[EM] BeatpathWinner/CSSD and FBC

[MIKE OSSIPOFF]

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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