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.

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 (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_{S} 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:

N_{S} = 2.5^{t}(MS_{B})^{1/6},

where N_{S} is the effective number of seat-winning parties (here, meaning the expectedN_{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 expectedN_{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’sDemocratic 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_{S} 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:

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.

[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 expectedN_{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 Studiesarticle 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_{S} (rev.)

N_{S} (Eq. 15.2)

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_{B} much greater than around 300 is not likely to be very common. My examples of MS_{B} =2,500 are motivated by the notion of S_{B}=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 N_{S} = –0.066 + 0.166log MS_{B} + 0.399t .

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:

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}_{B}. A fundamental law of compensation systems is that s_{1} ≤ s_{1}_{B}. (and N_{S} ≥ N_{SB}); 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 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}_{B}. If, on the other hand, it gets none of the upper-tier seats, it ends up with 20 of 100 seats, for s_{1}=0.2 = (1–t)*s_{1}_{B}. 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 s_{1}_{B} = 0.4. If compensation nets it 8 of the 20 compensation seats (t=0.2), it emerges with 40 of 100 seats, for s_{1}=0.4 = 1*s_{1}_{B}. If, on the other hand, it gets none of the upper-tier seats, it ends up with 32 of 100 seats, for s_{1}=0.32 = (1–0.2)*s_{1}_{B} = 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:

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_{S} than that of s_{1} is telling us that these systems have a different relationship of N_{S} to 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_{S} 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 s_{1} to N_{S} 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 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}_{B} 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 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:

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_{S} =1; anything else is absurd). If we suppress the constant, we get:

log N_{S} = 0.152log MS_{B} – 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 s_{1} and N_{S} in these two-tier systems. Just what remains to be seen.

[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 (N_{S}), the extended SPM formula (Shugart and Taagepera, 2017: 263) is:

N_{S}=2.5^{t}(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 MS_{B} 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 N_{S} (i.e., before taking account of the upper tier) is well explained by (MB)^{1/6}, while the multiplier term, 2.5^{t}, captures on average how much the compensation mechanism increases N_{S}. Perhaps most importantly of all, the extended seat product model’s prediction is closer to actually observed nationwide N_{S}, on average, than would be an estimate of N_{S} 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, N_{S}, and raise it to the power, 6. That is, if N_{S}=(MS)^{1/6}, it must be that MS=N_{S}^{6}. (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 effectiveM).

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 N_{S} 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 (E_{eff}) that is E_{eff} = E_{B}+tS, where E_{B} 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 N_{S}=3.65, from which we would derive an effective seat product, (MS)_{eff}=3.66^{6} =2,400. But it is MMM. So let’s calculate an effective FPTP seat product. E_{eff} = E_{B}+tS=300+180=480 (from which we would expect N_{S}=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 N_{S}=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 (N_{S}=3.17), but several other elections have had N_{S} 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 N_{S} 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, M_{eff}=(MS)_{eff}/S. For Japan, this would give us M_{eff}=2.25; for Korea M_{eff}=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.