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feanor1600 comments on Arrow's Theorem is a Lie - Less Wrong
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That's not what independence of irrelevant alternatives means; it means that, given the same relative ranking between A and B, the voting algorithm will always make the same decision, with or without the alternative. It doesn't mean that the voters will make the same decisions.
It says that if the voters make the same decisions once the alternative is added, then the voting algorithm does. See Arrow's definition of IIA on page 27 of the 2nd edition of Social Choice and Individual Values (you can find this part for free on Amazon).