# 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, NS = s1–4/3, applies equally well to two-tier systems as it does to simple, single-tier systems. This was a question I raised in the earlier planting on the revision of the extended Seat Product Model (incorporating two-tier systems without an empirical constant).

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

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

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

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

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

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

NS = 2.5t(MSB)1/6,

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

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

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

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

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

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

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

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

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

NS = (1–t)–2/3(MSB)1/6.

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

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

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

Notes

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

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

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

Further note

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

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

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.

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.