[EM] margins vs. winning votes

[Richard Moore]

This poll is forcing me to think about the difference between various
Condorcet methods,
which is something I've been putting off.

On the topic of margins vs. winning votes:

Draw an equilateral triangle. The top vertex represents all ties. The
lower right represents
winning votes, and the lower left represents losing votes.

Drop a vertical line from the top of the triangle to the base. Each
pairwise contest can
be mapped into a point on the right-hand side of the triangle. (Each
point can be labeled
with an order indicating the winner, so that the point representing A
vs. B is labeled AB
if A wins, BA if B wins).

For points in a cycle, dropping the contest with the smallest margin is
equivalent to
dropping the left-most point in the cycle. The left-most point is
closest to the vertical
line. If this point is XY, then it is the point in the cycle for which
the fewest votes would
have to change to move it to the left-hand side of the triangle (at
which point we replace
it with its mirror image point YX on the right-hand side). Another way
to picture this
is to imagine all the points in the cycle orthogonally projected onto
the triangle's base,
an operation equivalent to splitting the tie votes evenly between the
two candidates.

Using winning votes is equivalent to a projection down and to the left
along a 60
degree slope to the base of the triangle. This is equivalent to
converting all the tie votes
to losing votes.

I think the margins approach is better because it treats tie votes as
ambiguous, rather
than as losing votes. One other approach would be to allocate the tie
votes to the
winner and loser in proportion to the actual numbers of winning and
losing votes,
which is a projection onto the triangle's base from a focus at the top
corner of
the triangle.

Richard