I am trying to find ways that combine group preferences which hold for 3 out of the 4 Arrow's axioms. Here is what I have come up with so far:
Unrestricted Domain
Not sure what scheme would satisfying no dictator, Pareto and independence of irrelevant alternatives but not unrestricted domain.
No dictatorship
If we have a voting system whose individual preferences translate to the group preferences.
Pareto's condition
Not sure for this one. Require that individual preferences between options do not reflect onto group order.. how can we neglect individual rankings.
Independence of irrelevant alternatives
Borda count: voters rank candidates in order of preference. Removing rewards change group rankings.
Could someone help me find a scheme where unrestricted domain is not satisfied?
HINT: Require that there be at least three voters. Say that there are $n$ voters. If there is a ranking shared by at least $n-1$ voters, the method returns that as the societal ranking; otherwise, it does not yield a result. Check that this method has no dictator, and that when it applies at all, it satisfies both IIA and Pareto optimality.
To violate Pareto optimality while preserving the other three fairness conditions, make one voter an anti-dictator: if that voter prefers $x$ to $y$, the societal outcome is to prefer $y$ to $x$.