# Unrestricted domain vs complete

Arrow's impossibility theorem states that no social choice rule satisfies a certain list of desiderata. Amongst these are completeness and unrestricted domain. Could someone please explain the difference?

You could have completeness and give up unrestricted domain. A non social choice example:

The relation "larger than" or $$\leq$$ is not complete over the complex plane $$\mathbb{Z}$$, but it is complete if it is restricted to a subset of it, real numbers.

Similarly you could have unrestricted domain and give up completeness.

E.g. majority voting is not complete if you allow unrestricted domain.

Therefore these are separate properties. You can argue you need both, but perhaps you are willing to part with one or the other. Personally I agree that giving up completeness rather than unrestricted domain seems strange.

• Why is majority rule not complete if you allow unrestricted domain? Apologies if I am missing something obvious!
– user17900
Oct 25, 2018 at 20:22
• @freelunch Because it can result in no alternative getting 50%+ of the votes. It is the majority rule, not the plurality rule, so it is not enough to get the most votes (the plurality of the votes). Oct 26, 2018 at 6:13