Biproportional apportionment

Biproportional apportionment is a method to allocate seats into a party-list proportional representation respecting two characteristics. That is, for two different partitions each part receives the proportional number of seats within the total number of seats. For instance, this method could give proportional results by party and by region, or by party and by gender/ethnicity, or by any other pair of characteristics.

Process

Presume that the method is to be used to give proportional results by party and by region.

Each party nominates a candidate list for every region. The voters vote for the party lists of their region.

The results are computed in two steps:

In the so called upper apportionment the seats for each party (over all regions) and the seats for each region (from all parties) are determined.
In the so called lower apportionment the seats are distributed to the regional party list respecting the results from the upper apportionment.

This can be seen as globally adjusting the voting power of each party's voters by the minimum amount necessary so that the region-by-region results become proportional by party.

Upper apportionment

In the upper apportionment the seats for each party are computed with a highest averages method (for example the Sainte-Laguë method). This determines how many of all seats each party deserves due to the total of all their votes (that is the sum of the votes for all regional lists of that party). Analogical, the same highest averages method is used to determine how many of all seats each region deserves.

Note, that the results from the upper apportionment are final results for the number of the seats of one party (and analogical for the number of the seats of one region) within the whole voting area, the lower apportionment will only determine in which particular regions the party seats are allocated. Thus, after the upper apportionment is done, the final strength of a party/region within the parliament is definite.

Lower apportionment

The lower apportionment have to distribute the seats to each regional party list respecting both, the apportionment of seats to the party, and the apportionment of seats to the regions.

The result is obtained in an iterative process. Initially, for each region a regional divisor is chosen using the highest averages method in respect to the votes each regional party list in this region obtained. For each party a party divisor is initialized with 1.

Effectively, the objective of the iterative process is to modify the regional divisors and party divisors in a way, that

• the number of seats of each regional party list is the number of their votes divided by both, the regional and the party divisors, rounded by the rounding method of the highest averages method used, and
• the sum of seats of all regional party lists of one party is the number of seats computed in the upper apportionment for that party, and
• the sum of seats of all regional party lists of one region is the number of seats computed in the upper apportionment for that region.

The following two correction steps are executed until this objective is satisfies:

• modify the party divisors such that the apportionment within each party is correct with the chosen highest averages method,
• modify the regional divisors such that the apportionment within the region is correct with the chosen highest averages method.

Using the Sainte-Laguë method, this iterative process is guaranteed to terminate with appropriate seat numbers for each regional party list.

Example

There are three parties A, B and C and three regions I, II and III. 20 seats are to be distributed. And the Sainte-Laguë method is to be used. The votes for the regional party lists are as follows:

Party I II III total
A 123 45 815 983
B 912 714 414 2040
C 312 255 215 782
total 1347 1014 1444 3805

Upper apportionment

In the upper apportionment, the overall seat number for the parties and the regions are determined.

Using the divisor 190, the results for the distribution of the party seats are:

 Party A B C #votes 983 2040 782 #votes/divisor 5.2 10.7 4.1 #seats 5 11 4

Using the divisor 190, the results for the distribution of the region seats are:

 Region I II III #votes 1347 1014 1444 #votes/divisor 7.1 5.3 7.6 #seats 7 5 8

Lower apportionment

Initially, regional divisors have to be found to distribute the seats of each region to the regional party lists. In the tables, for each regional party list, there are two cells, the first shows the number of votes and the second the number of seats allocated.

region
Party I II III
A 123 1 45 0 815 5
B 912 4 714 4 414 2
C 312 2 255 1 215 1
total 1347 7 1014 5 1444 8
regional divisor 205 200 180

Now, the party divisors are initialized with ones and the number of seats within each party is checked (that is, compared to the number computed in the upper apportionment):

region
Party I II III total party divisor
A 123 1 45 0 815 5 983 6 1
B 912 4 714 4 414 2 2040 10 1
C 312 2 255 1 215 1 782 4 1
total 1347 7 1014 5 1444 8 3805 20
regional divisor 205 200 180

Since not all parties have the correct number of seats, a correction step have to be executed: For parties A and B, the divisors are to be adjusted. The divisor for A have to be raised and the divisor for B have to be lowered:

region
Party I II III total party divisor
A 123 1 45 0 815 4 983 5 1.1
B 912 5 714 4 414 2 2040 11 0.95
C 312 2 255 1 215 1 782 4 1
total 1347 8 1014 5 1444 7 3805 20
regional divisor 205 200 180

Now, the divisors for regions I and III have to be modified. Since region I has one seat too much (8 instead of the 7 seats computed in the upper apportionment), its divisor have to be raised; in opposite, the divisor for region III has to be lowered.

region
Party I II III total party divisor
A 123 1 45 0 815 4 983 5 1.1
B 912 5 714 4 414 3 2040 12 0.95
C 312 1 255 1 215 1 782 3 1
total 1347 7 1014 5 1444 8 3805 20
regional divisor 210 200 170

Again, the divisors for the parties have to be adjusted:

region
Party I II III total party divisor
A 123 1 45 0 815 4 983 5 1.1
B 912 4 714 4 414 3 2040 11 0.97
C 312 2 255 1 215 1 782 4 0.98
total 1347 7 1014 5 1444 8 3805 20
regional divisor 210 200 170

Now, the numbers of seats for the three parties and the three regions match the numbers computed in the upper apportionment. Thus, the iterative process is completed.

The final seat numbers are:

#seats region
Party I II III total
A 1 0 4 5
B 4 4 3 11
C 2 1 1 4
total 7 5 8 20

Usage

The method was proposed in 2003 by German mathematician Friedrich Pukelsheim. It is used for municipal elections in some cantons of Switzerland: e.g. Zurich (since 2006), Aargau and Schaffhausen (since 2008).

Fair majority voting

Fair majority voting is a biproportional apportionment method with single-member regions called "districts", so each district has exactly one representative. It was proposed in 2008 by Michel Balinski (who also invented the single-winner voting system called majority judgment) as a way to eliminate the power of gerrymandering, especially in the United States.

References

• Balinski, M. “Fair Majority Voting (or How to Eliminate Gerrymandering).” American Mathematical Monthly 115, no. 2 (2008): 97–113.