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 expectation5 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.

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Notes

  1. 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).
  2. 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.
  3. 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.
  4. 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).
  5. 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).

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

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

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

NS = 2.5t(MSB)1/6,

where NS is the effective number of seat-winning parties (here, meaning the expected NS), M is the mean district magnitude of the basic tier, SB is the total number of seats in the basic tier, and t is the “tier ratio” defined as the share of the total number of assembly seats allocated in the compensatory tier. In the case of a simple (single-tier) system, this reduces to the basic SPM: NS =(MS)1/6, given that for simple systems, by definition, t=0 and SB=S, the total size of the elected assembly.

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

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

NS = (1–t)–2/3(MSB)1/6.

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

Consider a few examples for hypothetical electoral systems.

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

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

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

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

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

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

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

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

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

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

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

            s1= (1–t)1/2(MSB)1/8,

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

            NS = (1–t)2/3(MSB)1/6.

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

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

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

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

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

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

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

______

Notes.

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

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

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

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

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

For effective number of seat-winning parties:

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

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

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

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

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

“Effective Seat Product” for two-tier PR (including MMP) and MMM

[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 of the exponent in the model for two-tier systems), although the basic framework remains the same. User beware! This also means that the datasets linked at the end of this post are not accurate. I will upload corrected ones at some point.]

The seat product for a simple electoral system is its assembly size (S) times its mean district magnitude (M) (Taagepera 2007). From this product, MS, the various formulas of the Seat Product Model (SPM) allow us to estimate the effective number of parties, size of the largest, disproportionality, and other election indicators. For each output tested in Shugart and Taagepera (2017), Votes from Seats, we find that the SPM explains about 60% of the variance. This means that these two institutional inputs (M and S) alone account for three fifths of the cross-national differences in party system indicators, while leaving plenty for country-specific or election-specific factors to explain as well (i.e., the other 40% of the variance).

The SPM, based on the simple seat product, is fine if you have a single-tier electoral system. (In the book, we show it works reasonably well, at least on seat outputs, in “complex” but still single-tier systems like AV in Australia, majority-plurality in France, and STV in Ireland.) But what about systems with complex districting, such as two-tier PR? For these systems, Shugart and Taagepera (2017) propose an “extended seat product model”. This takes into account the basic-tier size and average district magnitude as well as the percentage of the entire assembly that is allocated in an upper tier, assumed to be compensatory. For estimating the expected effective number of seat-winning parties (NS), the extended SPM formula (Shugart and Taagepera, 2017: 263) is:

NS=2.5t(MB)1/6,

where MB is the basic-tier seat product, defined as the number of seats allocated in the basic tier (i.e., assembly size, minus seats in the upper tier), and t is the tier ratio, i.e., the share of all assembly seats allocated in the upper tier. If the electoral system is simple (single tier), the equation reduces to the “regular” seat product model, in which MS=MB and t=0.

(Added note: in the book we use MSB to refer to what I am calling here MB. No good reason for the change, other than blogger laziness.)

We show in the book that the extended seat product is reasonably accurate for two-tier PR, including mixed-member proportional (MMP). We also show that the logic on which it is based checks out, in that the basic tier NS (i.e., before taking account of the upper tier) is well explained by (MB)1/6, while the multiplier term, 2.5t, captures on average how much the compensation mechanism increases NS. Perhaps most importantly of all, the extended seat product model’s prediction is closer to actually observed nationwide NS, on average, than would be an estimate of NS derived from the simple seat product. In other words, for a two-tier system, do not just take the basic-tier mean M and multiply by S and expect it to work!

While the extended seat product works quite well for two-tier PR (including MMP), it is not convenient if one wants to scale such systems along with simple systems. For instance, as I did in my recent planting on polling errors. For this we need an “effective seat product” that exists on the same scale as the simple seat product, but is consistent with the effect of the two-tier system on the effective number of parties (or other outputs).

We did not attempt to develop such an effective seat product in Shugart and Taagepera (2017), but it is pretty straightforward how to do it. And if we can do this, we can also derive an “effective magnitude” of such systems. In this way, we can have a ready indicator of what simple (hypothetical) design comes closest to expressing the impact of the (actual) complex design on the party system.

The derivation of effective seat product is pretty simple, actually. Just take, for the system parameters, the predicted effective number of seat-winning parties, NS, and raise it to the power, 6. That is, if NS=(MS)1/6, it must be that MS=NS6. (Taagepera 2007 proposes something similar, but based on actual output, rather than expected, as there was not to be a form of the seat product model for two-tier systems for almost another decade, till an initial proposal by Li and Shugart (2016).)

Once we do this, we can arrive at effective seat products for all these systems. Examples of resulting values are approximately 5,000 for Germany (MMP) in 2009 and 6,600 for Denmark (two-tier PR) in 2007. How do these compare to simple systems? There are actual few simple systems with these seat products in this range. This might be a feature of two-tier PR (of which MMP could be considered a subtype), as it allows a system to have a low or moderate basic-tier district magnitude combined with a high degree of overall proportionality (and small-party permissiveness). The only simple, single-tier, systems with similar seat products are Poland (5,161), with the next highest being Brazil (9,747) and Netherlands before 1956 (10,000). The implication here is that Germany and Denmark have systems roughly equivalent in their impact on the party system–i.e., on the 60% of variance mentioned above, not the country-specific 40%–as the simple districted PR system of Poland (S=460, M=11) but not as permissive as Brazil (S=513, M=19) or pre-1956 Netherlands (M=S=100). Note that each of these systems has a much higher magnitude than the basic-tier M of Germany (1) or larger assembly than Denmark (S=179; M=13.5). Yet their impact on the nationwide party system should be fairly similar.

Now, suppose you are more interested in “effective district magnitude” than in the seat product. I mean, you should be interested in the seat product, because it tells you more about a system’s impact on the party system than does magnitude alone! But there may be value in knowing the input parameters separately. You can find S easily enough, even for a complex system. But what about (effective) M? This is easy, too! Just take the effective seat product and divide it by the assembly size.

Thus we have an effective M for Germany in 2009 of 7.9 and for Denmark in 2007 of 36.9. These values give us an idea of how, for their given assembly sizes, their compensatory PR systems make district magnitude “effectively”–i.e., in terms of impact on the inter-party dimension–much larger than the basic-tier districts actually are. If we think low M is desirable for generating local representation–a key aspect of the intra-party dimension–we might conclude that Germany gets the advantages M=1 in local representation while also getting the advantages of the proportionality of 8-seat districts. (Best of both worlds?) By comparison, simple districted PR systems with average M around 8 seats include Switzerland and Costa Rica. (The Swiss system is complex in various ways, but not in its districting.) Eight is also the minimum magnitude in Brazil. Denmark gets whatever local representation advantages might come from an actual mean M of 13.5, yet the proportionality, for its assembly size, as if those districts elected, on average, 37 members. Actual districts of about this magnitude occur only in a relatively few districts within simple systems. For instance, the district for Madrid in Spain has M in the mid-30s, but that system’s overall average is only 6.7 (i.e., somewhat smaller than Germany’s effective M).

Now, what about mixed-member majoritarian (MMM) systems. Unlike MMP, these are not designed with a compensatory upper tier. In Votes from Seats, Taagepera and I basically conclude that we are unable to generalize about them. Each MMM system is sui generis. Maybe we gave up too soon! I will describe a procedure for estimating an effective seat product and effective magnitude for MMM systems, in which the basic tier normally has M=1, and there is a list-PR component that is allocated in “parallel” rather than to compensate for deviations from proportionality arising out of the basic tier.

The most straightforward means of estimating the effective seat product is to treat the system as a halfway house between MMP and FPTP. That is, they have some commonality with MMP, in having both M=1 and a list-PR component (not actually a “tier” as Gallagher and Mitchell (2005) explain). But they also have commonality with FPTP, where all seats are M=1 plurality, in that they reward a party that is able to win many of the basic seats in a way that MMP does not. If we take the geometric average of the effective seat product derived as if it were MMP and the effective seat product as if it were FPTP, we might have a reasonable estimate for MMM.

In doing this, I played with both an “effective FPTP seat product” from the basic tier alone and an effective FPTP seat product based on assuming the actual assembly size. The latter works better (in the sense of “predicting,” on average for a set of MMM systems, what their actual NS is), and I think it makes more logical sense. After all, the system should be more permissive than if were a FPTP system in which all those list-PR component seats did not exist. So we are taking the geometric average of (1) a hypothetical system in which the entire assembly is divided into a number of single-seat electoral districts (Eeff) that is Eeff = EB+tS, where EB is the actual number of single-seat districts in the basic tier and S and t are as defined before, and (2) a hypothetical system that is MMP instead of MMM but otherwise identical.

When we do this, we get the following based on a couple sample MMM systems. In Japan, the effective seat product becomes approximately 1,070, roughly equivalent to moderate-M simple districted PR systems in the Dominican Republic or pre-1965 Norway. For South Korea, we would have an effective seat product of 458, or very roughly the same as the US House, and also close to the districted PR system of Costa Rica.

Here is how those are derived, using the example of Japan. We have S=480, with 300 single-seat districts and 180 list-PR seats. Thus t=0.375. If it were two-tier PR (specifically, MMP), the extended seat product would expect NS=3.65, from which we would derive an effective seat product, (MS)eff=3.666 =2,400. But it is MMM. So let’s calculate an effective FPTP seat product. Eeff = EB+tS=300+180=480 (from which we would expect NS=2.80). We just take the geometric mean of these two seat-product estimates: (2400*480)1/2=1,070. This leads to an expected NS=3.19, letting us see just how much the non-compensatory feature reduces expected party-system fragmentation relative to MMP as well as how much more permissive it is than if it were FPTP.

How does this work out in practice? Well, for Japan it is accurate for the 2000 election (NS=3.17), but several other elections have had NS much lower. That is perhaps due to election-specific factors (producing huge swings in 2005 and 2009, for example). As I alluded to above already, over the wider set of MMM systems, this method is pretty good on average. For 40 elections in 17 countries, a ratio of actual NS to that predicted from this method is 1.0075 (median 0.925). The worst-predicted is Italy (1994-2001), but that is mainly because the blocs that formed to cope with MMM contained many parties (plus Italy’s system had a partial-compensation feature). If I drop Italy, I get a mean of 1.0024 (but a median of only 0.894) on 37 elections.

If we want an effective magnitude for MMM, we can again use the simple formula, Meff=(MS)eff/S. For Japan, this would give us Meff=2.25; for Korea Meff=1.5. Intuitively, these make sense. In terms of districting, these systems are more similar to FPTP than they are to MMP, or even to districted PR. That is, they put a strong premium on the plurality party, while also giving the runner-up party a considerable incentive to attend to district interests in the hopes of swinging the actual district seat their way next time (because the system puts a high premium on M=1 wins, unlike MMP). This is, by the way, a theme of the forthcoming Party Personnel book of which I am a coauthor.

(A quirk here is that Thailand’s system of 2001 and 2005 gets an effective magnitude of 0.92! This is strange, given that magnitude–the real kind–obviously has a lower limit of 1.0, but it is perhaps tolerable inasmuch as it signals that Thailand’s MMM was really strongly majoritarian, given only 100 list seats out of 500, which means most list seats would also be won by any party that performed very well in the M=1 seats, which is indeed very much what happened in 2005. The concept of an “effective” magnitude less than 1.0 implies a degree of majoritarianism that one might get from multi-seat plurality of the MNTV or list-plurality kind.)

In this planting, I have shown that it is possible to develop an “effective seat product” for two-tier PR systems that allows such systems to be scaled along with simple, single-tier systems. The exercise allows us to say what sort of simple system an actual two-tier system most resembles in its institutional impact on inter-party variables, like the effective number of seat-winning parties, size of the largest party, and disproportionality (using formulas of the Seat Product Model). From the effective seat product, we can also determine an “effective magnitude” by simply dividing the calculated effective seat product by actual assembly size. This derivation lets us understand how the upper tier makes the individual district effectively more proportional while retaining an actual (basic-tier) magnitude that facilitates a more localized representation. Further, I have shown that MMM systems can be treated as intermediary between a hypothetical MMP (with the same basic-tier and upper-tier structure) and a hypothetical FPTP in which the entire assembly consists of single-seat districts. Again, this procedure can be extended to derive an effective magnitude. For actual MMP systems in Germany and also New Zealand, we end up with an effective magnitude in the 6–8 range. For actual MMM systems, we typically get an effective magnitude in the 1.5–3 range.

I will post files that have these summary statistics for a wide range of systems in case they may be of use to researchers or other interested readers. These are separate files for MMM, MMP, and two-tier PR (i.e, those that also use PR in their basic tiers), along with a codebook. (Links go to Dropbox (account not required); the first three files are .CSV and the codebook is .RTF.) [As noted at the top of this article, these files should no longer be used. At some point I will upload corrections. Sorry for the inconvenience.]

Added note: In the spreadsheets, the values of basic-tier seat product (MB) and tier ratio (t) are not election-specific, but are system averages. We used a definition of “system” that is based on how Lijphart (1994) defines criteria for a “change” in system. This is important only because it means the values may not exactly match what you would calculate from the raw values at a given election, if there have been small tweaks to magnitude or other variables during an otherwise steady-state “system”. These should make for only very minor differences and only for some countries.