Mathematical utility expectation maximization in Approval

[Bart Ingles]

Martin Harper wrote:
> 
> Forwarded from Mike Ossipoff at his request.
> 
> MIKE OSSIPOFF wrote:
> 
> >
> >
> > I've posted this before, but let me describe mathematical strategy
> > in Approval, for maximizing utility expectation:
> >
> > Strategic value:
> >
> > Ui is the utility of candidate i, as judged by you.
> > Pij is the probability that i & j will be the 2 frontrunners.


Slight correction/clarification:  The precise meaning of Pij is usually
taken to mean the probability, given a tie exists for first place, that
i and j will be involved in that tie.

Thus in the thumbnail description of mathematical Approval strategy,
frontrunners are candidates those with the best chance of being involved
in a tie, not necessarily the best chance of winning.

Example:
- You have candidates A, B, C, D, and E,
- A has an 90% chance of winning,
- B and C each have a 4% chance of winning,
- D and E each have a 1% chance of winning,
then A, B, and C are considered the frontrunners, since they are equally
likely to be involved in a tie for first place (either 2-way or 3-way).


> > The strategic value of candidate i, for you, is the sum of
> > Pij(Ui-Uj), over all j.
> >
> > In Approval (if your an instrumental voter--but don't be one) vote
> > for every candidate whose strategic value is positive.
> >
> > In Plurality, vote for the candidate with highest strategic value.
> >
> > The Ui can be estimated directly, or can be estimated indirectly from
> > lottery choices, as suggested by Morgenstern & von Neuman.
> >
> > Of course we could estimate the Pij directly too. But Tideman has
> > suggested another way to estimate them: It's probably easier to guess
> > the win probabilities of the candidates. The probability that a
> > candidate will be one of the 2 frontrunners can justifiably be estimated
> > from the probability that he'll win. The probability that X will be
> > a frontrunner is proportional to the square root of the probability
> > that he'll win. Tideman shows a geometrical justification of that
> > assumption. At http://www.barnsdle.demon.co.uk/vote/sing.html, I
> > describe it, and a non-geometrical way of getting the same result.
> >
> > Of course, after you've determined candidate i's probability of being
> > one of the 2 frontrunners, and j's probability, then you can multiply
> > those to get an estimate of Pij.
> >
> > Of course you can also calculate Pij from a _reliable_ poll. As I
> > was saying, if the poll says that there's a certain probability that
> > the vote totals will be within a certain distance from the predicted
> > ones, then it's possible to calculate the Pij by assuming that the
> > probability distributions are normal, and that those of the various
> > candidates are similar.
> >
> > Mike Ossipoff