[EM] Why does Approval go to the voter-median? Median-estimate strategy.

[Michael Ossipoff]

First, I've just noticed that, when I wrote my previous posting, I must have
accidentally written from someone else's account. There have been times when
I've noticed when logging on that I was in someone else's account, and I
logged out and started over. This time I apparently didn't notice, and so
the message seems to have posted under someone else's name.

I included 2 topics in the subject line of this message.

1. Why does Approval go to voter median?: I've said that Approval will
quickly home in on the voter-median and then stay there. I should say
something about why that is. On EM, some years ago, several people
demonstrated why that is so. The demonstration that I give below might very
well be similar to one of those. Myerson & Weber demonstrated it in a
different way. The demonstration in this posting is brief:

Suppose that people are voting strategically. That means they're using
better-than-expectation strategy, of which the various Approval strategies
are special cases and implementations.

Suppose, at least at first, that a voter's perception of his expectation in
the current election is the utility of the winner of the previous election.
That's a reasonable first guess about what to expect. 

Say the winner, W, of the 1st Approval election is some distance to the
right of the voter-median. 

Of course each voter will approve everyone who is closer to hir than W is. A
majority of the voters are to the left of W. That means that a candidate a
little to the left of W will get more approvals than W will, and will get
more than anyone who is to the right of W.

For some candidate to the left of W, s/he'll of course get approvals from
everyone to the left of hir. And, to hir right s/he'll get approvals from
voters up to halfway to W. Obviously the nearest candidate to the left of W
will win.

So it looks as if the win will move leftward, one candidate per election,
eventually reaching the voter-median. It will get to the voter-median, but
it looks like a slow process.

But only if 1)  the continuum is thick with candidates, and 2) if voters
really consider the utility of W to be what they expect in the current
election.

When the win starts moving left, there will be no particular reason to
expect it to stay the same, from one election to the next. A person would
expect it to be somewhat to the left of where it was last time.  That
farther left expectation will mean that the win moves farther this time. And
that, in turn will make people expect a similar move next time, further
accelerating the leftward movement. So the movement toward the voter median
will keep accelerating, and the arrival at the voter-median will be much
sooner than the above two assumptions would imply.

Q.E.D.


2. Median-Estimate strategy

Now, consistent with, but more than, the better-than expectation strategy, a
person, instead of just voting according to hir feel for what s/he expects,
might want to go by hir feel about where the voter median is.  ...and
approve down to the voter median.  The voter median would be the best that
s/he could get, if everyone knew where it is. S/he doesn't really know where
it is, but s/he can guess.  If hir guess is accurate, and if others use that
strategy and make the same guess (as they would if it's correct and they're
as good guessers as s/he is), then that's where the win will be. That's why
I say that this median-estimate strategy is consistent with better than
expectation.

Mike Ossipoff