[EM] Trying to define "Later-no-harm for viable candidates criterion" (Re: full rankings, voter desire for)

[Rob Lanphier]

On Sun, 2005-10-16 at 22:47 +0200, Kevin Venzke wrote:
> I don't know of a way to weaken LNHarm which would still result in a guarantee
> that voters could "take to the bank."

My hope would be that we can come up with a system where voters could
feel comfortable ranking all but one of the viable candidates.  So, if
we end up in a situation like we were at one point in 1992, where
Clinton, Bush and Perot were all viable candidates, voters could feel
comfortable ranking two out of three of them, without worrying at all
about helping anyone defeat their first choice.  For such a system, we
could then recommend that voters do not rank anyone below their least
favorite viable candidate (which would be a very minimal amount of
strategy to impose).

So, the partial definition of Later-no-harm for viable candidates
criterion" (LNHarmVC) could be:
"Adding a /viable/ preference to a ballot must not decrease the
probability of election of any candidate ranked above the new
preference."

The trick, of course, is to define "viable" in mathematical terms in
such a way that matches the popular view of viability.

A simple, but probably incorrect, definition would be "any candidate who
is ranked on a majority of ballots".  I would hope we could come up with
a less stringent definition, because that would potentially mean that a
candidate in a close, polarized three way race might not be "viable" by
the definition.  An alternative definition might be "any candidate who
could win without violating Plurality".

I think working with the MMPO example you posted a while back may help
to arrive at an answer:

n A
m A=C
m B=C
n B

When n>2 and m=1, then C wins decisively, no matter how large n gets.

The horrifying thing about this particular example is that it seems
quite feasible for a fringe write-in candidate to win under this
example.  It's a gross Plurality violation, which is clearly
unacceptable.  More to my point above, a write-in candidate would very
rarely be considered "viable", so violating LNHarm for this candidate is
not a big concern.

However, there's probably a threshold for m which that result doesn't
look so bad.  Clearly, when m>n, it's hard to argue that anyone but C
should be the winner.  Is there a lower value for m relative to n where
the result is still defensible?  Is there anything mathematically
interesting about that threshold, that might lead us to a good
definition of "viable"?

Rob