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I have a hard time understanding completely Arrow’s Impossibility Theorem proof, even the very pedagogical one by Geanakoplos, that can be found for instance here : https://users.ssc.wisc.edu/~dquint/econ698/lecture%202.pdf

I don't understand the very last part (part 3b in this pdf) where, after proving that there is a "dictator over policies except a specific one we arbitrarily chose", we prove that in fact it is THE dictator (by repeating the previous process with an other arbitratily, different, policy) It's supposed to be the "easy part" so I suspect I don't quite understand something before this final part... The specific part that I don't get is when it is said that as "we already know that Bob’s preferences over b sometimes matter" then " the new dictator must also be Bob". These 2 phrases sum up the part where they prove that the "dictator over policies except a specific one we arbitrarily chose" we found before is in in fact a "dictator over all policies" (and does not depend on any policy, and is unique). The first quote in this paragraph seems very light to me...

I'll give my confused explanation so that someone can tell me if I'm talking nonsense or not, and explain to me where I'm wrong in that case. One point where I'm confused (and I guess that maybe it's why I don't get the final part) is how the "dictator over policies except a specific one we arbitrarily chose" has been proven to be unique ? In the proof (part 2 and 3a) I don't see where we prove its uniqueness (I understood how we show its existence in part 2). In fact, I have the feeling that it's the very definition of this "partial dictator" that underpins his uniqueness, since he alone determines society's choice independently of all the other voters, it's impossible that there could to be two of them, is that right ? Is this what explains the final part ? I guess my confusion is visible... Many thanks in advance. I can elaborate more if needed

NB : A more concise and "mathematical" version of this proof can be found here : https://www.ceremade.dauphine.fr/~vigeral/Memoire2017Elie.pdf If you prefer to discuss it based on this version.

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There cannot be two dictators over all alternatives, what would happen if one wanted alternative 1 and the other alternative 2? So if they prove that Bob is a dictator, they have proven that Bob is the unique dictator.

In other words: they don't have to prove uniqeness, it is inherent in the definition of dictator.

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