Double proportional allocation procedure

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The double proportional allocation procedure , or double proportional for short , is a method for distributing parliamentary seats to parties when there are several constituencies with proportional representation .


The procedure is called in detail the double proportional divisor method with standard rounding . The original official name was New Zurich Allocation Procedure . Colloquially, “ Doppelter Pukelsheim ”, a word created by the former Zurich Interior and Justice Director Markus Notter, has prevailed . The latter names go back to the fact that the mathematician Friedrich Pukelsheim concretized the procedure on behalf of the Directorate of Justice and the Interior of the Canton of Zurich . The method was originally based on the work of Michel Balinski and Peyton Young .


The double-proportional allocation procedure is used in some cantons and cities in Switzerland. The election of the Cantonal Council of the Canton of Zurich and the City Council of the City of Zurich has been based on this procedure since 2006 . On February 24, 2008, the Swiss cantons of Aargau and Schaffhausen also introduced a corresponding electoral system by referendum. On September 22, 2013, the electorate in the cantons of Nidwalden and Zug , and on March 8, 2015 in the canton of Schwyz , said yes to double proportional representation.


The constituencies at the cantonal level are largely based on the district boundaries, the constituencies of the municipal council elections of the city of Zurich on the districts. Due to internal migration, the constituencies have very different numbers of inhabitants and therefore a very different number of mandates to issue in elections. In cantonal elections, the number of mandates goes from four in the Andelfingen district to sixteen in the Horgen, Uster and Bülach districts. The same applies to the city of Zurich, where until the reform of the constituencies for the municipal council elections between two seats in the smallest constituency up to nineteen seats in the largest constituency of the city were to be allocated.

In the previous seat allocation procedure according to Hagenbach-Bischoff , each constituency was considered in isolation, the party votes in one constituency had no influence on the allocation of seats in another district. As a result, small parties in the small districts are severely disadvantaged. In a constituency with two seats to be allocated, this can mean that a party with just under a third of the votes cast goes empty-handed and the votes cast for it expire. This leads to the fact that the big parties in certain constituencies have de facto guaranteed seat claims and, in addition, certain voters, aware that votes to small parties will be worthless, may not vote for the party they prefer, but rather the big party that they most like.

After the municipal council elections in the city of Zurich in March 2002, the Green Party raised a voting rights complaint based on these considerations, which the Federal Supreme Court partially upheld and declared the previous electoral process to be unconstitutional.

Therefore, the canton of Zurich had to look for a new electoral process that removed the discrimination against small parties and reduced the number of weightless votes to a minimum. The discussion included the amalgamation 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 arithmetic construct, because the seats to be allocated are initially calculated on the basis of a constituency association and only then are allocated to the individual constituencies. By merging small constituencies to form an association, the disadvantages of small parties can be offset. The disadvantage is that the methods used are not very transparent. The amalgamation of constituencies, on the other hand, has the disadvantage that potential candidates have to campaign in significantly larger areas and may not be well anchored regionally afterwards.

A request from the mathematician Friedrich Pukelsheim led him to develop a process that allowed the previous constituencies to be retained and at the same time was intended to make the injustices disappear. The “double” proportionality refers to the fact that both the proportionality between the candidate parties and the proportionality between the existing constituencies are maintained, so that both the parties and the regions (or, in the case of municipal council elections, the individual city districts) are proportionally represented in parliament .

The procedure in detail

The procedure is divided into an upper allocation and a sub - allocation .


In the overall allocation, the votes cast are initially considered at canton level. Since the usual procedure in Switzerland allows voters to cast as many votes as there are seats to be allocated in their constituency, the votes cast must first be divided by the number of mandates to be allocated in the constituency so that they can be compared across the canton. For example, while a voter in the Meilen district can vote for thirteen candidates, a voter in the Andelfingen district only has four votes. So that the votes are comparable, the votes in Andelfingen are divided by four, in miles, however, by thirteen and are then weighted equally.

On this basis, the votes of the individual lists are added up across the canton. The seats are then distributed according to the Sainte-Laguë / Schepers procedure . This minimizes the so-called difference in success values ​​between the individual lists, i.e. the quotient of the votes cast divided by the number of mandates received is as high as possible for all parties. This removes the disadvantage of the small parties.


After the overall allocation, the seats are assigned to the various parties. In the case of sub-allocation, it must now be determined in which constituencies these seats will be realized. On the one hand, the procedure used must guarantee that each constituency receives as many seats as it is entitled to; on the other hand, that each party receives as many seats as it was awarded in the upper allocation.

An iterative algorithm is used, which is best carried out by a computer. However, the final result of this algorithm can then easily be checked for correctness with a calculator. First, a table is created from constituencies and parties, with each entry in the table showing the number of voters of the respective party in the respective constituency:

Constituency A (4 seats)   Constituency B (5 seats)   Constituency C (6 seats)
List group 1 (4 seats) 5100 9800 4500
List group 2 (5 seats) 6000 10,000 12000
List group 3 (6 seats) 6300 10200 14400
Step 1 Constituency A (4 seats)   Constituency B (5 seats)   Constituency C (6 seats)
List group 1 (4 seats) 1.25 => 1 1.48 => 1 0.87 => 1
List group 2 (5 seats) 1.47 => 1 1.51 => 2 2.33 => 2
List group 3 (6 seats) 1.54 => 2 1.54 => 2 2.80 => 3
Constituency divisor 4090 6635 5150
step 2 Constituency A (4 seats)   Constituency B (5 seats)   Constituency C (6 seats)   Lists groups divisor
List group 1 (4 seats) 1.39 => 1 1.64 => 2 0.97 => 1 0.9
List group 2 (5 seats) 1.47 => 1 1.51 => 2 2.33 => 2 1
List group 3 (6 seats) 1.50 => 2 1.50 => 1 2.73 => 3 1,025

In the first step, a suitable constituency divisor is sought in each constituency . This must have the property that it divides the numbers in its column in such a way that, if they are rounded to the next whole number (from .5 upwards, otherwise downwards), the sum of the column entries results in exactly the number of seats to be allocated in the constituency.

The next step is to proceed line by line. A suitable list group divisor is searched for for each line. This should divide the (not rounded) numbers calculated in the first step in such a way that the sum of the line entries (rounded to a whole number) corresponds exactly to the number of seats allocated to the corresponding list group (party).

In the example shown, the total seats of the individual list groups as well as those of the constituencies are already filled after the second step. If this is not the case, the two steps are now repeated alternately:

The third step then proceeds again in columns. The constituency divisors are adjusted where necessary as in the first step, in the fourth step one proceeds again line by line as in the second step, etc. It is mathematically guaranteed that this process terminates, i.e. H. at some point finds suitable constituency and list group divisors in which both the sum of the rounded table entries corresponds row by row to the seats allotted to the respective party and the sum of the column entries of the seats to be allocated in the corresponding constituency. As soon as this has been done, the table shows how many seats a party is entitled to in which district.

A suitable method for finding a suitable divisor in each work step is bisection .

Advantages and disadvantages

The great advantage of the procedure is that it can guarantee regionally proportional representation in parliament and the proportional distribution of the seats among the parties. The differences in the quotient ‹votes received divided by number of mandates› between the list groups are as small as possible, so the disadvantage of the small parties is eliminated despite the fact that the constituencies are retained.

The disadvantage of the procedure, on the other hand, is that within an electoral district, party preferences are no longer precisely mapped to the distribution of mandates in the constituency. This can be easily seen from the table shown in the previous section. The distribution would be exactly proportional within the constituencies if the list group divisors were 1 everywhere, which is of course not usually possible. A party can win a seat within a constituency even though another party has cast more votes (as in the 2007 Zurich cantonal elections in the Uster district: Here the FDP received 3 seats with a share of 14.6 percent of the vote, while the SP won 17.3 percent of the vote only received 2 seats). However, this will be compensated for across the entire electoral area.

Web links


Court judgments


  1. Adi Kälin: The Truth About People's Will - Friedrich Pukelsheim's joint skiing holidays and the person responsible in the Justice Department were not to blame for the emergence of the new Zurich electoral system. But coincidence played a part. Neue Zürcher Zeitung 11.7.17