Comparison of the Hare and Droop quotas

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. The largest remainder method of party-list proportional representation can also use Hare quotas or Droop quotas.

Under most STV systems it is possible to be elected by less than quota (at the end when the field of candidates is thinned to the number of remaining open seats) if not every ballot ranks all candidates, but the quota used (whether Hare or Droop) is the amount that guarantees election. A particular type of STV uses a "uniform quota" where anyone who exceeds that quota is elected. New York City used such a system during its use of STV in city elections. Its set quota was 75,000 and each borough got a number of members that depended on the voter turnout in the borough. One problem with the system was that the size of the city council varied from election to election depending on the voter turnout. (1.7M votes were cast and generally 20 to 27 councillors were elected.) ^[2]

The quota sets the number of surplus votes held by elected candidates. For fairness it is important that the number of votes held by the early successful candidate after surplus votes are transferred away is not surpassed later by another candidate who is unable to take a seat as they are already filled. That is the reason why the quota needs to be set at a point high enough to prevent more candidates achieving it than the total number of seats. ^[3]

General comparison

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 use of the Hare implies either there will not be any exhausted votes or one or more candidates must win with less than quota.

The Droop quota is smaller than the Hare quota, and was first suggested ^[4] because it is the smallest quota that ensures that the number of candidates who reach the quota will not be greater than the number of seats to be filled. The Hare quota also does this but is larger than it absolutely needs to be. Any quota smaller than the Droop quota carries a real, or at least theoretical, risk of more candidates being eligible for election than there are seats to be filled. The Droop quota is the next integer larger than V / (n+1) or a number just fractionally larger than V / (n+1), depending on the STV variant used.

In a ranked-vote 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, likely no candidate will take the Hare quota (due to exhausted votes) and such a target would necessitate vote transfers from eliminated candidates perhaps long after a candidate has a majority of the votes. it is more efficient to use the Droop quota, which will be equal to a simple majority of votes cast, meaning 50% plus one (one more than votes/(seats plus 1)). In single-winner elections, receiving a majority of the votes is considered to be a satisfactory marker of rightful success, and either Hare or Droop would achieve that result.

Election systems that use quota produce what is thought to be more fair results, where the elected member(s) represent the majority of votes cast, whether with a single winner or multiple winners. Systems that do not use quotas do not ensure such fairness - often under First-past-the-post voting and the Single non-transferable vote system the elected candidate(s) do not reflect the majority of votes cast.

In an STV election, where there are multiple winners, quota, not plurality, is used because it ensures that most successful candidates are elected with the same number of votes. If they receive surplus votes above the quota, the surplus votes are re-allocated, if possible. This equality ensure that a high proportion of votes cast are used effectively to elect someone, which in turn ensures that each party receives its due proportion of the seats.

Being of different sizes, the Hare and Droop quota produce different results in some situations.

• The Hare quota is generally kinder to small parties than the Droop quota Small parties have a better chance to win seats. This is because the most popular elected candidates under Hare does not transfer as many un-needed votes as it would under Droop. Smaller parties, especially parties that win just one seat, do not suffer from this fault as much as larger parties. The Hare therefore can mean more seats for small parties, sometimes even more than they are due proportionally. But this comes at the expense of majority representation.

  In an 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.

• Under certain circumstances, Droop quota may give a smaller party to have less chance to win seats. This comes at the expense of minority representation. In an election held under the Droop quota it is possible for a group of candidates to over-represent a small majority of voters and thus deny due representation to the remaining voters who make up a minority, see Scenario 2 below. In Scenario 3 below, we compare results under STV using Droop, and largest remainder method (where votes cannot cross party lines) using Hare and Droop. No party has majority of votes so majority/minority question does not enter into it. The most-popular party does not do as well under Hare as it does under Droop, under the largest remainder method. As well, Droop, being smaller than Hare, allows a small party running just one candidate to achieve quota easier. That party may have enough votes to take a seat under Droop but would not under Hare.

  A system using quota (and multiple winners) may leave nearly one quota's worth of votes held by unsuccessful candidates; these ballots are effectively ignored. Droop quota, being smaller, means there are fewer unrepresented voters than under Hare quota.

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 produces more proportional results for large parties and for small parties (although small parties as a whole are denied the over-representation produced by use of the Hare quota). Second, because a majority of votes in a group are more likely to fill a majority of the seats. The possibility under the Hare quota for 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

An example of an STV election where compared to the result under Hare, the result under the Droop quota would more closely reflect 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
Slate	Andrea Carter Brad	Carter Andrea Brad	Brad Andrea Carter	Delilah Scott Jennifer	Scott Delilah Jennifer	Jennifer Delilah Scott
Voters	31	30	2	20	19	18

Voters are voting for full slates. Supporters of the Alpha party all rank all three Alpha party candidates higher than any of the Beta party candidates. Similarly, voters who support the Beta party all give their first three preferences to Beta party candidates. The voters may not have indicated 4th, 5th and 6th preferences, but if they did or had to due to electoral rules, they are not shown above because they will not affect the result of the election.

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. The party's vote tally is spread over the party's three candidates.

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. (This is caused by hare being so large that two Alpha party winners take up so many votes that not enough are left over for a third Alpha candidate to avoid being eliminated as least-popular candidate. A less-even distribution of votes among the Beta party candidates might have saved Brad from this outcome.)

When the same election is conducted under the Droop quota, however, the Alpha party's majority is rewarded with a majority of seats. (More of the Alpha party winners' votes are seen as surplus and distributed. Brad receives the transfers and they give him quota, letting him win a seat.)

Count under the Hare quota

The Hare quota is calculated as 120/5 = 24.

1. When first preferences are tallied Andrea and Carter have both reached a quota and are declared elected.
2. Andrea has a surplus of 7 and Carter has a surplus of 6. Both surpluses are transferred to Brad (the next relevant preference indicated on the ballots) so the tallies become:
   • Brad (Alpha party): 15
   • Delilah (Beta party): 20
   • Scott (Beta party): 19
   • Jennifer (Beta party): 18
3. No candidate has reached a quota. There are just four candidates remaining and there are only three more seats to be filled. Brad is the candidate with the fewest votes and so he is declared defeated. Only Delilah, Scott and Jennifer remain and they are all declared elected.

Result: The elected candidates are: Andrea and Carter (from the Alpha party), and Delilah, Scott and Jennifer (from the Beta party).

Count under the Droop quota

The Droop quota is calculated as 21 (the next number higher than 120/6).

1. When first preferences are tallied Andrea and Carter have reached the quota and, as before, are declared elected.
2. Andrea has a surplus of 10 and Carter a surplus of 9. These surpluses transfer to Brad and the tallies become:
   • Brad (Alpha party): 21
   • Delilah (Beta party): 20
   • Scott (Beta party): 19
   • Jennifer (Beta party): 18
3. Brad has now reached a quota and is declared elected.
4. Brad has no surplus so no vote transfer is needed to deal with that. Three candidates remain and two seats remain open. Jennifer, who has the fewest votes, is declared defeated. Delilah and Scott are left in the count, and there are only two seats left to fill, they are both declared elected.

Result: The elected candidates are Andrea, Carter and Brad (from the Alpha party) and Delilah and Scott (from the Beta party). Thus the party with the majority of votes receives a majority of seats.

Scenario 2

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 Chris are members of the Alpha party; Jo is candidate for the Beta party; and Kim is candidate for the Gamma party.

There are 99 voters and they vote as follows:

	Alpha party	Beta party	Gamma party
Slate	Alex Bobbie Chris	Jo	Kim
Voters	50	25	24

Count under the Hare quota

1. The Hare quota is calculated as 33.
2. When votes are tallied Alpha party has more than one full quota so the first candidate on its list, Alex, is declared elected. Alpha party still has 17 not used.
   • Alpha Party (Bobbie and Chris) : 17
   • Beta Party (Jo): 25
   • Gamma Party (Kim): 24
3. No other candidate has reached the quota. Alpha is the party with the fewest votes and so its remaining candidates (Bobbie and Chris) are excluded. Because just two parties remain and there are only two more seats to be filled, the first (or only) candidate on the party list for each of those two parties, Jo and Kim, are declared elected.

Result: The elected candidates are: Alex (from the Alpha party), Jo (from the Beta party), and Kim (from the Gamma party). The Alpha party won a slight majority of votes but did not win a majority of seats. Taking one more seat would have given it a large amount of over-representation, which would have made the final result more dis-proportional than the equal distribution of seats to the three parties.

Count under the Droop quota

1. The Droop quota is calculated as 25.
2. When first preferences are tallied, Alpha party has two full quotas so Alex and Bobbie are declared elected, and Beta party has one full quota so Jo is declared elected.
3. With three candidates elected there are no more seats to fill.

Result: The elected candidates are Alex and Bobbie (from the Alpha party) and Jo (from the Beta party). The Gamma party is excluded from representation despite getting 24% of the votes. (This is an unusual case where one party did not have any un-used votes, and one party came up just one vote short of a full quota)

Scenario 3

October 2012 City of Melbourne, Australia municipal election

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, we will re-run the Melbourne eleciton as if it was conducted using largest remainder method (LRM).

The City of Melbourne council elections were held in October 2012, under the rules of STV, using the Droop quota, with 9 vacancies to be elected from 40 candidates representing 11 teams plus three independents. The system of vote counting was STV with a Gregory-style method used for transfer of surplus votes. ^[5]

For 9 seats, the Droop quota was 10% of votes, whilst the Hare quota would have been 11.11%. 63,674 votes were cast, so the Droop quota was 6,368, and the Hare quota would have been 7,075. The following table shows the percentage of first-preference votes received by each party's slate of candidates as a whole and the number of quotas this represents, under each quota, as if the largest-remainder system was in use. To calculate the final result under STV, back-up preferences must be taken into account as well. (They sometimes cross party lines.)

	First-preference votes
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
Total formal	63674	100.00%	10.00	9.00

Winning candidates (in the order of their winning in real life through STV in 2012):

Team	2012 Droop quota STV result	Hare quota result (LRM)	Droop quota result (LRM)
Team Doyle	Louey, Kevin	Louey, Kevin	Louey, Kevin
The Greens	Oke, Cathy	Oke, Cathy	Oke, Cathy
Gary Singer – John So Melbourne Living	Ong, Ken	Ong, Ken	Ong, Ken
Team Doyle	Wood, Arron	Wood, Arron	Wood, Arron
Team Doyle	Pinder-Mortimer, Beverley	Pinder-Mortimer, Beverley	Pinder-Mortimer, Beverley
Morgan Elliott – Prosperity for Liveability	Watts, Jackie	Watts, Jackie	Watts, Jackie
Our Melbourne	Foster, Richard	Foster, Richard	Foster, Richard
Stephen Mayne	Mayne, Stephen	Mayne, Stephen	Mayne, Stephen
Shanahan Chamberlin for Melbourne		Chamberlin, Kevin
The Greens	Leppert, Rohan
Team Doyle			Jetter, Carl

Five seats are allocated based on full quotas (3 Team Doyle, Gary Singer, Green) under both Hare and Droop. Three of the remaining seats, filled through vote transfers under STV or by largest remainders, are filled the same under both Droop and Hare.

One seat switches from Greens to Shanahan Chamberlin if Hare is used in largest remainder method versus STV.

Team Doyle (headed by Melbourne Lord Mayor Robert Doyle) received 37.5% of first-preference votes. Under Hare quota, they would elect three representatives, comprising 33.3% of the seats. Its remainder is not enough to give it a fourth seat under Hare but under Droop, the .75 quota left over is larger than almost every other fraction. so under Droop, it would take four seats.

The Greens, who received 15.1% of first-preference votes, elect two under STV but only one representative under largest remainder method, if either Droop quota or the Hare quota is used.

Kevin Chamberlin, who received 5.8% of votes, would have been elected if Hare LRM was used. Shanahan Chamberlin for Melbourne did not have enough first-preference votes to make quota under either system. But its fraction is enough to elect one candidate under Hare but not under Droop. (Under Droop, Team Doyle takes four seats, denying the seat to Chamberlin.)

Under the largest-remainder system, votes left over after full quotas fill seats may result in the winning of an additional seat. Seats are filled by comparing fractions and declaring elected the larger until seats are filled.) (This reflects how votes are transferred from a successful candidate to other candidates of the same party slate under STV.)

Under STV, under the Droop quota, the second Green candidate (Rohan Leppert) accumulated enough votes from transfers to win a seat, over either the first Chamberlin candidate or the fourth Doyle candidate. And so the Green party took a second seat under STV. ^[6]

But under the largest remainder method (with no transfers), using Droop, Team Doyle had enough votes to win a fourth seat.

The Hare quota requires a larger number of votes to elect a member, so reduces the number of votes available for the fourth Doyle member, putting Kevin Chamberlin ahead.

For awarding the final seat, the Droop quota is more favorable to the larger Team Doyle party, while the Hare quota is more favorable to the less-popular Shanahan Chamberlin slate. The Greens benefited from vote transfers when STV was used so took a second seat under STV. In addition, though, we see that the Hare quota, being larger than Droop, put the winning of a seat even farther out of reach for Residents First, Forward Together and other small parties.

Notes

1. Hill, I.D. (1987). "Algorithm 123 — Single Transferable Vote by Meek’s method".
2. "Proportional Representation in New York City, 1936-1947".
3. Harris, Practical Working of Proportional Representation (1930) https://babel.hathitrust.org/cgi/pt?id=mdp.39015030803426&view=1up&seq=7
4. 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.
5. "2012 council election results". Victorian Electoral Commission. Archived from the original on 2023-03-22. Retrieved 17 April 2023. Follow the link on this page to "Melbourne City Council". Archived 17 April 2023 at the Wayback Machine . For detailed results, follow the link on the previous page to the "Distribution Report" (Microsoft Excel spreadsheet), 28 October 2012. Archived 17 April 2023 at the Wayback Machine
6. Victorian Electoral Commission https://www.vec.vic.gov.au/results/general/MelbourneCouncillors2012.xls . Retrieved 18 May 2015.{{cite web}}: Missing or empty |title= (help)

See also

• List of democracy and elections-related topics