If all you care about are margins, then you can build up the pairwise array from "atomic" pairs of voters.
For example, I consider ABCD and DCAB to be an atom for A>B because all other pairwise races cancel out, yielding the following margins matrix: | A | B | C | D | ===+=====+=====+=====+=====+ A | 0 | +2 | 0 | 0 | ---+-----+-----+-----+-----+ B | -2 | 0 | 0 | 0 | ---+-----+-----+-----+-----+ C | 0 | 0 | 0 | 0 | ---+-----+-----+-----+-----+ D | 0 | 0 | 0 | 0 | ---+-----+-----+-----+-----+ But I don't know how this fits in with the way you calculate approval. ~ Andy Jennings On Mon, Oct 10, 2011 at 2:51 PM, Ted Stern <[email protected]>wrote: > Hi, > > Say I have a pairwise array that looks like > > | A | B | C | D | > ===+=====+=====+=====+=====+ > A | 60 | 45 | 46 | 60 | > ---+-----+-----+-----+-----+ > B | 55 | 55 | 55 | 49 | > ---+-----+-----+-----+-----+ > C | 54 | 45 | 54 | 52 | > ---+-----+-----+-----+-----+ > D | 40 | 51 | 48 | 51 | > ---+-----+-----+-----+-----+ > > For this example, I assume that a tie between candidates is counted as > one vote for each candidate, and the diagonal entry is equal to the > maximum non-diagonal entry on that row. This is a way to extract > Approval from the pairwise array. > > The exact numbers are not important. What really matters to me is > that the candidates in descending order of approval are A, B, C, D, > and the pairwise outcomes look like > > | A | B | C | D | > ===+=====+=====+=====+=====+ > A | - | L | L | W | > ---+-----+-----+-----+-----+ > B | W | - | W | L | > ---+-----+-----+-----+-----+ > C | W | L | - | W | > ---+-----+-----+-----+-----+ > D | L | W | L | - | > ---+-----+-----+-----+-----+ > > The reason I'm looking for a set of ranked ballots that lead to this > outcome is that I believe it might be a counterexample to Forest > Simmons' Enhanced DMC proposal. > > If there is a set of rankings that lead to this array, then B would be > the winner under Schulze, Ranked Pairs, River and DMC, but Enhanced > DMC would pick either A or C. > > Ted > -- > araucaria dot araucana at gmail dot com > > ---- > Election-Methods mailing list - see http://electorama.com/em for list info >
---- Election-Methods mailing list - see http://electorama.com/em for list info
