[EM] Help requested: rankings corresponding to pairwise array

[Andy Jennings]

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 <araucaria.araucana at gmail.com>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
>
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