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The first what if you suggested is the one that will most likely bite you in the ass. If two candidates are similar and overwhelmingly popular, the more people that vote only for their most preferred are each making it more likely that the one candidate(s) they don't approve of have a more competitive number of votes. It's simply a bad strategy. While I think plenty will think that way in the shirt term, I would argue that the incentive is actually to not push that strategy, which will better serve everyone in the long term.
As for the lesser of two evils vote, that's not a problem, that's a feature. You can still vote for the lesser of two evils for the candidates you think has a chance AND for the candidates you actually approve of. The point of approval voting is to find the candidate that is MOST approved of. Even if angry single person voted for all of the candidates except their idea of the worst one, at the very least, that still eliminates the possibility of the least popular candidate from being elected. There can definitely be some spoilage there, an extremist candidate makes an easy target to rally against and a slightly less extreme though still fringe candidate gets more votes as a result, but it is still an unlikely outcome for them to have enough support to win a race that they are unpopular in.
I disagree.
Let's say there are 4 candidates, A B C and D, and a large group of people have them in that order of preference, their (honest) acceptance would be A and B, but they'd much prefer C over D if those were the only two options.
A prominent forecast comes out and predicts a tossup between C or D. They all act in self-interest and strategically list A B and C as approved, to lower the chance of D winning over C.
Now that forecast was wrong about A's low chances for whatever reason and had they solely and honestly put down A and B, A would've barely won. All of them adding C doomed them to have to put up with someone they don't honestly approve of.
As you said before though, if we take this scenario into a single vote fptp system, we have all of them giving their single vote to C. Not only does this harm the chances of A winning even more, it also reinforces never voting for A as "A doesn't have a chance anyway and voting for A would be a wasted vote".
You can also construct a similar scenario the other way around for leaving out a candidate the group would approve of.
You're correct in all this. I simply think the real world application of approval is unlikely to typically end up with results like this. It's not impossible, but it could easily happen. The only thing ranked choice has on approval is that you can... well, rank your choice, providing weight to some candidates over others. But the standard ranked choice center squeeze effect is a pretty big problem to me. Also, sometimes the rank is arbitrary between two candidates you like equally well.
I have heard an interesting idea of using a scored approval instead. Where for n candidates, you rank the candidate you would most approve of with the number n. Then each other candidate you approve of in descending order (n-1, n-2, etc.). So in your ABCD example. If you are trying to make sure D doesnt win, you would rank A as 4, B as 3, and C (the lesser of two evils) as 2, and leave D blank. You then add those scores up and the winner is the one with the highest score. This would provide weighed results for the approval to your most preferred candidates, allowing you to give a measured amount of support to any given candidate. However, this still has the arbitrary ranking issue for similarly liked candidates. Maybe you could vote whatever rank any number of times you wish for any candidates? Idk. I'd have to try it or see some examples to think it through properly.