Hagenbach-Bischoff quota
Encyclopedia
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
' and vice versa (especially in connection with the Largest remainder method
) because the two are very similar. However, under the Hagenbach-Bischoff 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 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.
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.
Because there are 100 votes cast, and 2 seats, the Hagenbach-Bischoff is:
To begin the count the first preferences cast for each candidate are tallied and are as follows:
Andrea has more than 33+1/3 votes. She therefore has reached the quota and is declared elected. She has 11+2/3 votes more than the quota. These votes are transferred to Carter so the tallies become:
Carter has now reached the quota so he is declared elected. The winners are therefore Andrea and Carter.
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 Hagenbach-Bischoff quota. The problem is best illustrated by an example.
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 Hagenbach-Bischoff 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 Hagenbach-Bischoff quota, however, the Alpha party's majority is rewarded with a majority of seats.
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:
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).
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:
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).
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.
Proportional representation
Proportional representation is a concept in voting systems used to elect an assembly or council. PR means that the number of seats won by a party or group of candidates is proportionate to the number of votes received. For example, under a PR voting system if 30% of voters support a particular...
(PR). It is used in some elections held under the largest remainder method
Largest remainder method
The largest remainder method is one way of allocating seats proportionally for representative assemblies with party list voting systems...
of party-list proportional representation
Party-list proportional representation
Party-list proportional representation systems are a family of voting systems emphasizing proportional representation in elections in which multiple candidates are elected...
as well as in a variant of the D'Hondt method
D'Hondt method
The d'Hondt method is a highest averages method for allocating seats in party-list proportional representation. The method described is named after Belgian mathematician Victor D'Hondt who described it in 1878...
known as the Hagenbach-Bischoff system
Hagenbach-Bischoff system
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...
. The Hagenbach-Bischoff quota is named for its inventor, Swiss
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....
professor of physics and mathematics Eduard Hagenbach-Bischoff (1833–1910)
The Hagenbach-Bischoff quota is sometimes referred to as the 'Droop quota
Droop Quota
The Droop quota is the quota most commonly used in elections held under the Single Transferable Vote system. It is also sometimes used in elections held under the largest remainder method of party-list proportional representation . In an STV election the quota is the minimum number of votes a...
' and vice versa (especially in connection with the Largest remainder method
Largest remainder method
The largest remainder method is one way of allocating seats proportionally for representative assemblies with party list voting systems...
) because the two are very similar. However, under the Hagenbach-Bischoff and any smaller (e.g. the Imperiali
Imperiali quota
The Imperiali quota is a formula used to calculate the minimum number, or quota, of votes required to capture a seat in some forms of single transferable vote or largest remainder method party-list proportional representation voting systems....
) 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 Hagenbach-Bischoff quota should be used for elections under the Single Transferable Vote
Single transferable vote
The single transferable vote is a voting system designed to achieve proportional representation through preferential voting. Under STV, an elector's vote is initially allocated to his or her most preferred candidate, and then, after candidates have been either elected or eliminated, any surplus or...
(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:- 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.
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:45 voters
|
25 voters
|
30 voters
|
Because there are 100 votes cast, and 2 seats, the Hagenbach-Bischoff 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 33+1/3 votes. She therefore has reached the quota and is declared elected. She has 11+2/3 votes more than the quota. These votes are transferred to Carter so the tallies become:
- Carter: 36+2/3
- 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 quotaDroop Quota
The Droop quota is the quota most commonly used in elections held under the Single Transferable Vote system. It is also sometimes used in elections held under the largest remainder method of party-list proportional representation . In an STV election the quota is the minimum number of votes a...
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
Hare quota
The Hare quota is a formula used under some forms of the Single Transferable Vote system and the largest remainder method of party-list proportional representation...
but can still occur under the Droop quota in rare circumstances. It is a possibility that is only completely eliminated by use of the Hagenbach-Bischoff 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 comprising 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 Hagenbach-Bischoff 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 Hagenbach-Bischoff 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 Hagenbach-Bischoff quota
1. The Hagenbach-Bischoff 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 Hagenbach-Bischoff quota
Another example shows how too many candidates can be elected. Imagine an election with three candidates for two positions where the 300 votes are50 voters
|
150 voters
|
75 voters
|
25 voters
|
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.
See also
- CPO-STVCPO-STVCPO-STV, or the Comparison of Pairs of Outcomes by the Single Transferable Vote, is a preference voting system designed to achieve proportional representation. It is a highly sophisticated variant of the Single Transferable Vote system, designed to overcome some of that system's perceived...
- Schulze STVSchulze STVSchulze STV is a draft preference voting system designed to achieve proportional representation. It was invented by Markus Schulze who developed the Schulze method for resolving ties under the Condorcet method. It is similar to CPO-STV in that it compares possible winning sets of candidate...