= Two-voter dictatorship from binary choice
For two voters and at least three alternatives, pairwise unanimity, pairwise independence and transitivity force the <binary social ordering from strategyproof choice> to follow one fixed voter. If voter one wins a conflict $a$ against $b$, use profiles $(a>b>c,b>c>a)$ and $(c>a>b,b>c>a)$ to show it also wins $a$ against $c$ and $c$ against $b$. Repeating this implication yields both directions of every pair. If the initial conflict instead follows voter two, exchange the roles. The social choice, being the top of that ordering, is a <dictatorship in social choice>. This proves the two-voter case of the <Gibbard-Satterthwaite theorem>.
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