With three or more alternatives, no rule turning individual rankings into a social ranking can satisfy unrestricted domain, unanimity, independence of irrelevant alternatives and non-dictatorship at once.
Arrow proved in 1951 that no social welfare function can meet unrestricted domain, unanimity, independence of irrelevant alternatives and non-dictatorship once there are at least three alternatives. Every real electoral system therefore gives up at least one, and arguments about voting reform are arguments about which.
In practice
- Three shortlisted options are voted on pairwise, and the winner depends entirely on which pair is put first.
- A committee that "just wants a fair ranking" discovers that changing a bottom-ranked option reorders the top two.
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- Law & Politics
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- What can be proved and measured about power