[EM] WMDDA
Warren Smith
wds at math.temple.edu
Thu Oct 13 17:28:32 PDT 2005
Deluxe MDDA or DMMDA for short:
0. Votes are preference-rank-orderigs of the candidates, AND
each vote also includes an "approval threshhold" where all candidates
ranked at or above it are "approved" by that voter. Rankings can include both > and =
relationships, e.g. A>B=C>D=E>F=G.
1. A candidate is "disqualified" if another candidate is ranked over him/her
by a majority of the voters.
(Unless that rule would disqualify all the candidates, in which case no
one is disqualified.)
2. The winner is the most-approved un-disqualified candidate.
[end of DMDDA definition]
Here is the definition of a related voting method devided by Warren D Smith,
(actually originally due to a misunderstanding of DMMDA, but I'll take it:)
call it WMDDA:
1. A candidate is "disqualified" if not in the "Smith set"
(The "Smith set" is the candidates S such that each candidate in S is pairwise
winner over each candidate not in S. Different Smith - not me.)
2. The winner is the most-approved un-disqualified candidate.
[end of WMDDA definition]
I claim both DMMDA and WMDDA obey SFC, SDSC, and FBC.
PROOFS of above 3 DMDDA claims (by Venzke) with WMDDA add-on proofs by WD Smith:
* satisfies SDSC because if a majority prefer X to Y, and don't approve Y,
then Y will have a majority-strength defeat. If not all candidates have such
defeats, then Y can't win. If all candidates have such defeats, then Y still
can't win, because X has greater approval than Y.
WDS adds note: WMMDA also satisfies SDSC because of the same proof with this addition:
Y is either not in the SMith set (in which case Y cannot win) or is, in which case
X and Y ill be in the same Condorcet cycle. In that latter case Y cannot win
because X has greater approval than Y.
* satisfies SFC because if no majority prefers anyone to X, then X will
not be disqualified. If X has a majority-strength win over Y, then Y will
be disqualified, so that Y can't win.
WDS adds note: WMMDA also satisfies SFC because of the same proof.
* satisfies FBC because if X wins, and a faction has ranked Y insincerely
low, then if this faction raises X and Y to the first position, the winner
will then be either X or Y. This is because X and Y can only lose defeats in
this way, and other candidates can only gain them. Also, X and Y's
approval can only increase.
NOTE: this FBC proof only works if the "ranking" is allowed to include EQUALITIES not
just ">" relations, e.g. A>B=C=D>E=F>G=H.
WDS adds note: WMMDA also satisfies FBC because of the same proof.
wds
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