Question about the "No Dictator" Criterion in Arrow's Impossibility Theorem

Question

I have a question about Arrow's impossibility theorem. I'm not sure I understand exactly what is meant by the "no dictator" criterion. Does the presence of a "dictator" mean that across all possible combinations of individual preference orderings for N individuals, the same person's preferences will always match the group preference ordering exactly? Or does it simply mean that across all possible combinations of individual preference orderings for N individuals, there will be some individual for each one of those combinations whose preferences match the group preferences exactly, but that individual doesn't necessarily have to be the same for each combination?

If the former, I can clearly construct a counter-example that (I think) satisfies the other two criteria (unanimity and independence of irrelevant alternatives). For example:

Combination 1:

Person 1: A > B > C

Person 2: B > A > C

Person 3: C > A > B

Group preference: A > B > C

Combination 2:

Person 1: A > B > C

Person 2: B > A > C

Person 3: B > A > C

Group preference: B > A > C

In this example, Person 1 is the "dictator" in combination 1, but not in combination 2. Therefore, by the first definition above of "dictator" there is no dictator in this system (I'm also assuming I haven't violated the other two criteria: someone please point out if I have). If we're going by the second definition of "dictator" above, sure, this system has a "dictator" under each combination, but this doesn't seem like an appropriate definition to me. If the ballot changed with new issues to rank (e.g. A*, B*, and C*) these "dictators" might not hold sway anymore, so are they really "dictators" in the sense we normally think? Or is it just happenstance that their preferences happened to align with the group preferences one time? Obviously, there's something about the theorem that I'm not understanding, so if someone could please clarify that for me I'd be very appreciative. Thanks!

Answer 1

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.