In Approval Strategy: Summary and Mathematical Basis, we discussed the Best-Frontrunner strategy. There's a smooth, seamless gradation from that strategy to the estimation of the Pij probabilities defined in Approval Strategy I, and their use in the formula included there, for maximizing utility expectation.
That is, we can start with the best-frontrunner strategy, and add one elaboration, and then another, until we have all the Pij for the strategic value formula in Approval I, for Weber's method.
If we believe that a certain two candidates will be the top vote getters, then each of these elaborations takes into account a less likely possibility, by using some more estimates for the probability of those less likely possibilities. Obviously, even if one felt like using one of these elaborations, that doesn't mean that one would want to use all of them.
If we're really sure that a certain two candidates, X and Y, will be the top two vote getters, then we definitely want to vote for the better of those, and not for the worse. And if we're sure, then it's harmless, if profitless, to vote for anyone else. And obviously you can't do any harm by voting for those whom you prefer to the better of those two, even if they proved to not be the top two.
In this discussion, we'll call the likely top-two vote getters "the frontrunners". We'll call the other candidates "the low candidates", or "the lows". The names X and Y will be reserved for the two frontrunners. X is the frontrunner we prefer to the other. Z, or Z1, Z2, etc., will be used for the low candidates.
One could leave it at that, which is the plain Best-Frontrunner strategy. Or one could consider the possibility that one and only one of the frontrunners will be in a tie or near-tie if there is one. That's the first elaboration from best-frontrunner.
Obviously if some low, Z, is equally likely to tie each of the two frontrunners, and if he's better than the halfway point between those frontrunners, then we should vote for him, because if he takes victory from the worse frontrunner the improvement is greater than the loss if he takes victory from the better of the two frontrunners. And both of those possibilities are equally likely.
But say we have reason to believe that one frontrunner is more likely to outpoll the other than vice-versa, a more general situation.
Say Px is the probability that X will outpoll Y, and Py is the probability that Y will outpoll X.
Of course if one and only one frontrunner is in a tie or near tie for first place, then it will be the frontrunner that outpolls the other frontrunner.
Then, if Z is some other candidate, whose utility is between those of X and Y, vote for Z if Px(Ux-Uz) < Py(Uz-Uy).
Because that's evident, we state it here without demonstration. But it also can be obtained by considering expectation, as follows.
Px is also the probability that if Z ties or nearly ties X or Y, it will tie or nearly tie X, with our own ballot able to tip that balance. So if we vote for Z, our ballot expectation for doing so is
Px(Uz-Ux) + Py(Uz-Uy)
To justify our voting for Z, that expression should be greater than zero:
Px(Uz-Ux) = -Px(Ux-Uz)
so, to justify voting for Z, it should be true that
Py(Uz-Uy) > Px(Ux-Uz)
PyUz-PyUy > PxUx-PxUz
PxUz+PyUz > PxUx+PyUy
Uz(Px+Py) > PxUx+PyUy
Since Px + Py = 1, then
Uz > PxUx + PyUy
So we vote for Z only if Z is better than the lottery between X and Y, better than our expectation if the winner must be X or Y.
Of course if we're pretty sure that the winner will be X or Y, then the above strategy is a special case of the better-Than-Expectation strategy.
Of course if X and Y are equally likely to outpoll each other, and Px = Py = 1/2, then we vote for Z if
Uz > (Ux+Uy)/2
in which case we're voting for Z if Z is better than the halfway point between X and Y.
The next elaboration is to also consider ties and near ties between two lows, and the possibility of not voting for either
Let Pboth be the probability that if there's a tie or near tie for first place it will be between X and Y.
Let Pneither be the probability that if there's a tie or near tie for first place, neither X nor Y will be in it.
Given those numbers, we can derive estimates for all of the Pij:
Pxy = Pboth
(We've estimated that it's Pboth that if there's a tie or near tie it will be between X and Y. The fact that X and Y are apparently the likely top two vote getters makes it easier to estimate the probability that they will be -- asking the question "How sure am I that they'll be the top two?" Such an estimate would be more difficult for some arbitrarily chosen pair of candidates.)
The probability that one and only one of the putative frontrunners will be in the tie or near tie for first place if there is one is: 1 - Pboth - Pneither, since there must be either 1, 2, or 0 of the 2 expected frontrunners in the tie or near tie for first if there is one.
Pxz = (1-Pboth-Pneither) * Px * Pfz
where z is some low candidate.
Pfz is the probability that if one of the lows is in a tie with one of the frontrunners, that low will be Z.
So (1-Pboth-Pneither) * Px * Pfz is the probability that exactly one of the frontrunners is in the tie or near tie for first place if there is one, and that Px outpolls Py (so that X is the one in that tie), and that Z is the low candidate that ties a frontrunner if any low candidate does.
How to estimate the Pfz? Why not numerically rate the lows according to how likely they seem to be the one who can tie a frontrunner. These ratings needn't be probabilities or add up to 1. You might assign the rating of 1 to the least likely, or to the most likely, and then rate the others with respect to that one. Then, for each particular low candidate Z, divide Z's rating by the sum of all the lows' ratings. That's an estimate of Pfz.
We'll refer to those ratings again, and will call them "frontrunner tying ratings". (The initial ratings, as opposed to the resulting Pfz).
Next, the Pij for pairs of lows. It's a reasonable approximation to assume that a low's fitness to tie another low is proportional to its fitness to tie a frontrunner. So, for each possible pair of lows, whom we'll call Z and Z', multiply together the frontrunner tying probabilities of Z and Z'. Divide the product for the Z and Z' pair by the sum of the products for all the possible pairs of lows. That's the probability that if a tie or near tie is between two lows, Z and Z' will be those lows. We'll call that probability Pfzz'.
So Pzz', the Pij for Z and Z', is: Pneither * Pfzz. The probability that the tie or near tie, if there is one, will be between two lows, multiplied by the probability that, if the tie or near tie is between two lows, z and z' will be those lows.
Now we have estimates for all of the Pij, for use in the strategic value formula in Approval Strategy I, for use in maximizing utility expectation by the Weber strategy.
Again, you may not want to take Best-Frontrunner that far, but these optional elaborations are available for those who want to estimate probabilities of less likely possibilities, to refine the strategy.