Counting Single Transferable Votes

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There are many ways to count votes in Single Transferable Vote elections.

Contents

[edit] Voting

If a class of children were choosing representatives, say, they could line-up behind the candidate of their choice. Since they would all know that each candidate only needs a certain number of classmates to vote for them to be elected, those arriving last in line for a candidate who already has enough votes would choose to not waste their vote and would instead move to another line to help someone else to win. Likewise, those children whose candidate obviously could not win, would move to another line, and so on, until all the representatives are chosen.

When using an STV ballot, these preferences are set out in advance, as instructions to the counters.

Each voter ranks all candidates in order of preference. For example:

1 Andrea
2 Carter
3 Brad
4 Delilah

[edit] Setting the quota

[edit] Choice of quota

The quota (sometimes called the threshold) is the number of votes a candidate must receive to be elected. The Droop quota is preferred because it is the smallest number that ensures, if as many candidates as there are seats to be filled each have a quota of votes, no other candidate can have a quota. A candidate's surplus votes are transferred to other candidates according to the next available preference. In Meek's method the quota must be recalculated throughout the count.

With the Hare quota even if each voter expresses a preference for every candidate, at least one candidate is likely to be elected with less than a full quota. If each voter expresses a full list of preferences, the Droop quota guarantees that every candidate elected will meet the quota rather than be elected by being the last remaining candidate after lower candidates are eliminated.

[edit] Droop quota

The most common formula for the quota used now is the Droop quota which is most often given as:

\left({{\rm votes} \over {\rm seats}+1}\right)+1
The Droop Quota

Unlike the Hare quota, this does not require that all preferences must reach a final home. It is only necessary that enough votes be allocated to ensure that no other candidate still in contention could win. This leaves nearly a quota's worth of votes unallocated, but it is held that this quota simplifies voting, and that counting these votes would not alter the eventual outcome.

[edit] Hare quota

When Thomas Hare originally conceived his version of Single Transferable Vote, he envisioned using the quota:

\rm votes \over \rm seats
The Hare Quota

This has thus become known as the Hare quota. It would require that all votes cast be divided equally between the eventually successful candidates. The only differences, thus, between the votes received for each candidate would be based on the distribution of voters between constituencies (Hare's original proposal was for a single national constituency) and the number of non-continuing votes, i.e. people who did not express a preference for all candidates, meaning that some candidates would be elected with less than a quota as the last remaining.

[edit] Counting the votes

Process A: First preference votes are tallied. If one or more candidates receive a quota of votes, they are declared elected. After a candidate is elected, she may not receive any more votes (though see below for a modernisation).

The excess votes for the winning candidate are reallocated to the next-highest ranked candidates on the ballots for the elected candidate. There are different methods for determining how to reallocate excess votes. (See below)

Process A is repeated until there are no more candidates who have reached the quota.

Process B: The candidate with the least support is eliminated, and his votes are reallocated to the next-highest ranked candidates on the eliminated ballots. After a candidate is eliminated, he may not receive any more votes. (But see Meek's method below)

After each iteration of Process B is completed, Process A starts again if any candidates are elected, until all candidates have been elected or eliminated. Process B cannot be resumed while there is any elected candidate with excess votes to be transferred.

[edit] Surplus re-allocation

The votes an elected candidate receives in excess of the quota constitute a surplus. To minimise wasted votes, these are transferred to the remaining unelected candidates based on the next stated preference. Several methods exist to decide which of the candidate's votes to select for transfer. Some are usually only applied to the initial surplus (when an unelected candidate first exceeds the quota); others are also applied to subsequent surpluses (when an elected candidate receives further transferred votes).

[edit] Randomisation

Some of the surplus allocation methods rely on selecting a random sample of the votes. Ensuring randomness is done in various ways. In many cases, all the relevant ballot-papers are simply manually mixed together. In Cambridge, Massachusetts, votes are counted one precinct at a time, imposing a spurious ordering on the votes. To prevent all ballots for transfer being selected from the same precinct, every nth ballot paper is selected, where \begin{matrix} \frac {1} {n} \end{matrix} is the fraction to be selected.

[edit] Initial surplus

Suppose candidate X, at a certain stage of the count, has 190 votes, and the quota is 200. Now X receives 30 votes transferred from candidate Y (after Y was either elected or eliminated). This gives X a total of 220 votes, i.e. a surplus of 20 to be transferred. But which 20 votes will be transferred?

[edit] Hare's method

20 votes are drawn at random from the 30 received from Y's transfers. These 20 votes are each transferred to the next available preference after X stated on the ballot. In a manual count of paper ballots, this is the easiest method to implement; it was Thomas Hare's original proposal in 1857. It is used in all universal suffrage elections in the Republic of Ireland. However, there is no guarantee that the 30 transfers from Y express later preferences similar to those of the other 190 votes previously obtained by X. Thus it is potentially unfair, and leaves open the possibility of tactical voting. Also, exhausted ballots are excluded, so if more than 10 of the 30 votes have no preference stated after X, then it is impossible to select 20 to transfer and so some votes must be wasted.

[edit] Cincinnati method

20 votes are drawn at random from all 220 votes. This is used in Cambridge, Massachusetts (where every 11th vote ( (220-200)/220 = 1/11 surplus) would be selected for transfer). It is more likely than Hare's method to be representative, and less likely to suffer from too many exhausted ballots. However, the random element is still present. This may tip the balance in close elections; also, if a recount of the votes is required, it must be ensured that the same sample is used in the recount (i.e. the recount must only be to check for mistakes in the original count, not to try another random lottery selection of votes).
If a candidate exceeds the quota on the first count (i.e. purely with first preference votes), Hare's method and the Cincinnati method will be equivalent, since all the candidate's votes are in the "last batch received" from which the Hare surplus is drawn.

[edit] Clarke method

All 220 votes are divided into separate bundles, based on the next stated preference. An equal proportion of ballots is drawn at random from each bundle and transferred to the relevant candidate. The proportion is \begin{matrix} \frac {\mbox{surplus}}{\mbox{(total - exhausted)}} \end{matrix}.
In the example, if we assume there are 40 ballots expressing no preference after X, then the proportion is \begin{matrix} \frac {20} {220 - 40} \end{matrix} = \begin{matrix} \frac {1} {9} \end{matrix}. For example, if 54 of X's 220 votes state a next preference for Candidate A, 90 for B and 36 for C, then 6, 10 and 4 votes will be drawn from the 3 respective bundles and transferred to the respective candidate. This is the method used in Australia. If there are fractional votes, the Australian practice is to round down: if the A:B:C vote split was 52 : 88 : 40 (i.e. 5.8 : 9.8 : 4.4) the transfer would be 5:9:4, so only 18 votes are transferred instead of 20. The number of such "lost" votes is always less than the number of remaining candidates; in practice this is a very small proportion since the number of votes is much larger than the number of candidates. This method reduces but does not eliminate the randomisation issues of the Cincinnati method.

[edit] Gregory method

Another method is known variously as the Senatorial rules (after its use for most seats in Irish Senate elections), or the Gregory method (after its inventor in 1880, J.B. Gregory of Melbourne). This eliminates all randomness and in effect continues Clarke's method through all subsequent transfers. Instead of transferring a fraction of the votes at full value, one transfers all the votes at a fractional value. The relevant fraction is the same as in the Clarke method, i.e. \begin{matrix}\frac19\end{matrix} in the example. Note that part of X's 220 vote total may already be composed of fractions from earlier transfers; e.g. perhaps Y had been elected with 250 votes, 150 with X as next preference, so that the previous transfer of 30 votes was actually 150 ballots at a value of \begin{matrix} \frac15 \end{matrix} of a vote each. In this case, these 150 ballots would now be retransferred with a compounded fractional value of \begin{matrix} \frac15 \end{matrix} \times \begin{matrix} \frac19 \end{matrix} = \begin{matrix} \frac{1}{45} \end{matrix}. In practice, the transferred value of a ballot is usually expressed not as a fraction but as a decimal, rounded to 2 or 3 places. To simplify tallying, the initial votes may be given a nominal value of 100 or 1000 to remove the decimal point.
Calculating compound fractions is labour-intensive, so in the Republic of Ireland the method is used only for the Senate whose franchise is restricted to c. 1500 councillors and members of Parliament. However, in Northern Ireland the method has been used for all STV public elections since 1973, with up to 7 transfers (in 8-seat district council elections), and up to 700,000 votes counted (in 3-seat European Parliament elections).

[edit] Subsequent surplus

All the above methods apply only to the transfer of an initial surplus, when a previously unelected candidate exceeds the quota of votes for the first time during the count. A similar question arises where a ballot is to be transferred and the next stated preference is for an already-elected candidate. The common practice has been simply to ignore this preference and transfer the ballot instead to the next unelected (and uneliminated) candidate. This is in effect Hare's method and suffers from all the deficiencies of that method.

In principle it would be possible to apply one of the other methods. Suppose a previously-elected candidate X receives 20 transfers from a newly-elected candidate Y in addition to the quota of 200 previously retained: for the Cincinnati method, mix all X's 220 votes and select 20 at random for transfer from X. However, the problem with this is that some of these 20 ballots may transfer back from X to Y, creating recursion. This is messy; in the case of the Senatorial rules, since all votes are transferred at all stages, it will be an infinite recursion, with ever-decreasing fractions applicable.

[edit] Meek's method

In 1969, B.L. Meek devised an algorithm based on Senatorial rules, which uses an iterative approximation to short-circuit this infinite recursion. It requires computer counting. This system is currently used for some local elections in New Zealand.

All candidates are allocated one of three statuses - Hopeful, Elected, or Excluded. Hopeful is the default. Each status has a weighting, or keep value, which is the fraction of the vote a candidate will receive for any preferences allocated to them while holding that status.

The weightings are:

Hopeful 1
Excluded 0
Elected wnew = wold \times \left( \frac{\mathrm{Quota}} \mathrm{Candidate's\ votes} \right)

which is repeated till wnew = wold for all elected candidates

Thus, if a candidate is Hopeful they retain the whole of the remaining preferences allocated to them, and subsequent preferences are worth 0.

If a candidate is Elected they retain the value of their weighting and the remainder of the value of the vote is passed along fractionally to subsequent preferences depending on their weighting, with the formula

\left (1 -{\mathrm{nth Weighting}} \right)

being carried out at each preference.

This results in a fractional excess, which is disposed of by altering the quota, hence Meek's method is the only method to change quota mid-process. The quota is found by

\left({{\rm votes - excess} \over {\rm seats}+1}\right),

a variation on the Droop quota. This has the effect of also altering the weighting for each candidate.

This process continues until all the Elected candidates' vote values almost equal the quota (within a very close range, i.e. between 0.99999 and 1.00001 of a quota).[1]

[edit] An example

Suppose we conduct an STV election using the Droop quota where there are two seats to be filled and four candidates: Andrea, Brad, Carter, and Delilah. Also suppose that there are 57 voters who cast their ballots with the following preference orderings:

16 Votes 24 Votes 17 Votes
1st Andrea Andrea Delilah
2nd Brad Carter Andrea
3rd Carter Brad Brad
4th Delilah Delilah Carter

The threshold is: \left({57 \over (2+1)}\right) +1 = 20

In the first round, Andrea receives 40 votes and Delilah 17. Andrea is elected with 20 excess votes. Her 20 excess votes are reallocated to their second preferences. For example, 12 of the reallocated votes go to Carter, 8 to Brad.

As none of the remaining candidates have reached the quota, Brad, the candidate with the fewest votes, is eliminated. All of his votes have Carter as the next-place choice, and are reallocated to Carter. This gives Carter 20 votes and he is elected, filling the second seat.

Thus:

Round 1 Round 2 Round 3
Andrea 40 20 20 Elected in round 1
Brad 0 8 0 Eliminated in round 2
Carter 0 12 20 Elected in round 3
Delilah 17 17 17 Defeated in round 3

[edit] References

  1. ^ Hill, I. David; B. A. Wichmann and D. R. Woodall (1987). "Algorithm 123 — Single Transferable Vote by Meek's Method". The Computer Journal 30 (2): 277-281. 

[edit] See also

[edit] External links

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