Arrow's impossibility theorem

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Arrow's impossibility theorem is a key impossibility theorem in social choice theory, showing that no ranked voting rule^{[note 1]} can produce a logically coherent ranking of more than two candidates. Specifically, no such rule can satisfy a key criterion of rational choice called independence of irrelevant alternatives: that a choice between ${\displaystyle A}$ and ${\displaystyle B}$ should not depend on the quality of a third, unrelated outcome ${\displaystyle C}$.

The theorem is often cited in discussions of election science and voting theory, where ${\displaystyle C}$ is called a spoiler candidate. As a result, Arrow's theorem implies that a ranked voting system can never be completely independent of spoilers.

The practical consequences of the theorem are debatable, with Arrow himself noting "Most [ranked] systems are not going to work badly all of the time. All I proved is that all can work badly at times."^[1][2] Spoiler effects are common in some ranked systems (like instant-runoff (RCV) and plurality), but rare in majority-vote methods (see below).

While originally overlooked, a large class of systems called rated methods are not affected by Arrow's theorem or IIA failures.^[2][3][4] Arrow initially rejected these systems on philosophical grounds, but reversed his opinion on the issue later in life, referring to score voting as "probably the best".^[2]

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