Every now and then someone discovers Arrow’s Impossibility Theorem and gets all excited. “Democracy is impossible! Let’s have a dictator!” they declare from the rooftops, or words to that effect. I certainly found this fascinating when I first discovered it.
Eventually they calm down. There are a lot of commonly mentioned reasons why Arrow’s impossibility theorem doesn’t have massive real world consequences – it’s not like anyone thought they were using a perfectly fair voting system in the first place, and the mechanism described in the theorem doesn’t actually correspond that closely with real world votes, which are mostly just trying to elect a single winner and don’t require nearly so strong consistency.
One reason I haven’t seen mentioned is the following: If it were possible to create a voting system which satisfied the criteria of Arrow’s impossibility theorem, it would be a bad idea. Independence of irrelevant alternatives, that the ordering of A and B doesn’t depend on the introduction of C, is an appealing condition on the face of it, but it turns out that you don’t actually want real world voting systems to have it. Consider the following set of opinions:
A > B > C
B > C > A
C > A > B
B > C > A
The numbers work out as follows:
There is a 50-50 split on A > B.
75% of people think that B > C
75% of people think that C > A.
Therefore even though there is a tie between A and B, the only fair combined ordering is that B > C > A – any other ordering would make a lot more people unhappy. So the introduction of an apparently irrelevant alternative has taken a tie between A and B and broken it decisively.