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I'm trying to prove the Arrow's Theorem is not true when there are two candidates, however I'm having trouble trying to prove that there is no dictator. I have suggested that in a majority rules voting system, unanimity and IIA are satisfied but I have no idea how to prove that it isn't a dictatorship.

user2980566
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A dictator has his way whenever he expresses strict preferences.

Suppose that in the majority rule, there is a dictator $d$ and suppose that $x\succ_d y$ for some choices $x$ and $y$. If everybody else has the reverse preference: $y\succ_i x$ for all $i\neq d$, what does the majority rule say about the social choice? Does $d$ get his way in this situation?

The majority rule is the correct candidate for your situation. It fails transitivity in general but when there are only 2 candidates, transitivity is vacuously satisfied

Kim Jong Un
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  • so you're saying that it doesn't matter what the dictator votes he is treated like a single person and majority will always rule? – user2980566 Oct 09 '14 at 00:56
  • I'm saying that if a dictator exists and he votes $x$ strictly over $y$, the social choice should choose $x$ over $y$. But this breaks down when the remaining voter(s) vote $y$ strictly over $x$: in this case, the majority rule will not rank $x$ strictly over $y$. – Kim Jong Un Oct 09 '14 at 00:58
  • sorry what do you mean by the social choice should choose x over y? – user2980566 Oct 09 '14 at 01:01
  • The social choice (aka the social welfare function) is the mechanism that aggregates individuals' preferences. – Kim Jong Un Oct 09 '14 at 01:02
  • gotcha, thanks a lot! – user2980566 Oct 09 '14 at 01:02