> Ranked choice absolutely conveys more information than approval. It's not as much as score voting, but it's mathematically false to say it's the same amount as approval. Among five candidates, approval would be 2^5-2=30 possibilities (almost 5 bits of information, assuming you can't approve/disapprove all because that's the same as not voting), while ranked choice would be 5!=120 possibilities (almost 7 bits of information, assuming ties aren't permitted). It's only with 3 candidates that they have equal information in terms of possibilities (2^3-2=6=3!).

This is a common intuition trap, but it's wrong. Consider the difference between score voting on a 16-point scale vs a million-point scale vs arbitrary-precision score voting. A ballot for the first conveys 4 bits per candidate; the second, about 20 bits per candidate; the last conveys infinite bits of information. You ... probably don't expect a substantial difference between election results with each option.

The trick is that the aggregation rule destroys information, which is kind of the point of aggregation. Different aggregation rules destroy different information. The difference between aggregated approval voting and aggregated arbitrary-precision score voting eats LSBs, sure, but the obvious probabilistic constructions for extracting approval votes from arbitrary precision score votes actually approximate the arbitrary-precision votes quite well, which is not what your intuition predicts, because it turns out LSBs are, well, less significant.

Also, the flat fact is that most single-victor elections in practice don't have more than three "top" candidates that you're practically interested in extracting information about.

> And individuals don't have intransitive preferences, only populations can. (Assuming mental competence.)

That's one hell of a bad assumption.

Do I contradict myself?

Very well then I contradict myself,

(I am large, I contain multitudes.)

It's not a fantastic proxy, I agree, since the aggregation rule implicitly assumes linearity and it's almost certainly not. I think it's somewhat mitigated in practice by the kind of small intersection between situations where it's strategically safe for a party to push bullet voting and where it's useful to do so being pretty constrained to elections where the top candidates are mostly acceptable to the same voters, though? I find it pretty unintuitive to construct an approval vote election scenario with a rational spoiler effect, and that's probably less big a deal than the vote weighting issues that crop up with multi-point score votes, but that's definitely something you can quibble over.

The wider point imo is that compared to all forms of ranked choice voting, arbitrary-precision score voting is pretty obviously night-and-day superior (information-content-wise, at least), to which approval voting is a "close enough", at worst moderately inferior but at least theoretically equivalent, trade-off in favor of implementability.