[EM] Approval strategies

[MIKE OSSIPOFF]

Forest--

It's certainly true that the Approval strategies that I described
in my previous posting on the subject aren't the only possible strategies.

Of course that's because it depends on what the voter knows, or what the voter
has a feel about.

My posting covered a few kinds of voter perceptions or estimates. There could
be different ones too.

Candidates whose utility is between those of the two perceived frontrunners:

There too, it depends on what the voter knows or has a feel for.

If you have an estimate of the utilities of the two frontrunners, and of their
win-probabilities, then you can calculate, from that, your expectation for the
election, and then vote for candidates whose utility is greater than that.

The expectation for the election is Pw*Uw + Pb*Uw  (where those symbols have
the obvious meanings given below).

so vote for every candidate whose utility is greater than that.

Some time ago, we showed in at least two different ways that (provided that one
has the necessary information or estimates) one should vote for the better-than-
expectation candidates.

But it can be looked at in this way too:

Pw is the probability that the worse of the 2 frontrunners has more votes than the
best one.

Pb is the probability better of the 2 frontrunners has more votes than the worse one.

You're looking for the utility that makes equal the damage-expectation of the
inbetween candidate beating the better frontrunner (in the unlikely even that
s/he could), and the benefit-expectation of the inbetween candidate beating the
worse frontrunner.

Call that utility "U".

Uw is the utility of the worse frontrunner and Ub is the utility of the better frontrunner.

You want:

Pw(U-Uw) = Pb(Ub-U)

Multiply both sides out, and collect U terms on the left:

U(Pw+Pb) = PbUb + PwUw

U = (PbUb + PwUw)/(Pw+Pb)

But Pw + Pb is unity.

So U = PbUb + PwUw

Vote for the candidates whose utility is greater than that.

Those are the better-than-expectation candidates.

Of course, though you might have a feel for who the frontrunners will be,
you might not have an estimate of their win probabilities, or numerical
estimates of their utilities.

For that situation, in the article that I mentioned, which described a recommended 
best-frontrunners strategy, the authors (who included Brams, or
Fishburn, or both) suggested voting for all the candidates who seem better than
the perceived midpoint between the two frontrunners' merit.

That assumes an guess, for want of better information, that the two frontrunners
have equal probabilities of winning.

Why did Rob LeGrand say to put the Approval cutoff adjacent to one of the 
frontrunners' utilities?

Of course if you knew that one of the frontrunners was almost certain to outpoll the other,
then the election's expectation would be quite close to hir, because P
for that frontrunner would be much larger than for the other frontrunner.

So maybe Rob was assuming that it's a sure thing that the more winnable frontrunner
will win.

So when more accurate information isn't available, it's a question of which is a
better guess?:

1. Both frontrunners are equally likely to outpoll eachother.

or

2. The more likely front runner is sure to outpoll the other.

When one doesn't know, I'd tend to go along with Brams &/or Fishburn's 
guess, #1 above.

Of course, sometimes you might feel a lot of sureness that one frontrunner
or the other will outpoll the other.

Yes, if you feel sure enough that a particular one of the two frontrunners
will outpoll the other, then the Approval cutoff should be adjacent to hir.

Otherwise, though, guess #1 is the right one.

In these situations, lack of information seems more likely, and that makes
#1 the typically best guess.

Joe's suggestion for when you have a ranking but not ratings:

Sure. If you know nothing about the ratings differences, then you'd want to
at least equalize the probabilities of your pair-tie rival being above or
below in your ranking.

But that requires a probability estimate, and that can be difficult too.

Is the probability really what you have the best feel for? The direct
gut feeling? I don't think so. I think that feelings of threat, or
feelings of promise are the more basic feeling, more instinctive, from
our earliest ancestors.

So place your approval cutoff where the feeling of threat from worse
candidates feels as strong as the feeling of promise from better
candidates.

Or put it this way:

Which feels more: The fear that your vote for x will make hir take the
win from someone better, or your hope that s/he'll take the win from
someone worse.

It seems to me that, at the time, I called that the Threat/Promise strategy.

Mike Ossipoff