Arrow's Impossibility Theorem: example function

Question

I am looking for a social welfare function which satisfies "unrestricted domain", "Pareto efficiency", and "independence of irrelevant alternatives". One of the known proofs for Arrow's theorem argues by assuming the existance of such a function and follows that a dictator must exist. I wanted to execute this "construction" on an explicit function but noticed that I cannot actually find any online.

Answer 1

I have always liked the idea of delegated dictatorship, so you have a hierarchy of individuals with respective preferences meeting the standard requirements $\succeq_1,\succeq_2,\succeq_3, \dots$ such that

• $A \succ B\qquad$ if $A \succeq_1 B$, and $B \not \succeq_1 A$ (the dictator)
• $A \succ B\qquad$ if $A\succeq_i B$ for $i \le n$, and $B\succeq_i A$ for $i \lt n$, and $B \not \succeq_n A$ (delegated dictator)
• $A \asymp B\qquad$ if $A\succeq_i B$ and $B\succeq_i A$ for all $i$ (total indifference)

It is rather like ordering the complex numbers by their real parts, but if those are equal then by their imaginary parts.