Biproportional apportionment

The Double Pukelsheim is a seat allocation method for allocating parliamentary seats to parties in the presence of several constituencies in proportional representation. The method is actually called " Double Proportional divisor method with standard rounding " and was the mathematician Friedrich Pukelsheim developed on behalf of the Directorate of Justice and Home Affairs of the Canton of Zurich. It is since 2006 in the election of the Cantonal Council of the Canton of Zurich and the municipal councils of the cities of Zurich and Winterthur apply. The process is also under the name " New Zurich allocation method " known.

On February 24, 2008, the Swiss cantons of Aargau and Schaffhausen led by referendum a an appropriate electoral system. In contrast to Zurich for the time being no 5 - percent clause was inserted. On November 27, 2011, the Aargau voters an amendment to the Grand Council Law has accepted; Since then applies a 3 percent clause here. On September 22, 2013, the voters also said in the cantons of Nidwalden train and yes to the " double Pukelsheim ". The following remarks apply mutatis mutandis to these four cantons.

Starting point

The constituencies at the cantonal level are largely based on the district boundaries, the constituencies of the local elections the city of Zurich to the former borders of the independent communities. Due to the internal migration constituencies very different populations and thus in elections also have to assign a very different number mandates. At cantonal elections, the number of seats goes from four in the district Andelfingen up to sixteen in the districts Horgen, Uster and Bülach. The same is true for the city of Zurich, here are for the municipal elections between two seats in the smallest constituency up to nineteen seats in the largest constituency of the city to forgive.

In the previous seat allocation method according to Hagenbach - Bischoff each constituency was considered in isolation, the party votes in a constituency had no influence on the allocation of seats in another district. This means that small parties are seriously disadvantaged in the small districts. In a constituency with two seats to be awarded this can lead to a party with just under one-third share of the vote goes blank and void votes they. This means that the major parties in some constituencies have de facto guaranteed seat claims and also some voters in the knowledge that votes for small parties will be worthless, may not choose from them preferred party, but they still most likely congenial United Party.

After the municipal elections in the city of Zurich in March 2002, the Green Party raised from these considerations, a voting rights complaint, which the federal court partially approved of and the previous option method declared as unconstitutional.

Therefore, the Canton of Zurich had to search for a new election procedures, which abolished the discrimination of small parties and the number of weightless voices reduced to a minimum. For a discussion of his visit included the merging of constituencies or the establishment of constituency associations, as had already been introduced in the cantons of Bern and Basel-Landschaft. Constituency associations consist of one or more constituencies and are a purely mathematical construction, because the available seats are first calculated on the basis of a constituency association and will only be allocated to the individual constituencies. By joining smaller constituencies to an association the disadvantages smaller parties can be compensated. The disadvantage is that this methods utilized are little transparent. The merging of constituencies, however, has the disadvantage that potential candidates need to make in much larger areas election campaign and are then no longer be anchored well regionally.

An inquiry to the mathematician Friedrich Pukelsheim brought these to develop a method that allowed the retention of the existing constituencies and at the same time should bring the injustices disappear. The " double " proportionality refers to the fact that both the proportionality between the campaigning parties as well as the proportionality between the existing election districts is maintained, so that both the parties and the Regions (in case of local elections, the individual neighborhoods ) are proportionally represented in the Parliament.

The procedure in detail

The process is divided into an upper and a lower allocation allocation.

Upper allocation

In the upper allocation, the votes are first considered at the cantonal level. Since the customary in Switzerland procedure, voters can cast as many votes as there are to be awarded seats in their constituency, the votes cast must first be divided by the number assigned unit mandates in the constituency so that they are CANTON comparable throughout. While a voter in the district of miles may be, for example thirteen candidates his voice, a voter in the district Andelfingen has only four voices are available. So the voices are similar, the voices in Andelfingen be divided by four, in miles, by contrast, thirteen and are then equally weighted.

On this basis, the voices of the individual lists are added together canton far. Subsequently, the seats are distributed according to the Sainte-Laguë/Schepers. This minimizes the so-called success difference in value between the individual lists, that is, the quotient of Votes Cast is divided by the number of received mandates all parties are as high as possible. The disadvantage of the small parties is thus canceled.

Under allocation

After high allocation of the seats are allocated to the various parties. In the sub- allocation must now be defined in which constituencies these seats can be realized. The method used to this upcoming one hand must guarantee that each constituency gets as many seats as it was entitled, on the other hand, that each party gets as many seats as you were awarded in the upper allotment.

Using an iterative algorithm is applied, which one is best carried out by a computer. The end result of this algorithm can be then easily but check with a calculator on its accuracy. First, a table of constituencies and parties will be formed, each table entry represents the number of votes of each party in the corresponding constituency:

In the first step, a suitable Wahlkreisdivisor is first searched in each constituency. This must have the property that in his column so divides the numbers that, when rounded to the nearest integer (from .5 up, or down), the sum of the column entries exactly gives the number to be issued in the constituency seats.

The next step will now proceed line by line. In this case, a suitable Listengruppendivisor is searched for each row. This will divide the calculated in the first step numbers such that the sum of the line items ( rounded to integer ) this corresponds exactly to the number of the party awarded seats.

The third step then returns before columns. The Wahlkreisdivisoren be adjusted where necessary as in the first step, the fourth step, then goes back line by line before as in the second step, etc. It is mathematically guaranteed that this procedure terminates, ie eventually appropriate constituency and Listengruppendivisoren place, in which both the sum of the rounded table entries line by line the amounts attributable to that party seats as also corresponds to the sum of the column entries to be awarded in the appropriate constituency seats. Once this is done, can be read from the table, how many seats a party in which the district is entitled.

A suitable method to find in each step, an appropriate divisor, is the bisection.

Pros and Cons

The great advantage of the method is that it can guarantee a regional proportional representation in parliament and the proportional distribution of seats among the parties simultaneously. The differences divided by the number Mandate> between the groups in the list quotient

The disadvantage of the method is, however, that within a constituency party preferences no longer be mapped exactly to the distribution of seats in the constituency. This can easily be seen from the table shown in the previous section. Exactly proportional within the constituencies of the distribution would be just when the Listengruppendivisoren everywhere one would be, which usually is not but of course possible. Thus, a party can win a seat in a constituency, although another party has more votes ( so at the Zurich Cantonal elections in 2007 in the district of Uster: Here the FDP received a share of the vote from 14.6 percent allocated three seats, while the SP with a share of the vote from 17.3 percent only received 2 seats ). This is however balanced again seen over the whole electoral area of time.