[EM] Challenge - give an example where FBC is violated for Condorcet methods

[Peter Zbornik]

Dear all,

It seems to be the case that the prevailing opinion on this list is,
that Condorcet methods violate the favorite betrayal criterion (FBC).
Is that correct?
I have looked for a while in the archives election-methods list for an
example of the FBC violation for Condorcet methods, but didn't find
any.
Maybe I didn't look deep enough.
FBC is not listed on Wikipedia with the other criteria, so it seems
that this criterion is at a "pre-scientific" stage of formulation,
correct?
There is no example on the electrorama wiki either.
So I thought I'll turn to the experts on this list for some quick help
in this area.
Personally I think that Condorcet methods (take Schulze as reference)
with symmetrical ballot completion might satisfy FBC, but I am not
sure, so that's why I ask for some guidance here.

In order to avoid misunderstandings, below I:
A] first provide the relevant definitions,
B] then give questions for FBC violation examples and
C] finally specify how the examples of FBC violations should (could?)
look like in order to avoid misunderstandings.

A] DEFINITIONS (Alex Small):
https://sites.google.com/site/physicistatlarge/FBC_latest.pdf?attredirects=0
1. FBC: A voting method satisfies the Favorite Betrayal Criterion
(FBC) if there do not exist situations where a voter is only able to
obtain a more preferred outcome (i.e. the election of a candidate that
he or she prefers to the current winner) by insincerely listing
another candidate ahead of his or her sincere favorite.

2. SFBC: A voting method satisfies the Strong Favorite Betrayal
Criterion (SFBC) if there do not exist situations where a voter is
only able to obtain a more preferred outcome (i.e. the election of a
candidate that he or she prefers to the current winner) by insincerely
listing another candidate ahead of or equal to his or her sincere
favorite.

3. Generalized symmetrical completion is defined as follows:
(i) equalities between candidates on the ballot (A=B) are resolved as
0.5 votes for each candidate in the pairwise comparison.
(ii) Ballots are completed with the null candidate ("none of the
above" X), who is ranked strictly lower than all candidates explicitly
ranked on the ballot,
(iii) candidates not explicitly ranked on the ballot (i.e. truncated
ballots, excluding the null candidate) are ranked equally below the
null candidate.
Example: election with candidates A, B C, the bullet-vote ballot A is
completed as A>X>B=C, A>B>C is completed as A>B>C>X).

B] QUESTIONS
QUESTION 1. Do you know of an example which shows that FBC or SFBC is
violated for Condorcet elections (reference method Schulze) where
ballots are completed using generalized symmetrical completion?

QUESTION 1. Do you know of an example which shows that FBC or SFBC is
violated for other types of Condorcet elections (reference method
Schulze), than those where generalized symmetrical completion is used?

C] SPECIFICATIONS OF EXAMPLES
In order to avoid misunderstandings, please send your examples on the
following form.
Please submit two elections on the standard form (i.e. number of
voters for each distinct type of ranked ballot, equally ranked ballots
allowed)
In the first election some voters C give the first preference to a
candidate A. An other candidate than A wins the election
In the second election A wins the election and some of the voters in
group C give A a second preference or less (FBC) or give an other
candidate a shared first preference with A (i.e. A=B is the shared
first preference) (SFBC). No other changes occur, i.e. all other
rankings in group C and outside group C remain unchanged from the
first election.

I assert for the sake of argument, that FBC is satisfied for Condorcet
methods (at least Schulze) when generalized symmetrical completion is
used.
Please show me wrong :o)
If there is something I haven't understood, please give some advice.
I am a newbie on this list, so things are a bit new to me here.
Thanks.

Best regards
Peter Zborník