Binary social ordering from strategyproof choice

For an onto strategyproof rule on unrestricted strict preference orders, rank-raising monotonicity implies unanimity and respect for unanimous pairwise preference. Promote each alternative pair to the top two positions in all reports. The selected member depends only on each voter's comparison of that pair. These binary choices form a strict transitive social order: a directed three-cycle would contradict the rule's choice when that triple is placed above all other alternatives, because the chosen member must win both of its binary comparisons. The original outcome is the top of this order. This creates a social ordering with pairwise independence without presuming it as an axiom of the original choice rule.

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.