How does Arrow's Impossible Theorem show that the only aggregation rule that works is Dictatorship. Couldn't we have a Non-dictatorship with IIA or a Non-Dictatorship with Pareto Efficiency?

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    $\begingroup$ Why do you think this is what the Theorem shows? Please include a reference. $\endgroup$ – Giskard Jun 6 '19 at 13:48

Arrow's Impossibility Theorem merely states that (with a ranked voting method) we cannot simultaneously satisfy IIA, Pareto Efficiency, Non-dictatorship and unrestricted domain. You can satisfy any subset of those conditions, though. So Non-dictatorship with IIA is fine — so long as you're willing to drop Pareto Efficiency. Similarly, Non-dictatorship and Pareto Efficiency can be satisfied — if you drop IIA.

You can also use a system that is not ranked, such as Cardinal Voting although there is some debate as to how to adapt the IIA criterion in such cases, and some such systems nonetheless fail different criteria not addressed by Arrow, such as the Majority Criterion.

(Unrestricted domain has more to do with how to decide which candidates are eligible to appear on the ballot, and which voters are allowed to vote in general, and so isn't usually considered as one of the criteria that can be relaxed... it's more a formality in Arrow's theorem, than the other criteria.)


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