11 ms·
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 ensur
by kelvintran 11y ago
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).
- SilasX 11y agoYeah, but that takes the punch out of the theorem. It's saying, "hey, sometimes you have really screwy preferences, too bad." 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?
- gweinberg 11y agoNo, of course not. It's really easy to come up with a system that always comes up with "good" results if you rule out "screwy" voter preferences, with a sufficiently restrictive value of "screwy".
- SilasX 11y ago"Screwy" in this context means the sort of intransitive situations referred to in the parent of my last comment.
- reagency 11y agoThat is all the punch he theorem has. Arrows theorem shows that aggregate preferences have ties even when individuals don't, and strategic voting can tip the results in those cases.
- amelius 11y agoYou should update the wikipedia page with this explanation.