HagenbachBischoff quota
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The HagenbachBischoff 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 partylist proportional representation as well as in a variant of the D'Hondt method known as the HagenbachBischoff system. The HagenbachBischoff quota is named for its inventor, Swiss professor of physics and mathematics Eduard HagenbachBischoff (1833–1910)
The HagenbachBischoff quota is sometimes referred to as the 'Droop quota' and vice versa (especially in connection with the Largest remainder method) because the two are very similar. However, under the HagenbachBischoff and any smaller (e.g. the Imperiali) quota it is theoretically possible for more candidates to reach the quota than there are seats, whereas under the slightly larger Droop quota (see "Formula" below) this is mathematically impossible. Some scholars of electoral systems argue that the HagenbachBischoff 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 HagenbachBischoff quota may be given as:^{[1]}
where:
 Total votes = the total valid poll; that is, the number of valid (unspoilt) votes cast in an election.
 Total seats = the 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 HagenbachBischoff quota. If this happens the last seat is typically treated as a tie and a winner between the two candidates must be selected.
An example of use in STV
To see how the HagenbachBischoff quota would work in an STV election imagine an election in which there are 2 seats to be filled and 3 candidates: Andrea, Carter, Brad. There are 100 voters who vote as follows:
45 voters

25 voters

30 voters

Because there are 100 votes cast, and 2 seats, the HagenbachBischoff is:
To begin the count the first preferences cast for each candidate are tallied and are as follows:
 Andrea: 45
 Carter: 25
 Brad: 30
Andrea has more than 33⅓ votes. She therefore has reached the quota and is declared elected. She has 11⅔ votes more than the quota. These votes are transferred to Carter so the tallies become:
 Carter: 36⅔
 Brad: 30
Carter has now reached the quota so he is declared elected. The winners are therefore Andrea and Carter.
Advantage over the Droop quota
Some voting systems experts, such as Christine Cierra Danica, have observed that in an STV election held under the Droop quota it is sometimes possible for a group of candidates supported by a majority of voters to receive only a minority of seats. Such an outcome is far more likely under the older Hare quota but can still occur under the Droop quota in rare circumstances. It is a possibility that is only completely eliminated by use of the HagenbachBischoff quota. The problem is best illustrated by an example.
Scenario
Imagine an election in which there are 7 seats to be filled. There are 8 candidates standing, in two groups: Andrea, Carter, Brad and Delilah are members of the Alpha party; Scott, Jennifer, Matt and Susan are members of the Beta party. There are 104 voters and they vote as follows:
Alpha party  Beta party  

14 voters

14 voters

14 voters

11 voters

13 voters

13 voters

13 voters

12 voters

It can be seen that supporters of the Alpha party all rank all four Alpha party candidates higher than any of the Beta party candidates (the last four preferences of the voters are not shown above because they will not affect the result of the election). Similarly, voters who support the Beta party all give their first four preferences to Beta party candidates. Overall, the Alpha party receives 53 votes out of a total of 104. The Alpha party therefore has a majority of one. The Beta party receives a minority share of the vote.
Below the election results are shown first under the Droop and then under the HagenbachBischoff quota. It can be seen that under the Droop quota, despite having the support of a majority of voters, the Alpha party receives only a minority of seats. When the same election is conducted under the HagenbachBischoff quota, however, the Alpha party's majority is rewarded with a majority of seats.
Count under the Droop quota
1. The Droop quota is calculated as 14.
2. When first preferences are tallied Andrea, Carter and Brad (all from the Alpha party) have all reached a quota and are declared elected. However none of them has a surplus. The tallies of the remaining candidates are therefore:
 Delilah (Alpha party): 11
 Scott (Beta party): 13
 Jennifer (Beta party): 13
 Matt (Beta party): 13
 Susan (Beta party): 12
3. No candidate has reached a quota so Delilah, who is the candidate with the fewest votes, is excluded. Because there are only four seats left to fill, and only four candidates remain in the contest, all four are declared elected.
Result: The elected candidates are: Andrea, Carter and Brad (from the Alpha party), and Scott, Jennifer, Matt and Susan (from the Beta party).
Count under the HagenbachBischoff quota
1. The HagenbachBischoff quota is calculated as 13.
2. When the first preferences are tallied Andrea, Carter and Brad (from the Alpha party) and Scott, Jennifer and Matt (from the Beta party) have all reached the quota and all six are declared elected. However this time the three elected Alpha party candidates each has a surplus of 1. These surpluses all transfer to Delilah so the tallies of the remaining candidates become:
 Delilah (Alpha party): 14
 Susan (Beta party): 12
3. Delilah has now reached a quota and is declared elected.
Result: The elected candidates are Andrea, Carter, Brad and Delilah (from the Alpha party) and Scott, Jennifer and Matt (from the Beta party).
Disadvantage of HagenbachBischoff quota
Another example shows how too many candidates can be elected. Imagine an election with three candidates for two positions where the 300 votes are
50 voters

150 voters

75 voters

25 voters

The HagenbachBischoff 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 HagenbachBischoff 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+1way tie, and eliminating one of the candidates at random; still another solution would call for a runoff between the candidates.