▁the ▁assumption ▁that ▁the ▁rule ▁is ▁un anim ous , ▁in ▁the ▁sense ▁that ▁if ▁all ▁vot ers ▁prefer ▁the ▁same ▁candidate , ▁then ▁she ▁must ▁be ▁elected . ▁ ▁Sk etch ▁of ▁proof ▁ ▁The ▁Gib b ard – S atter th wa ite ▁theorem ▁can ▁be ▁proved ▁based ▁on ▁Ar row ' s ▁impos s ibility ▁theorem , ▁which ▁de als ▁with ▁social ▁ranking ▁functions , ▁i . e . ▁voting ▁systems ▁designed ▁to ▁yield ▁a ▁complete ▁preference ▁order ▁of ▁the ▁candidates , ▁rather ▁than ▁simply ▁choosing ▁a ▁winner . ▁We ▁give ▁a ▁sketch ▁of ▁proof ▁in ▁the ▁simplified ▁case ▁where ▁the ▁voting ▁rule ▁ ▁is ▁assumed ▁to ▁be ▁un anim ous . ▁It ▁is ▁possible ▁to ▁build ▁a ▁social ▁ranking ▁function ▁, ▁as ▁follows : ▁in ▁order ▁to ▁decide ▁whether ▁, ▁the ▁ ▁function ▁creates ▁new ▁prefer ences ▁in ▁which ▁ ▁and ▁ ▁are ▁moved ▁to ▁the ▁top ▁of ▁all ▁vot ers ' ▁prefer ences . ▁Then , ▁ ▁exam ines ▁whether ▁ ▁cho oses ▁ ▁or ▁. ▁It ▁is ▁possible ▁to ▁prove ▁that , ▁if ▁ ▁is ▁non - man ip ul able ▁and ▁non - dict atorial , ▁then ▁ ▁satisfies ▁the ▁properties : ▁un anim ity , ▁independence ▁of ▁irrelevant ▁alternatives , ▁and ▁it ▁is ▁not ▁a ▁dict ator ship . ▁Ar row ' s ▁impos s ibility ▁theorem ▁says ▁that , ▁when ▁there ▁are ▁three ▁or ▁more ▁alternatives , ▁such ▁a ▁ ▁function ▁cannot ▁exist . ▁Hence , ▁such ▁an ▁ ▁function ▁also ▁cannot ▁exist .