Editing Arrow's impossibility theorem (section)

{{Short description|Proof all ranked voting rules have spoilers}}
{{Electoral systems|expanded=Social and collective choice}}

'''Arrow's impossibility theorem''' is a key result in [[social choice theory]] showing that no [[Ordinal utility|ranked]]-choice procedure for group decision-making can satisfy the requirements of [[rational choice]].<ref name="plato.stanford.edu"/> Specifically, [[Kenneth Arrow|Arrow]] showed no such rule can satisfy [[independence of irrelevant alternatives]], the principle that a choice between two alternatives {{Math|''A''}} and {{Math|''B''}} should not depend on the quality of some third, unrelated option {{Math|''C''}}.<ref name="Arrow1950">{{cite journal |last1=Arrow |first1=Kenneth J. |author-link1=Kenneth Arrow |year=1950 |title=A Difficulty in the Concept of Social Welfare |url=http://gatton.uky.edu/Faculty/hoytw/751/articles/arrow.pdf |url-status=dead |journal=[[Journal of Political Economy]] |volume=58 |issue=4 |pages=328–346 |doi=10.1086/256963 |jstor=1828886 |s2cid=13923619 |archive-url=https://web.archive.org/web/20110720090207/http://gatton.uky.edu/Faculty/hoytw/751/articles/arrow.pdf |archive-date=2011-07-20}}</ref><ref name="Arrow 1963234">{{Cite book |last=Arrow |first=Kenneth Joseph |url=http://cowles.yale.edu/sites/default/files/files/pub/mon/m12-2-all.pdf |title=Social Choice and Individual Values |date=1963 |publisher=Yale University Press |isbn=978-0300013641 |archive-url=https://ghostarchive.org/archive/20221009/http://cowles.yale.edu/sites/default/files/files/pub/mon/m12-2-all.pdf |archive-date=2022-10-09 |url-status=live}}</ref><ref name="Wilson1972">{{Cite journal |last=Wilson |first=Robert |date=December 1972 |title=Social choice theory without the Pareto Principle |url=https://doi.org/10.1016/0022-0531(72)90051-8 |journal=Journal of Economic Theory |volume=5 |issue=3 |pages=478–486 |doi=10.1016/0022-0531(72)90051-8 |issn=0022-0531}}</ref>

The result is often cited in discussions of [[Electoral system|voting rules]],<ref name="Borgers2233">{{Cite book |last=Borgers |first=Christoph |url=https://books.google.com/books?id=u_XMHD4shnQC |title=Mathematics of Social Choice: Voting, Compensation, and Division |date=2010-01-01 |publisher=SIAM |isbn=9780898716955 |quote=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}}</ref> where it shows no [[ranked voting]] rule to eliminate the [[spoiler effect]].<ref>{{Cite journal |last=Ng |first=Y. K. |date=November 1971 |title=The Possibility of a Paretian Liberal: Impossibility Theorems and Cardinal Utility |url=https://www.journals.uchicago.edu/doi/10.1086/259845 |journal=Journal of Political Economy |volume=79 |issue=6 |pages=1397–1402 |doi=10.1086/259845 |issn=0022-3808 |quote="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."}}</ref><ref>{{Cite journal |last1=Kemp |first1=Murray |last2=Asimakopulos |first2=A. |date=1952-05-01 |title=A Note on "Social Welfare Functions" and Cardinal Utility* |url=https://www.cambridge.org/core/journals/canadian-journal-of-economics-and-political-science-revue-canadienne-de-economiques-et-science-politique/article/note-on-social-welfare-functions-and-cardinal-utility/653F2AEF0D2372DDE202BC7C3B0A231F |journal=Canadian Journal of Economics and Political Science |volume=18 |issue=2 |pages=195–200 |doi=10.2307/138144 |issn=0315-4890 |jstor=138144 |quote=The abandonment of Condition 3  makes it possible to formulate a procedure for arriving at a social choice. Such a procedure is described below |via= |accessdate=2020-03-20}}</ref><ref>{{cite web |last1=Hamlin |first1=Aaron |date=25 May 2015 |title=CES Podcast with Dr Arrow |url=https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |url-status=dead |archive-url=https://web.archive.org/web/20181027170517/https://electology.org/podcasts/2012-10-06_kenneth_arrow |archive-date=27 October 2018 |access-date=9 March 2023 |website=Center for Election Science |publisher=CES}}</ref>  This result was first shown by the [[Marquis de Condorcet]], whose [[voting paradox]] showed the impossibility of logically-consistent [[majority rule]]; Arrow's theorem [[Generalization|generalizes]] Condorcet's findings to include non-majoritarian rules like [[collective leadership]] or [[consensus decision-making]].<ref name="plato.stanford.edu">{{cite book |title=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |chapter=Arrow's Theorem |chapter-url=https://plato.stanford.edu/entries/arrows-theorem/ |first=Michael |last=Morreau |date=2014-10-13}}</ref>

While the impossibility theorem shows all ranked voting rules must have spoilers, the frequency of spoilers differs dramatically by rule. [[Plurality-rule family|Plurality-rule]] methods like [[First-past-the-post voting|choose-one]] and [[Instant-runoff voting|ranked-choice (instant-runoff) voting]] are highly sensitive to spoilers,<ref name="McGann2002">{{Cite journal |last1=McGann |first1=Anthony J. |last2=Koetzle |first2=William |last3=Grofman |first3=Bernard |date=2002 |title=How an Ideologically Concentrated Minority Can Trump a Dispersed Majority: Nonmedian Voter Results for Plurality, Run-off, and Sequential Elimination Elections |url=https://www.jstor.org/stable/3088418 |journal=American Journal of Political Science |volume=46 |issue=1 |pages=134–147 |doi=10.2307/3088418 |issn=0092-5853 |jstor=3088418 |quote=As with simple plurality elections, it is apparent the outcome will be highly sensitive to the distribution of candidates.}}</ref><ref name="Borgers223222">{{Cite book |last=Borgers |first=Christoph |url=https://books.google.com/books?id=u_XMHD4shnQC |title=Mathematics of Social Choice: Voting, Compensation, and Division |date=2010-01-01 |publisher=SIAM |isbn=9780898716955 |quote=Candidates C and D spoiled the election for B&nbsp;... With them in the running, A won, whereas without them in the running, B would have won.&nbsp;... Instant runoff voting&nbsp;... does ''not'' do away with the spoiler problem entirely, although it unquestionably makes it less likely to occur in practice.}}</ref> creating them even in some situations where they are not [[Condorcet cycle|mathematically necessary]] (e.g. in [[Center squeeze|center squeezes]]).<ref name="Holliday23222">{{cite journal|last1=Holliday |first1=Wesley H. |title=Stable Voting |journal=Constitutional Political Economy |date=2023-03-14 |volume=34 |number=3 |doi=10.1007/s10602-022-09383-9 |issn=1572-9966 |doi-access=free |pages=421–433 |arxiv=2108.00542 |quote=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. |last2=Pacuit |first2=Eric}}</ref><ref name="Campbell2000">{{cite journal |last1=Campbell |first1=D. E. |last2=Kelly |first2=J. S. |year=2000 |title=A simple characterization of majority rule |journal=[[Economic Theory (journal)|Economic Theory]] |volume=15 |issue=3 |pages=689–700 |doi=10.1007/s001990050318 |jstor=25055296 |s2cid=122290254}}</ref> By contrast, [[Condorcet method|majority-rule (Condorcet) methods]] of [[ranked voting]] uniquely [[Arrow's impossibility theorem#Minimizing|minimize the number of spoiled elections]]<ref name="Campbell2000"/> by restricting them to [[cyclic tie|voting cycle]]s,<ref name="Holliday23222"/> which are rare in ideologically-driven elections.<ref name="Gehrlein-2002">{{Cite journal |last=Gehrlein |first=William V. |date=2002-03-01 |title=Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences* |url=https://doi.org/10.1023/A:1015551010381 |journal=Theory and Decision |volume=52 |issue=2 |pages=171–199 |doi=10.1023/A:1015551010381 |issn=1573-7187}}</ref><ref name="VanDeemen">{{Cite journal |last=Van Deemen |first=Adrian |date=2014-03-01 |title=On the empirical relevance of Condorcet's paradox |url=https://doi.org/10.1007/s11127-013-0133-3 |journal=Public Choice |volume=158 |issue=3 |pages=311–330 |doi=10.1007/s11127-013-0133-3 |issn=1573-7101}}</ref> Under some [[Mathematical model|models]] of voter preferences (like the left-right spectrum assumed in the [[Black's median voter theorem|median voter theorem]]), spoilers disappear entirely for these methods.<ref name="Black-1948">{{Cite journal |last=Black |first=Duncan |date=1948 |title=On the Rationale of Group Decision-making |url=https://www.jstor.org/stable/1825026 |journal=Journal of Political Economy |volume=56 |issue=1 |pages=23–34 |doi=10.1086/256633 |jstor=1825026 |issn=0022-3808}}</ref><ref name="Black-1968">{{Cite book |last=Black |first=Duncan |author-link=Duncan Black |title=The theory of committees and elections |publisher=University Press |year=1968 |isbn=978-0-89838-189-4 |location=Cambridge, Eng.}}</ref>

[[Rated voting|Rated voting rules]], where voters assign a separate grade to each candidate, are not affected by Arrow's theorem.<ref>{{Cite journal |last=Ng |first=Y. K. |date=November 1971 |title=The Possibility of a Paretian Liberal: Impossibility Theorems and Cardinal Utility |url=https://www.journals.uchicago.edu/doi/10.1086/259845 |journal=Journal of Political Economy |volume=79 |issue=6 |pages=1397–1402 |doi=10.1086/259845 |issn=0022-3808 |quote="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."}}</ref><ref>{{Cite journal |last1=Kemp |first1=Murray |last2=Asimakopulos |first2=A. |date=1952-05-01 |title=A Note on "Social Welfare Functions" and Cardinal Utility* |url=https://www.cambridge.org/core/journals/canadian-journal-of-economics-and-political-science-revue-canadienne-de-economiques-et-science-politique/article/note-on-social-welfare-functions-and-cardinal-utility/653F2AEF0D2372DDE202BC7C3B0A231F |journal=Canadian Journal of Economics and Political Science |volume=18 |issue=2 |pages=195–200 |doi=10.2307/138144 |issn=0315-4890 |jstor=138144 |quote=The abandonment of Condition 3  makes it possible to formulate a procedure for arriving at a social choice. Such a procedure is described below |via= |accessdate=2020-03-20}}</ref><ref name="Poundstone, William.-2013232">{{Cite book |last=Poundstone, William. |title=Gaming the vote : why elections aren't fair (and what we can do about it) |date=2013 |publisher=Farrar, Straus and Giroux |isbn=9781429957649 |pages=168, 197, 234 |oclc=872601019 |quote=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}}</ref> 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.<ref>"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 [https://books.google.com/books?id=7ECXDjlCpB0C&pg=PA33 p.&nbsp;33] by {{citation |last=Racnchetti |first=Fabio |title=The Active Consumer: Novelty and Surprise in Consumer Choice |volume=20 |pages=21–45 |year=2002 |editor-last=Bianchi |editor-first=Marina |series=Routledge Frontiers of Political Economy |contribution=Choice without utility? Some reflections on the loose foundations of standard consumer theory |publisher=Routledge}}</ref> However, Arrow would later describe this as a mistake,<ref name="Hamlin-interview12">{{Cite web |last=Hamlin |first=Aaron |date=2012-10-06 |title=Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow |url=https://www.electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |url-status=dead |archive-url=https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |archive-date=2023-06-05 |accessdate= |work=The Center for Election Science}} {{Pbl|'''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.}}</ref><ref>{{Cite journal |last=Harsanyi |first=John C. |date=1979-09-01 |title=Bayesian decision theory, rule utilitarianism, and Arrow's impossibility theorem |url=http://link.springer.com/10.1007/BF00126382 |journal=Theory and Decision |volume=11 |issue=3 |pages=289–317 |doi=10.1007/BF00126382 |issn=1573-7187 |quote=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. |accessdate=2020-03-20}}</ref> admitting rules based on [[Cardinal utility|cardinal utilities]] (such as [[Score voting|score]] and [[approval voting]]) are not subject to his theorem.<ref>{{Cite web |last=Hamlin |first=Aaron |date=2012-10-06 |title=Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow |url=https://www.electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |url-status=dead |archive-url=https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |archive-date=2023-06-05 |accessdate= |work=The Center for Election Science}}<poem>'''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.</poem></ref><ref>{{Cite web |last=Hamlin |first=Aaron |date=2012-10-06 |title=Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow |url=https://www.electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |url-status=dead |archive-url=https://web.archive.org/web/20230605225834/https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |archive-date=2023-06-05 |accessdate= |work=The Center for Election Science}} {{pbl|'''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 voting|cardinal systems]]. So not within [[ranked voting|ranking systems]].
'''Dr. Arrow:''' And as I said, that in effect implies more information.}}</ref>