[EM] Election Methods and DanceSport

[Markus Schulze]

Dear Xavier Mora,

suppose "TBRC[i]" is the position of candidate i in the
"Tie-Breaking Ranking of the Candidates" (TBRC). Then
"TBRC[i] < TBRC[j]" means "candidate i is ranked higher
than candidate j in the TBRC".

Many different ways to rank pairwise defeats of otherwise
equal strength have been proposed. For example, Blake Cretney
suggested that when ij and mn each have the same strength
then ij should be ranked higher if and only if one of the
following conditions is met:

   1) pos[i] < pos[m].
   2) i = m and pos[n] < pos[j].

Blake Cretney wrote (22 Feb 2001):
> If you have two pairs of equal margin, you take the one whose winner
> is higher in the tie-breaking ranking first. (...) Where the winner
> is the same, you can order by loser. However, this is unnecessary,
> as pairs with the same winner can be processed in arbitrary order
> without affecting the result.

******

Rob LeGrand suggested that when ij and mn each have the same
strength then ij should be ranked higher if and only if one
of the following conditions is met:

   1) pos[n] < pos[j].
   2) n = j and pos[i] < pos[m].

Rob LeGrand wrote (27 Sep 2001):
> I realized, when a pairwise victory is locked, it usually hurts the loser
> much more than it helps the winner. So it seems more logical to order the
> pairs by loser than by winner. For example, if the tiebreaking ranking
> were A>B>C>D, I would lock a C>D before a B>A of equal margin. In my
> implementation, I gather together all the pairs (that are individually
> consistent with the locked pairs so far) with the highest margin that
> hasn't been locked or skipped. Of all the losers in that group, I pick
> the one lowest in the tiebreaking ranking, then I lock at once all the
> pairs in the group with that loser, which never causes a contradiction.

******

In your paper "Improving the Skating System", you refer to Rob LeGrand.
However, the method which is promoted in your paper is Cretney's one and
not LeGrand's one.

Markus Schulze