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 ---- Election-Methods mailing list - see http://electorama.com/em for list info
