Hagenbach-Bischoff quota

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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.

Formula

The Hagenbach-Bischoff quota may be given as:^[1]

{\frac {{\mbox{total}}\;{\mbox{votes}}}{{\mbox{total}}\;{\mbox{seats}}+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 Hagenbach-Bischoff 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 Hagenbach-Bischoff 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 Andrea Carter	25 voters Carter	30 voters Brad

Because there are 100 votes cast, and 2 seats, the Hagenbach-Bischoff is:

{\frac {100}{2+1}}=33{\frac {1}{3}}

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 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, 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 Andrea Carter Brad Delilah	14 voters Carter Andrea Brad Delilah	14 voters Brad Andrea Carter Delilah	11 voters Delilah Andrea Carter Brad	13 voters Scott Jennifer Matt Susan	13 voters Jennifer Scott Matt Susan	13 voters Matt Scott Jennifer Susan	12 voters Susan Scott Jennifer Matt

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 are

50 voters Andrea Brad	150 voters Andrea Carter	75 voters Brad Carter	25 voters Carter Brad

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.

References

↑ Council of Europe, ed. (2008). Electoral Law. p. 199. ISBN 978-92-871-6424-7.

Types of majority

Single member	Plurality Majority Supermajority Double majority

Multi-member (election threshold)	Hare quota Droop quota Imperiali quota Hagenbach-Bischoff quota

Voting systems

Part of the Politics and Election series

Single-winner voting system	Approval voting Borda count Bucklin voting Contingent vote Coombs' method Copeland's method Exhaustive ballot First-past-the-post voting Instant-runoff voting Kemeny–Young method Simple majoritarianism Minimax Condorcet Nanson's method Plurality Positional voting system Range voting Ranked pairs Schulze method Two-round system

Proportional representation	Mixed-member Party-list Single transferable vote Schulze STV CPO-STV Highest averages method Sainte-Laguë D'Hondt Largest remainder method Alternative vote Plus Closed list Open list Overhang seat Underhang seat

Semi-proportional representation	Parallel voting Single non-transferable vote Cumulative voting Limited voting Proportional approval voting Sequential proportional approval voting Satisfaction approval voting

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Voting system criterion	Condorcet criterion Condorcet loser criterion Consistency criterion Independence of clones Independence of irrelevant alternatives Independence of Smith-dominated alternatives Later-no-harm criterion Majority criterion Majority loser criterion Monotonicity criterion Mutual majority criterion Pareto efficiency Participation criterion Plurality criterion Resolvability criterion Reversal symmetry Smith criterion

Voting system quotas	Droop quota Hagenbach-Bischoff quota Hare quota Imperiali quota

Other	Ballot Election threshold First-preference votes Spoilt vote Sortition

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