OP seems dumb.
What's the deal with Arrow's Impossibility Theorem?

op doesn't know the meaning of "proof" or "assumption"
so what is the point of trying to explain it to himI think he doesn't know the statement of the theorem, and then talking about proof or assumption is meaningless.
I'm not saying you're wrong; I also think he doesn't know what a proof is. But the confusion arises before we get to that point.

I am also completely dumb and therefore think that the way they prove Arrow's theorem for instance in Jehle/Reny shows merely that there always "exist" preferences for which the pivotal agent will be a dictator in unconstraint preferences case.
It seems to me that the essence of the original theorem (as seen in MWG) shows a bit more. It shows that you cannot even begin to develop a social welfare function (the one that ranks the states) that would satisfy all the assumptions and still would not look like a dictatorial one.
To give an example, using Jehle/Reny's approach one could think that pairwise majority could work but would run the risk of sometimes producing a dictator (given some very specific preferences of the agents). A real student of MWG would however immediately recognize that since the pairwise majority is not dictatorship, it cannot possibly satisfy all the assumptions we required. In this case, pairwise majority does not always result in transitive ranking.
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Arrow's says that we can't have a voting rule satisfying:
1. It is pareto optimal
2. There is no dictator
3. It applies for any set of preferences that can be fed into it
... other stuffThe part of the proof you are becoming unhinged about is that while a Pareto optimal voting rule can be constructed for certain restrictions on the preferences fed into it, this is false if you consider the set of all preference orderings. So when Arrow goes looking for a dictator, what he's saying is that a particular agent's preferences become pivotal as a consequence of the other conditions, and this violates the no dictator condition. It's really not that distressing.

^Could you give an example of a SWF that would not have the format of a dictatorship but otherwise fulfilled all other conditions? You can't. Is this fact proved in the "pivotal method" of the proof? I doubt it. They just show that this SWF would result in a dictatorship for some profile of preference orderings of the voters.
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