in social choice theory, the gibbard–satterthwaite theorem is a result published independently by philosopher allan gibbard in 1973 and economist mark satterthwaite in 1975. it deals with deterministic ordinal electoral systems that choose a single winner. it states that for every voting rule, one of the following three things must hold:
- the rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or
- the rule limits the possible outcomes to two alternatives only; or
- the rule is susceptible to tactical voting: in certain conditions some voter's sincere ballot may not defend their opinion best.
while the scope of this theorem is limited to ordinal voting, gibbard's theorem is more general, in that it deals with processes of collective decision that may not be ordinal: for example, voting systems where voters assign grades to candidates.
gibbard's 1978 theorem and
hylland's theorem are even more general and extend these results to non-deterministic processes, i.e. where the outcome may not only depend on the voters' actions but may also involve a part of chance.