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Voting matters - Issue 13, April 2001

Difficulties with equality of preference

I D Hill

One of the things that some people do not like about STV is the fact that voters have to give a strict order of preference of those candidates whom they mention, where they would sometimes prefer to be allowed to express equality. Even where they are clear about the ordering of their first few preferences, and their last few, they may well wish to separate out their middle candidates from their high ones and their low ones without ordering those middle ones.

Instructions to voters

Difficulties arise in deciding how such equality is to be specified. Suppose candidate A is first preference, then B and C equal, followed by D. Some voters will wish to mark those first four candidates as 1, 2, 2, 3. Others will insist that logic requires 1, 2, 2, 4, while still others may want to use 1, 2½, 2½, 4. What is allowed has to be specified and made not too difficult to follow.

One way out of such difficulties is to say that any numbers the user may wish can be used, but only their order will be taken into account. But if such freedom is to be allowed to those who use equality, it must in fairness also be allowed to those not using equality. This disables some useful tests that can be made for correctness of data input to a computer file. Furthermore suppose someone uses 0; is this to be regarded as better than 1? Then suppose that there are 17 candidates in total and that one voter marks four candidates as 1, 2, 3, 4 while another marks four candidates as 1, 2, 3, 17. Did they both really mean the same thing? I doubt it.

Such difficulties are not fatal, but they need careful thought, and they may complicate the instructions to voters. If they lead to less secure input of data to the computer because of the checks that can no longer be made, that also matters.

Counting the votes

There are other difficulties though in how to count such votes. The basic idea is as set out by Brian Meek[1], that a vote for A(BC)D, where the brackets indicate equality of preference for B and C, should be treated as half a vote reading ABCD and half a vote reading ACBD, and similarly with equalities of more than two candidates. This needs careful handling to avoid a 'combinatorial explosion' if equality of large numbers of candidates is allowed.

However there is a difficulty of principle, rather than merely of the mechanics of the operation, that arises if voters choose to mention all candidates and to put two or more of them in equal last place. Meek's paper mentioned this possibility with approval, as allowing voters the option of indicating all remaining candidates as equal, as an alternative to not mentioning them at all. It is the one point in Meek's STV papers where I have to disagree with him, for allowing that option would mean having to explain to voters how to choose which method to use and what their different effects could be; not a task that I would wish on anyone. Or alternatively, just not to mention it, leaving voters uninformed about what they are doing.

The trouble is that there are two important principles in counting votes that are here in conflict:

  1. that a vote should be interpreted in accordance with what is actually written on it, and in no other way;
  2. that votes of identical meaning should be treated identically.
Now, with five candidates, for example, if one voter marks ABC as the first three preferences and stops there, while another voter marks ABC(DE), the strict interpretation of how to handle the two votes, once the fate of A, B and C has been settled, is different, but their meaning, in terms of preferences, is identical. If voters had been asked to express degrees of preference in some way, perhaps those two things might not be thought identical, but all that they have been asked for is an order of preference, and I cannot see how those two orders could possibly be thought different. This difficulty does not arise where equality is not allowed, since it so happens that two votes ABCD and ABCDE are treated identically by STV in any case, if those five are the only candidates.

There are three options: (1) to treat them differently even though their meanings are identical; (2) to treat both votes as if they had been ABC(DE); (3) to treat both votes as if they had been ABC. Of these I believe the third option to be the most satisfactory, in that there are cases where an abstention gives a better result than an equality of all remaining candidates, but I know of no case where the opposite can be claimed. (See Woodall's discussion of 'symmetrical completion'[2]). I have therefore adopted this approach in my STV computer program.

The difference comes out very clearly in the results of an actual election, that used my program and allowed equality. Some voters, believe it or not, put all the candidates (not merely enough to fill all seats) as equal first choice. The program did not blink an eyelid but put those votes at once into non-transferable, treating them merely as a new way of abstaining. Surely this is right, rather than the alternative of diluting the meaningful votes with this useless information.

Having decided on option (3) then, there arises yet another problem. One of the two fundamental principles on which the Meek system is based is 'If a candidate is eliminated, all ballots are treated as if that candidate had never stood'. Suppose then that we have 5 candidates and someone has voted AB(CD). The (CD) equality has to be included as these are not last places; it is important to the voter's wishes that C and D, though not differentiated from each other, are both preferred to the unmentioned E.

If E is now excluded, we must behave as if E had never been a candidate. With E gone, all four remaining candidates are mentioned and, in accordance with the option adopted above, the AB(CD) vote must now be treated as AB. Any part of the vote that was previously awarded equally to C and D now becomes non-transferable instead. This still treats them equally, of course, but it can have the odd effect that somebody's vote may go down in the course of the count, whereas normally votes can only go up until the candidate is elected or excluded. This is certainly an extra complication that one has to be ready to explain if it occurs.

Overall, my conclusion is that, although allowing equality has some advantages, and it can be implemented, the complications may be too many to be worth it. On the other hand, those bodies that have actually used it report no difficulties, and say that the facility is strongly valued by a significant number of electors.


  1. B L Meek, A new approach to the single transferable vote, paper II. Voting matters, Issue 1, 7-11.
  2. D R Woodall, Properties of preferential election rules. Voting matters, Issue 3, 8-15. (Elections 11 and 12 on page 14).

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