[EM] Strong FBC, at last

Alex Small asmall at physics.ucsb.edu
Thu Jan 30 13:22:38 PST 2003

Forest Simmons said:
> On the other hand, it would be nice to have it come out as a consequence
> of some natural requirement that non-mathematicians could appreciate.
> A monotone function can be non-differentiable only on a set of measure
> zero, so perhaps monotonicity would be sufficient.

I had thought about monotonicity in a somewhat different context: 
Excluding bizarre voting schemes.  In my proof that strong FBC is only
satisfied by methods equivalent to top-2 voting, I had to include the
weird "top and bottom" method.  It's clearly an absurdity, and requiring
monotonicity would eliminate it and similar methods for 4+ candidates,
making the analysis much simpler.

However, there are non-monotonic voting systems that are not absurd.  IRV
is a perfectly reasonable proposition, but a careful analysis finds that
there are better methods.  A more satisfactory answer to the queston of
strong FBC would include reasonable but non-monotonic methods, to firmly
rule out as many methods as possible.

I think it should be easy to show that monotonicity and majority rule are
incompatible with strong FBC for any number of candidates, now that some
techniques for tackling the problem of SFBC have been established. 
Monotonicity and strong FBC without majority may be a little trickier.

As for eliminating fractal boundaries, I guess it's reasonable to require
that the method can be defined by a set of linear inequalities.  All this
really does is restrict our attention to methods that might be
contemplated in public elections.  After all, neutrality and anonymity
already rule out a very wide category of decision rules that might be used
outside by other decision-making bodies (e.g. shareholder meetings where
votes are weighted by the number of shares owned, legislative bodies
choosing among possible bills when some bills require super-majorities).

Another thought on fractal boundaries:  When I tried to approach the
problem discretely (i.e. integer number of voters) rather than
continuously (the fraction of the electorate with each preference is a
real number), I found myself considering the case where a single voter
making a particular adjustment changes the outcome, and then another voter
with the same preferences making the same strategic adjustment changes the
outcome yet again.  I couldn't draw nice lines dividing electorate space
(or at least a 2D projection onto my piece of paper) into regions, I had
to draw discrete cells scattered everywhere.  That may not be a fractal,
but it's an apparently random method, it's very undesirable, and it's
intractable to study.


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