Thursday, April 20, 2017

Wonk Central: Why We Don't Use The Hare Quota In Hare-Clark (Or The Senate)

For this exciting episode of Wonk Central I turn to the question of the Hare Quota, and why it is deservedly extinct in Single Transferable Vote multi-member electoral systems like the ACT and Tasmanian parliaments, and also the federal Senate and various state upper houses.  A warning that as usual for Wonk Central articles, this piece is especially mathsy.  A more important warning: I strongly advise readers with the slightest interest in the merits of different quotas for STV to stay well away from Wikipedia coverage of the matter.  It is so bad that I can't work out where to start in attempting to improve it.

The Hare Quota is best known to the psephosphere through the output of one Anthony van der Craats (aka MelbCity, democracyATwork etc) who continues to argue that the current Droop Quota disenfranchises voters and distorts outcomes in favour of the major parties.  An example is here (PDF); I am not sure if it is a representative example and I don't really care; any random one is enough to get the flavour of it.

Recently the Hare Quota made a surprise appearance in Bob Day's failed challenge against the new Senate laws (a challenge brought at a time when Day was not actually eligible to even be in the Senate, as it turns out).  Day asked the court to find in favour of the Hare Quota and against the Droop Quota for the Senate, but not much was seen of this claim in the actual arguments.  The court concluded that Day's argument against the Droop quota was "elusive", and that even had it been valid, there was no codified principle of proportional representation for a quota system to offend against anyway.

The Hare quota is sometimes used in multi-member elections that do not employ preferences.  While it can cause problems there too (such as encouraging parties to deliberately split into multiple tickets), this article is only concerned with its use in STV elections.

I hope that this piece will save people the effort of arguing with Hare-quota supporters who complain about the Droop quota.  While these people are extremely few in number they can be extremely persistent with their pet theory.  All you need to do if you see this nonsense is to thwack it with the link to this article.

The quotas

If a candidate gets a quota in one of these sorts of multi-member STV elections, they are elected.  If they have over a quota then spare votes are passed on to other candidates and the elected candidate is brought down to a quota in this process.  Whether these spare votes consist of all the candidate's votes at reduced value, some of them at reduced value, a selection at full or even increased value (etc) will vary by the system and the stage of the election at which the candidate has crossed the line.  The key question for this article is what that actual quota should be.

The Hare Quota is simply the number of formal votes divided by the number of seats.  If there are 42000 formal votes and 6 seats then the Hare Quota is 7000 votes.  The Hare Quota was used in some early implementations of STV but recent examples of its use in serious legislatures seem to not even exist.

The Droop Quota, now generally used instead, is:

(votes/(seats+1))+1  , rounded down.

So for 42000 voters and 6 seats, the Droop Quota is 6001 votes.

The argument for Droop Quota

The argument for the Droop Quota is that, for an election for n seats, the Droop Quota is the smallest number of votes that n candidates can reach, but more than n candidates cannot.  If the quota was 6000 votes, then in theory seven candidates could get exactly that many each, so there might be a need to break ties.  If the quota was higher than the Droop Quota, then not only might the count take longer than it needs to (an issue with hand-counting) but also votes could flow pointlessly as preferences to candidates who are already mathematically sure to win.

(In electoral systems that allow votes to exhaust, a common modification is to have a progressively reducing Droop quota rather than one fixed for the whole count.  Now that computers can easily handle this, it would be nice to see a lot more of this in Australian systems.)

Hare Quota: failed arguments and other problems

The wasted quota argument

The major argument made for the Hare Quota is that if there are n seats and Droop Quota is used, then around 1/n of the vote never elects anyone, but sits as an allegedly "wasted quota" of supposedly "disenfranchised" votes.  Quite often these votes pool with a candidate who is neither elected nor excluded - the candidate who comes seventh in an election for six seats, and loses because he/she is in last place, everyone has been excluded and there are no other votes left to distribute.

If we apply this to a single-seat preferential election, we should immediately notice how silly this is.  For a single-seat election, the Droop Quota is 50% of votes, plus one, rounded down, which is also known as an absolute majority.  Does this mean those whose votes finished with the losing candidate out of the final pair were "disenfranchised" and their votes were "wasted" just because they did not elect anybody?  No, it just means that when only one seat is elected it isn't possible that everyone preferences the winner.  Elections have winners and losers.

Also, when someone wins big in a single seat election, the supposedly wasted vote isn't 1/n - it's actually much smaller.

Both these points apply in multi-seat elections too.  Just as somebody has to come second in a single seat election, so also somebody has to come sixth in a race for five seats.  In the Hare-Clark or Senate systems using Droop quota this person is left without quota while the others are elected, but it happens in the Hare quota too.  For instance for a Hare Quota six-candidate election with five seats and 5000 voters, suppose the six candidates poll 1000, 900, 850, 800, 750 and 700 votes.  The candidate with 700 votes is excluded immediately and loses.  As all the other candidates will automatically win, and distributing the excluded candidate's preferences will not change the result of the election, so effectively 14% of the votes have been "wasted".  This isn't some scandal of disenfranchisement, it just means that six into five doesn't go.

And the number of votes left with the final failed candidate under Droop Quota is usually not that close to 1/n anyway, unless the outcome for the final seat is very close.  In Tasmanian House of Assembly elections, in theory the wasted quota is 16.67%, but in practice in 2014 the rump candidate finished with 11.1-13.7%.  Exhaust plays a role here, but even without it, margins of 2% or so for the final seat are not uncommon.  In theory one could keep distributing votes once the last candidate got a quota and thereby "fill up" the wasted quota, but why bother.  After all, the votes coming in have helped elect someone.

And indeed, it's often not understood that the votes that make up the "wasted quota" also include fractions of votes that have been used to elect successful candidates.  Even if there is a "wasted quota" of vote values, that doesn't mean there is a quota of voters who have not helped elect anybody.

Hidden Vote Wastage

What the Hare Quota devotees won't tell you when they talk about the "wasted quota" is that it is actually their system that wastes vote-values, only the vote-wasting is hidden through the count rather than sitting somewhere obvious.  Once a candidate has reached the Droop Quota, they are certain to be elected no matter what the quota is even if they don't get any more votes for the rest of the preference distribution.  So all the votes they get between the Droop Quota and the Hare Quota are unnecessary to them, and will just reduce the value of their surplus below what it would be under Droop.

This is most starkly seen if we look at the Hare-Clark system, in which only the last bundle of preferences that puts a candidate over the line are thrown again as a surplus, rather than all votes they have received being thrown again as in the Senate.  Suppose there are four seats and candidate A tops the poll and polls the Droop Quota (20%+1) on primaries. Candidate B polls last and polls 4%, all of whom put candidate A second. In a Droop quota election, candidate A is elected immediately.  Candidate B is excluded, and B's votes flow at full value to whover else is in the race.  In a Hare Quota election, Candidate B's votes flow to Candidate A even though Candidate A was already certain to be elected, and flow no further from this point.  Although they technically help elect someone, they really make no difference.  B's votes may as well have been thrown away after the primary count, so it is really the Hare quota system that is destroying voter intention.

Over the course of a Hare quota STV election, this hidden vote wastage can waste (n-1)/(n*(n+1)) of the value of all votes before the final seat gets decided - 15% for 4 seats, 13.3% for 5 seats, 11.9% for 6 seats and so on.  And this is genuine pointless vote value wastage, not the spurious conception of "waste" that comes from the fact that not everyone can always vote for a winner.  It's a point completely missed by those who think the use of the Droop quota is just a convenience for manual processing.

Minority Rules

A well-known paradox with the use of Hare quota for STV goes like this.  In a very polarised electorate six candidates for five seats run in two teams, A, B and C versus D, E and F.  The first team's voters all vote in the order A-B-C, meaning that A gets a primary vote of 52%.  The second team's voters split their votes, with D getting 17% of primaries, E 16% and F 15%.  A is elected with a surplus of 32% which flows to B.  B is elected with a surplus of 12% which flows to C.  Now C is last with 12% and loses.  As a result team A-B-C wins two seats with 52% of the vote, and team D-E-F wins three with 48%.

In contrast using the Droop system: the quota is now 16.67% so all of A, D, B and C are elected with quota, and the surpluses of D (0.33%) and C (2%) will then decide the last seat between E and F.  The more popular team wins three seats, as it should be.  Some might say parties should be encouraged to pick a range of candidates who can poll healthy primary votes in their own right, but it's not a party's fault if it happens to have a candidate who its voters all think is especially good.

This objection is in itself widely and rightly considered fatal to the use of the Hare quota in STV.


Supporters of the Hare Quota frequently argue that it is more proportional.  That it is more helpful for small parties generally compared to the Droop Quota is pretty obvious because the major parties cop hidden vote wastage every time a surplus is thrown.  The table below shows what share of seats a party would be guaranteed in a five-seat election, assuming a 100% down-the-ticket preference flow (but no preferences from any other party) for various sizes of primary vote under each system:

In the Hare quota system parties are only guaranteed a seat share exceeding their primary vote share if they poll above 90% (in this case they are certain to win the fifth seat without quota).  But in the Droop system there are bands where this occurs at as low as just over a sixth of a vote, and these become commoner as the vote increases.

However, it may well be that the Hare quota system would be too favourable for leading minor parties.  A good example is the 1998 Tasmanian state election, in which Labor (44.8%) won 14 seats, the Liberals (38.1%) won 10 seats, the Greens (10.2%) won 1 seat and Tasmania First (5.7%) won nothing.  The Greens were clearly diddled on a proportionality basis (more so than has been normal in the 25-seat House). Using the Hare quota, however, the seat result would have been 10-10-5, with Labor under-represented for their vote share and the Greens getting almost twice as high a share of seats than votes.  Indeed, the result of this 1998 election using the Hare quota would be exactly the same as for the 2010 election at which the Greens actually did break 20%!

In conclusion

The Hare quota is simply not a valid option for STV elections.  The system fails to prevent the so-called "wasted" vote issue in the Droop system, which actually isn't an issue at all, but in the process wastes votes itself (and lots of them.)  The system creates a risk of a minority beating a majority unfairly.  While it has been claimed that it results in greater proportionality, that claim has not been tested enough using real electoral data, and even if it were true, the other defects mean that the system is unusable anyway.

Note: Normally I reject posts that advocate the Hare Quota from this site but this thread is of course an exception.  If anyone wants a go at defending the indefensible, go right ahead.


  1. Well put, Kevin. One further related mark against the Hare Quota is that a candidate in any STV system is guaranteed to win the final (or sole) seat as soon as he or she has more than half of the votes remaining in the count. (Putting aside for the moment the issue of optional preferences exhausting, because that affects both the Hare and the Droop quotas). Under the Droop quota, in a race for five seats, with the quota being 16.6667%, all five candidates need a full quota to be elected. A candidate with "only" 16.6665% could still be defeated in a nail-biter. By contrast, under the Hare quota, in a race for five seats, with the quota being 20.0000% or (in some versions) 20.0001%, the first four candidates need a full 20% to be elected, but whichever candidate wins the fifth and last seat is home and hosed as soon as he or she passes 10.0001%. In other words, the final candidate - who is, all else being equal, likely to have lower first-preference support than the other four - needs only half as many votes to win a seat!
    Sure, you could keep eliminating the lowest and distributing their preferences until the fifth candidate does officially pass 20%, but (a) that's only a victory lap and (b) by exactly the same token, you could do the same under Droop Quota STV and increase the likelihood that all or most of the five with 16.6667% will edge up to, or over, 20%.
    Thomas Hare deserves accolades for coming up with the idea of a single transferable vote, but his version was a first draft that has been improved upon in the 150 years since. Like the Julian calendar, it's been superseded by a more finely-tuned Mark 2.0, and for good reason. (STV advocates have also dropped Thomas Hare's and JS Mill's model of a write-in ballot with 650 lines for names of candidates for all 650 Commons seats, elected at large.) Arguing that the Hare Quota gives better representation is the psephological equivalent of arguing that printing more dollar notes makes everyone wealthier: it has a certain intuitive appeal at first glance but doesn't hold up when you study it more closely.
    PS: Some Hare-quota advocates cheat by arguing that most party-list systems employ the Hare and not the Droop quota. This is true for the minority that use largest remainders, but for the majority that use D'Hondt highest averages (eg, the Hagenbach-Bischoff method used in Switzerland), it makes no difference to the ultimate allocation of seats which quota is used as an initial short-cut.

  2. Hi Kevin,

    Just trying to get my head around this concept of a Hare quota. I noticed that you have accidentally linked to a document on your C: drive.

    1. I thought I had fixed that link! It should be fixed now.

  3. Excellent walk through. I will definitely refer to this in future.