[EM] Strong FBC, at last

Alex Small asmall at physics.ucsb.edu
Wed Jan 29 17:38:21 PST 2003

```Forest Simmons said:
> 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

I have an idea for generalizing:  add the Pareto condition and consider an
election where all voters place candidates 4, 5, 6... etc. place 1, 2, and
3.  I have to give it a little thought.  If you have ideas in this

> 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.

I think the Partial Decisiveness condition removes the possibility of
fractal boundaries, since I specified that the ties occur on a set of 5
dimensions (or N!-1 dimensions for N candidate races).  I don't know much
about fractal curves in a mathematical sense (although I know a tiny bit
about experimental studies on fractals in physics and materials science),
so I'm not certain that Partial Decisiveness removes fractal boundaries,
but that's my best stab at it right now.

> 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.

Although my methods certainly need simplifying, I think they might be
usable for showing that majority and SFBC are incompatible in 4+ candidate
elections.  Once again, if you have some ideas I'd love to hear them.

It should be easy to show that the generalized Condorcet criterion is
incompatible with SFBC for ANY number of candidates.  (The generalized
Condorcet criterion says the winner must come from the innermost unbeaten
set, AKA Smith Set, AKA Schwartz Set.)  A few months ago I posted a proof
that with 3 candidates SFBC and Condorcet are incompatible.  All we really
need to tackle is a case where 3 candidates are in the Smith set, although
we have to allow for the possibility that strategic manipulations may
expand the Smith set to 4 or more candidates.

Of course, this is more restrictive than the simple Majority criterion.

> 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.

I was thinking about MCA as well.  My arguments imply that only top-2
voting satisfies SFBC, at the expense of expressiveness.  By allowing
truncation we recover some expressiveness, since we can decide how many
candidates to support.  MCA goes one step further and allows us to make
support for a second choice conditional on how things unfold at the polls,
recovering a feature of IRV, Bucklin, and other ranked methods, but it
keeps the FBC characteristic of rated methods.

In the end, MCA may be the best method available if you want to balance
(weak) FBC, expressiveness, majoritarianism, simplicity, and the strategic
safeguard of making support for #2 conditional on how things unfold at the
polls.

Alex

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