Restricting the domain allows for weaker independence

Arrow’s classical axiom of independence of irrelevant alternatives may be more descriptively thought of as binary independence. This can then be weakened to ternary independence, quaternary independence, etc. It is known that under the full domain these are not real weakenings as they all collapse into binary independence (except for independence over the whole set of alternatives which is trivially satisfied). Here we investigate whether this still happens under restricted domains. We show that for different domains these different levels of independence may or may not be equivalent. We specify when and to what extent different versions of independence collapse into the same condition.