Question about the "No Dictator" Criterion in Arrow's Impossibility Theorem
Arrow's Theorem concerns a social preference function ---that is, a function that produces a group preference order for every possible profile of individual preferences.  The axioms "Nondictatorship", "Independence of Irrelevant Alternatives", etc. must then be satisfied by the function at every profile.  (To be more precise, an axiom like IIA involves a comparison between the behaviour of the function at two different profiles.  So it must be satisfied by the function for every pair of profiles.)   Your counterexample only shows me the outcome at two particular profiles;  it does not specify the function itself.  Until you specify a function, we can't know whether the axioms are satisfied.  Arrow's Theorem says that, when you try to extend your example to completely specify the outcome at every profile, sooner or later you will run into a situation where you must violate one of the axioms.