[EM] The special tie rule works with MinMax(wv) and (margins)
stepjak at yahoo.fr
Sun Jun 5 13:32:30 PDT 2005
MMPO has a couple of grave problems: It's indecisive and it fails
the Plurality criterion. It occurred to me only a couple of days ago
that one could use the "T matrix" with MinMax(wv) to avoid these
problems and still satisfy FBC.
MinMax(wv) can be defined like this: A candidate's score in a given
pairwise contest is either zero (if he doesn't lose the contest) or
the number of votes against him in that contest (if he does lose).
Then a candidate's overall score is his greatest score in any single
contest. The candidate with the lowest overall score is the winner.
In the "T matrix," where "T" stands for "tied at the top," t[a,b]
(which equals t[b,a]) represents the number of voters ranking A and B
tied in first place, possibly with other candidates.
To use this matrix, change the above definition so that even if a
candidate X loses a given pairwise contest with Y, if
v[x,y] + t[x,y] >= v[y,x], then X's score for the contest is zero.
(Also, necessarily, Y's score will be zero as well.)
To show that this satisfies FBC, I'll use the same demonstration that
I used for MMPO. Say that Y wins and some faction wishes to raise X
to the top position (because he is their favorite), and to play it
safe also raises Y to the top position. Let Z signify some arbitrary
candidate already ranked in the top position, and let W signify some
arbitrary candidate ranked below the top, possibly below X and/or Y.
Clearly, raising X and Y can't hurt their overall scores. No candidate
receives additional votes over them, and there is no way for X or Y
to obtain a new loss. t[z,x] might increase, but only at the expense
of v[z,x]. One of X or Y might lose votes over the other, but they are
added back to t[x,y], so that all that might happen is that X and Y
obtain a "special tie."
Raising X and Y also can't help anyone else's overall scores. It may
hurt some scores, as X and Y obtain votes over W. It doesn't affect Z:
v[z,x] decreases, t[z,x] increases. This might negate Z's win over X,
but it doesn't affect Z.
So the winner will still be either X or Y.
This is more indecisive than ordinary WV, since it can conceivably
find multiple CWs, but it's nowhere near as indecisive as MMPO.
Unfortunately, the T matrix seems useless with more complicated WV
methods that use a notion of path strength.
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