Goldfish (single-winner method)

[Blake Cretney]

Here's my entry for single-winner system of the week. It was motivated
by my desire to make a method that would be easy to program.  To this
end, it does not require cycles or the Smith set to be found.
I'll call it "Goldfish" until someone shows me a previous mention 
under a different name.  The idea of goldfish is that the candidates
seem to eat each other, becoming bigger and bigger, until
only one is left swimming in the electoral fish bowl.

Goldfish definition:
Successively find the worst defeat and eliminate the pair-wise loser.  
Any win achieved by the pair-wise loser is now scored as if it was
achieved by the pair-wise winner, provided it is larger than the
one already scored by him, or he is currently scored a loss.

Start by making a "victory" table.  For each row, enter the votes
against each column's candidate, if the row's candidate wins
pair-wise.  Otherwise enter a 0.

The best way to resolve ties is for a chairman, president, or random
voter to enter a special ballot.  This ballot must not be truncated.

Repeat until only one candidate is left:
    FIND:
    Find the highest value in the table.  Call this cell i,j.  If
    more than one row share this value, choose the row that is
    higher in the special ballot.
    MERGE:
    Here's where the big fish eats the little one.  For each cell
    in the i row, if there is a higher value for that column in
    the j row, copy it over.  For each cell in the i column, if
    there is a zero for that row in the j column, copy it over.
    Do not change the empty cells on the diagonal.
    ELIMINATE:
    Remove the j candidate and its row and column from
    consideration.

I'm going to use the word "beats" to mean "defeats pair-wise" and
"eats" to mean "is chosen to defeat and merge with".

MIIAC -- Candidates outside the Smith set are always beaten by
members of the Smith set.  When they eat them, the rows and
columns are merged, but this provides nothing of use for beating
other Smith members, because candidates outside the Smith set only
have losing scores against those inside, and the merge rule does
not copy losing scores.  This is because only 0 values are copied
from column to column.

GITC -- If someone outside a clone set eats a clone, all the
clones will be eaten on successive rounds, just as if there was
only one.  If a clone eats someone outside, the merge occurs.
Because the outsider loses to the clone, it can provide no help in
defeating other clones.  It does not matter which clone eats an
outsider, because eventually all clones will be eliminated, or one
will eat all the others, and merge with them.

GMC -- Because candidates are removed in order of votes against,
and because removal does not eliminate a majority vote against
a candidate, but merely copies it, candidates with a majority against 
will be removed first.

Elimination methods frequently have the problem that it is
possible to help elect a candidate by ranking it lower.  This
happens when you can reduce the amount by which a victory is
obtained, so that a candidate is not eliminated, and can carry on
to defeat your enemies.  The merge step in Goldfish makes this 
strategy unnecessary.  The winner ends up beating the same candidates 
as the loser, and by as much.  Lower losing values are not copied, 
but having another candidate in the race with lower losing values
is not helpful.

This seems like a pretty good system and is fairly easy to program.
With a couple of tweaks, it can be converted to Tideman.



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