Arrow's impossibility theorem

Arrow's impossibility theorem is a key result in social choice theory showing that no ranking-based decision rule for a group can satisfy the requirements of rational choice.^[1] Specifically, Arrow showed no such rule can satisfy 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 often cited in discussions of voting rules,^[5] where it implies that no ranked voting rule can eliminate the spoiler effect.^[6]^[7]^[8] 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, which shows the impossibility of consistently making decisions according to majority rule, and shows similar problems exist for every collective decision-making procedure based on relative comparisons.^[1]

While the impossibility theorem shows all ranked systems must have some spoilers, the frequency of spoilers can differ dramatically by method. Plurality-rule methods like first-past-the-post and ranked-choice (instant-runoff) voting are highly sensitive to spoilers,^[9]^[10] even in situations where they are not forced.^[11]^[12] By contrast, majority-rule (Condorcet) methods of ranked voting uniquely minimize the number of spoiled elections^[12] by restricting them to voting cycles,^[11] which are rare in ideologically-driven political elections.^[13]^[14] Under some models of voter behavior (e.g. the left-right spectrum of the median voter theorem), spoilers disappear entirely for these methods.^[15]^[16]

Rated voting rules, where voters assign a separate grade to each candidate, are not affected by Arrow's theorem.^[17]^[18]^[19] 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.^[20] However, Arrow would later describe this as a mistake,^[21]^[22] stating rules based on cardinal utilities (such as score and approval voting) are not subject to his theorem.^[23]^[24]

^ ^a ^b Morreau, Michael (2014-10-13). "Arrow's Theorem". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
^ 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.
^ 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. {{cite book}}: ISBN / Date incompatibility (help)
^ 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.
^ 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
^ Ng, Y. K. (November 1971). "The Possibility of a Paretian Liberal: Impossibility Theorems and Cardinal Utility". Journal of Political Economy. 79 (6): 1397–1402. doi:10.1086/259845. ISSN 0022-3808. In the present stage of the discussion on the problem of social choice, it should be common knowledge that the General Impossibility Theorem holds because only the ordinal preferences is or can be taken into account. If the intensity of preference or cardinal utility can be known or is reflected in social choice, the paradox of social choice can be solved.
^ Kemp, Murray; Asimakopulos, A. (1952-05-01). "A Note on "Social Welfare Functions" and Cardinal Utility*". Canadian Journal of Economics and Political Science. 18 (2): 195–200. doi:10.2307/138144. ISSN 0315-4890. JSTOR 138144. Retrieved 2020-03-20. The abandonment of Condition 3 makes it possible to formulate a procedure for arriving at a social choice. Such a procedure is described below
^ Hamlin, Aaron (25 May 2015). "CES Podcast with Dr Arrow". Center for Election Science. CES. Archived from the original on 27 October 2018. Retrieved 9 March 2023.
^ 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.
^ 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.
^ ^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.
^ ^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.
^ 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.
^ 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.
^ 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.
^ Black, Duncan (1968). The theory of committees and elections. Cambridge, Eng.: University Press. ISBN 978-0-89838-189-4.
^ Ng, Y. K. (November 1971). "The Possibility of a Paretian Liberal: Impossibility Theorems and Cardinal Utility". Journal of Political Economy. 79 (6): 1397–1402. doi:10.1086/259845. ISSN 0022-3808. In the present stage of the discussion on the problem of social choice, it should be common knowledge that the General Impossibility Theorem holds because only the ordinal preferences is or can be taken into account. If the intensity of preference or cardinal utility can be known or is reflected in social choice, the paradox of social choice can be solved.
^ Kemp, Murray; Asimakopulos, A. (1952-05-01). "A Note on "Social Welfare Functions" and Cardinal Utility*". Canadian Journal of Economics and Political Science. 18 (2): 195–200. doi:10.2307/138144. ISSN 0315-4890. JSTOR 138144. Retrieved 2020-03-20. The abandonment of Condition 3 makes it possible to formulate a procedure for arriving at a social choice. Such a procedure is described below
^ Poundstone, William. (2013). Gaming the vote : why elections aren't fair (and what we can do about it). Farrar, Straus and Giroux. pp. 168, 197, 234. ISBN 9781429957649. OCLC 872601019. IRV is subject to something called the "center squeeze." A popular moderate can receive relatively few first-place votes through no fault of her own but because of vote splitting from candidates to the right and left. [...] Approval voting thus appears to solve the problem of vote splitting simply and elegantly. [...] Range voting solves the problems of spoilers and vote splitting
^ "Modern economic theory has insisted on the ordinal concept of utility; that is, only orderings can be observed, and therefore no measurement of utility independent of these orderings has any significance. In the field of consumer's demand theory the ordinalist position turned out to create no problems; cardinal utility had no explanatory power above and beyond ordinal. Leibniz' Principle of the identity of indiscernibles demanded then the excision of cardinal utility from our thought patterns." Arrow (1967), as quoted on p. 33 by Racnchetti, Fabio (2002), "Choice without utility? Some reflections on the loose foundations of standard consumer theory", in Bianchi, Marina (ed.), The Active Consumer: Novelty and Surprise in Consumer Choice, Routledge Frontiers of Political Economy, vol. 20, Routledge, pp. 21–45
^ Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.
Dr. Arrow: Now there’s another possible way of thinking about it, which is not included in my theorem. But we have some idea how strongly people feel. In other words, you might do something like saying each voter does not just give a ranking. But says, this is good. And this is not good[...] So this gives more information than simply what I have asked for.
^ Harsanyi, John C. (1979-09-01). "Bayesian decision theory, rule utilitarianism, and Arrow's impossibility theorem". Theory and Decision. 11 (3): 289–317. doi:10.1007/BF00126382. ISSN 1573-7187. Retrieved 2020-03-20. It is shown that the utilitarian welfare function satisfies all of Arrow's social choice postulates — avoiding the celebrated impossibility theorem by making use of information which is unavailable in Arrow's original framework.
^ Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.

Dr. Arrow: Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes (in spite of what I said about manipulation) is probably the best.[...] And some of these studies have been made. In France, [Michel] Balinski has done some studies of this kind which seem to give some support to these scoring methods.
^ Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.
CES: Now, you mention that your theorem applies to preferential systems or ranking systems.
Dr. Arrow: Yes.
CES: But the system that you're just referring to, approval voting, falls within a class called cardinal systems. So not within ranking systems.
Dr. Arrow: And as I said, that in effect implies more information.

[plato.stanford.edu-1] Morreau, Michael (2014-10-13). "Arrow's Theorem". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.

[Arrow1950-2] 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.

[Arrow_1963234-3] 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. {{cite book}}: ISBN / Date incompatibility (help)

[Wilson1972-4] 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.

[Borgers2233-5] 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] Ng, Y. K. (November 1971). "The Possibility of a Paretian Liberal: Impossibility Theorems and Cardinal Utility". Journal of Political Economy. 79 (6): 1397–1402. doi:10.1086/259845. ISSN 0022-3808. In the present stage of the discussion on the problem of social choice, it should be common knowledge that the General Impossibility Theorem holds because only the ordinal preferences is or can be taken into account. If the intensity of preference or cardinal utility can be known or is reflected in social choice, the paradox of social choice can be solved.

[7] Kemp, Murray; Asimakopulos, A. (1952-05-01). "A Note on "Social Welfare Functions" and Cardinal Utility*". Canadian Journal of Economics and Political Science. 18 (2): 195–200. doi:10.2307/138144. ISSN 0315-4890. JSTOR 138144. Retrieved 2020-03-20. The abandonment of Condition 3 makes it possible to formulate a procedure for arriving at a social choice. Such a procedure is described below

[8] Hamlin, Aaron (25 May 2015). "CES Podcast with Dr Arrow". Center for Election Science. CES. Archived from the original on 27 October 2018. Retrieved 9 March 2023.

[McGann2002-9] 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.

[Borgers223222-10] 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.

[Holliday23222-11] 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.

[Campbell2000-12] 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.

[Gehrlein-2002-13] 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.

[VanDeemen-14] 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.

[Black-1948-15] 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.

[Black-1968-16] Black, Duncan (1968). The theory of committees and elections. Cambridge, Eng.: University Press. ISBN 978-0-89838-189-4.

[17] Ng, Y. K. (November 1971). "The Possibility of a Paretian Liberal: Impossibility Theorems and Cardinal Utility". Journal of Political Economy. 79 (6): 1397–1402. doi:10.1086/259845. ISSN 0022-3808. In the present stage of the discussion on the problem of social choice, it should be common knowledge that the General Impossibility Theorem holds because only the ordinal preferences is or can be taken into account. If the intensity of preference or cardinal utility can be known or is reflected in social choice, the paradox of social choice can be solved.

[18] Kemp, Murray; Asimakopulos, A. (1952-05-01). "A Note on "Social Welfare Functions" and Cardinal Utility*". Canadian Journal of Economics and Political Science. 18 (2): 195–200. doi:10.2307/138144. ISSN 0315-4890. JSTOR 138144. Retrieved 2020-03-20. The abandonment of Condition 3 makes it possible to formulate a procedure for arriving at a social choice. Such a procedure is described below

[Poundstone,_William.-2013232-19] Poundstone, William. (2013). Gaming the vote : why elections aren't fair (and what we can do about it). Farrar, Straus and Giroux. pp. 168, 197, 234. ISBN 9781429957649. OCLC 872601019. IRV is subject to something called the "center squeeze." A popular moderate can receive relatively few first-place votes through no fault of her own but because of vote splitting from candidates to the right and left. [...] Approval voting thus appears to solve the problem of vote splitting simply and elegantly. [...] Range voting solves the problems of spoilers and vote splitting

[20] "Modern economic theory has insisted on the ordinal concept of utility; that is, only orderings can be observed, and therefore no measurement of utility independent of these orderings has any significance. In the field of consumer's demand theory the ordinalist position turned out to create no problems; cardinal utility had no explanatory power above and beyond ordinal. Leibniz' Principle of the identity of indiscernibles demanded then the excision of cardinal utility from our thought patterns." Arrow (1967), as quoted on p. 33 by Racnchetti, Fabio (2002), "Choice without utility? Some reflections on the loose foundations of standard consumer theory", in Bianchi, Marina (ed.), The Active Consumer: Novelty and Surprise in Consumer Choice, Routledge Frontiers of Political Economy, vol. 20, Routledge, pp. 21–45

[Hamlin-interview12-21] Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.
Dr. Arrow: Now there’s another possible way of thinking about it, which is not included in my theorem. But we have some idea how strongly people feel. In other words, you might do something like saying each voter does not just give a ranking. But says, this is good. And this is not good[...] So this gives more information than simply what I have asked for.

[22] Harsanyi, John C. (1979-09-01). "Bayesian decision theory, rule utilitarianism, and Arrow's impossibility theorem". Theory and Decision. 11 (3): 289–317. doi:10.1007/BF00126382. ISSN 1573-7187. Retrieved 2020-03-20. It is shown that the utilitarian welfare function satisfies all of Arrow's social choice postulates — avoiding the celebrated impossibility theorem by making use of information which is unavailable in Arrow's original framework.

[23] Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.

Dr. Arrow: Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes (in spite of what I said about manipulation) is probably the best.[...] And some of these studies have been made. In France, [Michel] Balinski has done some studies of this kind which seem to give some support to these scoring methods.

[24] Hamlin, Aaron (2012-10-06). "Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow". The Center for Election Science. Archived from the original on 2023-06-05.
CES: Now, you mention that your theorem applies to preferential systems or ranking systems.
Dr. Arrow: Yes.
CES: But the system that you're just referring to, approval voting, falls within a class called cardinal systems. So not within ranking systems.
Dr. Arrow: And as I said, that in effect implies more information.

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