- unanimity, if everyone prefers A over B, then A will be ranked higher than B.
- independence of irrelevant alternatives, if everyone prefers A over B, if asked to choose between A, B, and C, A should still rank higher than B regardless of how C is ranked.
- transitivity, if everyone prefers A over B and B over C, then A should be preferred over C.
- non-dictatorship, there should be no individual whose choices determine the overall results for society.
The existence of such a dictator does not seem to be such a big problem. This is not a dictator who gets to decide the outcome. This is a dictator who is the straw that breaks the camel's back. Also, it is impossible to determine beforehand which voter will be the dictator. Finally, in a real election, there will generally be many people whose vote matches the final outcome, but that does not mean that only their vote determined the result.
Given that I don't really see why the existence of such an arbitrary dictator is a big problem, it does detract from the significance of Arrow's Impossibility Theorem. It's interesting, but not really relevant to anything (admittedly true of much of maths on first sight). If I were to design a voting method, I'd be more concerned about the transitivity property, which is much more troublesome just on its own.
Writing time: too long (I got distracted by cartoons)
Time since last post: (also too long, I'm going to try and write more, do something constructive with my unemployed time, and also get around to finishing my travel stuff)
Current media: I just turned the TV off