▁also ▁based ▁on ▁Ar row ' s ▁impos s ibility ▁theorem , ▁but ▁he ▁doesn ' t ▁expose ▁the ▁more ▁general ▁version ▁given ▁by ▁Gib b ard ' s ▁theorem . ▁Later , ▁several ▁authors ▁develop ▁variants ▁of ▁the ▁proof , ▁generally ▁shorter , ▁either ▁for ▁the ▁theorem ▁itself ▁or ▁for ▁the ▁cor ollary ▁and ▁weak ened ▁versions ▁we ▁mentioned ▁above . ▁ ▁Rel ated ▁results ▁ ▁Gib b ard ' s ▁theorem ▁de als ▁with ▁processes ▁of ▁collect ive ▁choice ▁that ▁may ▁not ▁be ▁ord inal , ▁i . e . ▁where ▁a ▁vot er ' s ▁action ▁may ▁not ▁consist ▁in ▁communic ating ▁a ▁preference ▁order ▁over ▁the ▁candidates . ▁Gib b ard ' s ▁ 1 9 7 8 ▁theorem ▁and ▁Hy ll and ' s ▁theorem ▁extend ▁these ▁results ▁to ▁non - det erm in istic ▁mechan isms , ▁i . e . ▁where ▁the ▁outcome ▁may ▁not ▁only ▁depend ▁on ▁the ▁ball ots ▁but ▁may ▁also ▁involve ▁a ▁part ▁of ▁chance . ▁ ▁The ▁D ug gan – Sch wart z ▁theorem ▁extend ▁this ▁result ▁in ▁another ▁direction , ▁by ▁dealing ▁with ▁determin istic ▁voting ▁rules ▁that ▁choose ▁a ▁non empty ▁subset ▁of ▁the ▁candidates ▁rather ▁than ▁a ▁single ▁winner . ▁ ▁Post er ity ▁ ▁The ▁Gib b ard – S atter th wa ite ▁theorem ▁is ▁generally ▁presented ▁as ▁a ▁result ▁belonging ▁to ▁the ▁field ▁of ▁social ▁choice ▁theory , ▁and ▁applying ▁to ▁voting ▁systems , ▁but ▁it ▁can ▁also ▁be ▁seen ▁as ▁the ▁sem inal