Arrow's Impossibility Theorem

Arrow's impossibility theorem is a result in social choice theory showing that no ranked-choice voting system can convert the ranked preferences of three or more individual voters into a single community-wide ranking while simultaneously satisfying a specific, individually reasonable-seeming set of fairness criteria all at once. Economist Kenneth Arrow first proved the theorem in his 1951 doctoral work, published the same year as the book Social Choice and Individual Values, for which he later shared the 1972 Nobel Memorial Prize in Economic Sciences, awarded partly in recognition of this contribution. The theorem shows that any ranked voting system covering three or more options must violate at least one of a small set of conditions Arrow specified, including that the system should not simply ignore some voters (non-dictatorship), should respect a unanimous preference (Pareto efficiency), should work for any possible pattern of individual preferences (unrestricted domain), and should rank two options based only on voters' relative preferences between those two options and not on their views of some unrelated third option (independence of irrelevant alternatives). The result is considered foundational to social choice theory and to the mathematical study of voting systems, since it demonstrates a genuine mathematical limit on designing any perfectly fair ranked voting mechanism, rather than merely a defect of some particular system in use.