when voting is on the mathstresstest

The German Chancellery in Berlin

Do you think stable looking democracy in Germany could crumble in a nearer future? No?
Actually this could take place on sunday.




We know that there is trouble in Syria and that was one reason why I had recently asked about the voting system in Syria. But I had already mentioned here that there was more that I got interested in voting systems and the trouble is this time in Germany.

That is I got serious doubts about the new election law which is currently valid in Germany and according to which the german federal elections will take place on sunday.
My doubts where initiated by comments from Oinki and Thorsten to a post about the new german voting system on the german blog Netzpolitik.

Tim and me then finally sat down for a couple of hours and tried to understand Germany’s new federal election system. The new system is mostly desribed in §6 of the federal election law. The law itself is written in socalled Juristendeutsch. So Tim and me had really a hard time to understand the text and hence it may of course be the case that we completely misunderstood the text, however what we understood is rather disturbing.

We both came to the conclusion that the new law seems to be not only unreconcilable with the german constitution but that in principle it would have been possible for rather small minorities to raise legally to power. This should not necessarily remind of the Machtergreifung and the elections of 1932 – last but not least we still have currently quite a different economical situation in Germany then in 1932 – however the chances for very disturbing election results seem not too small.

So what’s wrong? Basically we came to the conclusion that there is some problem with the math so let’s review very briefly, where the problem with the voting system is concretely.

German elections take place in about 300 electoral districts. There are two types of votes. One is for voting a party (the socalled “Zweitstimme”) and one is for voting a candidate (“Erststimme”) in a respective electoral district. All voted Erststimme-candidates are entering the Bundestag. The seats in the german parliament, the Bundestag are about 600. They are distributed according to the outcome of the party voting, i.e the Zweitstimme-votes. Every party which had won more then 5% of the Zweitstimme-votes or 3 Erststimme-candidates can enter the Bundestag. It may happen though that the number of seats that a party had gained via the Zweitstimme is smaller than the number of its Erststimme-candidates. In this case all seats of this party are filled up with Erststimme-candidates and the rest of the Erststimme-candidates gets extra seats, namely socalled Überhangmandate or Overhang seats.

If a party has won more seats from the Zweitstimme than can be filled with Erststimme-candidates then the rest of these seats will be filled with candidates from the socalled party lists. In the old law there was a kind of – I call it – rounding problem with the overhang seats so the supreme court ruled that the old election law had to be renewed. And it is this new regulation of how the overhang seats are regulated which seems to be a problem.

In this new regulation after a first distribution of votes the overhang seats have to be supplemented until the proportions of the Zweitstimme-votes is reached. This may however pose problems if the share of Zweitstimme-votes is very different from the share of the Erstimme-votes. That is if for example in all over Germany only one single person votes for a party with its Zweitstimme and if one has 43 Million Zweitstimme-votes then the share of Zweitstimme-votes would be 1/43million (i.e. 1 divided by 43 million). However if this party has won 3 Erststimme-candidates then as what we understood from the law text this would account for as Overhang-votes, because the share of seats for this party is according to Zweitstimme-votes zero if one rounds down and thus one would need to supplement the seats so that 3 seats correspond to a share of 1/43million or in other words this “refill of overhang seats” would need to consist of 3 times 43 million seats. Quite a bit.

There is however a restriction. That is refills are only taking place as long as there are candidates on the respective party lists. So for this case it is finally the share of the candidates on the party lists which rules the final proportions in such a vote. But as it seems there are no regulations about the size of the candidates on the party lists and in fact a brief look on some party lists display that sometimes small parties may have the same number of candidates or more than big parties.

In short: with a certain strategy a small party could try to win 3 districts and tell its voters not to vote for them in the Zweitstimme and thus depending on how many names they have put on the party lists, win rather many seats.

So will this happen? Very probably: no. That is if the elections are approximately similar as the last elections then the number of “refill seats” will stay considerably small and sofar no party, which is likely to have Erststimme candidates told its voters not to elect it with its Zweitstimme.

I write “likely to have” since it may not be so unambigously clear wether there won’t be any parties, which may unexpectedly win in some districts. So for example for the regional election in the region of Uecker-Randow (the region is however smaller than a Bundestagswahldistrict) the Erststimme-candidate of the socalled NPD missed majority by about 1600 votes. And it is unfortunately not so clear how the corresponding Bundestagswahl district would vote there, because the districts were reorganized for the new elections. Moreover this is just one example I found more or less by try and error. There may be more and it is not so easy to get a good overview. I was lucky to find the number of candidates on party lists of the corresponding region via the regional office for statistics, since this kind of information is not available on the official website for the elections.

On the other hand I could imagine that if there would have been a bigger change in residential status between districts then this would have been noticed by our authorities….

As said we could be wrong with our understanding but it looks very much to us that this new law is not in accordance with the german constitution and that there are small chances that the election on sunday could lead to very disturbing results. For this reason I filed a request to the Bundeswahlleiter via Frag den Staat.

Sofar I haven’t gotten an answer.

10 Responses to “when voting is on the mathstresstest”

  1. F. Siebrecht Says:

    It always possible that there are some highly constructed awkward cases and loopholes in some laws that is why we have courts. So what you say may be interesting from a mathematical view but I find one shouldn’t make a mountain out of a molehill.

  2. nad Says:

    Sure there can always be loopholes in laws but here we have a rather unambigous procedure, which decribes clearly how concrete actions (Here assignment of seats) have to be made and these actions may be in certain circumstances and if our understanding is right be quite out of a reasonable scheme.
    And sure there are sometimes people who actively watch out for loopholes in laws. But as I said it seems luckily that this is not the case this time.
    I should have also better explained that the example given above is of course rather extreme, but in our understanding of the description you have a whole scale of possibilities in between. That is if you have any party which wons 3 Erststimme-candidates and stays below 5% (Zweitstimme) there will already be quite some trouble. That is if the party is approximately at 5% 0.5% *** then the Bundestag will need to have about 1200 seats, because 3 seats is 5% of 600 0.5%, if it is about 2.5% 0.25% then the Bundestag will need to have about 1800 seats, because 3 seats is 2.5% 0.25% of 1200 and so on. It doesn’t currently look like as if there will be a party with 3 Erstimme candidates, which will stay below 5%, but on the other hand it is actually possible and far from very constructed. that this might happen.

    *** 28.9.13 correction: of course 3/600 = 0.005 = 0.5% and not 0.05 = 5%. Lesson learned: eventually use calculators
    at late voting-nights. Don’t know though what it means that nobody pointed this mistake out to me in the
    meantime.
    If one stays right below 5% the seats will thus be about 600+60 seats and not 600+600 =1200 seats, that is only at 0.5% of Zweitstimme one would have about 1200 seats. This could however still happen if like there is a strong regional concentration.

  3. F. Siebrecht Says:

    So now the results are out and as you see nothing of your strange mathematics panic-prognosis happened.
    It seems I need to repeat this: I find one shouldn’t make a mountain out of a molehill.

  4. nad Says:

    I didn’t claim that some of the above described cases will happen, but just that they could have happened.
    And by the way I haven’t seen yet any results for the Erststimme-candidates. So even if I don’t think that even the bigger small parties, which currently look as being slightly below 5%, like AFD or FDP will have 3 Erststimme-candidates this can’t sofar be fully excluded unless there is more information.

  5. F. Siebrecht Says:

    There are the seats on http://www.tageschau.de at
    http://www.tagesschau.de/wahl/bundestagswahl312.html

    Which is the major german news source and which you should know.
    And this shows that there will be most probably 598 seats.
    Enough of this panic nonsense!

  6. nad Says:

    These are no panic postings. I just say how I understood things.
    That what you call “panic”-part comes from the possible reality.

    598 is also the number of seats for the Zweitstimme-seats only before counting Überhangmandate. That is yes it looks that since the CDU had won so many Zweitstimme-votes that all it’s previous Überhangmandate are now swallowed within their Zweitstimme-seats and thus there are no Überhangmandate at all. And yes tageschau wrote “Sitzverteilung” that should mean seats for both: Erst- and Zweitstimme. But sometimes in the heat of news presentations things may be forgotten. So they could have forgotten to write that this is just “Sitzverteilung according to Zweitstimme”, even if this not so likely.
    We will probably know tomorrow.

  7. Christian Says:

    I am studying mathematics at TUM right now and was one of the many plaintiffs in the 2012 lawsuit before the Federal Constitutional Court dealing with the negative vote weight, which was possible with the old voting system.
    (For details on the negative vote weight, see the extensive German wikipedia article: https://de.wikipedia.org/wiki/Negatives_Stimmgewicht_bei_Wahlen)
    As there is no attachment listed at the answer of the Bundeswahlleiter on FragDenStaat, I would be really interested what his view was. Could you maybe attach his answer to the issue on FragDenStaat?

  8. nad Says:

    Hello Christian

    the answer is now at: https://fragdenstaat.de/files/foi/12538/20131021_Antwort_NAME_geschwaerzt.pdf

  9. Christian Says:

    Thank you very much for the upload. Unfortunately, the “Statistische Bundesamt” did not really review your request and only send links to press releases. More generally, I would prefer a more comprehensible electoral law, I even guess that our members of Parliament do not fully understand the voting system they voted for :-)

  10. nad Says:

    Christian wrote:

    I even guess that our members of Parliament do not fully understand the voting system they voted for

    Actually I wonder HOW they can understand this! That is Tim and me had quite some troubles to understand this law text. And we are still not sure wether we fully grasped everything in all detail. I wrote this also in my answer at “Frag den Staat” today.

    I think if I would have been a politician who would have been in charge for signing this or voting for I would have rightous refused to sign or vote for such a law unless it wouldn’t have been formulated in a more understandable fashion!

    That is usually one often has to believe and trust at least partially what people tell you about a law text, because one doesn’t have always have the time to go through all the fine print and the power to change it anyways, however for such a case, which heavily affects constitutional rights this seems to be a different case and in particular as a politician you should have the right to demand more clarity – do they have the right?….or if I would be a politician would I be identified as a megagoonfool for all my stupid questions….?

Leave a Reply


comments in german, french and russian will be translated into english.
you can use LaTeX in your math comments, by using the [latex] shortcode:
[latex] E = m c^2 [/latex]