# [EM] Strong FBC, at last

Forest Simmons fsimmons at pcc.edu
Wed Jan 29 16:54:47 PST 2003

```Good work, Alex.  I think the argument can be simplified so that
it will generalize easier, but nobody else has faced up to it like you
have.

BTW it seems like every N+3 candidate election has a three candidate
election embedded within it as far as each faction is concerned, since
each faction has its Favorite, along with the two front runners to worry

And if even one faction has incentive to dump favorite in even one
election, then the FBC is not satisfied by the method being used.

I have one further tangential (actually orthogonal) comment below:

On Wed, 29 Jan 2003, Alex Small wrote:

....

> Consider the boundary between the A region and the B region.  Call the
> normal to the boundary |Nab>.

If the boundary is a fractal, then there is no normal.

Rob LeGrand reported that his Cumulative Repeated Approval Balloting
simulations yielded graphs with (what appeared to be) fractal boundaries
separating the victory regions.

On the other hand, CRAB (in its simplest formulation) satisfies the
Majority Criterion, and I think we can prove that no method satisfying the
Majority Criterion can also satisfy the (Strong) FBC, whether or not the
boundaries are smooth.

>From a practical point of view, no method that requires expression of a
unique favorite will ever be adopted for public elections unless it
satisfies the Majority Criterion.  So it is sufficient, for public
proposal purposes, to show that the Majority Criterion precludes the
Strong FBC.

Unfortunately, this doesn't help to distinguish between IRV and Condorcet.

It does help to explain why it is hard to improve on Majority Choice
Approval:

If we restrict favorite status to one candidate per ballot, then we gain
the (unique majority version of the) Majority Criterion at the expense of
the FBC.

If we allow more than one candidate per ballot to have favorite status,
then we regain the FBC, but lose the ability to detect a unique majority
favorite.

Furthermore, we give up the chance of strong FBC, since there are
definitely cases where one should give Compromise equal billing with
Favorite when that is allowed.

This predicament is not just a result of our lack of ingenuity, but a
fundamental incompatibility between the majority criterion and the FBC.

[This is not intended to be a proof, but only to show an instructive
application of the result.]

Faced with this predicament, most of us prefer the version of MCA that
allows more than one favorite, so that more than one candidate can have a
fifty percent plus "majority."

It seems to me that if we were consistent in our thinking, then in the
context of Condorcet methods we should prefer ballots that allow equal
ranking of candidates at the top as well as at the bottom (a.k.a.
truncation), if not in between as well.

That's another reason why I prefer Grade Ballots (for example, grades A
through Z) over traditional preference ballots for Condorcet methods.

[Construct the pairwise matrices from the Grade ballots, and then use your
favorite Condorcet method to score the matrices.]

Forest

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