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Voting matters - Issue 7, September 1996

On the lack of Convexity in STV

C H E Warren

Hugh Warren is a retired scientist

If the voters in a constituency are divided into two districts and the ballots are processed separately and the results in the two districts are the same, then there is said to be convexity if processing the ballots of all voters together gives the same result.

As Woodall has pointed out, quoting an example of David Hill's, STV does not satisfy convexity. We give here a further example, in which the lack of convexity arises, not from the elimination of candidates as in David Hill's example, but from the transfer of surpluses. We assume that these transfers are made by the method currently recommended by the Electoral Reform Society, and which the Electoral Reform Society uses for its own elections - the Meek method.

There are four candidates A, B, C, D, and three seats to be filled. The voting is as follows:

          District 1    District 2    Constituency
ABD           10           -              10
BAD            -          10              10
AC             -           8               8
AD             -           1               1
BC             8           -               8
BD             1           -               1
D              1           1               2
Totals        20          20              40
Elected     A,B,C       A,B,C            A,B,D
This further example reinforces Woodall's comment in the article quoted that ... sadly, convexity is of no use to us, as this seemingly ideal property conflicts with a more desirable property.


  1. D R Woodall, Properties of Preferential Election Rules, Voting matters, Issue 3, pp 8-15, December 1994.
  2. B L Meek, A new approach to the Single Transferable Vote, reproduced in Voting matters, Issue 1, pp 1-10, March 1994.

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