Arrow's impossibility theorem

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Arrow's impossibility theorem is a key result in social choice theory, showing that no ranking-based decision rule can satisfy the requirements of rational choice theory.[1] Most notably, Arrow showed that no such rule can satisfy all of a certain set of seemingly simple and reasonable conditions that include independence of irrelevant alternatives, the principle that a choice between two alternatives A and B should not depend on the quality of some third, unrelated option C.[2][3][4]

The result is most often cited in discussions of voting rules.[5] However, Arrow's theorem is substantially broader, and can be applied to methods of social decision-making other than voting. It therefore generalizes Condorcet's voting paradox, and shows similar problems exist for every collective decision-making procedure based on relative comparisons.[1]

Plurality-rule methods like first-past-the-post and ranked-choice (instant-runoff) voting are highly sensitive to spoilers,[6][7] particularly in situations where they are not forced.[8][9] By contrast, majority-rule (Condorcet) methods of ranked voting uniquely minimize the number of spoiled elections[9] by restricting them to rare[10][11] situations called cyclic ties.[8] Under some idealized models of voter behavior (e.g. Black's left-right spectrum), spoiler effects can disappear entirely for these methods.[12][13]

Arrow's theorem does not cover rated voting rules, and thus cannot be used to inform their susceptibility to the spoiler effect. However, Gibbard's theorem shows these methods' susceptibility to strategic voting, and generalizations of Arrow's theorem describe cases where rated methods are susceptible to the spoiler effect.

  1. ^ a b Morreau, Michael (2014-10-13). "Arrow's Theorem". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
  2. ^ Arrow, Kenneth J. (1950). "A Difficulty in the Concept of Social Welfare" (PDF). Journal of Political Economy. 58 (4): 328–346. doi:10.1086/256963. JSTOR 1828886. S2CID 13923619. Archived from the original (PDF) on 2011-07-20.
  3. ^ Arrow, Kenneth Joseph (1963). Social Choice and Individual Values (PDF). Yale University Press. ISBN 978-0300013641. Archived (PDF) from the original on 2022-10-09.
  4. ^ Wilson, Robert (December 1972). "Social choice theory without the Pareto Principle". Journal of Economic Theory. 5 (3): 478–486. doi:10.1016/0022-0531(72)90051-8. ISSN 0022-0531.
  5. ^ Borgers, Christoph (2010-01-01). Mathematics of Social Choice: Voting, Compensation, and Division. SIAM. ISBN 9780898716955. Candidates C and D spoiled the election for B ... With them in the running, A won, whereas without them in the running, B would have won. ... Instant runoff voting ... does not do away with the spoiler problem entirely
  6. ^ McGann, Anthony J.; Koetzle, William; Grofman, Bernard (2002). "How an Ideologically Concentrated Minority Can Trump a Dispersed Majority: Nonmedian Voter Results for Plurality, Run-off, and Sequential Elimination Elections". American Journal of Political Science. 46 (1): 134–147. doi:10.2307/3088418. ISSN 0092-5853. JSTOR 3088418. As with simple plurality elections, it is apparent the outcome will be highly sensitive to the distribution of candidates.
  7. ^ Borgers, Christoph (2010-01-01). Mathematics of Social Choice: Voting, Compensation, and Division. SIAM. ISBN 9780898716955. Candidates C and D spoiled the election for B ... With them in the running, A won, whereas without them in the running, B would have won. ... Instant runoff voting ... does not do away with the spoiler problem entirely, although it unquestionably makes it less likely to occur in practice.
  8. ^ a b Holliday, Wesley H.; Pacuit, Eric (2023-03-14). "Stable Voting". Constitutional Political Economy. 34 (3): 421–433. arXiv:2108.00542. doi:10.1007/s10602-022-09383-9. ISSN 1572-9966. This is a kind of stability property of Condorcet winners: you cannot dislodge a Condorcet winner A by adding a new candidate B to the election if A beats B in a head-to-head majority vote. For example, although the 2000 U.S. Presidential Election in Florida did not use ranked ballots, it is plausible (see Magee 2003) that Al Gore (A) would have won without Ralph Nader (B) in the election, and Gore would have beaten Nader head-to-head. Thus, Gore should still have won with Nader included in the election.
  9. ^ a b Campbell, D. E.; Kelly, J. S. (2000). "A simple characterization of majority rule". Economic Theory. 15 (3): 689–700. doi:10.1007/s001990050318. JSTOR 25055296. S2CID 122290254.
  10. ^ Gehrlein, William V. (2002-03-01). "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*". Theory and Decision. 52 (2): 171–199. doi:10.1023/A:1015551010381. ISSN 1573-7187.
  11. ^ Van Deemen, Adrian (2014-03-01). "On the empirical relevance of Condorcet's paradox". Public Choice. 158 (3): 311–330. doi:10.1007/s11127-013-0133-3. ISSN 1573-7101.
  12. ^ Black, Duncan (1948). "On the Rationale of Group Decision-making". Journal of Political Economy. 56 (1): 23–34. doi:10.1086/256633. ISSN 0022-3808. JSTOR 1825026.
  13. ^ Black, Duncan (1968). The theory of committees and elections. Cambridge, Eng.: University Press. ISBN 978-0-89838-189-4.