Arrow's Impossibility Theorem

A formal social‑choice theorem stating that no deterministic rank‑order aggregation rule (voting system) converting individual complete preference orderings into a collective ranking can satisfy simultaneously a standard set of fairness axioms — unrestricted domain, paretian unanimity, independence of irrelevant alternatives, and non‑dictatorship — when there are at least three alternatives.

A formal result stating that when there are at least three alternatives, no social welfare function that maps every profile of individual strict preference orderings to a complete and transitive collective ordering can simultaneously satisfy the axioms of unrestricted domain, Pareto (unanimity), independence of irrelevant alternatives, and non‑dictatorship.