Arrow's non-dictatorship principle states that "no single voter possesses the power to always determine the group's preference."
I am unsure if I am missing some nuance from this definition, so I was wondering if the following would be an example of a dictator:
There exists a simple majoritarian democracy, where a referendum asks voters to vote between policy A or B, and there are 101 voters. If 50 vote for A then the 51st voter will be a dictator, as their choice of A or B will determine which policy results, regardless of what the other voters vote for. Would this be a correct formulation?
If this is correct then what significance does Arrows theorem really have; is it not quite obvious that even the most basic democratic systems will have a dictator? In addition, why does this even matter? In most real world elections, it is basically impossible to "find" this dictator, since the location of the dictator itself can only be determined retrospectively, as it is dependent on what everyone else has voted for. Even at its most basic, for instance, in my example, the 51st voter would no longer be a dictator, if someone who originally voted for B now votes for A; this voter would now be the dictator. It seems then quite inconsequential that a dictator does exist, if it cannot be pinpointed, but more importantly, the specific "dictator" voter is dependent on what everyone else has voted for in a specific election.
Any help fixing my misunderstanding of the significance of Arrow's theorem would be much appreciated, as I realise it is quite important in the field.