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”. 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’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 *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 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 much *N*_{S} 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.)

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.)

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.