[EM] Simple Tournament Proposal

[Forest Simmons]

Here's my suggestion for choice of tournament champion:

Lacking an undefeated team, elect the pairwise victor of the defensive and
offensive champs.

In the sports context the natural choices of offensive and defensive
champs, respectively are the team with the greatest point total, and the
team with the smallest opposing team points total, both totalled for the
entire tournament.

In the election context the natural choices of offensive and defensive
champs, respectively, are the candidate with the greatest vote total, and
the candidate with the smallest opposing vote total ... noth totalled for
all of the direct democratic matchups between pairs of candidates.

However, in the context of elections the presence of clone candidates will
unfairly distort these results. This distortion is easily overcome by the
following modification:

The Offensive Champ is the candidate that can honestly say "everybody else
had a matchup where they got fewer votes than I did in my worst matchup."

The Defensive Champ is the candidate that can honestly boast, "Every other
candidate had a matchup where their opponent got more votes than mine did
in my worst matchup."

How do we get the matchup votes for candidates X and Y in their direct
democratic compsrison?

They are simply the transferred votes that they would have if all of the
other candidates were eliminated.

Candidate X would get one vote from every ballot on which X is ranked ahead
of Y.

Similarly, Y's vote total in this contest is the number of ballots that
rank Y ahead of X.

Now, some technical notes for the election technicians ... everybody else
thanked with heartfelt appreciation for their participation ... and excused:

The Pairwise Support Matrix is the matrix whose entry PS(j, k) in column k
of row j ... is the number of ballots on which candidate j is ranked ahead
of candidate k.

In each row j, circle the smallest entry ... m(j) = min over k of PS(j,k).

In each column k, highlight its largest entry M(k) = Max over j of PS(j, k).

Let J be argMax m(j), obtained by looking at the row number of the largest
circled entry in the matrix.

Candidate J is the Offensive Champ.

Let K be argmin M(k), obtained by looking at the column number of the
smallest highlighted matrix entry.

Candidate K is the Defensive Champ.

If PS(J, K) is larger than
PO(J, K)=PS(K,J),  then candidate J, the Offensive Champ defeats the
Defensive Champ K.

If PS(J,K) is less than PO(J,K), then the Offensive Champ J is defeated by
the Defensive Champ K.

(Otherwise candidates J and K are tied)

-Forest
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