Dan Finlay on Mythryl: “One answer to "what is the shape of a preference?"is: Posts!…”

I think sometimes voting systems get a hard time because people will cite Arrow’s Theorem, like it’s proof that democracy can work. But does Arrow’s theorem really say something we can’t work around?

Vitalik Buterin

@vitalik.ca

3w

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… https://firefly.social/post/ff-dbd021708ab74d0e995422a0b7467d5a?s=bsky

firefly.socialContinue reading on Firefly.SocialArrow'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.

Dan Finlay

@danfinlay.com

3w

What a treat to have your reply right after I thought "I guess bluesky doesn't have that sort..." That makes sense. Same axiom I was thinking was sketchy, but highlighting the "must not"/allergy case is clarifying. bsky.app/profile/danf... Gets me wondering: What's the real type of a preference?

Dan Finlay@danfinlay.com

I think I could really compromise on “Independence of Irrelevant Alternatives”. 1. It’s mostly a problem of representative democracy, and might not matter as much if voters were able to vote on individual issues.