[EM] Puzzle example X 1000

[MIKE OSSIPOFF]

> > Anyway, as I said before, even if your above argument were valid,
> > it would be better to give us a more direct demonstration that
> > above-mean isn't valid with very few voters.
>
>Actually I started working out the case for three other voters, three
>candidates, and zero info, on paper. I didn't finish it. First, a few basic
>assumptions are needed: Are all three voters guaranteed to cast votes?

In an actual voting situation we can't rule out the possibility that
1 or more voters could abstain completely, and so, now that you mention
it, that situation  should be considered.

>Do we assume nobody will cast an ABC ballot?

That seems a reasonable assumption, since with very few voters it's
easy to tell everyone that they gain nothing from such a ballot.


>Do we assume they
>are using the above-mean strategy

That question hadn't occurred to me. If we assume they're using it,
and, when they do, it benefits everyone who uses it, including me,
then that shows that above-mean works in that example. We could assume
that everyone perceives the same position-scale and is positioned with
their favorite. I'd thought that we had to assume that we know nothing
at all about the other voters, but it does seem reasonable to assume
that they're using above-mean strategy, and then if that works for
everyone then above-mean is right, and if it doesn't, then above-mean
is wrong with that number of voters & candidates.

But it seems to me that I've shown that, with 3 candidates, even with
very few voters, above-mean strategy maximizes expectation--when
0-info means that we know nothing at all about how the other people
will vote. As you said, though, we can reasonably assume that they
don't vote ABC, and it's worthwhile checking what happens if they all
use above-mean strategy.

>(even though it may not be the best
>strategy for this small population)? Assuming yes for all three of these
>questions, there are 6x6x6 equally probable permutations to consider,
>so it's a non-trivial problem. Interestingly, these assumptions aren't
>particularly important when dealing with large populations, because the
>random actions of a large population (such as not making it to the
>polling place on election day) tend to cancel each other out.
>
>If I get some time to finish the solution I'll happily post it here. But 
>don't
>
>let that stop you from trying the same. Maybe you'll beat me to it.

I've meant to add up your benefit by voting for B in addition to A,
with all 729 of the ways that those 3 people could vote, to find out
if that summed benefit is positive or negative. But it's also worthwhile
checking whether above-mean strategy works for everyone if everyone uses it, 
and no one votes ABC, but some people might abstain completely.

But it seems to me that I've shown that above-mean strategy maximizes
utility with 3 candidates, even with very few voters, if we know nothing
about how those other people will vote.

Mike Ossipoff



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