A conceptual primer on the single transferable vote – 2: Andrae's and Hare's methods
At the end of part 1, we provided an outline of STV:
a voting system which mimics the effect of repeated SNTV, where each voter has only 1 vote at a time, but where the voting system automatically redirects wasted votes from candidates elected with more votes than they need, and from candidates with no hope of election, without voters needing to physically vote again
In this part, we turn this outline into a complete (albeit primitive) implementation of STV.
The quota: how many votes are needed to win?
In part 1 and the description above, we have made repeated reference to candidates being elected with ‘more votes than they need’. Before continuing, we should formalise the notion of how many votes is enough.
Consider the Suburbia Pet Owners' Association election from part 1, with 100 voters electing 3 winners. Recall that in SNTV, the winners are the 3 candidates with the most votes. A relevant question, then, is ‘How many votes does a candidate need to receive to guarantee their election, regardless of how many votes the other candidates receive?’
The answer, in the 3 winner case, is ‘more than one quarter’, or with 100 votes, ‘at least 26’. No complicated mathematics is required to see why – it is possible for 4 candidates to each have a quarter of the votes (25 of 100), but it is not possible for any more than 3 candidates to each have more than a quarter (26 or more).
An equivalent way of framing this question is to ask ‘What is the smallest number of votes, such that no more than 3 candidates can have that many?’. The answer is termed the Droop quota. In general, for an election with $V$ votes and $S$ winners, the Droop quota $Q$ is given by:
\[Q = \left\lfloor\frac{V}{S + 1}\right\rfloor + 1\]Where $\lfloor …\rfloor$ is the floor operator, meaning to round down, discarding any remainder after the division.
From here on in, we will consider that the Droop quota is the number of votes needed for a candidate to win.^{1} When a candidate in an STV election reaches or exceeds the Droop quota, that candidate will be declared elected. Any votes the candidate would receive in excess of the quota is ‘more votes than they need’.
Andrae's method
Let's return to the Suburbia Pet Owners' Association election. Instead of requiring each voter to identify only their favourite candidate, let us allow each voter to rank the candidates in order of preference, from most preferred to least preferred. Now, instead of simply sorting and counting the ballots all at once, let us apply the following procedure:
Take each ballot one by one. We begin by allocating each ballot to its firstpreference candidate, just as it would be in SNTV. However, eventually a candidate might be allocated enough ballots to reach the quota and be elected. As we know, any further ballots would now be wasted on this candidate, and so any further ballots with a first preference for that candidate we allocate instead to the secondpreference candidate. We continue analogously as more candidates reach the quota and are elected.
As an example, suppose that the ballots are cast as follows – D1 and D2 are Dog Owner candidates 1 and 2, C1 and C2 are Cat Owner candidates 1 and 2, and F is the Fish Owner candidate; X > Y means that the voter prefers candidate X to Y:
Votes  Preferences 

21  D1 > D2 > F > C1 > C2 
19  D2 > D1 > F > C1 > C2 
40  C1 > C2 > F > D1 > D2 
6  C2 > C1 > F > D1 > D2 
14  F > D1 > D2 > C1 > C2 
100  Total 
Let us first distribute the 21 votes with a first preference for D1 and the 19 votes with a first preference for D2. No candidates reach the quota, so this so far is the same as in SNTV:
Candidate  Votes 

Dog Owner 1  21 
Dog Owner 2  19 
Cat Owner 1  0 
Cat Owner 2  0 
Fish Owner  0 
Total  40 
Quota  26 
Now let us distribute the 40 votes with a first preference for C1. 26 of these will be allocated to Cat Owner 1, allowing Cat Owner 1 to reach the quota and be elected. Giving Cat Owner 1 any more votes now would be wasteful, so the remaining votes can instead be allocated to their second preference, Cat Owner 2:
Candidate  Votes  

Dog Owner 1  21  
Dog Owner 2  19  
Cat Owner 1  26  Elected 
Cat Owner 2  14  
Fish Owner  0  
Total  80  
Quota  26 
Finally, we can distribute the remaining 6 votes with a first preference for C2, and the 14 votes with a first preference for F:
Candidate  Votes  

Dog Owner 1  21  Elected? 
Dog Owner 2  19  
Cat Owner 1  26  Elected 
Cat Owner 2  20  Elected? 
Fish Owner  14  
Total  100  
Quota  26 
Now, the top 3 candidates comprise 1 Dog Owner and 2 Cat Owners: by transferring excess votes from elected candidates, we can avoid those votes being wasted, solving one of the problems of SNTV.
This method was first devised by Carl Christopher Georg Andrae, Danish mathematician and then Minister for Finance, and applied in public elections from 1856.^{2}
Note, however, that this system does not solve the second problem of SNTV, where votes are wasted on candidates with no hope of election; for example, the 14 votes with a first preference for the Fish Owner candidate. As a consequence, only 1 of the candidates reached a full quota, and the others did not (shown in the table with an question mark).
As in the SNTV case, it is clear that if the 14 Fish Owner voters had instead voted for one of the Dog Owner candidates, 2 Dog Owner candidates could have reached the quota and been elected – a result which would be more preferable to the Fish Owner voters than electing 2 Cat Owner candidates.
Hare's method
In order to solve this second problem, where votes are wasted on candidates with no hope of election, we must introduce an additional rule. First, like when formalising the notion of a candidate being elected with ‘more votes than they need’, we must also formalise the notion of a candidate having ‘no hope of election’.
Let us say that, once all the votes have been distributed, if fewer than the required number of candidates have reached the quota, we will regard the candidate with the fewest votes as having no hope of election, and exclude them from the count.^{3} We then reexamine the votes for the excluded candidate, and reallocate them to their second preference.
Consider again the above example, once all votes have initially been distributed:
Candidate  Votes  

Dog Owner 1  21  
Dog Owner 2  19  
Cat Owner 1  26  Elected 
Cat Owner 2  20  
Fish Owner  14  
Total  100  
Quota  26 
Now, because only 1 candidate has reached the quota, the Fish Owner candidate should be excluded. The 14 votes with a first preference for the Fish Owner candidate had a next preference for Dog Owner 1. After transferring 5 of these, Dog Owner 1 would reach the quota and be elected:
Candidate  Votes  

Dog Owner 1  26  Elected 
Dog Owner 2  19  
Cat Owner 1  26  Elected 
Cat Owner 2  20  
Fish Owner  9  Excluding 
Total  100  
Quota  26 
Applying the principle from earlier, any additional votes for Dog Owner 1 would now be wasted, and so the remaining votes from the Fish Owner candidate can instead be transferred to their third preference, Dog Owner 2. After transferring 7 of these votes, Dog Owner 2 also reaches the quota and is elected:
Candidate  Votes  

Dog Owner 1  26  Elected 
Dog Owner 2  26  Elected 
Cat Owner 1  26  Elected 
Cat Owner 2  20  
Fish Owner  2  Excluding 
Total  100  
Quota  26 
We now have our winners! Dog Owner 1, Dog Owner 2 and Cat Owner 1 have each reached the quota and been elected. It is not possible for any other candidate to reach the quota, and so the votes have been distributed optimally – just as they were in repeated SNTV, but without requiring voters to repeatedly adjust their votes.
It is imperative to stress that every voter has still had equal voting power in this process. At each stage of voting, every voter has had 1 and only 1 vote, which has been counted for 1 and only 1 candidate. Although the vote may have been transferred between candidates, this is exactly the same as in repeated SNTV, where voters could adjust their votes between rounds.
We have just described what could be said to be the first complete implementation of the single transferable vote. It was proposed by Sir Thomas Hare, British lawyer, politician and electoral reform advocate, in this form in 1873.^{4}
Where to from here?
Hare's method is a complete and workable implementation of the single transferable vote. However, it suffers from one major shortcoming that limits its usefulness in the form described. In this part, we have referred to examining the ballots ‘one by one’ – in what order should we do this? For the sake of demonstration, we distributed the votes in order of candidate, but clearly, choosing a different order would change the order that the candidates are elected. Indeed, a different order could – by changing which votes are allocated to elected candidates and which are transferred to further preferences – change which candidates are elected.
To avoid bias, STV systems in use which examine ballots one by one do so in a random order. However, this means that the result can be influenced by random chance – an unsavoury outcome. In part 3 and subsequent parts, we will explore how to reduce (and eventually eliminate) the effect of random chance on the result.
Footnotes

Historically, the methods detailed subsequently (Andrae's and Hare's methods) in fact used the older Hare quota. However, the Hare quota does not correctly answer the question we have posed here, and modern implementations of STV invariably use the Droop quota. For simplicity, we will limit our discussion to the Droop quota. ↩

Hoag CG, Hallett GH. Proportional representation. New York: Macmillan; 1926. p. 171–5. https://babel.hathitrust.org/cgi/pt?id=mdp.39015030802907 ↩

Note that this is not necessarily the only choice that could be made to determine who has ‘no chance of election’. While it is the principle behind all forms of STV discussed in this series, some atypical variants of STV, such as BTRSTV and Sequential STV, exclude candidates according to different criteria. Other variants, such as CPOSTV and Schulze STV, retain STV's surplus distributions but determine outcomes in a different way which does not involve exclusions. ↩

Hare T. The election of representatives, parliamentary and municipal: a treatise. 4th edn. London: Longmans, Green, Reader and Dyer; 1873. p. 188. https://archive.org/details/electionreprese00haregoog ↩