Arrow's impossibility theorem

Arrow's impossibility theorem is a key impossibility theorem in social choice theory, showing that no ranked voting rule 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." Spoiler effects are common in some ranked systems (like instant-runoff and plurality), but rare in majority-rule methods (see below).

While originally overlooked, a large class of methods, called rated methods, are not afflicted by Arrow's theorem or IIA failures. Arrow himself came to support Score voting later in life, saying it "probably is best".