Arrow's impossibility theorem

Arrow's impossibility theorem is a key result in social choice theory showing that no ranked-choice procedure for group decision-making can satisfy the requirements of rational choice. Specifically, Arrow showed no such rule can satisfy the independence of irrelevant alternatives axiom. This is the principle that a choice between two alternatives A and B should not depend on the quality of some third, unrelated option, C.

The result is often cited in discussions of voting rules, where it shows no ranked voting rule to eliminate the spoiler effect. This result was first shown by the Marquis de Condorcet, whose voting paradox showed the impossibility of logically-consistent majority rule; Arrow's theorem generalizes Condorcet's findings to include non-majoritarian rules like collective leadership or consensus decision-making.

While the impossibility theorem shows all ranked voting rules must have spoilers, the frequency of spoilers differs dramatically by rule. Plurality-rule methods like choose-one and ranked-choice (instant-runoff) voting are highly sensitive to spoilers, creating them even in some situations where they are not mathematically necessary (e.g. in center squeezes). In contrast, majority-rule (Condorcet) methods of ranked voting uniquely minimize the number of spoiled elections by restricting them to voting cycles, which are rare in ideologically-driven elections. Under some models of voter preferences (like the left-right spectrum assumed in the median voter theorem), spoilers disappear entirely for these methods.

Rated voting rules, where voters assign a separate grade to each candidate, are not affected by Arrow's theorem. Arrow initially asserted the information provided by these systems was meaningless and therefore could not be used to prevent paradoxes, leading him to overlook them. However, Arrow would later describe this as a mistake, admitting rules based on cardinal utilities (such as score and approval voting) are not subject to his theorem.