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.

Suppose 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 calculated in two steps:

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.

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 analogically 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.

The lower apportionment has to distribute the seats to each regional party list in a way that respects both the apportionment of seats to the party and the apportionment of seats to the regions.

The result is obtained by an iterative process. Initially, for each region a regional divisor is chosen using the highest averages method for the votes allocated to each regional party list in this region. 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 so that

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

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

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

...
Wikipedia