[EM] Name that Criterion

[Forest Simmons]

On Thu, 6 Feb 2003, Alex Small wrote:

> Forest Simmons said:
> > It seems reasonable that if S is a ballot set with a definite winner X,
> > and T is any other ballot set, then sufficiently many copies of S added
> > to T should result in a ballot set supporting X.
>
> Let me see if I can understand what you meant about the openness of the
> victory regions.  Say that S and T each have the same number of voters in
> them.  We can represent adding copies of S to T by writing the resulting
> electorate as
>
> E = q*S + T  (q is a real, positive number)
>
> For sufficiently large q, T is "drowned out" and we assume that X is the
> winner.  Now, since we assume that election methods shouldn't care how
> many voters have each preference, only what fraction of the voters have
> each preference, we can always normalize our new electorate as
>
> E(normalized) = (q*S + T)/(1+q) = S*q/(1+q) + T/(1+q)
>
> If we assume that X is the winner for an electorate that is purely S, and
> also for an electorate that is "mostly S" (finite q) then we see that
> regions in which X wins must have non-zero volume, as measured in whatever
> space we're using.  Is that equivalent to what you say below?

That's the right idea. But instead of saying "must have non-zero volume,"
say "must be open."

A set is open if each member of the set is shielded from non-members by
other members of the set, so that sufficiently small perturbations cannot
knock it out of the set.

In other words, each member is surrounded by a ball of positive radius
entirely contained within the set.

In this context, positive radius implies positive N!-1 dimensional volume,
but the reverse implication is not true.

To see this in a simpler setting consider in three dimensional space that
every ball of positive radius has a positive volume, and every open set
contains balls of positive radius, so every open set has positive volume.

On the other hand, a ball of positive radius is not itself open if it
contains its boundary sphere, yet it still has positive volume.

[The points on the boundary sphere are not shielded from the complement of
the ball.]


>
> > For those who have had a little topology, this condition can be
> > interpreted as openness of the victory regions in the space of all
> > possible elections associated with the method.
> ...
> > I believe that this condition and the Pareto condition taken together
> > might be sufficient to rule out the Strong FBC.
>
> Do tell more when you get the chance.  I also have some ideas for how to
> resuscitate my treatment of strong FBC for 3 candidates, and perhaps on
> how to generalize to 4+ candidates.
>

Note that the Pareto condition and the Openness condition together imply
that sufficiently many ballots with X in first place will swamp any other
set of ballots and allow X to win.

So these two conditions together yield a kind of "Overwhelming Majority
Condition"  which is weaker than the Majority Criterion, but (I believe)
still strong enough to rule out the Strong FBC.

To be continued ...

Forest

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