In Arrow's theorem, why not $n=2$?

Question

In the statement of Arrow's impossibility theorem, why do we omit the case $n=2$?

I will appreciate it if you can explain it in easy words. I'm by no means an expert in the area (I think it's very much clear from the question too :) )

Answer 1

It's false in the case $n=2$ - just pick the standard voting model (sum up the votes between the two candidates), and it satisfies all three of Arrow's axioms, which you can check.

Answer 2

When there are only two candidates, simple majority rules -- i.e., neither candidate can win with less than half the vote.

A little more technically, the three conditions of Arrow's theorem (unanimity, non-dictatorship, and independence of irrelevant alternatives) are not incompatible. In particular, independence of irrelevant alternatives is automatically satisfied: If alternatives $a$ and $b$ have the same rankings in $(R_1,\ldots,R_N)$ and $(S_1,\ldots,S_N)$, and if $a$ and $b$ are the only alternatives, then $(R_1,\ldots,R_N)=(S_1,\ldots,S_N)$ and hence $F(R_1,\ldots,R_N)=F(S_1,\ldots,S_N)$ no matter what $F$ is.