component, which constitutes just under three fifths of all the seats (and uses Hare quota and largest remainders). If the MPLA had really wanted to create a system to advantage itself, it could have done so by making this tier smaller, or by various other designs.

I do note that UNITA is somewhat underrepresented. Its 90 seats is 40.9%. Given 44% of the votes, its advantage ratio is 0.928. Across a subset of electoral systems fitting the criteria I referred to above, this is quite low. In fact, the average for second parties is 1.075. (Subset because my dataset does not currently have second party shares for all elections; there are 147 elections here.)

In this sense, the electoral system’s design did indeed punish the main opposition. So if this was the MPLA goal, mission accomplished. The malapportionment must be a main cause of this, combined with the parallel (non-compensatory) allocation of the national seats. It should be noted as well, however, that with only two big parties, if one is overrepresented even a little bit (as the MPLA was), the second will probably be more underrepresented than would be the case in a multiparty system more typical of PR electoral systems.

Interestingly, much of the disadvantage to UNITA went to the advantage of smaller parties instead of to MPLA. There were three other parties, each of which won 2 seats. Two seats is 0.91% of the assembly; these parties had from 1.14% to 1.02% of the votes apiece. These small parties won only in the national district, where the only threshold was that a party could not win a seat by remainder unless it had already won a seat.^{2} Given that the national district is 130 seats, it could easily have supported even more parties than the five that won at least 2 seats. The largest party to win no seats had 0.75%. A simple quota for this district would be 0.769%, so this party was below the weak threshold anyway.

The effective numbers of parties were 2.20 by votes and 2.06 by seats–note not much difference there.^{3} The deviation from proportionality (Gallagher’s “least squares index”) was 4.44%. The latter figure, using again my set of simple non-FPTP systems, is not much different from average (4.87%). So all in all, despite the unusual electoral system, it is not a terribly remarkable result in terms of election indices.

As far as the effective seat product is concerned, for a parallel system I have found the satisfactory method is to take the geometric mean of what we would get if the basic tier were the entire system and what we would get if the system were compensatory. The seat product of the basic tier of this system is straightforward: district magnitude of 5, times tier size of 90 gives us 450. The formula for compensatory based on these parameters (an update and slight modification of a method I have shown here before) would yield an effective seat product of 3844. But because it is actually parallel, we take the geometric average of these values, which is 1315.

An effective seat product of 1315 is in the general range of the simple seat product Norway had (1297) before it adopted a small compensatory tier after 1985, or Peru’s in 1980 or 1985 (1296), and also not much smaller than Switzerland’s (1540).^{4}

The disproportionality we should expect from an effective seat product of of 1315 would be around three percent; the actual 4.4% is thus not too much higher. The seat share of the largest party in this election is about 1.4 times expectation^{5} from such a seat product and the effective number of seat-winning parties is about 0.62 the expectation. Obviously, this is due to MPLA political dominance. Or perhaps due to unfair vote reporting. That I can’t say. What I can say is that, despite a fairly unusual combination of extreme malapportionment in one tier and a greater than 50% parallel national tier, the impact this electoral system had on the seat allocation and disproportionality was not anything too out of the ordinary.

Finally, an interesting question but one I will not attempt to answer is whether, had UNITA won a narrow plurality of the nationwide vote, could the MPLA have retained a plurality or even majority of the assembly seats? Given the malapportionment and parallel allocation, I will say *maybe*. However, once again, I will point out that if they had wanted to ensure they could “win by losing,” the design they came up with was perhaps a little too “fair” to really be in their best (presumed to be anti-democratic) interest. On the other hand, if they are open to a gradual transition to democracy, and perhaps losing a fair election in five or ten years’ time, the system isn’t too bad. It plays to the MPLA’s regional strength yet does not overrepresent it greatly, and it creates space for the opposition, both UNITA and other parties, to operate.

____

Notes

- MPLA won 4-1 in Cuanza Sul, Moxico, Namibe, Huíla, and Cuando Cubango. It won 5-0 in Cunene (where the votes split 82.9%–14.4%). It is really striking that most of these strong MPLA districts are in the south, where UNITA was most present in the civil war. Meanwhile, the UNITA pluralities are Luanda (the capital and largest by far), Cabinda (the non-contiguous oil-rich enclave in the far north which has had a separatist movement) and Zaire (also in the northwest).
- It is not clear to me if this means a party could have won a provincial seat and thus been eligible for a remainder seat in the national district, or it had to have won a quota of nationwide votes. In any case, as all provincial seats were won by MPLA or UNITA, this detail would not have affected the results of this election.
- If I knew nothing other than that the effective number of vote-earning parties in some election was 2.2, I would expect the effective number of seat-winning parties to be around 1.72, based on logically derived, and empirically supported, formulas in
*Votes from Seats*. - By comparison, if we used the “as if compensatory” estimate of 3844, we would be in roughly the range of single-tier systems like Finland (3076 in 2019) or another former Portuguese colony, East Timor (4225). Indonesia is also in this seat-product neighborhood (4134), as was the French PR system of 1986 (3174).
- A ratio of actual to expected of 1.38 is near the 90th percentile for over a thousand elections, simple and complex, in the dataset (and would be about the same if I looked at just the simple non-FPTP subset).

# France 2022: Assessing the honeymoon election and towards a model of the impact of election timing on the president’s party’s seats

Was the French 2022 honeymoon election one that defies the usual impact of such election timing? Not to offer a spoiler, but the answer is yes and no.

Back around the time of the presidential runoff, I restated what I often say about elections for assembly held shortly after a presidential election: they are not an opportunity for the voters to “check” the president they have just chosen; presidential and semi-presidential systems just do not work that way. Well, usually. It seems hard to escape the notion that voters did just that–by holding Emmanuel Macron’s allies in Ensemble to less than a majority of seats, and by delivering bigger than expected seat totals to the Mélenchon-led united left (Nupes) and even to Le Pen’s National Rally (RN).

There will not be cohabitation, which was what I really meant in the French context when saying that honeymoon elections were not an opportunity to check the president. The results have not offered up any conceivable assembly majority that would impose its own choice for premier on Macron. I was also generally careful to say that I thought Macron’s allies would win a majority of seats, *or close to it*. They are relatively close, but considerably farther away that I expected, on about 42%. So, how does this outcome compare to honeymoon elections generally?

I have prepared an updated version of a graph I have shared before. An earlier version appears in *Votes from Seats*, as Figure 12.2. The x-axis is elapsed time, *E*, defined as the share of the period between presidential elections at which the assembly election occurs. The y-axis is the presidential seat ratio, *R _{P}*, calculated by dividing the vote share of the party (or pre-electoral alliance) supporting the president by the president’s own vote share in the first or sole round. The diagonal line is a regression best fit on the nonconcurrent elections (those with E>0), and is

*R*=1.2–0.7

_{P}*E*.

I added the France 2022 data point and label a little larger than the others, to call attention to it. The most notable thing is that this is the only case of a really extreme honeymoon–defined loosely as those with *E*<.05 but *E*>0–to have a value of *R _{P}*<1.00. So in that sense, it is a poor performance. There are other honeymoons for which

*E*≤0.1 that are below

*R*=1.00, including Chile 1965 and Poland 2001. In the Chilean case, the result obtains simply because the right did not present its own presidential candidate, but ran separately in the congressional election. Although this post is focused on honeymoon and other nonconcurrent elections, I also added labels to the two cases of concurrent elections (

_{P}*E*=0) that have unusually low presidential vote ratios. Note that on average,

*R*in concurrent elections tends to be a bit below 1.00, as a combination of strategic voting and small-party abstention from the presidential contest leads assembly voting to be more fragmented than presidential voting, hence lowering

_{P}*R*. However, in very early term elections, the president’s party/alliance almost always gains. So France 2022 is unusual, but not a massive outlier. In fact, in terms of distance from the regression line, it is about equivalent to France 1997 or El Salvador 2006 (labelled).

_{P}We see that the 2022 election also features the lowest *R _{P}* of any of France’s six honeymoon elections to date. The 2002 election (Chirac) produced an especially huge boost, whereas the 2017 election, when Macron had just been elected the first time, is almost on the regression line. (The regression does not include elections after 2015 because the dataset was collected around then; I added these more recent ones to the graph directly.) I also want to call attention to Volodomyr Zelenskyy’s 2019 honeymoon result in Ukraine for Servants of the People, as it is also among the most extreme honeymoon vote surges recorded anywhere as expected, perhaps aided by how uninstitutionalized that country’s party system has been. (If I wanted to be provocative, I’d say that factor also has been present in France, given frequent realignments on the right, the emergence of Macron, etc.)

(As an aside, I was somewhat surprised that an outlier, the one case of *E*>0.6 to have *R _{P}*>1 is the French late-midterm election of 1986. This is remembered as the election that produced the first cohabitation of the French Fifth Republic. But the vote share of the Socialists was still considerably higher than Mitterrand’s own vote share in the presidential first round of 1981, when the Communists had presented their own candidate.

^{1})

So much for the votes. I was wondering what happens if we look at **seats**? Strangely I had never done this before (at least with this dataset). This graph has as its y-axis the *seat* share of the president’s party (or alliance) divided by the president’s own first or sole-round *votes*, which I will call *R _{Ps}*. The x-axis is the same. In addition to plotting a best fit line, the diagonal, I also added the 95% confidence intervals from the regression estimates to this graph. There is also a lowess (local regression) plotted as the very thin grey line. Note how flat it is for a long portion of the term, a fact related to a point I will come to at the end (and also suggesting a more complex than linear fit may be more accurate, but I want to keep it simple for now).

The regression line here is very close to ** R_{Ps}=1.5–E**, which is a wonderfully elegant formula! It says that at a midterm election, a president’s party’s seat share would be, all else equal, the same as his or her own vote share half a term earlier. At a truly extreme honeymoon election–imagine one held the day after the president was elected, but with the result known–the seat share would be about 1.5 times the president’s vote share. At an extreme counter-honeymoon it would drop to around 0.5. So where did Macron’s Ensemble come out in the election just concluded? His

*R*=

_{Ps}**1.52**! So the party actually did about what the average trend says to expect. It was his 2017 surge that was higher than we perhaps should have expected (although, again, not as high as Chirac’s in 2002).

The result in the second figure is obviously holding constant the electoral system, so it should be taken with a grain of salt, given the importance of variation in electoral systems in shaping the size of the largest party (which is usually the president’s party, at least until we get to midterms and beyond).

What I find particularly elegant about the equation is its suggestion that midterm elections are no-effect elections, in terms of seat share for the president’s party. This was presumably what major party leaders were going for in the Dominican Republic when they shifted to the world’s only ever case (to my knowledge) of an all-midterm cycle. Both president and congress were elected to four-year terms, each at the halfway point of the other. (Actual outcomes during were not always no-effect, though on average they were close^{2}; they have since changed back to their former concurrent elections.) This may seem a surprise to readers who know the American system and its infamous midterm decline, but actually the midterm-election median in the US is 0.969. In an almost pure two-party system, anything below 1.00 might look bad, and be both politically consequential and also somewhat over-interpreted. But 0.969 is not really that much below 1.00! Okay I am cheating just a little by reporting the median. The mean is 0.943; it is brought down by a few major “shellackings” like 2010 (0.891), although 1990 was worse (0.719, in this case because G.H.W. Bush had won such a big landslide of his own).^{3}

In concurrent elections, the regression suggests also that on average, *R _{Ps}* is around 1.00. For the US, the median is 0.979, and the mean is 1.009. Note how it is higher than the midterm average, but perhaps not as much as one might expect.

^{4}

At this point, both these equations are just empirical regression best fits, not logical models. There is logic behind the general effects of electoral cycles on a presidential party’s performance, but not a logical basis for the specific parameters observed. I would very much like to have such a logical basis, but I have not hit upon it. Yet.

(Considerably nerdier and some rather half-baked stuff the rest of the way.) Such a logical model may be closer now that there is a simple and elegant empirical connection between presidential votes and seats. Seat shares are more directly connected to parameters of the electoral system than votes shares are–even vote shares for assembly parties, but vote shares for presidential candidates are a good deal more remote from the assembly electoral system. Nonetheless, in *Votes from Seats* we do derive a predictive formula for the effective number of presidential candidates, based on the assembly’s seat product. A regression reported in the book confirms its plausibility, but with rather low R^{2}. From that formula one could get an expected relationship for the leading presidential candidate’s vote total, *v _{p}*. It would be

*v*= 2

_{p}^{–3/8}[(

*MS*)

^{1/4}+1]

^{–1/4}. We already have, for the seat share of the largest party,

*s*

_{1}=(

*MS*)

^{–1/8}. It so happens that these return the same value at around

*MS*=175. Expectations of

*v*<

_{p}*s*

_{1}or

*s*

_{1}<

*v*would then depend on whether

_{p}*MS*(mean district magnitude times assembly size) is higher or lower than 175; for most presidential systems it is a good deal higher (the median in this sample of elections, including semi-presidential, is 480). Tying this observation to the one about midterm elections (

*E*=0.5) yielding actual (not predicted)

*s*

_{p}=

*v*and accepting for simplification that the president’s party seat share (

_{p}*s*

_{p}) is also the largest party seat share, at least in elections that are not after the midterm, might be a path towards a model. But that may take a while yet. Below I will copy a table of what the formulas for

*v*and

_{p}*s*

_{1}yield at various values of seat product,

*MS*, for simple systems. These values of

*s*

_{1}are without regard to elapsed time when the assembly election takes place.

Table of expected values of presidential vote shares (pv) and largest assembly party seat share (s1)

MS | pv | s1 | ratio_s1_pv |

1 | 0.65 | 1.00 | 1.54 |

10 | 0.60 | 0.75 | 1.26 |

25 | 0.57 | 0.67 | 1.16 |

50 | 0.56 | 0.61 | 1.10 |

100 | 0.54 | 0.56 | 1.04 |

150 | 0.53 | 0.53 | 1.01 |

175 | 0.52 | 0.52 | 1.00 |

200 | 0.52 | 0.52 | 0.99 |

225 | 0.52 | 0.51 | 0.98 |

256 | 0.51 | 0.50 | 0.97 |

300 | 0.51 | 0.49 | 0.96 |

500 | 0.50 | 0.46 | 0.92 |

1000 | 0.48 | 0.42 | 0.88 |

10000 | 0.42 | 0.32 | 0.75 |

25000 | 0.40 | 0.28 | 0.70 |

50000 | 0.39 | 0.26 | 0.67 |

100000 | 0.37 | 0.24 | 0.64 |

200000 | 0.35 | 0.22 | 0.61 |

*MS*due to system disproportionality, but higher as

*MS*increases beyond 175, presumably because of strategic behavior being different around the majoritarian presidential election and the more permissive assembly electoral system. The smallest

*MS*observed in this dataset for a (semi-)presidential system is 124 (Sierra Leone, 2002, 2007). The largest is 202,500 (Ukraine, 2006, 2007). For nonconcurrent elections, the minimum

*MS*is 240 (Chile, 1997, 2001).

Footnotes

- Also, Mitterand himself had finished second in the first round, with 25.9% of the votes (the incumbent, Giscard, had 28.3%). The Communist candidate had 15.4%. In the 1986 election, Socialists won 31% of the votes, for
*R*=1.2. (I am not counting the Communists as part of Mitterrand’s alliance by then, as he had fired the Communist ministers that were in his initial cabinet.)_{P} - The values for
*R*in these Dominican elections were: 0.587 in 1998, 0.975 in 2002, 0.945 in 2006, and 1.067 in 2010. So other than that first run, if the no-effect was what they wanted, they basically got it._{Ps} - [Added, 21 June.] I somehow forgot that my first publication on this topic, in the APSR in 1995, also used seats as its outcome of interest–but it was change in seat percentage for the president’s party from the prior assembly election (with president’s vote share as a control). Looking back on that pub, I see that my regression there would agree with my updated analysis here in suggesting that midterm elections, all else constant, are
*no-effect elections*. The regression line clearly passes very near the*change*=0,*E*=0.5 point in the article’s Figure 1. And, yes, in that article I commented on this as a “particularly striking feature” (p. 332). - The way I set up the regression, its constant term would be the
*R*when_{Ps}*E*=0, a concurrent election. This constant is actually 0.95, but its 95% confidence interval includes 1.00 (it is 0.844–1.057). The coefficient on the nonconcurrent dummy is 0.552, from which I get the approximation, 1.5, in the equation in the second figure (summing this coefficient and the constant). The coefficient on*E*is –1.072. R^{2}=0.215.

# France assembly 2022: Putting the prospects for NUPES in context

The first round of the French 2022 National Assembly election is on 12 June. As readers of this blog recognize, this is an extreme *honeymoon election*, owing to the short time that has elapsed since the presidential election. In that two-round contest in April, Emmanuel Macron was reelected, winning 27.9% of the vote in the first round and 58.6% in the runoff.

The runner-up in the presidential contest was Marine Le Pen of the extremist National Rally, with 23.2% in the first round and 41.5% in the runoff. In a close third place was the leftist Jean-Luc Mélenchon, with 22.0%. In the period since the runoff results were known, Mélenchon has led the formation of a left alliance known as the New Ecologic and Social People’s Union (NUPES). (See the series of very helpful comments from Wilf at an earlier post, where he shared news stories about the coalition bargaining as it was taking place.) Mélenchon has not been shy about his goal, proclaiming that he is *running to be premier*. If this happened, it could usher in a period of cohabitation, defined as president and premier from opposing parties and the president’s party not in the cabinet. (I say “could usher in” because there’s always the possibility Macron’s party would be in a cabinet headed by Mélenchon, although if the latter actually were premier–and especially if NUPES won a majority of seats–that would be rather unlikely.)

As readers of this space will know, I find such an outcome extremely unlikely. Honeymoon elections do not work that way. They are not a second chance for voters to “check” the president. They confirm the mandate the voters have just conferred on the new (or newly reelected) president. Or do they? Maybe this will be a special case. That is what I am setting out to explore in this post.

Regarding “normal” honeymoon elections, see the post on France that I wrote in 2017, just before the presidential runoff, suggesting that Macron’s then-new party would get around 29% of the vote, and be the largest party. It actually won almost exactly that, 28.2%, and given both allies and the majoritarian two-round electoral system, Macron ended up with a large assembly majority. See the graph in that post, which also appears in *Votes from Seats*, and shows how nearly all elections early in a presidential term result in rather significant surges for the president’s party. The graph shows something called “Presidential Ratio” graphed against “Elapsed Time.” The ratio, *R _{P}*, is simply the vote share of the president’s party, divided by the president’s own (first or sole round) vote share in the preceding presidential election. The elapsed time,

*E*, is the percentage of the time between presidential elections at which the assembly election takes place.

For all non-concurrent elections, a best fit shows a steep slope starting at about 1.2 if the honeymoon election is immediately after the presidential election, and dropping steadily as assembly elections occur later in the period between presidential elections. It crosses the 1.00 line (indicating identical assembly and presidential vote shares) at around *E*=0.28, or just past the quarter mark, then drops to around 0.84 when *E*=0.5, encompassing the well known midterm-decline phenomenon. Given that for France in 2022 (as in 2017 and some previous cycles), *E*=0.017, we expect *R _{P}*=1.19. Taking Macron’s first-round vote of 27.9%, his party should win around 33.1% of the votes. Presumably that would be a plurality and would again be sufficient to win a majority (or close to it) in the assembly when the two-round process is all said and done. Or should we be sure that would be a plurality this time? Let’s see.

Please remember that the equation of this line for presidential vote ratio is not a logical model (like the Seat Product Model or the Cube Root Law), and in any case, even logical model predictions get tripped up by real politics at times! Maybe this honeymoon election will be different. Macron won many voters in the runoff who would have preferred Mélenchon but felt they had to vote to stop Le Pen. There may be much more energy on the side of NUPES than is normal for an alliance that backed a loser.

So how surprising would a good performance be? I decided the best way to put a potential answer to this question in context was to go back to my dataset and augment it with votes data from runners-up and third-place presidential candidates. I have never looked into this before! So here we go…

First, let’s see what it looks like for the party of the candidate who finished second in the second or sole round of presidential voting.

We see that honeymoon elections are really bad for your party if you just lost the presidential election as the runner-up! All data points are below the 1.00 line until nearly *E*=0.3. The dashed curve is just a lowess (local regression) curve. I did not continue it much past the midterm, because the data get rather sparse late in the term. Not because there are no such elections (again, see the graph for presidential parties), but because the farther you go into the term, the more likely the runner-up’s party does not exist in a recognizable form. Presidential and semi-presidential systems can be that way.

In France 2022, it was Le Pen who finished second, and I do not think anyone would be surprised if her party got less than two thirds of what she won (in other words, around 15%). In fact, it will probably be much worse than that for her.

The topic of interest here, though, is **the third presidential candidate’s party**. Here is what that graph looks like:

Interestingly, the party backing the candidate who came in third quite often increases its support in a honeymoon election. In most cases, that probably comes predominantly at the expense of the second candidate’s party. But there is probably no reason why it could not come from the winner’s, in a case where there was a good deal of strategic voting in the presidential election (or specifically, in a runoff).

The curve is pretty level until *E*=0.2, with a mean of almost 1.5. Given how sparse the data are–there are lots of presidential elections with no third candidate or where the third had no party–I would not draw too much of a conclusion from this. However, note that 1.5 times Mélenchon’s vote would reach 33%, or almost exactly what we “predict” for Macron’s La République En Marche! (The exclamation point is in the party name, although you should be as excited about this convergence of their potential shares as I am!) If one were to add in the votes of the other presidential candidates whose parties since have joined NUPES, perhaps we would “predict” a voting plurality for Mélenchon.

So, while I still do not think Mélenchon is going to become premier, this data exploration has led me to believe it would not be as shocking a development as I initially assumed. It could be that this is the honeymoon election that has the ideal convergence of factors to generate an upset. And make no mistake, if a just-reelected president were to be forced to appoint as premier someone opposed to him, it would be an upset. On the other hand, polls do show it will indeed be close, at least in the first round.

# 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.

R=1.20–0.725E,_{p}

where *R _{p}* 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}

____

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 party^{2}), 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

- 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.
- 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%.
- 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 (*N _{S}*) sometimes being higher than the effective number of vote-earning parties (

*N*), 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.

_{V}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 cast up to *M* votes (*M* being the district magnitude) for individual candidates, 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 criteria^{2} 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.

____

- 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.
- 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”!) - 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.

No reason here to doubt that the logical model, **N****_{S} = s_{1}^{–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 *N _{S}* =

*s*

_{1}

^{–4/3},

*on average*. And, by the way, for those who care about such things, the R

^{2}=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 Seat*s 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 (*N _{S}*) 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

*N*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:

_{S}*N _{S}* = 2.5

*(*

^{t}*MS*

_{B})

^{1/6},

where *N _{S}* is the effective number of seat-winning parties (here, meaning the

*expected*

*N*),

_{S}*M*is the mean district magnitude of the basic tier,

*S*

_{B}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.

It always seemed unlikely that the design of the Austrian electoral system was such that *expected* *N _{S}* 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 *MS*_{B}=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 *MS*_{B}=17.5*(183–25)=17.5*158=2765, and *t*=0.137. This changes the “expected” *N _{S}* (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 *MS*_{B} 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 *N _{S}*” depicted in Figure 17.2 as 6.3 for recent elections to a more reasonable 3.83. And, in fact actually observed

*N*in recent years has tended to be in the 3.4–4.2 range.

_{S}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:

*N _{S}* = (1–

*t*)

^{–2/3}(

*MS*

_{B})

^{1/6}.

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 *N _{S}* 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 *s*_{1} model for two-tier works a little better than the one for *N _{S}*. Moreover, it was on

*s*

_{1}that the revised logic was based.

Note that the data plots show a light horizontal line at *s*_{1}=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:

*N _{S}* = 2.5

*(*

^{t}*MS*

_{B})

^{1/6},

where *N _{S}* is the effective number of seat-winning parties (here, meaning the

*expected*

*N*),

_{S}*M*is the mean district magnitude of the basic tier,

*S*

_{B}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:

*N*=(

_{S}*MS*)

^{1/6}, given that for simple systems, by definition,

*t*=0 and

*S*

_{B}=

*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:

*N _{S}* = (1–

*t*)

^{–2/3}(

*MS*

_{B})

^{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.

MS_{B} | t | 1-t | (1–t)^{–}^{2/3} | 2.5^{t} | N (rev.) _{S} | N (Eq. 15.2)_{S} |

100 | .5 | .5 | 1.59 | 1.58 | 3.42 | 3.40 |

100 | .25 | .75 | 1.21 | 1.26 | 2.61 | 2.71 |

250 | .3 | .7 | 1.27 | 1.32 | 4.68 | 4.85 |

250 | .4 | .6 | 1.41 | 1.44 | 3.53 | 3.62 |

250 | .6 | .4 | 1.84 | 1.73 | 4.62 | 4.35 |

2500 | .3 | .7 | 1.27 | 1.32 | 4.68 | 4.85 |

2500 | .15 | .85 | 1.11 | 1.15 | 4.11 | 4.23 |

It may not work especially well with very high *MS _{B}*, 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,

*S*much greater than around 300 is not likely to be very common. My examples of

_{B}*MS*=2,500 are motivated by the notion of

_{B}*S*=300 and a decently proportional basic-tier

_{B}*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 *N _{S}* = –0.066 + 0.166log

*MS*+ 0.399

_{B}*t*.

Logically, we expect a constant of zero and a coefficient of 0.167 on the log of *MS _{B}*; 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 *N _{S}* = –0.059 + 0.165log

*MS*– 0.654 log(1–

_{B}*t*) .

Again we expect a constant at zero and 0.167 on log* MS _{B}* . 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, **s**_{1}, as it was easier to conceptualize how it would work. First of all, we know that for simple systems we have *s*_{1}= (*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 *s*_{1}). 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 (*s*_{1B}). It can have no effect, in which case it is 1** s_{1B}*. 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*)*

*s*_{1}

*. A fundamental law of compensation systems is that*

_{B}*s*

_{1}≤

*s*_{1}

*. (and*

_{B}*N*≥

_{S}*N*); by definition, they can’t enhance the position of the largest party relative to its basic-tier performance.

_{SB}^{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 *s*_{1}*_{B}* = 0.4. If compensation also nets it 20 of the 50 compensation seats, it emerges with 40 of 100 seats, for

*s*

_{1}=0.4 = 1*

*s*_{1}

*. If, on the other hand, it gets none of the upper-tier seats, it ends up with 20 of 100 seats, for*

_{B}*s*

_{1}=0.2 = (1–

*t*)*

*s*_{1}

*. For a smaller*

_{B}*t*example… Suppose there are 100 seats, 80 of which are in the basic tier, and the largest gets 32 seats, so again

*s*_{1}

*= 0.4. If compensation nets it 8 of the 20 compensation seats (*

_{B}*t*=0.2), it emerges with 40 of 100 seats, for

*s*

_{1}=0.4 = 1*

*s*_{1}

*. If, on the other hand, it gets none of the upper-tier seats, it ends up with 32 of 100 seats, for*

_{B}*s*

_{1}=0.32 = (1–0.2)*

*s*_{1}

*= 0.8*0.4=0.32.*

_{B}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:

*s*_{1}= (1–t)^{1/2}(*MS _{B}*)

^{–}^{1/8},

and then because of the established relationship of *N _{S}* =

*s*

_{1}

^{–4/3}, which was also posited and confirmed by Taagepera (2007) and further confirmed by Shugart and Taagepera (2017), we must also have:

*N _{S}* = (1–t)

^{–}^{2/3}(

*MS*)

_{B}^{1/6}.

Testing of the *s*_{1} formula on the original data used for testing Equation 15.2 is less impressive than what was reported above for *N _{S}*, 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

*N*than that of

_{S}*s*

_{1}is telling us that these systems have a different relationship of

*N*to

_{S}*s*

_{1}, 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 s_{1} model turns out much better than with the original data**. Given that

*s*

_{1}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

*s*

_{1}that fits revised expectations best! On the other hand, the

*N*model ends up being a little more off.

_{S}^{3}Again, this must be due to the compensation mechanism of at least some of these systems affecting the relationship of

*s*

_{1}to

*N*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.

_{S}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 *s*_{1} than expected from the equation. (Never mind that this observed *s*_{1} 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 *s*_{1} to the basic-tier *s*_{1}*_{B}* 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

*s*

_{1}to the basic-tier

*s*_{1}

*here is 0.755, which is approximately (1–*

_{B}*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 *s*_{1} or *N _{S}* 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,

*MS*).

_{B}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 * s_{1}* = 0.047 – 0.126log

*MS*+ 0.433 log(1–

_{B}*t*) .

(Recall from above that we expect a constant of zero, a coefficient of –0.125 on log* MS _{B}* and 0.5 on log(1–

*t*).)

For effective number of seat-winning parties:

log *N _{S}* = –0.111 + 0.186log

*MS*– 0.792 log(1–

_{B}*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 * MS_{B}*=1 and

*t*=0, there is an anchor point at which

*N*=1; anything else is absurd). If we suppress the constant, we get:

_{S} log *N _{S}* = 0.152log

*MS*– 0.713 log(1–

_{B}*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 * s_{1}* and

*N*in these two-tier systems. Just what remains to be seen.

_{S}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 (*N _{S}*) in both Germany and New Zealand. For the latter country, it includes the pre-reform FPTP system. In the case of Germany, it plots

*N*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 systems

_{S}^{1}; 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 *N _{S}* 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 *N _{S}* 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 *N _{S}*” 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”

*N*of 0.36.

_{S} 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 *N _{S}*=3.08. This would be somewhat better for much of the earlier period in Germany. In fact, from 1953 through 2002, mean observed

*N*=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

_{S}*N*has been 3.14.

_{S}