[EM] From top-2 to Weber

[MIKE OSSIPOFF]

There's a smooth, seamless gradation from the top-2
(or best-frontrunner) strategy to Weber's method. In this message
I'd like to describe that, including some things I'd left out in
previous messages.

Top-2 strategy:

Vote for whichever of the 2 most likely top votegetters that you
prefer to the other, and for everyone whom you like better than him/her.

If you're really sure they're the frontrunners, then you definitely
want to vote for the better one, but not the worse one. And if you're
sure they're the only candidates who'll be in a tie for 1st, then
it's harmless (if profitless) to vote for other candidates too.

The next most likely kind of tie would include one of the probable
top-2, instead of one. Ties involving two nonfrontrunners can be
ignored for now. If a candidate's merit is between those of the
putative frontrunners, then one strategy suggested is to vote for
him if he's better than the average merit of the frontrunners. That
assumes that he's equally likely to tie each of them.

But say you have reason to believe that one frontrunner is more likely
to outpoll the other than vice-versa. Say that X & Y are the expected
frontrunners, and it's 70% that X will outpoll Y, and 30% that
Y will outpoll X.

Then, if Z is some other candidate, whose utility is between X & Y,
vote for Z if (.7)(Ux-Uz) < (.3)(Uz-Uy).

That also gives: Vote for Z if Uz > .7Ux+.3Uy.

Since .7 & .3 are the probabilities that X & Y will win, assuming
that they'll be the top votegetters, that means voting for Z if
he/she is better than the election's expected utility.

One could consider other possible ties between X or Y and other
candidates, but, since they involve considering voting for Y or
not voting for X, we shouldn't take the top-2 strategy that far.
Considering those other ties means going on to Weber.

Say that, additionally, it's 80% that X & Y will be the top votegetters
and will be in a tie for 1st if there is one. (Isn't it reasonable
to have those probabilities the same if we have no reason to expect
the likelihood of a tie to depend on who the frontrunners are?). And
say that you estimate a 2% probability that the top votegetters and
candidates who'll tie if anyone will are neither X nor Y.

Then, for Weber,

Pxy = .8
Pxz = (.18)(.7)*Pfz
(where Pfz is Pz divided by the sum of the Pi of the candidates other
  than X & Y. Pi is the probability that i will be in a tie for 1st,
  and is gotten from Wi, by Tideman's estimating method).
Pij, where i & j are 2 candidates other than X & Y, = (.02)PfiPfj

To get the Pi of the candidates other than X & Y, just call the
most winnable one's winnability 1. Then assign numbers to the others
accoring to how much less winnable they are. Take the square roots
of those numbers to get the Pi. Pfi is i's Pi divided by the sum
of the Pi of the other candidates other than X & Y.

This way of estimating Weber's Pij is probably more reliable than
estimating the Wi and using Tideman's estimating method.

Mike Ossipoff





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