However, since Arrow deals with social welfare functions which take a profile of preferences as input and outputs a full preference ranking, it really says something about aggregating a set of preferences into a single group preference.
I’m going to nitpick here—it’s possible to write down forms of Arrow’s theorem where you do get a single output. Of course, in that case, unlike in the usual formulation, you have to make assumptions about what happens when candidates drop out—considering what you have as a voting system that yields results for an election among any subset of the candidates, rather than just that particular set of candidates. So it’s a less convenient formulation for proving things. Formulated this way, though, the IIA condition actually becomes the thing it’s usually paraphrased as—“If someone other than the winner drops out, the winner stays the same.”
Since Arrow and GS are equivalent, it’s not surprising to see intermediate versions. Thanks for pointing that one out. I still stand by the statement for the common formulation of the theorem. We’re hitting the fuzzy lines between what counts as an alternate formulation of the same theorem, a corollary, or a distinct theorem.
I’m going to nitpick here—it’s possible to write down forms of Arrow’s theorem where you do get a single output. Of course, in that case, unlike in the usual formulation, you have to make assumptions about what happens when candidates drop out—considering what you have as a voting system that yields results for an election among any subset of the candidates, rather than just that particular set of candidates. So it’s a less convenient formulation for proving things. Formulated this way, though, the IIA condition actually becomes the thing it’s usually paraphrased as—“If someone other than the winner drops out, the winner stays the same.”
Edit: Spelling
Since Arrow and GS are equivalent, it’s not surprising to see intermediate versions. Thanks for pointing that one out. I still stand by the statement for the common formulation of the theorem. We’re hitting the fuzzy lines between what counts as an alternate formulation of the same theorem, a corollary, or a distinct theorem.