A Question About Arrow's Impossibility Theorem

Arrow's Impossibility Theorem is a theorem about voting methods that says that any voting method devised cannot have all of the following properties

• 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.

Now, I certainly agree that the first three are essential for a good voting method, put I'm uncertain against the fourth. Obviously, having a voting system that just follows the choices of one voter is a bad idea, but this is not the way the criteria is generally considered. The proofs I've read through has shown that if one starts changing voters preferences in a systematic but arbitrary way, there will be some voter who when they change their vote, the final result will change as well, and this voter in fact acts as a dictator.

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.

End Post
Writing time: too long (I got distracted by cartoons)
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