The Welfare Theorems

Society has $N$ individuals and a finite set of at least three alternatives. Each individual has a preference relation. A social welfare function (SWF) assigns to each profile of preference relations $\succeq=(\succeq_1,\ldots,\succeq_N)$ an aggregate or social preference relation.

• The domain of the SWF is all possible preference profiles.
• A SWF respects unanimity if it ranks $A$ above $B$ whenever every individual ranks $A$ above $B$.
• It respects independence of irrelevant alternatives if the relative ranking of $A$ and $B$ depends only on their relative ranking by every individual.
• A SWF $F$ is a dictatorship by individual $n$ if for all $\succeq$, $F(\succeq)={}\succeq_n$.

Arrow's Theorem. Any SWF that respects universal domain, independence of irrelevant alternatives, and unanimity is a dictatorship.