[EM] FBC (Favorite Betrayal Criterion) Definition

[Forest Simmons]

Craig Carey has been (patiently?) awaiting a formal definition of the FBC. 

I'll give one below, but I'm afraid it will impart understanding only to
those who have already made an attempt to grasp the concept as presented
informally in Mike Ossipoff's writings.

Informally, the FBC says that a method shouldn't be susceptible to
strategic incentives for showing a preference of some other candidate
above your favorite on your ballot.

The Borda Count, for example, fails the FBC because in a multicandidate
race in which there are two clear front runners among which your favorite
candidate A is not found, you have a clear strategic incentive to vote
your preferred (C for compromise) of the two front runners ahead of your
favorite candidate A. 

So any method that fails the FBC has the well known "lesser evil" problem
to one degree or another.

I'm not sure why Craig is so interested in a formal definition, but one
good reason would be to try to get away from the psychological aspect of
the phrase "strategic incentive".

So let's start with definitions of two boolean valued functions that will
be of use:

First the "winning function" W(C,S,M) whose three respective arguments
represent a Candidate, a Set of ballots, and a voting Method.

If method M gives the win to candidate C when S is the set of ballots cast
by the voters, then W(C,S,M)=True, otherwise False.

Next the "most favored candidate status" function mfcs(C,B)=True if and
only if no candidate A is ranked higher than candidate C on ballot B.



Now with these two auxiliary definitions in place we are ready to define
the FBC:


A voting method M satisfies the FBC if and only if ...

For each set of ballots S and for each candidate C ...

if W(C,S,M) is True, then ...

for each ballot B in S and each candidate A ...

there exists a ballot B' such that ...

mfcs(A,B') is True, and ...

if W(C,S-{B}+{B'},M) is False, then W(A,S-{B}+{B'},M) is True.



That's it.

To understand this you have to know enough set theory to understand that
S-{B}+{B'} is just the set S with B replaced by B' .

So this definition says that if C wins when the set S is fed into the
method, then any ballot can be replaced by some ballot that gives
candidate A most favored status without causing candidate C to lose unless
it causes A to win.

Think of candidate A as your favorite and candidate C as your compromise.

You don't want to jeopardize C's chances by giving A most favorite status
on your ballot unless C's loss will be replaced by A's win.

I hope this is sufficiently precise to satisfy all but the pickiest of
readers, while retaining the original intention of Mike's informal
definition. 

Forest