[EM] B must be more than just above mean

[Richard Moore]

MIKE OSSIPOFF wrote:

> It's occurred to me that if Pbc(Ub-Uc) > Pab(Ua-Ub), and Pab > Pbc,
> that doesn't must mean that Ub-Uc > Ua-Ub. It means that it must
> be greater by a factor of at least Pab/Pbc. And so, if, as seems to be
> the case, Pab > Pbc, then B must be more than just above the mean,
> when there are very few voters.

Good observation, and that may be another way of getting to where I
was going. It's probably not too hard to show Pab > Pbc, though as I
said I'll forego looking into it for the time being; I've got other pressing
matters. I'll get back to it eventually.

> So the fact that you're voting for A, & not for C, does seem to mess up the
> above-mean strategy when there are very few voters.
>
> If that's right, and if we don't even have the above-mean strategy
> when there are very few voters, then it's especially important to
> conduct a preliminary balloting, by Approval, Plurality, or rankings
> (depending on the conditions) and make that information available
> to the voters before the 2nd balloting. Fortunately, as I said,
> a preliminary balloting is especially feasible when there are very
> few voters. If we can take its results as a sure prediction, then we
> avoid the need to try to maximize expectation based on probability
> estimates.

I suspect that may be the case for small groups, like a committee.
But for application of Approval to large-scale elections there's
nothing to worry about, since above-mean ZI will work nicely. And
those who prefer non-ZI can use the polls, but the number of voters
who vote ZI vs. those that use non-ZI should have little effect on the
final outcome. I thought of a physics analogy today: A small number
of particles will behave very chaotically, but the ideal gas law is still
very good at predicting the behavior of a gas consisting of a large
number of molecules which are individually unpredictable.

 -- Richard