Mathematical utility expectation maximization in Approval

[Martin Harper]

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.
> 
> 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