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,DThis 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.