The single transferable vote (STV) is a family of multiple-winner voting systems which provide proportional representation by allowing voters to rank candidates in order of preference.

This series provides a primer on the principles of STV, from its conceptual foundations through to contemporary developments in STV systems (such as the weighted inclusive Gregory and Meek methods of STV).

Single non-transferable vote

Suppose it is election season at the Suburbia Pet Owners' Association, and 3 members are to be elected to their committee. Imagine that voters are allowed to vote for one and only one candidate, and the successful candidates are to be the 3 candidates with the most votes.

This voting system is the single non-transferable vote (SNTV). It is used in legislative elections in countries such as Japan, and is not infrequently seen in the association or company context.

Imagine now that there are 3 broad groupings of opinion among the voters and candidates – the Dog Owners, the Cat Owners and the Fish Owners. Suppose that the votes are cast as follows:

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

In this scenario, the elected candidates under SNTV would be 2 Dog Owners and 1 Cat Owner – the 3 candidates with the most votes.

SNTV is compelling for its conceptual simplicity. At face value, every voter has the same voting power (1 vote), and all candidates are treated equally. Directions to voters are nominally simple: ‘vote for your favourite candidate’. Ballot papers are easy to count – just sort them by candidate and count them up. Results are simple to report and easy to interpret.

Beneath this veneer of simplicity, however, lie a number of fundamental problems, which in actual effect substantially complicate the voter experience in SNTV.

Problems with SNTV

In the above example, notice that if we add the votes across all of the Dog Owners and all of the Cat Owners, the Cat Owners in total in fact had more votes than the Dog Owners:

Candidate Votes
Dog Owners 40
Cat Owners 46
Fish Owners 14
Total 100

Let us proceed on the basis that Dog Owner voters prefer each of the Dog Owner candidates to any other candidate, and likewise for the Cat Owner voters.1 The natural comparison here is to political parties, but analogous reasoning applies to any other characteristic the voters attach importance to – gender, region, etc.

This demonstrates the first problem of SNTV: votes are wasted on candidates who are elected with more votes than they need. Notice that Cat Owner 1 received a whopping 40 votes, but they would have needed only 19 (like Dog Owner 2) to ensure they placed in the top 3. Cat Owner 1's extra 40 − 19 = 21 votes were effectively wasted – once Cat Owner 1 received enough votes to win, receiving more votes on top of that had no effect.

If some of these extra votes had instead been cast for Cat Owner 2, Cat Owner voters would have been able to secure the election of a second Cat Owner candidate. For example:

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  

In this way, SNTV encourages Cat Owner 1 voters to vote tactically for Cat Owner 2, whose election is uncertain, rather than their true favourite candidate, Cat Owner 1, who is already certain to be elected.

In this situation, now Fish Owner voters suffer from the second problem of SNTV: votes are wasted on candidates with no hope of election. Notice that in neither scenario is the Fish Owner candidate elected – they do not have enough votes to reach the top 3 compared with the other groupings. Fish Owner candidate's 14 votes were effectively wasted – because they had no chance of being elected, the Fish Owner voters may as well not have voted, or could have voted for someone else, and the Fish Owner candidate would be no worse off.

Conversely, the competition for the 3rd seat is quite close, as the Dog Owners and Cat Owners have similar levels of total support. Because their votes are wasted on the Fish Owner candidate, their votes may well be better spent on one of the Dog or Cat Owners. Suppose Fish Owners generally prefer the Dog Owners to the Cat Owners (cats and fish are not natural allies, after all!). Had the Fish Owner voters foreseen this outcome and cast their votes instead for the Dog Owners, they might have been able to tip the result the other way:

Candidate Votes  
Dog Owner 1 28 Elected
Dog Owner 2 26 Elected
Cat Owner 1 26 Elected
Cat Owner 2 20  
Fish Owner 0  
Total 100  

In this way, SNTV encourages Fish Owner voters to vote tactically for a Dog Owner, rather than for their true favourite candidate. In common parlance, the Fish Owner candidate is said to have split the votes, taking votes away from the Dog Owner candidates and advantaging the Cat Owners. In other words, the Fish Owner was a spoiler candidate.

To summarise, there are 2 major issues with SNTV:

  • votes are wasted on candidates who are elected with more votes than they need
  • votes are wasted on candidates with no hope of election

Each problem encourages affected voters to engage in tactical voting to avoid wasting their votes, and instead vote for a candidate who is not their true first preference.

Repeated SNTV

Naturally, in the above analysis we were only able to identify which candidates had ‘no hope of election’ or were going to be ‘elected with more votes than they need’ after seeing the result. In a real election, where the result is of course not known beforehand, it is quite difficult for voters to accurately work through this analysis, and decide how to optimally cast their votes.

One conceptually simple solution to this problem is to repeat the election an indefinite number of times, and allow voters to change their votes in between each round.

For example, in the first round, the votes might be cast as they were in the original example:

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

Cat Owner voters would then be able to see that Cat Owner 1 has more votes than needed, and coordinate amongst themselves to instead cast some of their votes for Cat Owner 2. Fish Owner voters would be able to see their candidate is unlikely to win, and instead cast their votes somewhere else.

Through repetition of this process, voters would eventually converge on a way to optimally cast their ballots, and produce a result which is the most representative on the whole.

Repeated SNTV maintains the conceptual simplicity of SNTV, and clearly remains fair to voters – in each round of counting, each voter has the same voting power (1 vote).

Repeated SNTV may be practical in the setting of our Suburbia Pet Owners Association, if all members were meeting together and had the time to conduct multiple rounds of voting and discuss in between. A similar process is used in United States politics when a political party is selecting its nominee at a ‘contested convention’.

However, repeated SNTV is clearly inappropriate when applied to general elections with thousands or millions of voters, where it is would be unconscionably costly to repeat the voting and counting multiple times. A system is needed which achieves the same result, without requiring voters to vote multiple times.

Outlining the single transferable vote

In part 2, we will outline the single transferable vote – 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.

Footnotes

  1. The technical term for this is a solid coalition. ↩