Arrow's impossibility theorem

Arrow's impossibility theorem states that, with at least three alternatives, no ranked-voting rule can satisfy unrestricted domain, Pareto efficiency, independence of irrelevant alternatives, and non-dictatorship while producing complete, transitive social rankings.

Social welfare function: A rule that maps individuals' preference rankings to a social preference ranking. Arrow's theorem applies to this kind of rule and specifies the conditions it cannot jointly satisfy.
Preference relation: A relation that represents how an agent ranks or compares alternatives. Arrow's conditions constrain how individual preference relations combine into a social one.
Borda count: A ranked voting method that assigns points according to each alternative's position on each ballot. It can violate independence of irrelevant alternatives, avoiding Arrow's conclusion by dropping a condition.
Condorcet paradox: A cycle in pairwise majority preferences in which each of three alternatives is defeated by another. It provides a concrete way majority judgments can fail to form a transitive social ranking.
Unrestricted domain: A condition allowing every logically possible profile of individual rankings as input to a social choice rule. It prevents the theorem from being avoided by excluding inconvenient preference profiles.
Complete relation: A relation in which every pair of elements is comparable in at least one direction. Completeness ensures the social ranking can compare every pair of alternatives.
Plurality voting: A voting rule that selects the alternative with the most first-place votes. It uses only top rankings rather than producing the complete social ranking required by the theorem.
Voting paradox: A situation in which plausible voting rules produce outcomes that conflict with intuitive expectations of collective choice. Arrow's theorem gives a general impossibility behind several such conflicts.
Pareto efficiency: A condition requiring a social ranking to favor one option over another when everyone ranks it higher. This unanimity requirement is one of the theorem's jointly incompatible conditions.
Transitive relation: A relation where ranking one option above a second and the second above a third implies ranking the first above the third. Transitivity makes the social ranking coherent across chains of pairwise comparisons.

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