| This is how I was taught it (or understood I was taught it) - at law school, so it might have been dumbed down. The impossibility is the impossibility of ensuring rational (transitive) outcomes amongst ranked preferences and adhering to a set of fair and democratic norms. A rational transitive outcomes is one in which votes result in option A being preferred over option B and option B being preferred over option C, such that A is preferred over C (eg, A > B > C). Option A is known as the Condorcet winner. But there may be cases where the vote yields no Condorcet winner (eg, A > B > C > A). This is illustrated by the following table: Preferences
Voter 1 Choc Vanil Strwb
Voter 2 Vanil Strwb Choc
Voter 3 Strwb Choc Vanil Two voters prefer C over V and two prefer V over S, but two also prefer S over C. To ensure transitivity, we can introduce voting rules, but it is impossible to introduce rules that do not violate the fair and democratic norms (referred to as the pre-specified criteria in the Wikipedia article: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives). |
Realistically, that kind of situation doesn't break a voting system. We can say "we don't care about that case -- just pick a random winner then", but it's no longer deterministic.
Is there a stronger version of the theorem that says there's no sane procedure even ignoring those cases?