[EM] Nonzero-info, few voters, more candidates

[MIKE OSSIPOFF]

With nonzero-info, few voters, and just 3 candidates, there's still no
problem, because it's assured that we're voting for A & not for C.

Say there are 4 candidates, A, B, C, & D.

We want to find out Pbc. As before, if we're voting for C, then
Pbc is the probability that C's vote total is equal to B's vote total
or one vote less, and if we aren't voting for C, then Pbc is the
probability that B's vote total is equal to C's vote total or one
vote less.

But since it isn't zero-info, these aren't necessarily equal.

If there are lots of voters, then it seems safe to assume that they're
equal. (But you could calculate separate estimates from them by
Hoffman's method, if you wanted to).

If there are very few voters, maybe that can't be reliably assumed.

Then we can't calculate each candidate i's Pij independently of
whether or not we're voting for various other candidates.

We now simply use both of those different versions of Pij (depending
on whether we're voting for j) to determine how much we benefit
from each possible combination of which candidates to vote for.
Much more computation, but a computer program could easily & quickly do it.

The number of computations is reduced if we ignore combinations that
"skip". Skipping means voting for someone without voting for everyone
whom we like more.

It's been pointed out that situations where someone could benefit
from skipping are very unlikely.

Of course all this is moot unless we have a good way to estimate those
Pij. I don't know if Weber's, Hoffman's, or Cranor's methods for
estimating Pij are useful when there are very few voters, and so
I wouldn't use them under those conditions.

But that's ok, because when there are very few voters, some other
kinds of information become easier to obtain. Polls of favorites or
rankings are easy when there are few voters. In fact, it needn't
be a random sample poll. It could be a preliminary balloting of every voter.

Tideman, with another author, who might have been Merrill, Brams,
or Fishburn (but was probably Merrill) suggested that form of Approval: Hold 
2 Approval ballotings. The 1st doesn't elect anyone
unless he gets a vote total at least equal to half the number of voters. The 
2nd Approval balloting elects someone.

They suggested several possible strategies. The one that I remember
was, for the 2nd balloting, to vote for whichever frontrunner of the
1st balloting one likes better.

Actually, they said to also vote for everyone whose merit is
closer to that more preferred frontrunner than to the less preferred
frontrunner. I guess the justification for that is that, by
voting for compromises halfway (meritwise) to the less preferred
frontrunner, we increase the liklihood that we can give someone
enough votes to beat the less preferred frontrunner.

The circumstances would give someone a basis to choose one of those
2 strategies in the above paragraph, when information from a previous
Approval election is available.

But if voters & alternatives are all positioned in a 1-dimensional
issue-space, then it seems better to just use Plurality in the
1st balloting, advising people to vote for their favorite. That
would clearly show how far we must compromise in the 2nd balloting.

Or maybe, if there isn't a 1-dimensional issue space, it would
be useful to collect rankings in the 1st balloting, which could
be used to judge whom you can make win in the 2nd balloting.

I don't know if it would be a good idea to use Approval voting
_and_ rankings in the 1st balloting, because uncertainty about
which of those other voters are basing their strategy on could
make it more difficult to know how to vote in the 2nd balloting.

When I mentioned rankings in the 1st balloting, I meant rankings
that are made available to the other voters, but which aren't
used for electing anyone. I'm assuming that either people haven't
accepted Condorcet's method, or that there are lots of alternatives
and no computer, making Approval more feasible.

I mentioned some problems of the Weber approach when there are
very few voters, nonzero-info, and more than 3 candidates or
alternatives. It would be nice, and surprising, if Richard's approach
doesn't have those problems too.

Mike Ossipoff




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