[EM] Re: Richard's criteria
Richard Moore
moore3t1 at cox.net
Fri Feb 13 09:40:02 PST 2004
Last night I wrote that I would add a provision to my definitions to
ensure they apply to more than rank methods. Unfortunately I became so
focused on interpreting the allowable strategies in the original
definitions that I forgot to include that phrasing. I realized my
error only after shutting down for the night. D'oh!
The condition I initially suggested, "The method allows full ranking
of all candidates", is too strong. Something more complicated is
required. Here are the modified definitions:
SFC:
For any set of ballots that either are ranked ballots or allow
rankings to be uniquely calculated, let X be the Condorcet winner of
this set of ballots. Let Y be any other candidate. Transform all of
the ballots into ballots that can be accepted in the method under
consideration, subject to the following two constraints: (1) If X is
voted higher than Y on a ballot, then for any two candidates A and B,
A is voted higher than B on the new ballot if and only if A is voted
higher than B on the original ballot, and (2) If X is not voted higher
than Y on a ballot, then for any two candidates A and B, A is voted
higher than B on the new ballot only if A is voted higher than B on
the original ballot. However the ballots are transformed subject to
these constraints, Y cannot win an election with the resulting set of
ballots in the method under consideration.
GSFC:
For any set of ballots that either are ranked ballots or allow
rankings to be uniquely calculated, let X be a candidate in the Smith
set of this set of ballots. Let Y be a candidate outside of that Smith
set. Transform all of the ballots into ballots that can be accepted in
the method under consideration, subject to the following two
constraints: (1) If X is voted higher than Y on a ballot, then for any
two candidates A and B, A is voted higher than B on the new ballot if
and only if A is voted higher than B on the original ballot, and (2)
If X is not voted higher than Y on a ballot, then for any two
candidates A and B, A is voted higher than B on the new ballot only if
A is voted higher than B on the original ballot. However the ballots
are transformed subject to these constraints, Y cannot win an election
with the resulting set of ballots in the method under consideration.
SDSC:
For any set of ballots that either are ranked ballots or allow
rankings to be uniquely calculated, transform all of the ballots into
set ballots that can be accepted in the method under consideration,
subject to the following two constraints: (1) If X is not voted higher
than Y on a ballot, then for any two candidates A and B, A is voted
higher than B on the new ballot if and only if A is voted higher than
B on the original ballot, and (2) If X is voted higher than Y on a
ballot, then for any two candidates A and B, if A is voted higher than
B on the original ballot then either A is voted higher than B on the
new ballot, or no candidate is voted lower than A or B on the new
ballot and A is voted equal to B on the new ballot. However the
ballots are transformed subject to these constraints, Y cannot win an
election with the resulting set of ballots in the method under
consideration.
WDSC:
For any set of ballots that either are ranked ballots or allow
rankings to be uniquely calculated, transform all of the ballots into
set ballots that can be accepted in the method under consideration,
subject to the following two constraints: (1) If X is not voted higher
than Y on a ballot, then for any two candidates A and B, A is voted
higher than B on the new ballot if and only if A is voted higher than
B on the original ballot, and (2) If X is voted higher than Y on a
ballot, then for any two candidates A and B, if A is voted higher than
B on the original ballot then A is voted higher than or equal to B on
the new ballot. However the ballots are transformed subject to these
constraints, Y cannot win an election with the resulting set of
ballots in the method under consideration.
-- Richard
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