Hagenbach-Bischoff system

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The Hagenbach-Bischoff system is a variant of the D'Hondt method, used for allocating seats in party-list proportional representation. It usually uses the Hagenbach-Bischoff quota for allocating seats, and for any seats remaining the D'Hondt method is then applied so that the first and subsequent divisors (number of seats won plus 1) for each party list's vote total includes the number of seats that have been allocated by the quota. The system gives results identical to the D'Hondt method and it is often referred to as such in countries using the system e.g. Switzerland and Belgium. Luxembourg uses the Hagenbach-Bischoff method to allocate seats in its European Parliament elections.

1. Step: Basic Distribution

   The total number of valid votes cast in the election is divided by the number of seats to be allocated + 1. The result, rounded up to the next whole number, forms the distribution number (also election number). Each party or list is allocated as many seats as the distribution number is completely contained in its vote count. Thus, the number of seats for a party is:

   ${\displaystyle {\text{number of Seats}}=\left\lfloor {\frac {\text{number of Votes}}{\left\lfloor {\frac {\text{Total number of Votes}}{{\text{Total number of Seats}}+1}}+1\right\rfloor }}\right\rfloor }$

2. Step: If there is still a seat to be allocated:

   For each party, the quotient ${\displaystyle \textstyle {\frac {\text{number of Votes}}{{\text{already allocated seats}}+1}}}$ is calculated, and the next seat is allocated to the party with the largest quotient (highest number).

3. Step: If there is still a seat to be allocated, step 2 is repeated.

and so on.