Comparison of the Hare and Droop quotas

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In elections that use the single transferable vote (STV) method, quotas are used (a) for the determination of candidates considered elected; and (b) for the calculation of surplus votes to be redistributed.[1] Two quotas in common use are the Hare quota and the Droop quota.

General comparison[edit]

The earliest versions of STV used the Hare quota. The Hare quota is equal to the total valid poll (V) divided by the total number of seats (n), or V / n.

The Droop quota is smaller than the Hare quota, and was first suggested [2] because it is the smallest quota that, like the Hare quota, ensures that the number of candidates who reach the quota will not be greater than the number of seats to be filled. Any quota smaller than the Droop quota carries a real, or at least theoretical, risk of more candidates being elected than there are seats to be filled. The Droop quota is the next integer larger than V / (n+1).

The difference between the two quotas comes down to what the quota implies. Winners elected under a Hare system represent that proportion of the electorate; winners under a Droop system were elected by that proportion of the electorate.

In an STV election in which there is only one seat to be filled (in other words an Instant Run-off Voting election) it is possible to use the Hare quota, which will simply be equal to 100% of votes cast. However, it is more efficient to use the Droop quota, which will be equal to an absolute majority of votes cast, meaning 50% plus one, and both quotas will achieve the same result. When voters have only one vote—the single non-transferable vote system—a candidate is sure to win if reaching the Droop quota.

In an STV election in which there are multiple winners the situation is slightly different, particularly with respect to the final seat.

  • The Hare quota is generally kinder to small parties than the Droop quota because they have a better chance to win the final seat. Elected winners with the Hare quota more closely represent the proportionality of the electorate, and this can mean more proportional results for small parties. But this comes at the expense of emphasising the principle of majority rule. In an open list election held under the Hare quota it is possible for a group of candidates supported by a majority of voters to receive only a minority of seats if those voters do not disperse their vote relatively evenly across all their supported candidates, see Scenario 1 below. In contrast, such an outcome will not happen in an election held under the Droop quota unless voters in the majority do not rank all their preferred candidates or not enough preferred candidates seek office.
  • The Droop quota is generally kinder to large parties because they have a better chance to win the final seat. This comes at the expense of emphasising the principle of proportional representation. In an election held under the Droop quota it is possible for a group of candidates to over-represent a proportion of voters even though a majority of the remaining voters support a minor party, see Scenario 2 below.

The Droop quota is today the most popular quota for STV elections - and almost universal for government STV elections - for two reasons[citation needed]. First, because it can more efficiently elect candidates in the each round of distribution of seats (whether STV or list PR) than is the case with the Hare quota. Second, because the possibility under the Hare quota that a group of candidates supported by a majority of voters to receive only a minority of seats is considered undemocratic[citation needed].

Examples of the different outcomes between the Hare and the Droop quotas follow:

Scenario 1[edit]

An example with STV where the result under the Droop quota more closely reflects the support that voters have for a party, irrespective of the support they have for individuals within the party.

Imagine an election in which there are 5 seats to be filled. There are 6 candidates divided between two groups: Andrea, Carter and Brad are members of the Alpha party; Delilah, Scott and Jennifer are members of the Beta party. There are 120 voters and they vote as follows:

Alpha party Beta party

31 voters

  1. Andrea
  2. Carter
  3. Brad

30 voters

  1. Carter
  2. Andrea
  3. Brad

2 voters

  1. Brad
  2. Andrea
  3. Carter

20 voters

  1. Delilah
  2. Scott
  3. Jennifer

20 voters

  1. Scott
  2. Delilah
  3. Jennifer

17 voters

  1. Jennifer
  2. Delilah
  3. Scott

It can be seen that supporters of the Alpha party all rank all three Alpha party candidates higher than any of the Beta party candidates (the final three 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 three preferences to Beta party candidates. Overall, the Alpha party receives 63 votes out of a total of 120 votes. The Alpha party therefore has a majority of about 53%. The Beta party receives a 47% share of the vote.

Below the election results are shown first under the Hare quota and then under the Droop quota. It can be seen that under the Hare quota, despite receiving 53% of the vote, the Alpha party receives only a minority of seats. When the same election is conducted under the Droop quota, however, the Alpha party's majority is rewarded with a majority of seats.

Note that this issue can be avoided under the Hare quota if the Robson Rotation is used to randomly order the party candidates.

Scenario 2[edit]

An example with a closed list using the largest remainder method.

Imagine an election in which there are 3 seats to be filled. There are 5 candidates divided between 3 groups: Alex, Bobbie and Charlie are members of the Alpha party; Jo is a member of the Beta party; and Kim is a member of the Gamma party. There are 99 voters and they vote as follows:

Hare quota scenario 2.jpg
Alpha party Beta party Gamma party

50 voters

  1. Alex
  2. Bobbie
  3. Chris

25 voters

  1. Jo

24 voters

  1. Kim

Scenario 3[edit]

October 2012 City of Melbourne, Australia Municipal Election[edit]

As a real life example of the implementation of the two quota systems and the impact it has on the outcome of the election results.

The City of Melbourne Council Elections were held in October 2012 with 9 vacancies to be elected from 40 candidates representing 11 teams plus three independents.

Under the Droop quota system the quota (x/(9+1)) was 10% with a wasted quota whilst using the Hare quota (x/9) the quota is 11.11% with no wastage

The total number of formal votes cast was 63,674, The Droop Quota = 6,468 (10.00%)and Hare Quota = 7,074 (11.11%). The distribution of surpluses and excluded candidate votes is determined by the voters order of preferences allocated to each candidate.

Table showing the percentage vote allocation and number of quotas under each system

10.00% 11.11%
Team No of votes Vote % Droop Quotas Hare Quotas
OUR MELBOURNE 3953 6.21% 0.62 0.56
STEPHEN MAYNE 3828 6.01% 0.60 0.54
RESIDENTS FIRST:STOP THE RATES RIP-OFF! 1929 3.03% 0.30 0.27
SHANAHAN CHAMBERLIN FOR MELBOURNE 3686 5.79% 0.58 0.52
COMMUNITY AND BUSINESS LEADERSHIP 1267 1.99% 0.20 0.18
FORWARD TOGETHER 528 0.83% 0.08 0.07
THE GREENS 9942 15.61% 1.56 1.41
TEAM DOYLE 23864 37.48% 3.75 3.37
MORGAN ELLIOTT- PROSPERITY FOR LIVEABILITY 6114 9.60% 0.96 0.86
GARY SINGER - JOHN SO MELBOURNE LIVING 8314 13.06% 1.31 1.18
Ungrouped 249 0.39% 0.04 0.04
Wasted Quota 0.99
Total Formal 63674 100.00% 10.00 9.00

Winning Candidates

Droop (x/(y+1))+1 Hare (x/y)
Team Candidate Candidate
TEAM DOYLE LOUEY, Kevin LOUEY, Kevin
THE GREENS OKE, Cathy OKE, Cathy
GARY SINGER - JOHN SO MELBOURNE LIVING ONG, Ken ONG, Ken
TEAM DOYLE WOOD, Arron WOOD, Arron
TEAM DOYLE PINDER-MORTIMER, Beverley PINDER-MORTIMER, Beverley
MORGAN ELLIOTT- PROSPERITY FOR LIVEABILITY WATTS, Jackie WATTS, Jackie
OUR MELBOURNE FOSTER, Richard FOSTER, Richard
STEPHEN MAYNE MAYNE, Stephen MAYNE, Stephen
SHANAHAN CHAMBERLIN FOR MELBOURNE CHAMBERLIN, Kevin
THE GREENS LEPPERT, Rohan


Team Doyle (Headed by Melbourne Lord Mayor Robert Doyle) polled 37.5% of the vote. Under the Droop Quota they elected only three representatives and the remaining 7.5% were disenfranchised by being allocated to the wasted quota.

The Greens, who polled just 15.1%, elected two representatives with major teams increasing disproportionally the overall value of their representation.

Using the Hare Quota system Team Doyle's surplus votes would have been distributed and allocated to elect community based candidate Kevin Chamberlin. Every vote is counted and allocated to form part of a candidate's quota in equal proportion.

Opponents of the Droop quota argue that each vote should be counted equally and voters that form part of the wasted quota should not be disenfranchised. In a manual count the Droop quota allows for a result to be determined without having to fully distribute all votes. Under a Hare quota all votes would need to be distributed in order to obtain a result. With computerized counting systems the full distribution and the time required is no longer a limiting factor.

In the City of Melbourne scenario it is argued that Greens vote was inflated resulting in them increasing disproportionally their vote and electing two candidates as opposed to just one under a pure proportional Hare system. It is further argued that there is no merit justifying the exclusion of the Team Doyle surplus in determining the results of the proportional ballot.

Using the Droop Quota up to 25% of the total vote in a three member electorate can be ignored, locked into what is referred to as the redundant "Wasted Quota" with voters disenfranchised as a result. In the case of the nine member City of Melbourne Council election the wasted quota is just below 10%.

Notes[edit]

  1. ^ Hill, I.D. (1987). "Algorithm 123 — Single Transferable Vote by Meek’s method".
  2. ^ Henry Richmond Droop, "On methods of electing representatives" in the Journal of the Statistical Society of London Vol. 44 No. 2 (June 1881) pp.141-196 [Discussion, 197-202], reprinted in Voting matters Issue 24 (October 2007) pp.7–46.

See also[edit]