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Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem in social choice theory that states that when voters have three or more distinct alternatives , no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide ranking while also meeting the specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem. The theorem is named after economist and Nobel laureate Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book Social Choice and Individual Values. The original paper was titled "A Difficulty in the Concept of Social Welfare".
In short, the theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria:
Cardinal voting electoral systems are not covered by the theorem, as they convey more information than rank orders. However, Gibbard's theorem shows that strategic voting remains a problem.
The axiomatic approach Arrow adopted can treat all conceivable rules within one unified framework. In that sense, the approach is qualitatively different from the earlier one in voting theory, in which rules were investigated one by one. One can therefore say that the contemporary paradigm of social choice theory started from this theorem.