[EM] Beatpath winner

[Andrew Myers]

In Condorcet elections with the beatpath winner criterion, the
computation of the beatpath winner involves finding the strongest
beatpath connecting two candidates. To compute this one can run the
Floyd-Warshall algorithm on the vote matrix, but with the matrix
entries corresponding to a loss zeroed out. This zeroing out seems hard
to justify in some ways and I'd like to understand why it's really
necessary.

If candidate A directly beats candidate B 101 to 100, then the 100 votes
are thrown out even though there are 100 people who think that B beats
A. If candidate A ties with B 100 to 100, then both sets of votes are
thrown out. If A loses 100 to 101, then A's vote are tossed out.  This
computation seems very unstable to me because a small change in how
voters behave causes a potentially large change in the result.

What I'd like to understand is what bad things happen if you compute
beatpaths adding the links that go in the "losing" direction, with their
appropriate vote count. Clearly you get different answers for some
elections, but is there a good example that shows that the answers you
get are inferior?

-- Andrew Myers