Bavarian electoral system

Last Sunday, we had the election for the federal state of Bavaria. Since the electoral system is kind of odd (but not as odd as first past the post), I would like to analyse how some variations (assuming the actual distribution of votes) in the rule would have worked out. So, first, here is how actually, the seats are distributed: Each voter gets two ballots: On the first ballot, each party lists one candidate from the local constituency and you can select one. On the second ballot, you can vote for a party list (it's even more complicated because also there, you can select individual candidates to determine the position on the list but let's ignore that for today).

Then in each constituency, the votes on ballot one are counted. The candidate with the most votes (like in first past the pole) gets elected for parliament directly (and is called a "direct candidate"). Then over all, the votes for each party on both ballots (this is where the system differs from the federal elections) are summed up. All votes for parties with less then 5% of the grand total of all votes are discarded (actually including their direct candidates but this is not of a partial concern). Let's call the rest the "reduced total". According to the fraction of each party in this reduced total the seats are distributed.

Of course the first problem is that you can only distribute seats in integer multiples of 1. This is solved using the Hare-Niemeyer-method: You first distribute the integer parts. This clearly leaves fewer seats open than the number of parties. Those you then give to the parties where the rounding error to the integer below was greatest. Check out the wikipedia page explaining how this can lead to a party losing seats when the total number of seats available is increased.

Because this is what happens in the next step: Remember that we already allocated a number of seats to constituency winners in the first round. Those count towards the number of seats that each party is supposed to get in step two according to the fraction of votes. Now, it can happen, that a party has won more direct candidates than seats allocated in step two. If that happens, more seats are added to the total number of seats and distributed according to the rules of step two until each party has been allocated at least the number of seats as direct candidates. This happens in particular if one party is stronger than all the other ones leading to that party winning almost all direct candidates (as in Bavaria this happened to the CSU which won all direct candidates except five in Munich and one in Würzburg which were won by the Greens).

A final complication is that Bavaria is split into seven electoral districts and the above procedure is for each district separately. So there are seven times rounding and adding seats procedures.

Sunday's election resulted in the following distribution of seats:

After the whole procedure, there are 205 seats distributed as follows

• CSU 85 (41.5% of seats)
• SPD 22 (10.7% of seats)
• FW 27 (13.2% of seats)
• GREENS 38 (18.5% of seats)
• FDP 11 (5.4% of seats)
• AFD 22 (10.7% of seats)

You can find all the total of votes on this page.

Now, for example one can calculate the distribution without districts throwing just everything in a single super-district. Then there are 208 seats distributed as

• CSU 85 (40.8%)
• SPD 22 (10.6%)
• FW 26 (12.5%)
• GREENS 40 (19.2%)
• FDP 12 (5.8%)
• AFD 23 (11.1%)

You can see that in particular the CSU, the party with the biggest number of votes profits from doing the rounding 7 times rather than just once and the last three parties would benefit from giving up districts.

But then there is actually an issue of negative weight of votes: The greens are particularly strong in Munich where they managed to win 5 direct seats. If instead those seats would have gone to the CSU (as elsewhere), the number of seats for Oberbayern, the district Munich belongs to would have had to be increased to accommodate those addition direct candidates for the CSU increasing the weight of Oberbayern compared to the other districts which would then be beneficial for the greens as they are particularly strong in Oberbayern: So if I give all the direct candidates to the CSU (without modifying the numbers of total votes), I get the follwing distribution:

221 seats

• CSU 91 (41.2%)
• SPD 24 (10.9%)
• FW 28 (12,6%)
• GREENS 42 (19.0%)
• FDP 12 (5.4%)
• AFD 24 (10.9%)

That is, there greens would have gotten a higher fraction of seats if they had won less constituencies. Voting for green candidates in Munich actually hurt the party as a whole!

The effect is not so big that it actually changes majorities (CSU and FW are likely to form a coalition) but still, the constitutional court does not like (predictable) negative weight of votes. Let's see if somebody challenges this election and what that would lead to.

The perl script I used to do this analysis is here.

Postscript:
The above analysis in the last point is not entirely fair as not to win a constituency means getting fewer votes which then are missing from the grand total. Taking this into account makes the effect smaller. In fact, subtracting the votes from the greens that they were leading by in the constituencies they won leads to an almost zero effect:

Seats: 220

• CSU  91 41.4%
• SPD  24 10.9%
• FW  28 12.7%
• GREENS  41 18.6%
• FDP  12 5.4%
• AFD  24 10.9%

Letting the greens win München Mitte (a newly created constituency that was supposed to act like a bad bank for the CSU taking up all central Munich more left leaning voters, do I hear somebody say "Gerrymandering"?) yields

Seats: 217

• CSU  90 41.5%
• SPD  23 10.6%
• FW  28 12.9%
• GREENS  41 18.9%
• FDP  12 5.5%
• AFD  23 10.6%

Or letting them win all but Moosach and Würzbug-Stadt where the lead was the smallest:

Seats: 210

• CSU  87 41.4%
• SPD  22 10.5%
• FW  27 12.9%
• GREENS  40 19.0%
• FDP  11 5.2%
• AFD  23 11.0%