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*.

*N _{S}* = 2.5

*(*

^{t}*MS*

_{B})

^{1/6},

where *N _{S}* is the effective number of seat-winning parties (here, meaning the

*expected*

*N*),

_{S}*M*is the mean district magnitude of the basic tier,

*S*

_{B}is the total number of seats in the basic tier, and

*t*is the “tier ratio” defined as the share of the total number of assembly seats allocated in the compensatory tier. For Germany, basic-tier

*M*=1 and

*S*

_{B}=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

*N*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

_{S}*t*=436/735=0.593. Plug all this into the formula, and you get:

*N _{S}* = 2.5

^{0.593}299

^{1/6}=1.72*2.59=

**4.45**.

Now, what was the actual *N _{S}* 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-allocation

^{1}), 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 *N _{S}* =

**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

*N*to expected

_{S}*N*. Here are some elections in the dataset used for

_{S}*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” *N _{S}*=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 *N _{S}*=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

*N*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.

_{S}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 *N _{S}* =(

*MS*

_{B})

^{1/6}= 299

^{1/6}=

**2.59**. In fact, the basic-tier

*N*in this election was

_{S}**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=2.5

^{0.593}. 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.

____

Notes

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 *N _{S}*=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.

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