Hagenbach-Bischoff quota

The Hagenbach-Bischoff quota is a formula used in some voting systems based on proportional representation (PR). It is used in some elections held under the largest remainder method of party-list proportional representation as well as in a variant of the D'Hondt method known as the Hagenbach-Bischoff system. The Hagenbach-Bischoff quota is named for its inventor, Swiss professor of physics and mathematics Eduard Hagenbach-Bischoff (1833-1910)

The Hagenbach-Bischoff quota is sometimes referred to as the 'Droop quota' (especially in connection with the Largest remainder method) because the two are very similar. However, under the Hagenbach-Bischoff quota it is theoretically possible for more candidates to reach the quota than there are seats, whereas under the slightly larger Droop quota this is mathematically impossible. Some scholars of electoral systems argue that the Hagenbach-Bischoff quota should be used for elections under the Single Transferable Vote (STV) system, instead of the Droop quota, because in certain circumstances it is possible for the Droop quota to produce a seemingly undemocratic result. In practice the two quotas are so similar that they are unlikely to produce a different result in anything other than a very small or very close election.

Formula

The Hagenbach-Bischoff quota is given by the formula:

:$m\; votes\; over\; m\; \{seats+1\}$

*Votes = Total valid poll; that is, the total number of valid (unspoilt) votes cast in an election.
*Seats = Total number of seats to be filled in the election. The Droop quota's formula is slightly different in that the quotient arrived at by dividing the total vote by the number of seats plus 1 is rounded up if it is fractional, or if it is a whole number, 1 is added, so that in either case the quotient is increased to the next whole number.

As noted above, while under the Droop quota it is impossible for more candidates in an election to reach the quota than there are seats to be filled, this can theoretically occur under the Hagenbach-Bischoff quota. If this happens it is treated as a kind of tie and a candidate is chosen at random for exclusion.

Use in STV elections

An example

To see how the Hagenbach-Bischoff quota would work in an STV election imagine an election in which there are 2 seats to be filled and 4 candidates: Andrea, Carter, Brad and Delilah. There are 100 voters who vote as follows:

The Hagenbach-Bischoff quota is 300/(2+1) = 100. In the first round Andrea is elected with 200 preferences, while Brad (75) and Carter (25) remain in contention. Andrea's surplus of 100 is transferred: 25 to Brad and 75 to Carter, bringing each of them to 100. So all three have achieved the quota and so should be elected even though there are only two positions to fill.

One way of resolving this is to take as the quota the Hagenbach-Bischoff figure plus the smallest positive fraction which the counting system will allow. A quota of 100.01 in this example and of 13.01 in the earlier example would have prevented the problems identified.

Alternatively, B. L. Meek proposed treating the result as an n+1-way tie, and eliminating one of the candidates at random; still another solution would call for a runoff between the candidates.

ee also

* CPO-STV