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Arrow's theorem is mathematics, but even citing it is greatly misleading. All Arrow proved was that, sometimes, voting becomes rock-paper-scissors between the

Arrow's theorem is mathematics, but even citing it is greatly misleading. All Arrow proved was that, sometimes, voting becomes rock-paper-scissors between the three most popular candidates. That isn't "unfair". It especially doesn't mean that all voting systems are equally bad, and yet the most common reason for people to cite Arrow's theorem is to try and dispute the idea that we can do better.

thaumasiotes 9y ago
Unlike your parent comment, your comment is total nonsense. Arrow proved that, given a set of ordinal preference rankings held by several individuals, the concept of an aggregate preference ranking describing the overall "will of society" is not well defined; subject to four or five unobjectionable constraints, no function determining such an overall ranking exists. Voting systems which obey the assumptions of the theorem frequently break down in ways that cannot be described as "rock-paper-scissors between the three most popular candidates", as when strategic voting caused the papal conclave of 1334 to unanimously elect the least popular candidate to the papacy. Were you perhaps thinking of Condorcet's work when you referred to rock-paper-scissors? He, not Arrow, famously wrote about how voter preferences were not necessarily transitive.
- YokoZar 9y ago
  That "rock-paper-scissors" cycle among the top preferences is the Smith set, and every sensible voting "violates" Arrows theorem by saying that sometimes it's ok to acknowledge that cycle. One of Arrows assumptions, "Irrelevance of Independent Alternatives", is far too strong. If you relax it to "Local Independence of Irrelevant Alternatives", then Arrows theorem no longer applies. It's just not a relevant theorem. Claiming that allowing a voting system to recognize a rock-paper-scissors cycle among the top candidates is the same as allowing dictators is obvious rubbish. But if you interpret Arrow's theorem naively, you might be tempted to make such a claim.

thaumasiotes 9y ago

Unlike your parent comment, your comment is total nonsense. Arrow proved that, given a set of ordinal preference rankings held by several individuals, the concept of an aggregate preference ranking describing the overall "will of society" is not well defined; subject to four or five unobjectionable constraints, no function determining such an overall ranking exists. Voting systems which obey the assumptions of the theorem frequently break down in ways that cannot be described as "rock-paper-scissors between the three most popular candidates", as when strategic voting caused the papal conclave of 1334 to unanimously elect the least popular candidate to the papacy. Were you perhaps thinking of Condorcet's work when you referred to rock-paper-scissors? He, not Arrow, famously wrote about how voter preferences were not necessarily transitive.

YokoZar 9y ago
That "rock-paper-scissors" cycle among the top preferences is the Smith set, and every sensible voting "violates" Arrows theorem by saying that sometimes it's ok to acknowledge that cycle. One of Arrows assumptions, "Irrelevance of Independent Alternatives", is far too strong. If you relax it to "Local Independence of Irrelevant Alternatives", then Arrows theorem no longer applies. It's just not a relevant theorem. Claiming that allowing a voting system to recognize a rock-paper-scissors cycle among the top candidates is the same as allowing dictators is obvious rubbish. But if you interpret Arrow's theorem naively, you might be tempted to make such a claim.