Arrow's theorem assumes preferences are ordinal: we prefer A over B, but have no idea whether A is slightly better than B, or life-alteringly better than B.
This is not how our preferences work in reality. Of course naive…
firefly.social/post/ff-dbd021708ab7…
firefly.social
Continue reading on Firefly.Social
Arrow's theorem assumes preferences are ordinal: we prefer A over B, but have no idea whether A is slightly better than B, or life-alteringly better than B.
This is not how our preferences work in reality. Of course naive versions of cardinality also have problems (cf. the effective altruist thought experiments that everyone is talking about), but ordinality is much more blatantly wrong.
And if you allow cardinal preferences, lots of voting mechanisms work and have excellent properties (eg. QV)
The specific axiom of Arrow's Theorem that _feels_ intuitive, but is actually very sketchy, is independent of irrelevant alternatives (IIA). The axiom says: if a voting mechanism chooses A over B, then introducing some third choice C should not lead to a choice that lifts B above A. It feels right, but it's sketchy because "irrelevant alternatives" actually provide lots of meaningful information about cardinality (how much you prefer A over B)
Here's a toy example.
Suppose you have a collective meal, and 15 people prefer A > B, and 5 people prefer B > A. It seems to be, you choose A, easy.
But then you also introduce C, D ... Z and ask everyone to make rankings over those. You learn that there are 15 rankings of the form
... -> X -> Y -> Z -> A -> B -> C -> D -> ...
(B always right behind A, and let's say on average A and B are both uniformly near the front so the other letters are not even better choices)
And 5 rankings of the form
... -> X -> Y -> Z -> B -> C -> D -> .... -> A
(A always at the end)
Now, what's the actual situation? A contains substances that five people are allergic to.
Suddenly this looks very different, and the obvious choice becomes B. To the majority it seems only slightly worse than A, and to a minority the meal being B basically means that they go hungry to protect their health.
What's happening here? The "irrelevant alternatives" C ... Z are serving as _proxy evidence for the extent_ to which A is preferred over B or the reverse. And so an IIA-violating algorithm captures a relevant fact - a fact which a cardinal voting mechanism would be able to capture even more effectively.