Society has $N$ individuals and a finite set $X$ of at least three alternatives. Each individual
has a preference relation. A social welfare function maps each utility
profiles into a preference relation.
- The domain of the SWF is all utility functions on $X$.
- Unanimity.
If for utility profiles $u$ and $v$, for all $n$ $u_n(x)=v_n(x)$ and $u_n(y)=v_n(y)$, then the
social orders $F(u)$ and $F(v)$ agree on the ranking of $x$ and $y$.
This is IIA for utility functions.
- Utility is cardinal. If each element of utility profile $v$ is a
positive affine transformation of the corresponding element of $u$, then $F(u)=F(v)$.
Theorem. Any SWF that satisfies universal domain, unanimity, IIA, and
cardinality is a dictatorship.
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