You can take your input rankings, map them into scores, use your process on the numbers, and get output scores. These output scores satisfy the 4 criteria. You can then map these output scores back into rankings. Because the 4 criteria don’t depend on whether you use rankings or scores, they must still be satisfied, except for determinism. Determinism might not be satisfied because the mapping of rankings into scores is one-to-many.
Does Arrow’s theorem allow us to find solutions to the original ranking problem if we throw out the determinism criterion? I don’t see how determinism is useful.
By Arrow’s Theorem, any process by which rankings are mapped to scores must violate one of these criteria, once the scores are processed using my algorithm. This is not terribly surprising, as you lose all the information about relative importance in doing so.
but can we find a process which violates only determinism?
That’s exactly what tommccabe’s process does. It violates determinism in Arrow’s sense of the word, but it satisfies the other conditions. Moreover, it violates Arrow-style determinism in a way that we arguably want anyway.
You can take your input rankings, map them into scores, use your process on the numbers, and get output scores. These output scores satisfy the 4 criteria. You can then map these output scores back into rankings. Because the 4 criteria don’t depend on whether you use rankings or scores, they must still be satisfied, except for determinism. Determinism might not be satisfied because the mapping of rankings into scores is one-to-many.
Does Arrow’s theorem allow us to find solutions to the original ranking problem if we throw out the determinism criterion? I don’t see how determinism is useful.
By Arrow’s Theorem, any process by which rankings are mapped to scores must violate one of these criteria, once the scores are processed using my algorithm. This is not terribly surprising, as you lose all the information about relative importance in doing so.
Yes; but can we find a process which violates only determinism? Because we don’t need determinism.
That’s exactly what tommccabe’s process does. It violates determinism in Arrow’s sense of the word, but it satisfies the other conditions. Moreover, it violates Arrow-style determinism in a way that we arguably want anyway.