Negative votes and conflicting criteria

Yesterday, Matthijs Bogaards and Dierk Schleicher ran a session on the electoral system for the upcoming general election we are going to have on Sunday in Germany. I had thought I I know how it works but I was proven wrong. Before I was aware that there is something like Arrow's impossibility theorm which states that there is a certain list of criteria your electoral system is supposed to fulfill but which cannot hold all at the same time for any implementation. What typically happens are cyclic preferences (there is a majority for A over B and one for B over C and one for C over A) but I thought all this is mostly academic and does not apply to real elections. I was proven wrong and there is a real chance that there is a paradoxical situation coming up.

Before explaining the actual problem, I should explain some of the background. The system in Germany is quite complicated because it tries to accomodate a number of principles: First, after the war, the British made sure the system contains some component of constituency vote: Each local constituency (electoral district for you Americans) should send one candidate to parliament that is in principle directly responsible to the voters in that district so voters have something like "their representative". Second, proportional vote, that is the number of seats for a party should reflect the percentage of votes for that party in the popular vote. Third, Germany is a federal republic, so the sixteen federal states should each send their own representatives. Finally, there are some practical considerations like the number of seats in parliament should be roughly 600 and you shouldn't need a PhD in math and political science to understand your ballot.

So this is how it works. Actually, it's slightly more complicated but that shall not bother us here. And I am not going into the problem of how to deal with rounding errors (you can of course only have integer seats) which brings with it its own paradoxes. What I am going to cover is how to deal with the fact, that the number of seats has to be non-negative:

The ballot has two columns: In the first, you vote for a candidate from your constituency (which is nominated by its party). In the second, you vote for a party for the proportional vote. Each voter makes one cross in each column, one for a candidate from the constituency and one for a party in the proportional vote. There are half as many constituencies as there are seats in parliament and these are filled immediately according to majority vote of the first column.

The second step is to count the votes in the second column. If a party neither gets more than five percent of those nor wins three or more constituencies their votes are dropped. The rest is used to work out how many of the total of 600 seats each of the parties gets.

Now comes the federal component: Let's consider party A and assume the popular vote says they should get 100 seats. We have to determine how these 100 seats are distributed between the federal states. This is again done proportionally: Party A in federal state (i) gets that percentage of the 100 seats that reflects the percentage of the votes for party from state (i) of the total votes for party A in all of Germany. Let's say this is 10. Further assume that A has won 6 constituencies in federal state (i). Then, in addition to these 6 candiates from the constituencies, the top four candidates from party A's list for state (i) are send to Berlin.

So far, everything is great: Each constituency has "their representative" and the total number of seats for each party is proportional to its share of the popular vote.

Still, there is a problem: The two votes in the two columns are independent. And as the constituencies are determined by majority vote, except in a few special cases (Berlin Kreuzberg where I used to live before moving to Cambridge being one with the only constituency winner from the green party) it does not make much sense to vote for a constituency candidate that is not nominated by one of the two big parties. Any other vote would likely be irrelevant and effectively your only choice is between the candidate of SPD or CDU.

Because of this, it can (and in fact often does for the two big parties) happen that a party wins more constituencies in a federal state than it is entitled to for that state according to the popular vote. In that case (because there are no negative numbers of candidates from the list to balance this) the rule is that all the constituency winners go to parliament and none from the list of that party. The parliament is enlarged for these "excess mandates". So that party gets more seats than their proportion of the popular vote.

This obviously violates the principle of proportional elections but it gets worse: If that happens in a federal state for party A you can hurt this party by voting for it: Take the same numbers as above but assume A has won 11 constituencies in (i). If there are no further excess mandates, in the end, A gets 101 seats in the enlarged parliament of 601 seats. Now, assume A gets an additional proportional vote. It is not impossible that this does not increase A's total share of 100 votes for all of Germany but increases to proportional share for the A's candidates in federal state (i) from 10 to 11. This does not change anything for the represenatives from (i), still the 11 constituency candidates go to Berlin but there is no excess mandate anymore. Thus, overall, A sends only 100 representatives to a parliament of 600, one less than with the additional vote!

As a result, in that situation the vote for A has a negative weight: It decreases A's share in the parliament. Usually, this is not so much of a problem, because the weights of votes depend on what other people have voted (which you do not know when you fill out your ballot) and chances are much higher that your vote has positive weight. So it is still save to vote for your favourite party.

However, this year, there is one constituency in Dresden in the federal state of Saxony where one of the candidates died two weeks before election day. To ensure equal chances in campaining, the election in that constituency has been postponed for two weeks. This means, voters there will know the result from the rest of the country. Now, Saxony is known to be quite conservative so it is not unlikely that the CDU will have excess mandates there. And this might just yield the above situation: Voters from Dresden might hurt the CDU by voting for them in the popular vote and they would know if that were the case. It would still be democratic in a sense, it's just that if voters there prefer CDU or FDP they should vote for FDP and if they prefer SPD or the Greens they should vote for CDU. Still, it's not clear if you can explain that to voters in less then two weeks... I find this quite scary, especially since all polls predict this election to be extremely close and two very different outcomes are withing one standard deviation.

If you are interested in alternative voting systems, Wikipedia is a good starting point. There are many different ones and because of the above mentioned theorem they all have at least one drawback.

Yesterday, there was also a brief discussion of whether one should have a system that allows fewer or more of the small parties in parliament. There are of course the usual arguments of stability versus better representation of minorities. But there is another argument against a stable two party system that is not mentioned often: This is due to the fact that parties can actually change their policies to please more voters. If you assume, political orientation is well represented by a one dimensional scale (usually called left-right), then the situation of icecream salesmen on a beach could occur: There is a beach of 4km with two competing people selling icecream. Where will they stand? For the customers it would be best if they are each 1km from the two ends of the beach so nobody would have to walk more than 1km to buy an icecream and the average walking distance is half a km. However, this is an unstable situation as there is an incentive for each salesman to move further to the middle of the beach to increase the number of customers to which he is closer
than his competitor.

So, in the end, both will meet in the middle of the beach and customers have to walk up to 2km with an average distance of 1km. Plus if that happens with two parties in the political spectrum they will end up with indistinguishable political programs and as a voter you don't have a real choice anymore. You could argue that this has already taken place in the USA or Switzerland (there for other reasons) but that would be unfair to the Democrats.

I should have had many more entries here about politics and the election like my role models on the other side of the Atlantic. I don't know why these never materialised (vitualised?). So, I have to be brief: If you can vote on Sunday, think of where the different parties actually have different plans (concrete, rather than abstract "less unemployment" or "more sunshine") and what the current government has done and if you would like to keep it that way (I just mention the war in Iraq and foreign policy, nuclear power, organic food as a mass market, immigration policy, tax on waste of energy, gay marriage, student fees, reform of academic jobs, renewable energy) your vote should be obvious. Mine is.

Update:
The election is over and everybody is even more confused than before. As the obvious choices for coalitions do not have a majority one has to look for the several colourful alternatives and the next few weeks will show us which of the several impossibilities will actually happen. What will definitely happen is that in Dresden votes for the CDU will have negative weight (linked page in German with an excel sheet for your own speculations). So, Dresdeners, vote for CDU if you want to hurt them (and you cannot convince 90% of the inhabitants to vote for the SPD).