“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:

NS=2.5t(MB)1/6,

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

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

We show in the book that the extended seat product is reasonably accurate for two-tier PR, including mixed-member proportional (MMP). We also show that the logic on which it is based checks out, in that the basic tier NS (i.e., before taking account of the upper tier) is well explained by (MB)1/6, while the multiplier term, 2.5t, captures on average how much the compensation mechanism increases NS. Perhaps most importantly of all, the extended seat product’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.

Iraq electoral system change

The Iraqi parliament has passed a new election law. That is interesting in itself, but what really prompted me to “plant” about it was this stunning line from the caption to the photo accompanying the Al Monitor article, saying that the new law would establish:

a first-past-the-post system to replace the complex mix of proportional representation and list voting.

I’ve often remarked in the past about how journalists who clearly do not get electoral systems just call any PR “complex.” But a “complex mix” of PR and list voting? That is a new one on me. The current system is not such a remarkable variety among the larger orchard of electoral systems–it’s a districted list-PR system in which lists are open and the governorates serve as electoral districts.

Moreover, the new system is not going to be FPTP. As I understand it from a couple of contacts, it will be single non-transferable vote (SNTV). In terms of how most electoral-system experts tend to think of these things, that would be a substantial retrogression, adopting what most specialists consider one of the worst of all systems.

In connection with the change, the number of districts will be increased. The consequence thus would be a lowering of mean district magnitude. At least the reformers got that part right; if you must use SNTV, use small districts. The article, however, is confusing as to how the number of districts is being determined (to be honest, it is confusing about almost everything).

The political blocs agreed Sept. 14 to divide each of the country’s governorates into a number of electoral constituencies that reflect the number of seats allocated for women in parliament under the Constitution, which is 25.

For example, the capital, Baghdad, which has about 71 seats, including 17 seats reserved for women, will turn into 17 electoral constituencies.

I guess this just means the existing number of women’s set-aside seats is being used and, presumably, one winner in each new district will need to be a woman. But I can’t say for sure if my interpretation is correct. As for the new mean magnitude will be, in Baghdad the numbers cited imply it will be just over 4 (=71/17). However, if the size of the parliament (329) is staying the same and there will be just 25 districts, that would imply an overall mean magnitude of 13. This can’t be right. Surely there will not be 17 districts in Baghdad and only 8 in the rest of the country. So, who knows!

The article also offers some overview of opposition from groups who fear–probably for good reason–that they will be more poorly represented under the new electoral system.

(Note: The caption refers to the parliament having passed the law on Dec. 24; however, the news story is dated Nov. 2, 2020.)

UPDATE: Apparently the average magnitude indeed will be around 4; the article apparently has the total number of districts wrong. Not 25 districts, but existing women’s representation target (on which districting will be based) of 25%. See comments. If the assembly size is staying constant, then the number of districts should be 329/4=82.

Some thoughts on Peru’s midterm election

After the Constitutional Tribunal ruled them legal, Peru held extraordinary legislative elections on 26 January. President Vizcarra dissolved Congress on the grounds that Congress had voted no-confidence in his cabinet (although not directly) twice. This was the first use of this provision since Peru’s 1992 constitution was promulgated, and as such it was the first time when legislative and presidential elections were not held concurrently.

However, the election did not merely lack a presidential contest. Almost uniquely, President Vizcarra, despite having been elected as part of former President Pedro Pablo Kuczynski’s party (previously Peruanos por el Kambio, now Contigo), chose not to endorse any party for the elections, merely advising voters to inform themselves. This reluctance was seemingly not due to any concern that Vizcarra’s endorsement would be a weakness for any party: at the time of the election, his approval rating stood at 58%.

Peru’s unicameral Congress is elected using open party-list proportional representation in 26 regions, with a 5% threshold applied at the nationwide level. The average district magnitude of 5 makes this a relatively moderate form of proportional representation, which explains why Keiko Fujimori’s Fuerza Popular was able to win a comfortable majority of 56% of the seats in Congress at the 2016 election despite only winning 36% of the vote.

The results of this election, however, were extraordinarily fragmented. The largest party, Accion Popular, got only 10% of the vote, and nine parties made it above the 5% threshold to enter Congress. More than a quarter of votes went to parties below the threshold, and in four provinces the leading party will receive no representation in Congress.

I will leave it to Peruvian experts, which I most certainly am not, to discuss what this result means for Vizcarra’s ability to pass his agenda. However, the results are interesting for other reasons.

Since the promulgation of the 1992 Constitution, Peru’s party system has remained quite stable (at least in terms of numbers, the identity of the parties has changed quite a lot). It has also remained quite close to the number of parties that the Seat Product Model (Shugart and Taagepera, 2017) would predict.

These elections are thus extremely unusual, and are perhaps indicative of the high importance of presidential elections and presidential endorsements in imposing structure on legislative elections in presidential countries. A fact particularly suggestive of this is the disastrous result for the two leading parties in 2016, both of which were affiliated to presidential candidates. Keiko Fujimori’s Fuerza Popular fell from 36% of the vote and 78 seats to 7% and 15 seats, while Peruvanos por el Kambio/Contigo fell from 16% and 18 seats to 1% and no seats.

Reminder from UK 2019 result: Electoral systems matter

Keep this in mind about the UK result. The Conservatives won less than 44% of the vote. Polling has consistently shown that if there were another referendum on Brexit, a majority would vote for Remain. But the Conservatives won 56% of the seats, so Johnson is banging on about his great “mandate” to “get Brexit done”.

You see, electoral systems matter.

Even if you add in the Brexit Party votes (which got no seats), the combined votes cast for parties still advocating outright for leaving the EU do not reach a majority. In fact, it barely breaks 45%.

Meanwhile, the SNP has won 81% of the Scottish seats, with 45% of the votes cast in Scotland. And their leader, Nicola Sturgeon, is going on and on about the mandate for Scotland to decide on independence. It’s a fishy claim.

Which party gained the most in votes, relative to the last general election? That would be the Liberal Democrats. But the party suffered a net loss of one seat (and its leader was defeated).

The first-past-the-post (FPTP) electoral system makes a country seem more divided than it is, and often leads to policy outcomes a majority of voters actually oppose.

FPTP certainly is not very representative. But it can produce a decisive government, and Boris Johnson now looks like he could take his place among the significant Prime Ministers in the country’s recent history.

At least this result means my old lectures about British majoritarianism do not to be heavily caveated as they’ve been for the past several years.

Election days and early voting

Sometimes an orchard is all laid out in neat rows, and then “volunteers” pop up in unexpected places. (For those not familiar with the gardening use of that term, a volunteer is a plant that grows where the gardener did not intend. It is a nicer term than “weed” and does not connote anything undesirable.) This happens in the virtual orchard from time to time.

Case in point: There is a fun and interesting discussion of election days and early voting sprouting and even thriving productively over in the row originally planned for Australian Senate reform. Many readers might find it interesting, so I am calling it out with this link to where it starts. By this point, if you have something to contribute, you might as well add it there, and so I will not have comments open on this guide to unexpected virtual-orchard plantings.

New Zealand to have referendum questions on 2020 ballot, potentially including “tweaks” to MMP

Earlier in December, the Justice Minister of New Zealand, Andrew Little (Labour) announced that there would be a binding referendum on recreational cannabis use concurrent with the 2020 general election. There may also be a question on euthanasia, and–of core interest to this blog–electoral reform.

Earlier, Little had said:

It has been floating around that if we’re going to do a bunch of referenda, why wouldn’t we put this question about whether we want to make those final tweaks to MMP, reduce that 5 per cent threshold to 4 per cent, get rid of the one-seat coat-tailing provision.

These proposals were part of the Electoral Commission’s MMP Review, but the government at the time (National-led) did not act on them.

The multiparty nature of the New Zealand political system that MMP has institutionalized is apparent in these issues being on the table. Having a referendum on cannabis use was a provision of the confidence and supply agreement that Labour signed with the Green Party after the 2017 election. In addition, Labour’s other current governing partner, New Zealand First, has indicated support for a bill on euthanasia sponsored by the leader of ACT, another of the smaller parties (a right-wing partner to opposition National).

Both provisions that the MMP Review recommended changing have had past impacts on current parties. The ACT has depended for its representation in parliament on the so-called coat-tailing provision (a term I do not like for the alternative threshold) in several elections. The New Zealand First once was left out of parliament for having a vote share between 3.5% and 5%, despite other parties (including ACT) being represented, due to winning a single district (electorate) plurality. (Obviously, 4% would not have helped NZF in 2008, as it had only 3.65%. But the point is that the current provisions produce potential anomalies; I have suggested before that the two thresholds should be brought closer to one another.)

Also of note: Little said that the cabinet had discussed, but decided against, having a citizen’s assembly to deliberate issues related to cannabis (and perhaps also euthanasia).

Add the OHES to your electoral-systems book collection!

Now you can have them both!

You already know about Votes from Seats, which was published by Cambridge University Press in October, 2017. Now (as of March, 2018), you can also buy The Oxford Handbook of Electoral Systems. The description:

No subject is more central to the study of politics than elections. All across the globe, elections are a focal point for citizens, the media, and politicians long before–and sometimes long after–they occur. Electoral systems, the rules about how voters’ preferences are translated into election results, profoundly shape the results not only of individual elections but also of many other important political outcomes, including party systems, candidate selection, and policy choices. Electoral systems have been a hot topic in established democracies from the UK and Italy to New Zealand and Japan. Even in the United States, events like the 2016 presidential election and court decisions such as Citizens United have sparked advocates to promote change in the Electoral College, redistricting, and campaign-finance rules. Elections and electoral systems have also intensified as a field of academic study, with groundbreaking work over the past decade sharpening our understanding of how electoral systems fundamentally shape the connections among citizens, government, and policy. This volume provides an in-depth exploration of the origins and effects of electoral systems.

You can find more information, including the table of contents, at the above link.

Italy 2018: Assessing the electoral-system effect

[Note: data calculations in this post are based on preliminary results. For some updated information, see the comments by Manuel below.]

The Italian election of 4 March produced an “inconclusive” result, as the media (at least English-language) are fond of saying when no party wins a majority. However, there are many aspects of the Italian result that are being reported with considerable confusion over how the electoral system works. In this post, I want to try to offer a corrective, based on the results published in La Repubblica.

These summaries will apply to the Chamber of Deputies only. The interested reader is invited to perform the equivalent calculations on the Senate and report them to the rest of us.

One common note of confusion I have seen in media accounts is insufficient clarity about the distinction between alliance (or “coalition”) and party. The design of the electoral system is fundamentally one that works on pre-election alliances, each consisting of one or more parties. Obviously, if an “alliance” consists of only one party, it is just that–a party. Rather than invent some encompassing term, I will use “alliance” when referring to the set of vote-earning entities (that would be a “more encompassing term”!) that includes pre-electoral coalitions, and “party” only when looking at the sub-alliance vote-earning entities. In the case of the Five State Movement (M5S), the “alliance” and “party” are the same thing. In the case of the other two main entities, they are different. Centrodestra (Center-right, or CDX) is a pre-electoral alliance consisting of the Lega, Forza Italia, and other parties. Centrosinistra (Center-left or CSX) is a pre-electoral alliance consisting of the Democrats (PD) and other parties.

No alliance has achieved a majority of seats. The M5S is the biggest party, while the CDX is the biggest alliance. As the table below shows, CDX leads with 263 seats, with M5S second on 222. The CSX has 118.

The breakdown is as follows, showing the three main alliances, plus a fourth one, Liberi e Uguale, which was the only other to clear the 3% threshold for individual parties or 10% for multiparty alliances:

Alliance % votes seats % seats
Centrodestra 37.0 263 42.5
M5S 32.7 222 35.9
Centrosinistra 22.8 118 19.1
Liberi e uguali 3.4 14 2.3
others 4.1 2 0.3

(There are two other seats indicated as being won by “Maie” [Associative Movement Italians Abroad] and “Usei” [South American Union Italian Emigrants]; no vote totals are given.)

The total comes to 619. Another summation from the same sources yields 620. I will not worry about the small discrepancy.

As an aside, I have seen at least two accounts of the result that have had phrasing referring to no party having won the 40% “required” to form a majority. There is no such requirement. It is true that no alliance or party attained 40% of the overall votes cast. However, the understanding that some authors (even one Italian political scientist writing on a UK blog) seem to have is that had someone cleared 40%, that alliance or party would have been assured of a majority of seats. That is incorrect. In fact, given the way the system is designed (more below), it is highly unlikely that an alliance with just over 40% could have won more than half the seats. Possible, but very unlikely (and we might say not significantly less likely had it won 39.99%). This “40%” idea floating around is just totally wrong.

The presentation of the overall result leads me to a second key point: the outcome is not terribly disproportional. However, it would be wrong to conclude from this observation that the electoral system was “proportional”. It is not designed to be such, and the disproportional elements of the design have significant consequences that I shall explain.

In terms of the Gallagher index of disproportionality (D), the result, based on alliances, yields D=5.40%. That is slightly greater than the median for my set of over 900 elections, and somewhat less than the mean of the same set (4.9 and 7.1, respectively). It is very slightly greater than the mean for PR systems (4.6; median 3.8).

Thus, based on the outcome measure of disproportionality, the Italian system looks like a moderately disproportional variant of PR. however, it is not a PR system! We do not ordinarily classify electoral systems based on their outputs, but on their rules. By that common standard, the Italian system is not PR, it is mixed-member majoritarian (MMM). It consists of two components–one that is nominal and the other than is list. The nominal component is plurality rule in single-seat districts, while the list component is nationwide PR (for alliances or parties that clear the threshold). Crucially the list seats are not allocated in compensatory fashion, but in parallel; this is the feature that makes it MMM, not MMP.

Unusually for MMM, but not disqualifying it from that category, the list-PR component is a good deal larger than the nominal (plurality) component. The nominal component is only around 35% of the total. However, the lack of compensation means that any alliance (or party) that can win pluralities in a substantial number of single-seat districts (SSDs) will be over-represented even after adding on all those list-PR seats. And such over-representation is precisely what happened.

If we look at the 398 list-PR seats and their allocation to parties (and here I do mean parties), we see a substantially more proportional output than overall. The Gallagher index is D=3.93%. This is, as reported above, right near the mean and median for pure PR systems. Just as we would expect! And most of the disproportionality comes from parties below the threshold, not from disparities among the over-threshold alliances. Around 4% of the vote was cast for alliances (or individual parties) that did not qualify for any seats. Some other votes are lost due to a provision that sub-alliance parties that get under 1% of the vote also have their votes wasted. If a party is between 1% and 3%, its votes are still credited to the alliance of which it is a part, even though such a party is barred from winning any seats in the list component.

Focusing on some of the major parties, we see that the major CDX partners were not much over-represented in the list component of the system: Lega has 17.4% of the vote and 73 seats (18.3%) for an advantage ratio (%seats/%votes) of A=1.05. Forza Italia has 14% of votes and 59 seats (14.8%) for A=1.06. The second largest alliance, the stand-alone party M5S has 32.7% of votes and 33.7% of seats for A=1.03. In the CSX, the PD is more over-represented, with 18.7% of the votes but 91 seats (22.9%), and A=1.22. I suppose this is because its partners mostly failed to qualify for seats, but the votes still get credited to the alliance (as explained above), and hence to the PD.

We see from these results that, with the partial exception of the PD, the parties are represented quite proportionally in the list-PR component of the MMM system. What gets us from D=3.93% in the list component to D=5.40% overall is precisely the fact that the nominal tier of SSDs exists and favored, as one would expect, the larger alliances. The following tables shows just how dramatic this was.

Nominal result
seats % seats % votes
Centrodestra 109 49.1 37.0
Centrosinistra 24 10.8 22.8
M5S 89 40.1 32.7
total 222 100.0

The vote percentages are the same as those shown in the first table, because there is no ticket-splitting between the two components. Each alliance presents a single candidate in each district, and the voter can vote for either a party list or an alliance candidate. Votes for a list are attributed to the candidate, and a vote for the candidate is proportionally divided among the lists in the alliance that nominated the candidate (with the previously noted caveat about parties whose national vote is in the 1-3% range).

The seats in the nominal component are distributed quite disproportionally: the largest alliance, CDX has nearly half of them, despite only 37% of the vote. The M5S is also over-represented, with about 40% of seats on just under a third of the votes. As is typical under SSDs with plurality, the third-place finisher, CSX, is significantly underrepresented, with a percentage of seats not even half its votes percentage.

Also as is typical, candidates often won their district seats on vote percentages in the low 40s or less. The mean district winner had 43.9% of the vote. For the M5S the mean was 45.4%, while for CDX it was 43.7%. As might be expected for a third force winning some seats, the CSX tended to benefit most of all from fragmented competition, with its mean winner having 39.2%. The lowest percentage for any SSD winner was 24.1% (M5S in Valle d’Aosta). Four winners had over 60%, including two from M5S and two from CSX; the maximum was 65% (CSX in Trentino-Alto Adige/Südtirol).

The media focus is on the “inconclusive” result, and many are blaming “PR” and the failure of any party (or alliance) to reach 40% of the votes for the lack of a “clear” verdict. However, we have seen here that the system is not proportional, even if the overall level of disproportionality is modest. If the entire system had been based on the allocation used in the list-PR component, we would be looking at CDX with 38.7% of seats, M5S with 33.7%, and CSX with 23.6%. However, given the actual MMM system, and its inherent disproportionality, the result is CDX 42.5%, M5S 35.9%, and CSX 19.1%. The non-PR aspect of the system thus has made a difference to the seat balance. The bargaining context would be difficult either way, but the two largest alliances are both boosted somewhat by features of the electoral system. Had the leader reached 40%, it would have netted only slightly more seats, surely still short of a majority, because–contrary to some claims circulating–there was no guarantee of a seat majority for reaching any given vote percentage. To form a majority of parliament, an alliance would have to win a very large percentage of the single-seat districts as well as some substantial percentage of the votes (probably a good deal higher than 40%). That the outcome is “inconclusive” says more about the divisions of the Italian electorate than it does about the supposed problems of a proportional system that Italy doesn’t actually have.


Thank you to Gianluca Passrrelli for sharing the link from which I based my calculations and for his excellent chapter in the forthcoming Oxford Handbook of Electoral Systems.

Chile 2017: First round

Chile has presidential and congressional elections 19 November. Unfortunately, an article at AS/COA does something that is far too common in media coverage of Latin American elections: It ignores the congressional elections.

That is especially unfortunate in this case, as this year’s elections in Chile are particularly interesting due to changes in the electoral systems for both houses of congress. (Details in a previous planting.)

The presidential election requires the leading candidate to obtain 50%+1 of the valid votes cast in Sunday’s first round. Otherwise, the top two advance to a runoff, which will take place on the 17th of December.This is the electoral system known as “two-round majority” or “majority runoff.”

As for the congressional electoral system, it remains open-list PR with D’Hondt divisors, as has been the case since the current democratic regime was established in the late 1980s. However, the seat product for the Chamber of Deputies has been increased moderately. Previously, it was 240 (120 assembly seats times 2 per district), which is a highly restrictive system. Now it will be 852.5 (155 seats times a new mean of 5.5 per district). That is only modestly proportional, but still a substantial increase. (For the central importance of the seat product, see Votes from Seats.)

The Senate seat product is also being increased, but only half that chamber is elected at a time, so the new system will not be fully implemented till four years hence.

The new systems (both houses) will create more political space both for minor parties and alliances that currently have few or no seats, and for the representation of more of the member parties in the alliances that already are a hallmark of the Chilean party system’s adaptation to the more restrictive system that has been in place. In the sense of being a system of open alliance lists, it is essentially the same allocation formula as in Finland and Brazil. The crucial difference is district magnitude–formerly two (the second lowest possible!) and now to be increased, although still well short of what those other two countries have–and, in comparison to Brazil, with a much smaller assembly size. [Click here for an important correction on the intra-list allocation.]

As shown in a table of polling trends for the presidential election (first link), there is more of a contest for second place and thus entry into the runoff than there is for first place. Former president Sebastián Piñera is leading but not likely to clear 50% of the valid vote. Two leftist candidates are vying to face him in the expected runoff.

It might not seem obvious, but the congressional electoral-system changes could be influencing presidential competition. In fact, that is one of the findings of Votes from Seats: We can predict the average trend in the “effective” number of presidential candidates from the assembly seat product. (This is in contrast to conventional “coattails” arguments that claim we can understand assembly-election fragmentation only by knowing how many viable presidential candidates there are.)

In the past in Chile, there was strong pressure for parties to coalesce in order to be viable participants in the highly restrictive congressional electoral system. While parties in a common alliance for assembly seats could run separate presidential candidates–see the 2005 case of unusual alliance behavior on the right–usually they would not. (And the 2005 case did not work out that well for the right, at least in the Chamber.)

Now, the pressure to join forces for assembly elections is reduced, which should be expected to push up the number of viable contenders for presidential-runoff slots as well. The candidates vying for that second slot are Beatriz Sánchez, backed by an alliance called the Broad Front (Humanist Party, Liberal Party, Green Ecologists, and others), and Alejandro Guiller, backed by Fuerza Mayoría (including the Socialist Party of the outgoing incumbent, Michelle Bachelet, as well as the Communists, Democrats, and others). Which one will make it, and how will it affect the left’s combined chances of blocking a victory for Piñera in the runoff? And how will the candidates help (or not) their alliances’ electoral process in the new congressional election?

More parliamentarism in Central Asia

The Venice Commission has published an generally positive opinion on the Georgian government’s proposal for constitutional reforms. The reforms were proposed after the governing Georgian Dream party won 115 seats in the 150 member legislature in elections, slightly more than the three-quarters majority required to amend the document.

Specifically, the amendments propose repealing direct elections to the Presidency, replacing it with election by a 300-member electoral college composed of members of the national legislature and local councillors. In addition, most of the powers of the Presidency are stripped. This creates a parliamentary system, with a Prime Minister only removable through a constructive vote of no confidence.

The previously unicameral legislature will be replaced, nominally, with a bicameral legislature, comprised of a Senate and a Chamber of Deputies. However, the Senate specifically includes members elected from Abkhazia, currently under the control of a separatist government, and is only to be created after “appropriate conditions have been created throughout the territory of Georgia”. This would seem to imply that the chamber can only be created when Abkhazia returns to government control, and the Venice Commission’s report confirms that they understand its creation will be delayed.

In addition, there are changes to the electoral law. The existing mixed-member majoritarian system with a roughly even split between single-member constituencies elected using the two-round system and party-list PR with a 5% threshold will be replaced with a system of list PR only, still with a 5% threshold. While there is little elaboration, the document does specify that seats shall be allocated by the Hare quota, but instead of allocating seats by largest remainders, all remaining seats are allocated to the largest party (a method used in Greece in one of their endless electoral system changes).

The change bears some resemblance to the relatively recent amendments in Armenia. Like Georgia, a semi-presidential system with a legislature elected with a mixed-member system transitioned into a parliamentary one with a legislature elected under a list system with a bonus (though Armenia’s bonus is somewhat more elaborate, and guarantees a majority government in one form or another). While drawing broad conclusions off two examples is obviously bound to be, these two results may suggest that there is a shift away from politics centred around an all-powerful directly elected presidency, and towards more party-based politics.

A more tenuous argument along these lines could be made in relation to the electoral system. In both cases (along with Kyrgyzstan, which actually moved from single-member districts to MMM to party list), a system in which individual candidates were an important part of legislative elections (especially in the years shortly after independence) has been replaced by a system in which parties are the dominant actors. On the other hand, the pendulum has moved the other way elsewhere in the region, in Russia and the Ukraine.

The President, though endorsed by the Georgian Dream party at the 2013 election, does not appear to have been overly enthusiastic about the landslide victory. The Venice Commission did express some concerns about the power of a government with an overwhelming parliamentary majority, but that seems less likely in Georgia than in Armenia, owing to the more proportional system.

UK election 2017-any hope for electoral reform?

One of the most remarkable things about the UK election was the total number of votes cast for the top two parties, which increased from 67.3% in 2015 to 82.8% this year. It has been less noticed, however, that this has come with a substantial drop in the Gallagher (least squares) Index of disproportionality. From 15.02 in 2015, it has more than halved to 6.41, a figure actually larger than that in Ireland in 2011, Germany in 2013, or Poland in 2015.

UK_Gallagher_1966-2017

Source: Michael Gallagher

This is largely due to the absence of a large third party discriminated against by the single-seat plurality system. No established minor party outside Northern Ireland gained votes, with UKIP and the Greens both falling to below 2% and even the Liberal Democrats slipping back slightly in terms of votes despite picking up four seats. The (unusual, as Shugart points out) absence of an unearned majority for either party also contributed to the low score.

There also seems to be some evidence from past UK election results that two-party elections tend to be more proportional. Between the 1950 and 1970 elections, when support for the Liberals (the only substantial third party) never exceeded 11.2% (and was generally substantially below this), the Gallagher Index averaged only 6.41; since then, with the until-recent presence of the SDP-Liberal Alliance/Liberal Democrats, it has stayed high as can be seen above.

UK_margins_2015-17

The number of very marginal seats has also increased substantially at this election, as can be seen above. While other readers may disagree with me on this, I personally view a higher number of marginals as good for the status quo, given that it means more voters view their votes as having an impact upon the result. This is not necessarily a result that would appeal to the major parties, but it would seem to act to quell public concern over the current system.

The growth of the top two creates a problem for electoral reform advocates. Which party, exactly, benefits from implementing proportional representation? The Labour and Conservative parties obviously have no personal interest in PR. The Liberal Democrats, at present, are far too weak to extract such a concession (even with 57 seats in 2010, the best they could do was a referendum on AV) in a coalition, as are the Great British regionalist parties (which would probably be heavily expected to go with Labour anyway, weakening their leverage) and the Greens (much the same reason).

Labour’s manifesto makes no specific mention of proportional representation, while the Conservative manifesto goes further, calling for single-seat plurality to be adopted for mayoral and Police and Crime Commissioner elections. Shadow Chancellor John McDonnell has gone on record in the past as being in favour of proportional representation, but I suspect now Labour are at 40% that proposal will be soon forgotten.

While the past two months has demonstrated conclusively that much can change in a short time in UK politics, it’s difficult to imagine that proportional representation could gain any traction in a political system dominated by the top two parties after failing to gain traction (at least for the House of Commons) in nearly thirty years of three-party politics.

Remember: Honeymoon elections matter

On 23 April, when many commentators were lamenting how weak (then-expected) President Emmannuel Macron’s support might be in the National Assembly, I offered an estimate of 29% of the vote for his newly formed party. I based this solely on the mean surge that presidents’ parties tend to have when an assembly election occurs early in their terms–a honeymoon election.

Maybe that was an underestimate. While one poll (OpinionWay/ORPI) has Macron’s party, La République en marche! (LRM), on 27%, Harris Interactive sees it on 32%. Both agree this will be the biggest party (Reuters). Given the electoral system, such a share puts Macron well within reach of having a majority in the Assembly.

And what a party it is!

Half of the LRM preliminary list of 428 candidates for the 577-member National Assembly are women and 52 percent are civil society figures.

Better yet, 95% are not current MPs and one of them is a “rockstar mathematician”! (France24)

Macron has also named his cabinet. The premier will be Edouard Philippe, mayor of Le Havre and a member of the Les Republicans (the party of defeated and discredited presidential candidate François Fillon). Reuters reports:

A leading French conservative accused President Emmanuel Macron of “dynamiting” the political landscape on Tuesday as he put together a government that is expected to include former rivals on both left and right.

In other words, he is being “accused” of doing precisely what he won nearly two thirds of the vote (in the runoff) saying he would do.

Furthermore,

over 20 LR members of parliament, including some party heavyweights and former ministers, issued a joint statement on Monday urging the party to positively respond to the “hand extended by the president”.

All of the above should serve as a reminder of two things: (1) the purpose of the upcoming election is to ratify the new executive’s direction, not to be a second chance for an alternative vision; (2) the honeymoon electoral cycle matters.

What if Canada adopts MMP?

Among the electoral system types to be considered by Canada’s upcoming reform-proposal process is the mixed-member proportional system (MMP). What might we expect Canada’s effective number of seat-winning parties (Ns) to be under MMP?

As noted in the previous post on comparing the Alternative Vote to FPTP, Canada has had almost exactly the Ns, on average, that we should expect it to have, given its assembly size (around 2.6). Thus I will start from the premise here that Canada would continue to have about what we expect if it had MMP staring in 2019 (or whenever). That is, recent elections the country are neither surprisingly under- or over-fragmented, so there is no reason to think the country would over-shoot the expectation under a new system. (It might be more realistic that it would under-shoot, but that depends on how much we believe there is pent-up demand for new parties or for growth of existing smaller ones. I will leave that aside here.)

The answer to this “what if” depends on the precise MMP model. What Li and Shugart (2016) have shown is that a minor addition to the Seat Product Model (of Taagepera, 2007) captures two-tier compensatory systems well. MMP is a type of two-tier compensatory system, so let’s apply that model to a hypothetical Canadian MMP. The formula is:

Ns=2.5t(MSB)1/6.

In this formula, MSB is the “basic-tier seat product”, defined as the mean magnitude of the basic tier, times the total number of seats in that tier. For a typical MMP system, we retain M=1 in the basic tier, so the basic-tier seat product is simply the number of seats elected in that tier. The t in the formula stands for “tier ratio”, which is the share of the total assembly that is elected in the compensatory tier. This is an exponent on a constant term that is empirically determined to be 2.5. (Determined from the broader set of cases on which this is tested.*)

If a country’s electoral system is “simple”, meaning there is just a single tier, as with FPTP, then the above formula reduces to Taagepera’s (2007) seat product model:

Ns=(MS)1/6.

(In a system with no upper tier, t=0, and MSB=MS.)

For purposes of estimating, I am going to assume the assembly size will remain the same, currently S=338. I will further assume that it would be politically difficult to reduce the number of districts (ridings) in the basic tier of an MMP system too much below the current number (which is, of course, 338). I will adopt two thirds of the current number as my estimate of “not too much”. In such a scenario, we get a basic tier of 225 seats, giving us t=.33:

Ns=2.5.33(225)1/6=1.35(2.47)=3.33.

I did not plan my scenario to get to three-and-a-third, but it has a nice ring to it. And seems pretty reasonable. For comparison purposes, this is not much different form what Canada had in 2006 (3.22) or Germany, an actual MMP system, had in 1998 (3.31).

Based on another formula in Taagepera (2007), which is empirically very accurate, we can also derive an expectation for the seat share of the largest party (s1):

s1 = Ns-.75.

For our hypothetical MMP system in Canada, this implies the largest party with just over 40% of the seats.

We can tinker with the scenarios. For instance, suppose the assembly size were increased to 400, with half the seats in the basic tier. Then we get:

Ns=2.5.5(200)1/6=1.58(2.42)=3.82.

As would be expected intuitively, the fragmentation of the House goes up due to the larger compensation tier, and in spite of the basic-tier seat product being correspondingly lower. This scenario would have an expected s1=.366.

Note that I have ignored thresholds here. My ongoing research with Taagepera suggests that thresholds matter, but unless the threshold is very high (more than 5%) or the seat product, MS, is extremely high, the value of the threshold has much less impact than the parameters discussed here.

In conclusion, under common Mixed-Member Proportional (MMP) designs, Canada could expect its House to have an effective number of parties ranging from 3.33 to 3.8, compared to a recent average of around 2.6 under FPTP. Its largest party could be expected to have around 36% to 40% of the seats, on average, compared to majorities or nearly so in recent elections.


* Note that this parameter’s empirical derivation raises the risk that it could be an artifact of the particular sample we have, and thus not reliable for determining an expectation value, as I am doing here. Fortunately, unless it is off by a wide amount, it does not make a large difference. For instance, 2.33 =1.26, whereas 3.33 =1.44. This variance in our prediction is much less than “normal” fluctuation from election to election in many countries.

Does AV mean higher or lower effective number of parties?

There may be a conventional wisdom among people who study comparative electoral systems that the Alternative Vote (also known as Instant Runoff or Majority Preferential) tends to suppress the effective number of parties, compared to plurality (First Past the Post, or FPTP). Or maybe it is just me, but I will admit to having such a notion. After all, Australia is a pretty strict two-party system, isn’t it?

The correct way to approach the question of whether AV means a higher or lower effective number of parties (N) than FPTP is to ask: What we should expect N to be, given the country’s seat product?

As explained by Taagepera (2007) and further elaborated and tested by Li and Shugart (2016), the seat product is a country’s mean district magnitude (M), times its assembly size (S). The Seat Product Model says that the effective number of seat-winning parties (Ns) tends to be the sixth root of this product: Ns=(MS)1/6.

The model is logical, not a mere product of empirical regression work, although regression tests confirm it almost precisely (Li and Shugart, 2016).

When all districts elect just one member, thus M=1, the Seat Product is just the assembly size, S. Hence we take the sixth root of S to get an expectation for Ns. What if we do this for Australia’s House of Representatives? We get an expectation of 2.31.

The actual Ns for Australia’s elections since 1984, the year S was increased from 125 to 148 (subsequently it has increased to 150, a minor change) is… 2.53. However, I believe that figure (I am using Gallagher’s Election Indices) treats the Coalition parties as one in elections before 2010.

In the two most recent elections, Ns has been 2.92 and 3.23. The notes to Gallagher’s Election Indices indicate that for these elections the Liberal Party, the Nationals, and the Liberal National Party of Queensland are treated as separate parties. In my opinion they should be so treated, although I suppose one could have a debate about that.

The actual mean is thus above the expectation for a hypothetical FPTP of the same size assembly. If we use the figure of 2.53, it is obviously not much higher than 2.31 (the ratio is 1.10). However, if we consider the value, at least in recent elections, to be around 3.0, it is about 1.30 times the expectation value.

Contrast this with the UK, where elections of the same period (1987-2010) have a mean Ns=2.30. This is just what we expect for FPTP, right? Not much over 2.0. Not so fast! The UK has a huge assembly, and with S=650 (aprpox., as it varies over the period), we should expect Ns=2.94. The UK actually has one of the more under-fragmented assemblies, according to the Seat Product Model, with this recent-period average being only 78% of expectation.

So how about Canada, where AV is one of the potential reforms being considered? Over a similar period (1984-2011) we get Ns=2.63. With S around 300 during this time, we should get Ns=2.59. So Canada pretty much nails the expectation of the model.

So, should we expect Ns to go down if Canada were to adopt AV, as (what I characterized as) the conventional wisdom would have it? Or should we expect it to go up?

I would not be inclined to say ‘down’. I will just leave it at that for now.

‘Seat Product Model’–audio version

The audio-slides version of Li and Shugart (2016) is now available!

As previously announced, the publication details and abstract are as follows:

The Seat Product Model of the effective number of parties: A case for applied political science

Yuhui Li, Matthew S. Shugart

Electoral Studies 41, March 2016, pp. 23–34.

Abstract

This paper extends Taagepera’s (2007) Seat Product Model and shows that the effective number of seat-wining parties and vote winning parties can both be predicted with institutional variables alone, namely district magnitude, assembly size, and upper-tier seat share. The expected coefficients are remarkably stable across different samples. Including the further information of ethnic diversity in the models hardly improves the estimate of the effective number of parties, and thus the institutions-only models are preferable on the grounds of parsimony and the applicability to electoral-system design or “engineering”.