MMP as sub-category of two-tier PR–some basis for doubt

In yesterday’s review of the German election outcome, I used the extended Seat Product Model (SPM) formula for two-tier PR systems. I have done this many times, and Rein Taagepera and I (in our 2017 book, Votes from Seats) do explicitly include mixed-member proportional (MMP) in the category of two-tier PR systems.

However, there is one problem with that characterization. All other two-tier PR systems that I can think of entail a single vote, which is then used both for allocating seats in the basic tier and pooled across districts for national (or sometimes regional) compensation.

MMP, of course, usually entails two votes–a nominal (candidate) vote used only in the basic tier, and a second, party-list, vote used for determining overall proportionality. (In MMP, the basic tier is a “nominal tier” because the vote there is cast for a candidate, and the district winner earns the seat solely on votes cast for him or her by name.) This two-vote feature is a complex feature of MMP that is actually emphasized in my more recent coauthored book, Party Personnel Strategies, but which I may have tended to underplay in my comparative work on modeling the effects of electoral systems on party systems. Of course, by being two-tier, it is already a non-simple system, as Taagepera and I define that term. But we also say that two-tier PR, including MMP, is as simple as an electoral system can be and still be included in the complex category (see p. 263 and 299 of Votes from Seats).

Maybe that is not an accurate statement for two-vote MMP. Our definition of simple (pp. 31-36) concentrates on two features: (1) all seats allocated within districts, and (2) adherence to the rank-size principle, such that the largest party gets the first seat in a district, and remaining seats are allocated in a way that respects their relative sizes (i.e., by any of the common PR formulas). We further say that for simple PR, “the vote for candidate and for party is one act” (p. 35). This latter condition still holds for any two-tier list-PR system, because there is a list vote that applies both for allocating seats within a district, and also for the “complex” feature of the supra-district compensation mechanism. Obviously, however, MMP as used in Germany violates the principle that “the vote for candidate and for party is one act.” So maybe it is not “simple enough” to qualify as an almost-simple complex system. (Yes, that was a complex statement, but that’s kind of the point.)

If MMP were to tend to produce a party system more fragmented than expected from the extended SPM, it might be due to the “second” vote, i.e., the list vote. To test this, one could aggregate all the nominal votes and use them as the notional list votes in a simulated compensation. (This is how MMP in Germany worked in 1949, albeit with compensation only at state level. It is also how MMP now works in Lesotho.) The aggregation of basic-tier votes should work better from the standpoint of modeling the party system impact of the key features of a given MMP system–the size of the basic tier and the share of seats in the compensation tier.

The catch in all this is that, of course, till quite recently German MMP was under-fragmented, according to the SPM, despite using a separate list vote. Thus the issue did not arise. The New Zealand MMP system also has matched expectations well, after the first three post-reform elections were over-fragmented relative to model prediction. The graph below shows the relationship over time between the expectations of the SPM and the observed values of effective number of seat-winning parties (NS) in both Germany and New Zealand. For the latter country, it includes the pre-reform FPTP system. In the case of Germany, it plots NS alternately, with the CDU and CSU considered separately. As I noted in the previous discussion, I believe the “correct” procedure, for this purpose, is to count the “Union” as one party, but both are included here for the sake of transparency. In both panels, the dashed mostly horizontal line is the output of the extended SPM for the countries’ respective MMP systems1; it will change level only when the electoral system changes. (For New Zealand, the solid horizontal line is the expectation under the FPTP system in use before 1996.)

The German party system from 1953 through 2005 was clearly fitting quite poorly, due to how under-fragmented it was for the electoral system in use. The old CDU/CSU and SPD were just too strong and overwhelmed the considerable permissiveness of the electoral rules.2 So clearly the question I am raising here–whether the two-vote feature of MMP means it should not be modeled just like any (other) two-tier PR system–is moot for those years. However, perhaps it has become an issue in recent German elections, including 2021. The underlying feature of voter behavior pushing the actual NS to have risen to well above “expectation” would be the greater tendency of voters towards giving their two votes to different parties. At least that would be the cause in 2021, given that we saw in the previous post that the basic tier produced almost exactly the degree of fragmentation that the SPM says to expect. It is the compensation tier that pushed it above expectation, and the problem here (from a modeling perspective) is that the formula implicitly assumes the votes being used in the compensation mechanism are the same votes being cast and turned into seats in the basic (nominal) tier. But with two votes, they are not, and with more voters splitting tickets, the assumption becomes more and more untenable.

The previous planting on this matter emphasized that the SPM is actually performing well, even in this most recent, and quite fragmented, election. I am not trying to undermine that obviously crucial point! However, the marked rise in NS since 2009–excepting 2013 when the FDP failed to clear the threshold–may suggest that the model’s assumption that the two votes are pretty similar could be problematic.

Maybe two-vote MMP is more complex after all than its characterization as a two-tier PR system–the simplest form of complex electoral system–implies. In fact, maybe I should stop referring to MMP as a sub-category of two-tier PR. Yet for various reasons, it is a convenient way to conceptualize the system, and as yesterday’s discussion of the recent German election showed, it does work quite well nonetheless. It could be based on a flawed premise, however, and the more voters cast their nominal and list votes differently, the more that flaw becomes apparent.

A work in progress… in other words (fair warning), more such nerdy posts on this topic are likely coming.


1. The “expected NS” line for Germany takes the tier ratio to be 0.5, even though as I argued in the previous entry, we really should use the actual share of compensation seats in the final allocation. This would have only minimal impact in the elections before 2013; in 2021, it makes a difference in “expected” NS of 0.36.

2. Partly this is due to the 5% list-vote threshold, which is not a factor in the version of the SPM I am using. In Votes from Seats, we develop an alternate model based only on a legal threshold. For a 5% threshold, regardless of other features, it predicts NS=3.08. This would be somewhat better for much of the earlier period in Germany. In fact, from 1953 through 2002, mean observed NS=2.57. In the book we show that the SPM based only on mean district magnitude and assembly size–plus for two-tier PR, tier ratio–generally performs better than the threshold model even though the former ignores the impact of any legal threshold. This is not the place to get into why that might be, or why the threshold might have “worked” strongly to limit the party system in Germany for most of the postwar period, but the permissiveness of a large assembly and large compensation tier is having more impact in recent times. It is an interesting question, however! For New Zealand, either model actually works well for the simple reason that they just happen to arrive at almost identical predictions (3.08 vs. 3.00), and that for the entire MMP era so far, mean NS has been 3.14.

The Germany 2021 result and the electoral system

The German general election of 2021 has resulted in a situation in which neither major party can form a government without either the other, or more likely, a coalition that takes in both the liberal FDP and the Greens. With the largest party, the social-democratic SPD, under 30% of seats, it is an unusually fragmented result compared to most German elections. Naturally, this being Fruits & Votes, attention turns to how much more fragmented this outcome is than expected, given the electoral system. The answer may be a bit of a surprise: not all that much. I expected this outcome to be a significant miss for the Seat Product Model (SPM). But it is really not that far off.

For a two-tier PR system, of which Germany’s MMP can be thought of as a subtype, we need to use the extended version of the SPM developed in Votes from Seats.

NS = 2.5t(MSB)1/6,

where NS is the effective number of seat-winning parties (here, meaning the expected NS), M is the mean district magnitude of the basic tier, SB is the total number of seats in the basic tier, and t is the “tier ratio” defined as the share of the total number of assembly seats allocated in the compensatory tier. For Germany, basic-tier M=1 and SB=299. The tier ratio could be coded as 0.5, because the initial design of the system is that there are 299 list tier seats, allocated to bring the result in line with the overall party-list vote percentages of each party that clears the threshold. However, in Germany the electoral law provides that the list tier can be expanded further to the extent needed to reach overall proportionality. Thus t is not fixed; we should probably use the ratio that the final results are based on, as NS would necessarily be lower if only 299 list seats had been available. In the final result, the Bundestag will have 735 seats, meaning 436 list seats, which gives us a tier ratio of t=436/735=0.593. Plug all this into the formula, and you get:

NS = 2.50.5932991/6=1.72*2.59=4.45.

Now, what was the actual NS in the final result? We have to ask ourselves whether to count to two Christian “Union” parties, the CDU and the CSU, as one party or two. The answer really depends on the question being asked. They are separate parties, with distinct organization, and they bargain separately over portfolios and policy when they are negotiating a coalition with another party. However, for purposes of the SPM, I firmly believe that when two or more parties in a bloc do not compete against each other (or, alternatively, do so only within lists over which votes are pooled for seat-allocation1), they should be treated as one. The SPM does not “care” whether candidates of the bloc in question are branded as CSU (as they are in Bavaria) or as CDU (the rest of Germany). It simple estimates the effective number of “agents of the electorate” given the electoral rules. In terms of national politics, these are the same “agent”–they always enter government together or go into opposition together, and they jointly nominate a leader to be their Chancellor candidate.

Taking the CDU/CSU as a “party” for this purpose, we get actual NS =4.84 in the 2021 election. So, given an expectation of 4.45, the actual outcome is just over 8.75% higher than expected. That is nothing too extraordinary. For comparison purposes, we can just take the ratio of actual NS to expected NS. Here are some elections in the dataset used for Votes from Seats that are in the same range of over-fragmentation as Germany 2021:

      country   year   simple   Ns   exp_Ns   ratio 
     Barbados   1981        1    1.87   1.735597   1.077439  
       Norway   1965        1    3.51   3.255616   1.078137  
    Sri Lanka   1970        1    2.49   2.307612   1.079037  
Dominican Rep   1990        1    3.05   2.810847   1.085082  
     Trinidad   2002        1    1.98   1.824064   1.085488  
      Iceland   1963        0    3.33   3.060313   1.088124  
       Israel   1961        1    5.37   4.932424   1.088714  
     Trinidad   2001        1       2   1.824064   1.096452  
     Trinidad   2000        1       2   1.824064   1.096452  
      Iceland   1999        0    3.45   3.146183   1.096567  
      Denmark   1950        0    3.98   3.624933   1.097951  

(The table indicates as ‘simple’ those with a single tier; others are two-tier.)

The ratio variable has a mean of 1.021 in the full dataset and a standard deviation of 0.359. Its 75th percentile is 1.224 (and 25th is 0.745). So the German election of 2021 is actually very well explained by this method. The degree of fragmentation we saw in this election is not too surprising. It is about what should be expected with MMP consisting of 299 nominal-tier M=1 seats and a very generous and flexible compensation tier.

As an aside, if we used the initial tier size (299, so t=0.5) in the formula, we would get an “expected” NS=4.09. This would mean a ratio of 1.183, still short of the 75th percentile of the 584 elections included in the book’s main statistical test. Here is the company it would be keeping in that neighborhood:

            country   year   simple   Ns   exp_Ns   ratio 
            Germany   2009        0    4.83   4.121066   1.172027  
St. Kitts and Nevis   2000        1    1.75   1.491301   1.173472  
         Luxembourg   2009        1    3.63   3.077289    1.17961  
             Canada   2004        1    3.03   2.560218   1.183493  
            Denmark   1998        0    4.71   3.965222   1.187828  
          Venezuela   1963        0    4.32    3.63006   1.190063  
        Korea South   1988        0    3.55   2.981969   1.190488  
     Czech Republic   2010        1    4.51   3.767128   1.197199  
            Iceland   1991        0    3.77   3.146183   1.198277  

This would put the German 2021 election about as “over-fragmented” as the Canadian election of 2004. In other words, still not a big deal. If we count the two “Union” parties separately, obviously the degree of over-fragmentation goes up considerably. As I have said already, I think for this purpose counting them as one is the correct decision.2

As far as size of the largest seat-winning party is concerned, the SPD has 206 seats, for 28.03%. The SPM would predict, given expected NS=4.45, that the largest should have 32.6% (240 seats out of 735); that’s a ratio of 0.860 (which is a slightly bigger miss than the NS ratio of 1.088, the reciprocal of which would be 0.919). It is worth pausing on this for a bit. Polling before the election said the largest party might be only on a quarter of the votes. This was accurate, as the SPD won 25.7%. The advantage ratio (%seats/%votes) is 1.09, which is rather high for an electoral system that promises as near-perfect proportionality as Germany’s current system does, with its compensation for overhangs (cases in which a party has won more nominal-tier seats in a state than its list votes would have entitled it to). This bonus is a result of a rather high below-threshold vote. Not as high in 2013, of course, when two parties (FDP and AfD) narrowly missed the nationwide 5% threshold. But still considerably high, at 8.6% combined for all parties that failed to win a seat.

It is also worth asking whether the logic behind the extended SPM for two-tier systems holds for this German election. The formula says that the basic tier produces an initial allocation of seats consistent with the SPM for simple systems, and then inflates it based on the size of the compensation tier. So we can ask what the effective number of seat-winning parties is in the basic tier alone. It should be NS =(MSB)1/6= 2991/6= 2.59. In fact, the basic-tier NS in this election was 2.51 (as before, taking CSU/CSU as one party). The ratio of 0.969 is a pretty trivial miss. We should expect the largest party to have won 0.490 of these seats (about 146). Actually the Union parties, which together won the most single-seat districts, won 143 (0.478). Thus Germany’s MMP system, in the 2021 election, actually did produce a basic-tier (nominal-tier) party system pretty much just like it should, given 299 seats and M=1 plurality, and then augmented this through a large compensatory national tier. The actual inflator is a factor of 1.93=4.84/2.51, rather than the expected 1.72=2.50.593. Had it been 1.72 instead, the final effective number of seat-winning parties would have been 4.32, about “half a party” less than in reality, implying almost exactly one third of seats to the SPD instead of just 28%.

This surprised me (pleasantly, of course). When I saw that the Greens and AfD each had won 16 seats in the nominal tier, I thought that was too many! But in fact, it works out. Maybe sometimes even I think Duverger had a law, or something. But given 299 single-seat districts, this is pretty much in line with expectations.

The outcome is interesting in the many ways that it serves as a primer on details of the electoral system. Here I mean not only the substantial expansion of the Bundestag from 598 to 735 seats, due to the way the compensation mechanism works, but also the thresholds. One of the best known features of the German electoral system is the 5% nationwide threshold. But of course, the threshold is more complex than that. It is 5% of the national party-list vote or three single-seat wins, except if a party is an ethnic-minority party. All these provisions were on display. For instance, the Linke (Left) party fell below the 5.0% threshold, yet is represented at full proportionality. That is because it won three individual mandates, thus fulfilling the “or” clause of the threshold. There was a point on election night when it looked as if the Linke might hold only two single-seat districts. In that case, with less than 5% of the list votes nationwide, it would have held only those seats as its total. By winning three, it is entitled under the law to full proportional compensation, and as a result it was awarded 36 list seats. Then, for the first time in a very long time, an ethnic party has won a seat. The South Schleswig Voters’ Association (SSW), which had not contested federal elections in decades, ran in this one and was able to win a single (list) seat, because as a representative of the Danish and Frisian minorities, it is exempt from the usual threshold provisions, as long as its votes are sufficient to qualify it for a seat when the threshold is ignored. Its 0.1% of the national vote was good enough. The SSW has had some renewed success in state elections in Schleswig-Holstein recently, and now it has scored a seat in the federal parliament for the first time since 1949. In 1949, the MMP system was a bit different, in that the 5% threshold was determined state-by-state, rather than nationwide. If the threshold had been state-by-state in this election, one other party would have earned seats. The Free Voters won around 7.5% of party-list votes in Bavaria. However, they managed only 2.9% nationwide (and no district seat), so they are shut out.

Now attention turns to what the coalition will be. Two options are on the table: SPD+Greens+FDP (“traffic light”) or CDU/CSU+Greens+FDP (“Jamaica”). The possibility of a broad left coalition has been ruled out by the election results: SPD+Green+Linke is not a majority. It was never likely anyway; the SPD and Greens did not spend recent years convincing voters they were safe options near the center of German politics to team up with the far left. Nonetheless, had it been mathematically possible the SPD might have used it as leverage against the FDP. My guess is that the traffic light coalition will form. Despite some serious policy differences between the FDP and the other two, it would be a government made up of the winners of the election, as these three parties all gained votes compared to 2017. On the other hand, one led by the CDU/CSU would be led by a pretty big loser, even though it is mathematically possible and the Greens seem to have been positioning for it over the last several years.3 Following the election, the DW live blog has been reporting on comments by various prominent CDU and CSU politicians that could be interpreted as saying the bloc needs some time in opposition, after the disappointing result. I suspect this is the view that will prevail, and after a lot of intense and difficult bargaining, Germany will be led by a traffic light coalition for the first time.



1. Here I am thinking of cases like Chile, where alliance lists contain candidates of different parties, but for purposes of how the electoral system assigns seats between competing teams of candidates, we should count the alliances, not the component parties. The same condition applies in Brazil and Finland, only there it is essentially impossible to aggregate to a meaningful national alliance category because the combinations of parties are not always the same across districts. In Chile, and also in the FPTP case of India–as well as in the current case of Germany–there is no such problem, as the alliances are nationwide in scope and consistent across districts.

2. For the record, counting them separately yields NS=5.51 in this election, which would put the ratio just barely above the 75th percentile.

3.To be clear, they are much happier working with the SPD, but what I mean is that their positioning for the possibility of a coalition with the CDU/CSU should make finding common ground with the FDP easier than it otherwise would have been.

Elections in September, 2021–campaigns matter

It won’t be quite like September, 2005, back when the virtual orchard was just a sapling, but somewhat like that September sixteen years ago featured several interesting elections, this month also looks great for election watchers.

In September, 2005, we had elections in three major examples of mixed-member systems: Japan, New Zealand, and Germany. (As I look back, I see I wrote several times about Afghanistan’s election that month; I am guessing there will no longer be a need for new Afghanistan elections plantings. I also see lost of posts about a hurricane disaster in New Orleans. Some things do recur, though fortunately in this case not on as horrific a scale, though bad enough.)

In September, 2021, we have the California recall. I don’t have anything at the moment to say about that beyond what I’ve already said. We have Germany, again, with its general election on 26 Sept. And we have Canada, on 20 Sept. There are also various other elections this month, but these are the two I will focus on here.

The German election for this September has been known about for a long time, as it is occurring on schedule (unlike the one in 2005). The Canadian one, on the other hand, is a snap election as one would not have been due till 2023. Both of these elections are going to become case studies in how campaigns really can matter.

For months it has seemed certain that in Germany, the ruling Christian Democratic/Christian Social “Union” bloc (CDU/CSU) would again come out on top, with the Greens in second place and a likely new coalition partner. Then a funny thing happened: the Social Democratic Party (SPD), for years seemingly finished as a major party, started to surge and seemingly now has pulled ahead of the CDU/CSU. The range of coalition possibilities is suddenly rather large, with some novel possibilities in the cards.

The election is also notable compared to past German elections in that the incumbent Chancellor (prime minister), Angela Merkel (CDU) is not a candidate to remain head of government. In fact, it seems likely that the campaign has caused voters to reckon with the with the less than inspiring leadership of the party’s chosen successor to Merkel (Armin Laschet), to feel less than sure they are ready to make the Greens potentially the largest or even second largest party, and to have turned to the SPD and its new leader, Olaf Scholz as the potential safe pair of hands to lead the government. These three parties seem certain to be the big three, but with the largest still below 25% in polls, it will probably take three parties to forge a majority coalition (taking the Union as one for purposes of government-formation, even though it is actually two parties). Unless all three govern together, or the current no-so-grand coalition clears a majority fo seats and continues in power, only with the SPD on top, it is going to take some combination involving the Free Democrats (FDP) to have a majority coalition. The post-election bargaining will be interesting (FDP with either Greens or SPD is not “natural”), and thus the election will really matter for which combines are possible and how much bargaining power each party has.

In Canada, the Liberal Party of incumbent Prime Minister Justin Trudeau looked safe to be not only the largest party, but also to win a majority of parliamentary seats. In fact, given that the election timing was the government’s choice, that is precisely why it is happening–to convert a minority Liberal government into a majority Liberal government. Then Trudeau called the election and a funny thing happened: his party started sinking in the polls and the Conservatives appear to have caught up. In votes, that is.

As in 2019, the Conservatives could lead in votes and still come second in seats. The Conservatives probably need to win the votes by more than a couple percentage points to have a reasonably good chance at a plurality of seats, due to their inefficient vote distribution across the country. A majority Conservative government probably requires that party to keep adding support at a rapid pace, and that may be happening. Yet if that continues, some voters would probably dessert the New Democratic Party (NDP), currently running at around 20%, in favor of the Liberals. Given the FPTP electoral system, of course, such NDP desertion for the Liberals would not be guaranteed to help the latter at the aggregate seat-winning level, although it probably would do so. (Canada’s Green Party, meanwhile, is in a shambles, largely because its black, female, Jewish leader was not sufficiently anti-Israel for others in the party.) One thing seems safe to say at this point: A Liberal majority has become rather unlikely.

That’s the funny thing about campaigns. Sometimes they actually matter.