Perhaps obelos is referring to: https://en.wikipedia.org/wiki/Cardinal_voting

Correct. Methods that use “rating” instead of “ranking.”

I’m not sure what you mean by “avoiding constraints”, but Arrow’s theorem very much applies to cardinal methods too. Typically these methods (e.g. the Borda count) do not satisfy IIA.

Cardinal methods induce an order, and once a method creates an order, anything said about ordinal methods applies.

I'm no mathematician, but Arrow himself said he was speaking of ordinal methods: “And in my theorem I’m assuming that the information is a ranking. Each voter can say of any two candidates, I prefer this one to this one.”[1] That's not to say that cardinal methods don't have failure modes, but the particular set of interdependent failures and how pathologically one or more failure of them can appear is not described generally by Arrow's theorem.

How can a ballot capturing cardinal values be reduced to ordinal ones? I don't understand what you're saying here. In an election using a cardinal method, the slate of candidates can be ordered ultimately when summing results, but that's not the same information as the collective mass of ballot data.

1. https://electionscience.org/commentary-analysis/voting-theor...

I think you’re misinterpreting Arrow there. Even a voting method that requires cardinal information entails that voters be able to say which one they prefer between two candidates.

As for how a ballot capturing cardinal methods be reduced to ordinal ones: it depends on the ballot. Can you give an example of a cardinal ballot that does not induce an order or otherwise implies something about a voter’s preferences (assuming they were voting sincerely)?