[EM] FBC (Favorite Betrayal Criterion) Definition

[Forest Simmons]

As the moose used to say to the squirrel, "This time for sure!"
 
I left out the phrase that expresses the fact that A is a favorite on the
sincere ballot B.  So here's the definition without that omission:


As before W(C,S,M) is True if and only if candidate C wins when the set S
of ballots is processed by method M.

R(C,A,B) is True if and only if candidate C is ranked or rated higher than
candidate A on ballot B.

mfcs(A,B) is True iff for each candidate C, R(C,A,B) is False, i.e. no
candidate is preferred above A on ballot B.


The FBC is satisfied by method M iff

for each pair of ballots B and B'
and each candidate A such that mfcs(A,B)

there exists a ballot B'' such that
mfcs(A,B'') is True, and ...

for each set S of ballots
and each candidate C

whenever W(C,S+{B'},M) is True
there exists a candidate C' such that

W(C',S+{B''},M) is True and
R(C,C',B) is False.


That's it.

Think of B as your sincere ballot.

B' is the one you are tempted to cast so as to help your compromise C win.

But the FBC says you can safely vote some ballot B'' which gives most
favored status to your favorite candidate A.

If B' would have made C win, then B'' would make some candidate C' which
is no worse than C (according to sincere ballot B) win.

C' could be C or A or some candidate in between.

I think that's more like what Mike had in mind.

A method satisfies a stronger version of FBC if the sincere ballot B will
always serve as B'' .

Full blown Dyadic Approval satisfies the stronger version of FBC.

Forest