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Arrow's Theorem has to be one of the most over-celebrated yet uninteresting bits of math out there. I myself even thought it was a cool way of challenging alternative voting systems when I first started studying them.

But, as you say, Arrow's Theorem is simply too broad with it's definition of "fair". It's entirely reasonable for the voters to have a list of candidates they hate and a list of 3 candidates they would prefer, and for there to be a rock-paper-scissors situation among those top three candidates. It's entirely possible to have a voting system that only chooses from among the rock-paper-scissors options, and indeed every Condorcet system will make such a choice. Those options are the Smith set.

The tragedy is that Arrow's Theorem is often used to justify systems (like plurality or range voting) that often won't choose from among those top three, and might even choose the Condorcet loser! I'm not sure what a good definition of "fair" is, but I'd settle for not picking the candidate that would lose to literally every other candidate in a one-on-one election.




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