# Rawlsian SWF and Arrow Impossibilty Theorem

given Arrow impossibility theorem the only social welfare function that satisfies unrestricted domain, pareto and the independence of irrelevant alternatives is Dictatorship. However I was wondering which one of these criteria the Rawlsian SWF fails. I was suspecting Independence of irrelevant alternatives, however I can't come up with a proof.

Let's say there are individuals 1 and 2, and alternatives A, B, C, and D. Society uses the Rawlsian SWF and thus ranks alternatives according to their maximal rank within individuals' rankings. Denote society's preferences by $$\succ^*$$.

If the individual rankings are:

1: A $$\succ$$ B $$\succ$$ C $$\succ$$ D

2: B $$\succ$$ D $$\succ$$ A $$\succ$$ C,

then B $$\succ^*$$ A.

If you change the ranking of 1 such that

1: C $$\succ$$ D $$\succ$$ A $$\succ$$ B

2: B $$\succ$$ D $$\succ$$ A $$\succ$$ C,

then A $$\succ^*$$ B. Thus, society's ranking of A vs. B has changed, while each individual's ranking of A vs. B stayed the same. Hence this SWF fails IIA.

Independence of irrelevant alternatives prevents you from using the information needed to implement a Rawlsian SWF; the information who is society's worst-off cannot be used.

Indeed, the relevant information is not even specified in a profile of preferences.