In part 2, we described the original 1873 implementation of STV by Thomas Hare, noting that the method then proposed was highly influenced by the effect of random chance. In this part, we will introduce some strategies to reduce this effect somewhat.

For the sake of illustration, we will slightly adjust the votes in the Suburbia Pet Owners' Association election to be slightly less homogeneous:

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
12 F > D1 > C1 > C2 > D2
2 F > D1 (and no further preferences)
100 Total

Moving from ballots to parcels

In part 2, we began by examining the ballots one by one, and allocating each to its first-preference candidate, immediately declaring elected any candidate who reached the quota after the transfer of any single ballot. How this unfolds would naturally then be subject to the order that the ballots are taken.

Instead of doing that, let us say that we will distribute all of the ballots to their first preferences, exactly as in SNTV. Only once all ballots have been distributed will we check to see if any candidate has reached the quota. For reasons that will later become apparent, we will refer to all of the ballots so distributed to each particular candidate as a parcel.

By checking to see if any candidates have reached the quota only once all ballots are distributed, this first stage then becomes deterministic – independent of any random effects. This is a desirable quality – and also simplifies hand-counting, as the ballots can simply be independently sorted according to first preference.

For the Suburbia Pet Owners' Association election, the standings after the count of first preferences would be:

Candidate Votes  
Dog Owner 1 21  
Dog Owner 2 19  
Cat Owner 1 40 Elected
Cat Owner 2 6  
Fish Owner 14  
Total 100  
Quota 26  

Dealing with surpluses on first preferences

A problem arises now, however. Because we check for candidates reaching the quota only once all ballots are distributed, Cat Owner 1 has been elected on more than a quota. As discussed for SNTV, this is a problem because the extra votes received by the elected candidates in excess of the quota are effectively wasted.

In Andrae's and Hare's methods from part 2, these excess ballots would have been automatically redirected to their second preferences. So let's do that now. Cat Owner 1 has 40 − 26 = 14 votes more than the quota – this is said to be Cat Owner 1's surplus. We will randomly select 14 of Cat Owner 1's votes, and transfer them to their second preferences – in this case, Cat Owner 2:

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  

We will refer to the 14 votes transferred in this stage from Cat Owner 1 to Cat Owner 2 collectively as a parcel, noting they were received by Cat Owner 2 separately to the parcel of first-preference votes from earlier.

It is again imperative to stress that, throughout this process, every voter has still had equal voting power. Notice that the standings at this point are exactly the same as they were in part 1 in an analogous SNTV example, and each voter's vote counts at each stage for 1 and only 1 candidate. The votes have simply been transferred between candidates, just as voters could change their votes in repeated SNTV.

Dealing with consequential surpluses

At this point, there are no more surpluses to distribute (no candidates have more votes than they need). Just like in Hare's original method, as fewer candidates have been elected than there are vacancies to fill, we must exclude the candidate with the least votes, who is regarded as having no hope of election.

Again, just like in Hare's original method, we transfer Fish Owner's 14 votes to their next preferences – in this case, Dog Owner 1. However, unlike Hare's original method, we will apply the same principle from earlier in this part, and transfer all of those votes before checking to see if anyone has reached the quota.

After transferring Fish Owner's 14 votes to Dog Owner 1, Dog Owner 1 now has 35 votes. This exceeds the quota, and so Dog Owner 1 is now elected:

Candidate Votes  
Dog Owner 1 35 Elected
Dog Owner 2 19  
Cat Owner 1 26 Elected
Cat Owner 2 20  
Fish Owner 0 Excluded
Total 100  
Quota 26  

Dog Owner 1 now has more votes than required to be elected, with a surplus of 35 − 26 = 9 votes. We should, then, transfer 9 of Dog Owner 1's ballots to their next preferences.

However, note that Dog Owner 1's votes comprise 2 parcels: one parcel of 21 first-preference votes, and one parcel of 14 votes transferred during the exclusion of Fish Owner. Should the 9 votes to be transferred be drawn from both parcels, or only one?

Note that in Hare's original method, where ballots were considered one by one, all of Dog Owner 1's first-preference votes would have been allocated to Dog Owner 1 and would never have had an opportunity to be redirected to their second preferences, as Dog Owner 1 was not elected during the initial distribution of ballots. Only during the exclusion of Fish Owner would Dog Owner 1 have reached the quota, and further ballots from Fish Owner's exclusion be redirected to their next preferences. Therefore, it seems sensible to say that only the parcel of 14 votes last received by Dog Owner 1 – the parcel which put Dog Owner 1 over the quota – should contribute to the transfer of the surplus.

There is another, more pragmatic, reason for making this choice: When counting an election by hand, it would be more efficient to limit the number of ballots we must examine at any time. Examining only the last parcel of votes achieves this goal.1

Within the 14 ballots received from Fish Owner, however, there is further distinction. Only 12 of those ballots listed a next available preference (for Cat Owner 2, skipping Cat Owner 1 who has already been elected and needs no more votes) – these ballots are said to be transferable. The other 2 ballots did not list any further preferences – they are said to be non-transferable. Should the 9 ballots to transfer be drawn from all 14 in the parcel, or only the 12 transferable ballots?

Again, applying the pragmatic logic of hand-counted elections, it would be efficient to limit the number of ballots to examine, so let us answer that we will consider only the 12 transferable ballots.

We can now proceed to distribute Dog Owner 1's surplus. We randomly select 9 of the 12 transferable ballots and reallocate them to their next preference – in this case, Cat Owner 2:

Candidate Votes  
Dog Owner 1 26 Elected
Dog Owner 2 19  
Cat Owner 1 26 Elected
Cat Owner 2 29 Elected
Fish Owner 0 Excluded
Total 100  
Quota 26  

Cat Owner 2 now meets the quota and is elected. Because there were 3 vacancies, and 3 candidates have been elected, the count is now complete.

Reflections so far

In this part, we explored a method for reducing the effect of random chance (due to the order that ballots are considered) present in Hare's original method. This refined method remains in use in some STV elections today. Most notably, it is the method used (with minor variations) to elect the Seanad Éireann,2 and the New South Wales Legislative Council.3

However, this method does not completely remove the effect of random chance.4 When choosing which ballots to transfer during a surplus distribution, a random selection is still involved.

In part 4, we will introduce the Gregory method, which completely removes randomness from the surplus distribution process, and hence from the entire count.

Footnotes

  1. Another way to think about this choice is that we have tried as far as possible to keep different ballots separate to each other – an exclusive approach, keeping the parcel of first preference ballots separate to the parcel received from Fish Owner's exclusion. Some exclusive methods go even further with this approach, performing exclusions parcel by parcel (rather than all at once, as we have described), and keeping all such parcels forever separate during the count. 

  2. Electoral Act 1992 (Ireland), Part XIX. https://www.irishstatutebook.ie/eli/1992/act/23/enacted/en/print 

  3. Constitution Act 1902 (NSW), Schedule 6. https://legislation.nsw.gov.au/view/html/inforce/2022-04-13/act-1902-032 

  4. Some systems attempt to further reduce this effect; for example, by performing a stratified random sample of the ballots according to next preference, such that the number of surplus votes transferred to each next preference is deterministics. Nevertheless, some random effect necessarily remains in the preference flows thereafter.