Arrow's impossibility theorem
Arrow's impossibility theorem is a key result in social choice showing that no rank-order decision procedure can behave rationally when there is more than one voter. Specifically, any such rule violates independence of irrelevant alternatives, the principle that a choice between and should not depend on the quality of a third, unrelated option .[1][2]
The result is most often cited in election science and voting theory, where is called a spoiler candidate. In this context, Arrow's theorem can be restated as showing that no ranked-choice voting rule[note 1] can eliminate the spoiler effect.[3][4][5]
The practical consequences of the theorem are debatable, with Arrow 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."[4][6] Some methods are more susceptible to spoilers than others. Plurality and instant-runoff in particular are highly sensitive to spoilers,[7][8] often manufacturing them even in situations where they are not forced.[9][10] By contrast, Condorcet methods uniquely minimize the possibility of spoilers,[11] limiting them to rare[12][13] situations called Condorcet paradoxes.[9]
While originally overlooked by Arrow, rated methods are not affected by Arrow's theorem or IIA failures.[14][3][5] Arrow initially asserted the information provided by these systems was meaningless,[15] and therefore could not be used to prevent paradoxes. However, he and other authors[16] would later recognize this to have been a mistake,[17] with Arrow admitting systems based on cardinal utility (such as score and approval voting) are not subject to his theorem.[18][19][20]
Theorem[edit]
Intuitive argument[edit]
If we are willing to make the stronger assumptions that our voting system guarantees the principles of one vote one value and a free and fair election, the proof of Arrow's theorem becomes much simpler, and was given by Condorcet.[27] May's theorem implies that under the assumptions above, the only Pareto-efficient rule for choosing between two candidates is a simple majority vote.[28] Therefore, the requirement that all social preferences only depend on individual preferences effectively says that if and only if most voters prefer to . Given these assumptions, the existence of the voting paradox is enough to show the impossibility of rational behavior for ranked-choice voting.[27]
The above is sufficient to show Arrow's theorem for all "practical" voting systems, as systems that violate the above assumptions are unlikely to be considered meaningfully democratic.[27][28] However, Arrow's theorem can be made substantially more general.
Formal statement[edit]
Let A be a set of outcomes, N a number of voters or decision criteria. We denote the set of all total orderings of A by L(A).
An ordinal (ranked) social welfare function is a function:[29]
Common misconceptions[edit]
Arrow's theorem is not related to strategic voting, which does not appear in his framework.[31][71] The Arrovian framework of social welfare assumes all voter preferences are known and the only issue is in aggregating them.[71]
Contrary to a common misconception, Arrow's theorem deals with the limited class of ranked-choice voting systems, rather than voting systems as a whole.[71][72]