Is that something like this?:
(+1 for A 1st; +1 for B 2nd; -1 for C 3rd; ...)
That could be written as (+A, +B, -C).
The leading +s can be missed:
If C loses and no other information is given, then that can be written as (A B- -C)
A generalized montonicity rule would imply:
Prohibited: (A C B+) <---> (A B- C)
(It says: shifting B to the left doesn't cause it start losing)
For the negatively orientated preference this could be done:
. . . .Allowed: (A C +B+) <---> (A C -B-) Hence: Allowed: (A C B+ ) <---> (A -B- C)
For the power<=1 rule that enforces One Man One Vote:
The set of Winners intersected with {A,B,C} when this paper is added:
1(A -B C)
can be reproduced when these papers are added:
w.(A C) + x.(A) + y.(C) - z.(B)
for some w+x+y+z=1, 0<=(w,x,y,z).
This could be defined x.(A B C D) = -x.(-A -B -C -D).
So the negation of a paper p is paper p with the same weight but with the preferences negated.
The IFPP Duality principle (which produces the same method when switching winners with losers whilst also negating weights) has an interpretation.
The mix of the power<=1 (no-corruptness) power rule and duality, could lead to unconvincing alteration-examples.
It is the strange problems with power that may arise that could be of concern. I have not considered the problem.
When all the preferences are positive and there are only 3 candidates, power has nothing to impose after monotonicity has been imposed. It is not obvious at the outset if examples showing a failure to constrain power would show up if there are only 3 candidates. The 3 candidate case can't be that difficult.
For P2, the smudging of the weight over the next preference could have the added next preferences be either all +1 preferences or else all -1 type preferences.
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At 03\03\06 09:45 +1030 Thursday, Chris Benham wrote: >This is my first posting here. I have just composed a very expressive >(elect-one person) election method , and not as a joke. >Voters number candidates in order of preference, and also mark >candidates as Approved and up to >number of candidates minus 2 as Disapproved (or Unacceptable). >1. A candidate who is ranked number 1 on more than half the ballots wins. >2. If no result, if there is one and only one candidate who marked as >Approved by the majority, then that candidate wins. If the set of >candidates Approved by the majority has more than one member, then all >non-members are eliminated and their preferences are transferred. If a >candidate now has a majority then that candidate wins. If not, then >the numbered rankings are abandoned, and the winner is the candidate >with the highest Approvals minus Disapprovals tally. >3. If there were no candidates who were Approved by the majority, then >eliminate all candidates marked as Disapproved by the majority and >transfer their preferences and proceed as before. >If a candidate has a majority they win. If no result, then abandon the >numbered rankings and elect the candidate with the highest Approvals >minus Disapprovals score. >A vote with as little as just the number 1 or a single Approval or >Disapproval will be valid. The restriction on the number of > Disapprovals is mainly to make it impossible for all the candidates to >be Disapproved by the majority. >I think that there should be few strategy problems. I am interested >in any comments, but I don't promise to (immediately) understand them. > Chris Benham >
That is not all that clear and an example could have got more to take a meaning from the text.
Craig Carey
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