# [EM] FBC definition revisited

Richard Moore rmoore4 at home.com
Tue Jan 15 21:08:30 PST 2002

```One thing on my mental to-do list has been to go back and review Forest's
FBC definition, which stated (#8042 in the archives):

---------begin Forest's definition------------
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.
-----------end Forest's definition-----------

I think this definition is correct. I also think a shorter definition
is possible. Forest has come up with a nice way to incorporate the
concept of voter preferences, using the hypothetical ballot B and
the ranking function R. Perhaps that technique could even be
extended to come up with a workable definition of "sincere ballot".
However, I think FBC can be defined without defining any voter's
preferences. Here is my definition:

A method M passes FBC iff there is no set of ballots S such that,
for some subset R of the set of all candidates, and some candidate
C not found in R, both of the following statements are true for
every ballot B:

(1) M({B}) = C implies that M(S+{B}) is in R,
and
(2) there is a ballot B' for which M(S+{B'})is not in R.

In English:

A method (M) passes FBC if and only if there is no situation (S) such
that a voter can cast a ballot (B') that causes a result that is
different from every result (the candidates in R) that the voter
could get in situation S by voting a particular candidate (C) in first
place.

The definition relies solely on the actual ballots. Once S, R, C, and B'
are found that violate the criterion, the hypothetical existence of a
voter who prefers C to all other candidates, and who prefers M(S+B') to
any of the candidates in R, is all that is needed to show that this
definition is equivalent to the preference-based definition.

The only glitch I see is that, for methods that allow multiple first-place
votes (such as Approval), there may not be a unique winner for the single-
ballot election M(B), particularly if the tie-breaker has a random
component (actually the problem also exists for purely random methods
as well, regardless of whether multiple first-place votes are possible).
For such cases, "M({B}) = C" should be read as: "Method M applied to the
single ballot B could elect candidate C (either uniquely or by
tie-breaker)"; and "M(S+{B}) is in R" should be read as: "Any candidate
that method M can select from the ballots S+{B}, either uniquely or by
tie-breaker, is a member of R".

I suspect other strategic voting criteria could also be worded to avoid
references to voter preference, but I haven't looked into it.

-- Richard

```