I haven�t found an FBC failure for wv Condorcet with 3 candidates. I�ve checked the following scenario: Three candidates: A, B, & C. B is CW.
The names of the candidates are also used to represent the number of people who have that candidate as favorite. Except that C representst the number of people who have C as favorite and don�t use defensive strategy, and C1 represents the number of people who have C as favorite and do use defensive strategy.
The A voters use offensive order-reversal against B. The C voters, as I said, don�t use any defensive strategy, and rank sincerely. The C1 voters use defensive strategy to try to save B�s win.
Sincere rankings:
A: ABC B: BAC C: CBA
The B voters, in the scenario that I checked, rank sincerely, BAC.
Voted rankings in the scenario that I checked:
1.
A: ACB B: BAC C: CBA C1: (C=B)A
2.
A: ACB B: BAC C: CBA C1: BCA
Though it�s possible that I made an error, I found that if the C1 voters can�t save B by equally ranking B and C in 1st place, then they can�t save B by ranking B over C. So that scenario doesn�t give an FBC violation for wv Condorcet.
At least 2 of the inequalities that have to be satisfied for the C1 voters to not be able to save B by equal ranking, but to be able to save him by favorite-burial, contradict eachother. Unless I made an error.
I tried the same scenario for margins Condorcet, and again found that if the C1 voters can�t save B by equally ranking B with C, then they can�t save B by ranking B over C.
I emphasize that even if it turns out that wv and margins pass FBC, margins still fails WDSC and margins is still falsifying. And wv, including PC, passes SFC, WDSC, & Nonfalsifyingness. And SSD, BeatpathWinner/CSSD, SD, and RP additionally pass GSFC, SDSC, and probably Expressiveness.
As I said, there could be an error, but probably not, but even if it�s true that wv and margins can�t fail FBC in that scenario, that doesn�t necessarily mean that they can�t fail FBC with 3 candidates. Maybe if the B voters would defensively truncate, ranking only B, it would be possible for the C1 voters to save B by ranking B over C, even when they couldn�t save B by ranking B equal to C.
And even if it turns out that that scenario too doesn�t give an FBC failure for wv &/or margins, that doesn�t mean that one couldn�t be found when there are more candidates.
Could it be shown that if the failure can�t be found with N candidates then it can�t be found with N + 1 candidates--or, more specifically, that if it can�t be found with 3 candidates then it can�t be found at all?
Mike Ossipoff
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