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

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 (NS), the extended SPM formula (Shugart and Taagepera, 2017: 263) is:


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’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 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 much NS 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.)

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.

MMP in NZ: An example of “best of both worlds” in action

In Shugart and Wattenberg (2001) we ask if mixed-member systems offer a “best of both worlds.” That is, do they allow simultaneously for the benefits of local representation and individual-member accountability that are the (supposed) advantages of single-seat plurality (FPTP) and the representation of smaller national parties that might struggle to win districts but would be represented under proportional representation (PR).

There was a question mark in the book’s subtitle. Over time, I have come to believe that indeed the proportional type (MMP) does have a strong tendency to offer the best of both worlds. The reason is that members elected in districts have incentives to behave as local representatives at the time that there is close approximation between party vote and seat shares (assuming compensation is carried out nationwide or in large regions). The majoritarian type (MMM, as in Japan and Taiwan) probably does not; it is much closer in its overall incentive structure to FPTP, even though it does indeed permit smaller national parties to win seats.

For MMP, the “best of both worlds” argument assumes that parties nominate dually–meaning many elected members will have run in a district and had a (realistically electable) list position simultaneously. If they do, then even the list-elected members will have a local base, and should have incentives to act as the local “face” of the party, including possibly by offering constituent services. Both prior anecdotes I have shared from New Zealand (e.g., “shadow MPs” who win from the list and maintain a local office) and my forthcoming coauthored book, Party Personnel, offer further evidence that MMP does indeed work in this way.

Now comes a terrific anecdote from New Zealand’s 2020 election. In this election, Labour won a majority of seats (64/120) with 49.1% of the nationwide party list vote. In the nominal tier of single-seat districts (electorates) it won 43 of the 72 available seats. Its win included some districts that are normally strongholds of the center-right National Party (which won 35 seats overall and just 26 districts).

Commenting on some of the Labour wins in mostly rural districts, Federated Farmers president Andrew Hoggard said:

in some “flipped” electorates Labour list MPs had worked hard to raise their profile and get involved with the community and this had paid off when they campaigned for the electorate.

This is an ideal description of how the “best of both worlds” argument works: list-elected members have incentives to attend to local needs of the district in which they ran for the nominal seat (but “lost”) in hopes of capturing the local plurality in the next election.

Of course, there were other factors at work as well. I will offer another planting about one of those factors separately. There is also some uncertainty at this stage just exactly the degree to which rural voters flipped, as the wins may have come in significant part from very large swings in the town areas within districts that also include large rural areas. Regardless, MMP offers the key advantage of giving most elected members, if dually nominated, a tie to a local constituency while ensuring close approximation of overall seat totals to party-list votes.

South Korea 2020

South Korea had its assembly election on 15 April, with various covid-19 precautions in place. The Democratic Party of President Moon Jae-in (elected in 2017) won a majority of seats.

As discussed previously at F&V, the electoral system was changed from mixed-member majoritarian (MMM) to, at least partially, mixed-member proportional (MMP) prior to this election. It is only partially MMP not mainly because the number of compensatory list seats is so small (30 out of 300 total), but because there remain 17 seats that are, apparently, allocated in parallel (i.e., as if it were MMM).

There was some discussion in various media accounts (and in the previous thread) of the major parties setting up “satellite” parties to “game” the MMP aspect of the system. Under such a situation, a big party will contest the nominal tier seats and use a separate list to attract list votes and seats. By not linking its victorious nominal candidates with a same-party list, a party can gain extra seats, vitiating the compensation mechanism that defines MMP. This is what happened in Lesotho in 2007, for example. (That thread has an interesting series of comments about the issue, including why German parties do not do this in their MMP system.)

The Democratic Party set up a Together Citizens Party to compete for list seats and the main opposition United Future Party set up a Future Korea Party to do the same.

However, if I understand the results correctly (at Wiki), it seems the satellite was not necessary for the Democratic Party to win its seat majority. The Democrats won 163 constituency seats on 49.9% of the (nominal) vote; with 300 total seats, this is a majority no matter what happens with the list seats. Their satellite won 17 seats on 33.4% of the list votes. The United Future won 84 nominal seats on 41.5% of the nominal vote; their satellite won 19 seats on 33.8% of the list votes. I am finding these numbers hard to understand! Maybe someone else can figure this out for us.

South Korea moving to MMP?

South Korea’s National Assembly appears close to passing an electoral reform bill. It seems that it would change the existing mixed-member majoritarian (MMM) system to mixed-member proportional (MMP).

I always take media reports about important details of electoral systems with caution, but it seems the list seats will be made compensatory: “Under MMP, parliamentary seats are tied to the percentage of voters’ support for political parties.”

The current system (as of 2016) has 47 non-compensatory list seats, in a 300-member assembly.

However, there is a catch. The article says, “The number of PR posts to be allocated under the MMP representation scheme will be capped at 30.” Yet there are to remain 47 list seats; how are the other 17 allocated? To the largest party, or based on vote shares without taking district wins into account (as under MMM)? I wish it were clear, as such details would make quite a difference.

Regardless, proportionality will be quite limited.

An earlier provision of the reform bill that would have provided for 75 list seats was turned down.

Maybe we can call the new system MMp. Maybe.

Thanks to FairVote Vancouver and Kharis Templeman for the tip.

Lesotho (MMP) & Malta (STV) hold early elections on the same day

Lesotho and Malta will hold early elections this Saturday, June 3rd. Both have parliamentary systems and each one uses a different (and interesting!) type of proportional representation – each having a certain following among readers of this blog.

Lesotho uses a one-vote variant of MMP, with 80 single-seat districts in the nominal tier and 40 in the list tier. There is no threshold, and no seats are added in case of overhang, so a party can win a majority by taking more than 60 districts.

Malta uses STV, with a twist: if I understand correctly, in case one party receives an absolute majority of first-preference votes, seats are added to ensure that party has a majority, and that the majority is in proportion to its majority of the vote.

The elections were also called in different ways. Lesotho’s parliament (election not required before February 2020) was dissolved after the government lost a confidence vote in March – the prime minister could have handed over power to the coalition that ousted him, but chose instead to ask the king for an early election. Malta’s early election (originally not due until March 2018) was called by the prime minister.

New Zealand split-vote results released

The New Zealand Electoral Commission has released the split-voting statistics from the 2014 general election.

This is a great service provided by the Electoral Commission, showing in each electorate (district) what percentage of voters for each party list cast their vote for that party’s candidate or any other candidate in the electorate. To make it even better for those who like analyzing voting statistics, they offer CSV files.

The NZ Herald offers a summary of key electorates.

Piggyback MPs, part 2

With apologies to New Zealanders’ somewhat complicated memory of Robert Muldoon, I am sticking to my “piggyback MPs” as a preferred term for members elected under MMP via an alternative threshold to the one based on party-list votes.

Here I want to address briefly the question of whether allowing an alternative threshold, by which a party qualifies for list seats through the winning of one (or more) district seats, is itself a problem in electoral-system design. I have been wanting to address this issue for some time, and some of my thoughts are anticipated by a comment left by Rob at the previous thread.

Up front, let me state that I see no problem with the principle of an alternative threshold. If mixed-member proportional systems are to have a chance of delivering on the “best of both worlds” promise, then one really should allow both worlds to coexist simultaneously. One of those worlds is one in which local concentrations of support for particular parties or candidates are able to attain representation. The other world is one in which only nationwide levels of support for particular parties are worthy of representation. Any one of us might prefer one conception of representation over the other, but MMP is explicitly designed to promote both.

Now, one might respond that one need not have the alternative threshold in oder to obtain both of these worlds. Parties could still exist to target one or a few district seats, and earn their representation that way, without being entitled to any list seats.* I concede that this is a perfectly valid argument, and it seems to be the position taken by the New Zealand Electoral Commission in its MMP Review. That is fine; they have thought much more about these issues, and the needs of New Zealand society, than I ever can do.

However, I think it is a perfectly valid “best of both worlds” provision to say that we want to give incentives to smaller parties to attract support outside their district-based strongholds, while still being able to win representation based on their regional concentration. A very small party may have supporters around the country, but be concentrated in one area. Voters outside the areas of strength have little reason to vote for the list of such a party if it won’t win seats; by the same token, voters in a single district where the party has local strength may have little reason to vote for the party if it lacks any chance to win further seats via list votes obtained elsewhere. (If one seat is expected to affect the balance of power, the second consideration vanishes, of course.)

It seems to me that the decision whether to abolish the alternative threshold should be made not on the basis of disliking particular parties that take advantage of it. (Search on “Key cup of tea” if you are unfamiliar with the debate.) Rather, it should be taken after considering what minimal size of party is considered optimal in a given country’s proportional system. (One can never squeeze out all one-seat parties, as they are at least a latent possibility in any system that has single-seat districts, including mixed-member PR, but one can eliminated the opportunity for such parties to exist to seek additional seats via the list.)

What is the optimal minimum size for parties that win more than just a given district (or two or more), but also win list seats? Continue reading

MMP and dual candidacy in Wales

The question of dual candidacy in the Welsh Assembly mixed-member proportional (MMP)* system is being debated again. “Dual candidacy” refers to a provision permitting candidates to run simultaneously in a nominal (district) race and on their party’s list for the proportional component of the system.

Roger Scully offers an overview of the history and debate.

Wales permitted dual candidacy in 1999 and 2003 and banned it for elections of 2007 and 2011. Now a bill is in the House of Commons (yes, this decision is taken in London) to ban dual candidacy again.

Scully mentions various other reforms that have been debated, including an increase in the size of the assembly, from 60 to 80 or 100. As he notes, such an increase would have an impact on the proportionality of the system (independent of dual candidacy).

For instance: the easiest way to change from 60 to 80 AMs would be to raise the number of list AMs in each region (from 4 to 8). But with list AMs now comprising half of the Assembly’s membership, rather than one-third, the proportionality of the electoral system would be changed substantially. An 80-seat Assembly where 40 members came each from the constituency and list ballots would be more-or-less a fully proportional system, rather than the semi-proportional system we have at present.

This is an important point. Because the compensation in the Welsh MMP is carried out in regions instead of Wales-wide, and because the number of seats per region is relatively low, the proportionality is indeed modest. Michael Gallagher‘s Election Indices shows values on the Least Squares (Gallagher) index of disproportionality in the four elections of 8.61, 10.39, 11.36, and 10.47. By contrast, New Zealand, with nationwide proportionality in its MMP system and a 5% threshold, has had index values ranging from 1.13 to 3.84. The UK, with only single-seat districts, has averaged 16.53 on the index over the elections of the same period.

Alternatively, the number of constituencies for the nominal tier could be increased. To keep the same ratio between tiers as is current practice would require 53 constituencies, which “would require the drawing of new constituency boundaries, and losing ‘co-terminosity’ between Westminster and Assembly constituencies.”**

Scully’s preference is for STV, which would resolve the dual candidacy question by reverting to a single tier, while keeping the level of proportionality about the same (potentially). A commission proposed STV a decade ago. Scully notes that there have been two main proposals: grouping the current 40 constituencies into 20 pairs that each elect 4 assembly members or using local authority boundaries as districts (which, I assume, would mean district magnitude varying by municipality population).

As for the dual-candidacy issue, many readers of this blog will know my position. Dual candidacy is an essential feature of mixed-member systems, especially MMP systems, without which many of the main benefits of the system are unrealized. Sure, it does not affect proportionality, but the system also delivers benefits on the intra-party dimension, by encouraging more constituency focus of members elected from party lists than would be the case under pure PR.*** This benefit is likely lost if parties refrain from nominating their best personnel in districts where they are unsure of victory and instead nominate them only on the list. Thus the “legitimacy” problem of list members that underlies the charge against dual candidacy (“entering through the back door,” “zombies”, etc.) is actually made worse by eliminating dual candidacy and thus severing the constituency link of list candidates. The MMP Review in New Zealand extensively commented on this issue and came clearly down in favor of retaining the right to dual candidacy. Wales should do the same–if it retains MMP.

* In Wales it is called the Additional Member System (AMS). I very much dislike this name, as it treats the list-elected members as mere add-ons, rather than an integral (in fact, the decisive) component of the system. In fact, the name would fit better for the other main category, MMM (mixed-member majoritarian).

** I think “co-terminosity” is a new word for me. I like it.

*** And also without the direct intra-party competition of STV or OLPR, or the partisan incentive for “vote management” and “friends and family” voting/clientelism concerns that STV is especially prone to.