Say there are some candidates who are completely unacceptable, but winnable. Then obviously you should vote for all the acceptable candidates and for none of the unacceptable ones. Personally, it seems to me that that situation obtains in all of our national and state political elections, which greatly simplifies Approval strategy. But not everyone would agree with me on that, which is why strategies for non-zero-info elections without completely unacceptable but winnable candidates are important.
Of course most of these Approval Strategy articles are about less obvious situations in which there aren't completely unacceptable but winnable candidates.
I spoke of zero-info elections, where there's no information about candidates' winnability, or the various tie-probabilities. National and state elections are rarely if ever of that type, but organizational elections could be, if there hasn't been a lot of discussion. As I said, in a zero-info election, vote for all the above-mean candidates.
So far, I've only given two strategy instructions for when it isn't zero-info, and there are no completely unacceptable but winnable candidates:
Estimate the likely 2 top votegetters, and vote for whichever of those 2 you like better--and also vote for everyone whom you prefer to him/her.
That can also be worded: Vote for the candidate for whom you'd vote in Plurality, and also for everyone whom you like better.
Of course best-frontrunner is how most people vote in Plurality. (If it's a Plurality election with completely unacceptable but winnable candidates, you should vote for whichever acceptable candidate is most likely to be able to be able to be helped by you to take victory from an unacceptable).
Vote for every candidate whom you'd rather put in office directly, if you could, instead of holding the election.
Those are good strategies, but there are more too.
Before I continue, though, I should point out that often you'll know how you want to vote, without any kind of mathematical strategy, even in a non-zero-info election without completely unacceptable but winnable candidates, even if there are no clear frontrunners. Sometimes how you choose to vote is about principle, sometimes it's a way to make a comment to candidates or voters. The strategies that I describe are for those times when you don't already have a clear preference about how to vote in such an election.
In particular, for one thing, the Best-Frontrunner strategy leads to more elaborate strategies:
There's a smooth, seamless gradation from the best-frontrunner strategy to the estimation of the Pij that I defined in Approval Strategy I, and their use in the formula that I included there, for maximizing utility expectation. By the way, sometimes I'll refer to that use of that formula to calculate strategic value, for maximizing utility expectation, as "the Weber method", because the first mention of it that I know of was in an article by Weber, the mathematician who was the initial proponent of Approval.
By that, I mean that you can start with best-frontrunner, and add one elaboration, and then another, till you have all the Pij for the strategic value formula in Approval I, for Weber's method.
But I emphasize that, often or usually, you're likely to be satisifed with the un-elaborated best-frontrunner strategy. And if you use the 1st elaboration, you likely won't feel a need to go on to the 2nd elaboration, etc. I merely list these elaborations for the sake of completeness, and to show how "best-frontrunner" leads to estimates of the Pij for Weber's method.
If you're really sure that a certain 2 candidates, X & Y, will be the top 2 votegetters, then you definitely want to vote for the better of those, and not for the worse. And if you're sure, then it's harmless, if profitless, to vote for anyone else. Obviously you can't do any harm by voting for those whom you prefer to the better of those 2.
In this discussion, I'll call the 2 candidates expected to be the likely top 2 votegetters "the frontrunners". I'll call the other candidates "the low candidates", or "the lows". The names X & Y will be reserved for the 2 frontrunners. X is the frontrunner whom you prefer to the other. Z will be used for the low candidates.
Though you could leave it at that, which is the plain best-frontrunner strategy, you 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.
For the time being, we ignore ties between 2 lows.
If a low's merit is somewhere between those of the frontrunners, then one suggested strategy is to vote for him if his utility is better than the average utility of the frontrunners. That assumes that you have no information about which of the frontrunners low is more likely to be in a tie with (or, to say it differently, which of the frontrunners will outpoll the other).
Obviously if some low, Z, is equally likely to tie each of the 2 frontrunners, and if he's better than the halfway point between those frontrunners, then you should vote for him, because if he takes victory from the worse frontrunner the improvment is greater than the loss if he takes victory from the better of the 2 frontrunners. And both of those things are equally likely.
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 also 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. That's a special case of the better-than-expectation strategy discussed in Approval Strategy II.
The next elaboration is to also consider ties and nearties between 2 lows.
Say that it's 80% that X & Y will be the top votegetters and will be in a tie for 1st if there is one. And say that you estimate a 2% probability that if there's a tie or neartie, neither of its 2 members will be our putative frontrunners, but that they'll both be "lows".
I'm expressing this as an example, using example numbers such as 70%, 30%, 80%, & 2%, instead of naming those probabilities with letters, because it seems clearer to say it this way.
So, given those numbers, we can derive estimates for all of the Pij:
Pxy = .8
(We've estimated that it's 80% that if there's a tie or neartie it will be between X & Y. The fact that X & Y are apparently the likely top 2 votegetters makes it easier to estimate the probability that they will be. Such an estimate would be more difficult for some arbitrarily chosen pair of candidates).
Pxz = (.18)(.7)*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.
I'll talk about how to estimate Pfz in a minute.
The factor of .18 comes from the fact that we've estimated that it's 80% that, if there's a tie or neartie it will be between the frontrunners. And that it's 2% that such a tie will include neither frontrunner. That adds up to 82%. That leaves .18 as the probability that if there's a tie or neartie it will include one frontrunner and one low.
The factor of .7 in that expression comes from the fact that we've estimated that X will outpoll Y, so that, if a low ties or nearties a frontrunner, that frontrunner will be X.
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 some particular low candidate Z, divide Z's rating by the sum of all the lows' ratings. That's an estimate of Pfz. I'll refer to those ratings again, and will call them "frontrunner tying ratings".
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 & Z', multiply together the frontrunner tying ratings of Z & Z'. Divide the product for the Z & Z' pair by the sum of the products for all the possible pairs of lows. That's the probability that if a tie or neartie is between 2 lows, Z & Z' will be those lows. I'll call that probability Pfzz'.
So Pzz', the Pij for Z & Z', is: (.02)Pfzz'
...the probability that the tie or neartie, if there is one, will be between two lows, multiplied by the probability that, if the tie or neartie is between 2 lows, z & z' will be those lows.
Now we have estimates for all of the Pij, for use in the strategic value formula in Approval I, for use in maximizing utility expectation by the Weber strategy.