Editing Arrow's impossibility theorem (section)

== Interpretation and practical solutions ==
Arrow's theorem establishes that no ranked voting rule can ''always'' satisfy independence of irrelevant alternatives, but it says nothing about the frequency of spoilers. This led Arrow to remark that "Most systems are not going to work badly all of the time. All I proved is that all can work badly at times."<ref name="Hamlin-interview1" /><ref name="ns1222">{{cite journal |last=McKenna |first=Phil |date=12 April 2008 |title=Vote of no confidence |url=http://rangevoting.org/McKennaText.txt |journal=New Scientist |volume=198 |issue=2651 |pages=30–33 |doi=10.1016/S0262-4079(08)60914-8}}</ref>

Attempts at dealing with the effects of Arrow's theorem take one of two approaches: either accepting his rule and searching for the least spoiler-prone methods, or dropping one or more of his assumptions, such as by focusing on [[rated voting]] rules.<ref name="Pearce"/>

=== {{Anchor|Minimizing}}Minimizing IIA failures: Majority-rule methods ===
{{Main|Condorcet cycle}}

[[File:Italian_food_Condorcet_cycle.png|thumb|383x383px|An example of a Condorcet cycle, where some candidate ''must'' cause a spoiler effect]]
The first set of methods studied by economists are the [[Condorcet methods|majority-rule, or ''Condorcet'', methods]]. These rules limit spoilers to situations where majority rule is self-contradictory, called [[Condorcet cycle]]s, and as a result uniquely minimize the possibility of a spoiler effect among ranked rules. (Indeed, many different social welfare functions can meet Arrow's conditions under such restrictions of the domain. It has been proven, however, that under any such restriction, if there exists any social welfare function that adheres to Arrow's criteria, then [[Majority rule|Condorcet method]] will adhere to Arrow's criteria.<ref name="Campbell2000"/>) Condorcet believed voting rules should satisfy both independence of irrelevant alternatives and the [[Condorcet winner criterion|majority rule principle]], i.e. if most voters rank ''Alice'' ahead of ''Bob'', ''Alice'' should defeat ''Bob'' in the election.<ref name="McLean-1995"/>

Unfortunately, as Condorcet proved, this rule can be intransitive on some preference profiles.<ref>{{Cite journal |last=Gehrlein |first=William V. |date=1983-06-01 |title=Condorcet's paradox |url=https://doi.org/10.1007/BF00143070 |journal=Theory and Decision |language=en |volume=15 |issue=2 |pages=161–197 |doi=10.1007/BF00143070 |issn=1573-7187}}</ref> Thus, Condorcet proved a weaker form of Arrow's impossibility theorem long before Arrow, under the stronger assumption that a voting system in the two-candidate case will agree with a simple majority vote.<ref name="McLean-1995" />

Unlike pluralitarian rules such as [[Instant-runoff voting|ranked-choice runoff (RCV)]] or [[first-preference plurality]],<ref name="McGann2002"/> [[Condorcet method]]s avoid the spoiler effect in non-cyclic elections, where candidates can be chosen by majority rule. Political scientists have found such cycles to be fairly rare, suggesting they may be of limited practical concern.<ref name="VanDeemen" /> [[Spatial model of voting|Spatial voting models]] also suggest such paradoxes are likely to be infrequent<ref name="Wolk-2023">{{Cite journal |last1=Wolk |first1=Sara |last2=Quinn |first2=Jameson |last3=Ogren |first3=Marcus |date=2023-09-01 |title=STAR Voting, equality of voice, and voter satisfaction: considerations for voting method reform |url=https://doi.org/10.1007/s10602-022-09389-3 |journal=Constitutional Political Economy |volume=34 |issue=3 |pages=310–334 |doi=10.1007/s10602-022-09389-3 |issn=1572-9966}}</ref><ref name="Gehrlein-2002"/> or even non-existent.<ref name="Black-1948" />

==== {{Anchor|Single peak}}Left-right spectrum ====
{{Main|Median voter theorem}}
Soon after Arrow published his theorem, [[Duncan Black]] showed his own remarkable result, the [[median voter theorem]]. The theorem proves that if voters and candidates are arranged on a [[Political spectrum|left-right spectrum]], Arrow's conditions are all fully compatible, and all will be met by any rule satisfying [[Condorcet winner criterion|Condorcet's majority-rule principle]].<ref name="Black-1948" /><ref name="Black-1968"/>

More formally, Black's theorem assumes preferences are ''single-peaked'': a voter's happiness with a candidate goes up and then down as the candidate moves along some spectrum. For example, in a group of friends choosing a volume setting for music, each friend would likely have their own ideal volume; as the volume gets progressively too loud or too quiet, they would be increasingly dissatisfied. If the domain is restricted to profiles where every individual has a single-peaked preference with respect to the linear ordering, then social preferences are acyclic. In this situation, Condorcet methods satisfy a wide variety of highly-desirable properties, including being fully spoilerproof.<ref name="Black-1948" /><ref name="Black-1968"/><ref name="Campbell2000"/>

The rule does not fully generalize from the political spectrum to the political compass, a result related to the [[McKelvey-Schofield chaos theorem]].<ref name="Black-1948" /><ref>{{Cite journal |last1=McKelvey |first1=Richard D. |author-link=Richard McKelvey |year=1976 |title=Intransitivities in multidimensional voting models and some implications for agenda control |journal=Journal of Economic Theory |volume=12 |issue=3 |pages=472–482 |doi=10.1016/0022-0531(76)90040-5}}</ref> However, a well-defined Condorcet winner does exist if the [[Probability distribution|distribution]] of voters is [[Rotational symmetry|rotationally symmetric]] or otherwise has a [[Omnidirectional median|uniquely-defined median]].<ref>{{Cite journal |last1=Davis |first1=Otto A. |last2=DeGroot |first2=Morris H. |last3=Hinich |first3=Melvin J. |date=1972 |title=Social Preference Orderings and Majority Rule |url=http://www.jstor.org/stable/1909727 |journal=Econometrica |volume=40 |issue=1 |pages=147–157 |doi=10.2307/1909727 |jstor=1909727 |issn=0012-9682}}</ref><ref name="dotti2">{{Cite thesis |title=Multidimensional voting models: theory and applications |url=https://discovery.ucl.ac.uk/id/eprint/1516004/ |publisher=UCL (University College London) |date=2016-09-28 |degree=Doctoral |first=V. |last=Dotti}}</ref> In most realistic situations, where voters' opinions follow a roughly-[[normal distribution]] or can be accurately summarized by one or two dimensions, Condorcet cycles are rare (though not unheard of).<ref name="Wolk-2023" /><ref name="Holliday23222"/>

==== Generalized stability theorems ====
The Campbell-Kelly theorem shows that Condorcet methods are the most spoiler-resistant class of ranked voting systems: whenever it is possible for some ranked voting system to avoid a spoiler effect, a Condorcet method will do so.<ref name="Campbell2000" /> In other words, replacing a ranked method with its Condorcet variant (i.e. elect a Condorcet winner if they exist, and otherwise run the method) will sometimes prevent a spoiler effect, but can never create a new one.<ref name="Campbell2000" />

In 1977, [[Ehud Kalai]] and [[Eitan Muller]] gave a full characterization of domain restrictions admitting a nondictatorial and [[Strategyproofness|strategyproof]] social welfare function. These correspond to preferences for which there is a Condorcet winner.<ref>{{Cite journal |last1=Kalai |first1=Ehud |last2=Muller |first2=Eitan |year=1977 |title=Characterization of domains admitting nondictatorial social welfare functions and nonmanipulable voting procedures |url=http://www.kellogg.northwestern.edu/research/math/papers/234.pdf |journal=Journal of Economic Theory |volume=16 |issue=2 |pages=457–469 |doi=10.1016/0022-0531(77)90019-9}}</ref>

Holliday and Pacuit devised a voting system that provably minimizes the number of candidates who are capable of spoiling an election, albeit at the cost of occasionally failing [[Monotonicity criterion|vote positivity]] (though at a much lower rate than seen in [[instant-runoff voting]]).<ref name="Holliday23222"/>{{clarify|reason=Needs a quote saying what is claimed, for instance how it has fewer spoilers than other Smith methods.|date=November 2024}}

=== Going beyond Arrow's theorem: Rated voting ===
{{main|Spoiler effect}}
As shown above, the proof of Arrow's theorem relies crucially on the assumption of [[ranked voting]], and is not applicable to [[Graded voting|rated voting systems]]. This opens up the possibility of passing all of the criteria given by Arrow. These systems ask voters to rate candidates on a numerical scale (e.g. from 0–10), and then elect the candidate with the highest average (for score voting) or median ([[graduated majority judgment]]).<ref name=":mj2">{{cite book |last1=Balinski |first1=M. L. |title=Majority judgment: measuring, ranking, and electing |last2=Laraki |first2=Rida |date=2010 |publisher=MIT Press |isbn=9780262545716 |location=Cambridge, Mass}}</ref>{{rp|4–5}}

Because Arrow's theorem no longer applies, other results are required to determine whether rated methods are immune to the [[spoiler effect]], and under what circumstances. Intuitively, cardinal information can only lead to such immunity if it's meaningful; simply providing cardinal data is not enough.<ref name="x031">{{cite web | last=Morreau | first=Michael | title=Arrow's Theorem | website=Stanford Encyclopedia of Philosophy | date=2014-10-13 | url=https://plato.stanford.edu/entries/arrows-theorem/#ConAga | access-date=2024-10-09 | quote=One important finding was that having cardinal utilities is not by itself enough to avoid an impossibility result. ... Intuitively speaking, to put information about preference strengths to good use it has to be possible to compare the strengths of different individuals’ preferences. }}</ref>

Some rated systems, such as [[range voting]] and [[majority judgment]], pass independence of irrelevant alternatives when the voters rate the candidates on an absolute scale. However, when they use relative scales, more general impossibility theorems show that the methods (within that context) still fail IIA.<ref name="w444">{{cite journal | last=Roberts | first=Kevin W. S. | title=Interpersonal Comparability and Social Choice Theory | journal=The Review of Economic Studies | publisher=[Oxford University Press, Review of Economic Studies, Ltd.] | volume=47 | issue=2 | year=1980 | issn=0034-6527 | jstor=2297002 | pages=421–439 | doi=10.2307/2297002 | url=http://www.jstor.org/stable/2297002 | access-date=2024-09-25 |quote=If f satisfies U, I, P, and CNC then there exists a dictator.}}</ref> As Arrow later suggested, relative ratings may provide more information than pure rankings,<ref>{{Cite journal |last1=Maio |first1=Gregory R. |last2=Roese |first2=Neal J. |last3=Seligman |first3=Clive |last4=Katz |first4=Albert |date=1 June 1996 |title=Rankings, Ratings, and the Measurement of Values: Evidence for the Superior Validity of Ratings |journal=Basic and Applied Social Psychology |volume=18 |issue=2 |pages=171–181 |doi=10.1207/s15324834basp1802_4 |issn=0197-3533 |quote=Many value researchers have assumed that rankings of values are more valid than ratings of values because rankings force participants to differentiate more incisively between similarly regarded values ... Results indicated that ratings tended to evidence greater validity than rankings within moderate and low-differentiating participants. In addition, the validity of ratings was greater than rankings overall.}}</ref><ref name=":feelings22">{{cite journal |last1=Kaiser |first1=Caspar |last2=Oswald |first2=Andrew J. |date=18 October 2022 |title=The scientific value of numerical measures of human feelings |journal=Proceedings of the National Academy of Sciences |volume=119 |issue=42 |pages=e2210412119 |bibcode=2022PNAS..11910412K |doi=10.1073/pnas.2210412119 |issn=0027-8424 |pmc=9586273 |pmid=36191179 |doi-access=free}}</ref><ref name="The Possibility of Social Choice2" /><ref name="Hamlin-interview1">{{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 name="Arrow">Arrow, Kenneth et al. 1993. ''Report of the NOAA panel on Contingent Valuation.''</ref> but this information does not suffice to render the methods immune to spoilers.

While Arrow's theorem does not apply to graded systems, [[Gibbard's theorem]] still does: no voting game can be [[Dominant strategy|straightforward]] (i.e. have a single, clear, always-best strategy).<ref>{{Cite book |last=Poundstone |first=William |url=https://books.google.com/books?id=hbxL3A-pWagC&q=%22gibbard%22%20%22utilitarian%20voting%22&pg=PA185 |title=Gaming the Vote: Why Elections Are not Fair (and What We Can Do About It) |date=2009-02-17 |publisher=Macmillan |isbn=9780809048922}}</ref>

==== {{Anchor|Meaning|Cardinal|Validity|Meaningfulness}}Meaningfulness of cardinal information ====
{{Main|Cardinal utility}}
Arrow's framework assumed individual and social preferences are [[Ordinal utility|orderings]] or [[Ranked voting|rankings]], i.e. statements about which outcomes are better or worse than others.<ref>{{Cite journal |last=Lützen |first=Jesper |date=2019-02-01 |title=How mathematical impossibility changed welfare economics: A history of Arrow's impossibility theorem |url=https://www.sciencedirect.com/science/article/pii/S0315086018300508 |journal=Historia Mathematica |volume=46 |pages=56–87 |doi=10.1016/j.hm.2018.11.001 |issn=0315-0860}}</ref> Taking inspiration from the [[Behaviorism|strict behaviorism]] popular in psychology, some philosophers and economists rejected the idea of comparing internal human experiences of [[Cardinal utility|well-being]].<ref name="Racnchetti-2002">"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><ref name="Pearce" /> Such philosophers claimed it was impossible to compare the strength of preferences across people who disagreed; [[Amartya Sen|Sen]] gives as an example that it would be impossible to know whether the [[Great Fire of Rome]] was good or bad, because despite killing thousands of Romans, it had the positive effect of letting [[Nero]] expand his palace.<ref name="The Possibility of Social Choice2">{{cite journal |last1=Sen |first1=Amartya |date=1999 |title=The Possibility of Social Choice |url=https://www.aeaweb.org/articles?id=10.1257/aer.89.3.349 |journal=American Economic Review |volume=89 |issue=3 |pages=349–378 |doi=10.1257/aer.89.3.349}}</ref>

Arrow originally agreed with these positions and rejected [[cardinal utility]], leading him to focus his theorem on preference rankings.<ref name="Racnchetti-2002" /><ref name="Arrow 1963234" /> However, he later stated that cardinal methods can provide additional useful information, and that his theorem is not applicable to them.

[[John Harsanyi]] noted Arrow's theorem could be considered a weaker version of his own theorem<ref>{{Cite journal |last=Harsanyi |first=John C. |date=1955 |title=Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility |journal=Journal of Political Economy |volume=63 |issue=4 |pages=309–321 |doi=10.1086/257678 |jstor=1827128 |s2cid=222434288}}</ref>{{Failed verification|reason=Paper seems to argue that if we can estimate others' utilities, then the decision function must be total utilitarianism - it doesn't say that Arrow's theorem is a corollary.|date=December 2024}} and other [[utility representation theorem]]s like the [[Von Neumann–Morgenstern utility theorem|VNM theorem]], which generally show that [[Coherence (philosophical gambling strategy)|rational behavior]] requires consistent [[Cardinal utility|cardinal utilities]].<ref name="VNM2">[[John von Neumann|Neumann, John von]] and [[Oskar Morgenstern|Morgenstern, Oskar]], ''[[Theory of Games and Economic Behavior]]''. Princeton, NJ. Princeton University Press, 1953.</ref>

==== Nonstandard spoilers ====
[[Behavioral economics|Behavioral economists]] have shown individual [[irrationality]] involves violations of IIA (e.g. with [[decoy effect]]s),<ref>{{cite journal |last1=Huber |first1=Joel |last2=Payne |first2=John W. |last3=Puto |first3=Christopher |year=1982 |title=Adding Asymmetrically Dominated Alternatives: Violations of Regularity and the Similarity Hypothesis |journal=Journal of Consumer Research |volume=9 |issue=1 |pages=90–98 |doi=10.1086/208899 |s2cid=120998684}}</ref> suggesting human behavior can cause IIA failures even if the voting method itself does not.<ref>{{Cite journal |last1=Ohtsubo |first1=Yohsuke |last2=Watanabe |first2=Yoriko |date=September 2003 |title=Contrast Effects and Approval Voting: An Illustration of a Systematic Violation of the Independence of Irrelevant Alternatives Condition |url=https://onlinelibrary.wiley.com/doi/10.1111/0162-895X.00340 |journal=Political Psychology |language=en |volume=24 |issue=3 |pages=549–559 |doi=10.1111/0162-895X.00340 |issn=0162-895X}}</ref> However, past research has typically found such effects to be fairly small,<ref name="HuberPayne20142">{{cite journal |last1=Huber |first1=Joel |last2=Payne |first2=John W. |last3=Puto |first3=Christopher P. |year=2014 |title=Let's Be Honest About the Attraction Effect |journal=Journal of Marketing Research |volume=51 |issue=4 |pages=520–525 |doi=10.1509/jmr.14.0208 |issn=0022-2437 |s2cid=143974563}}</ref> and such psychological spoilers can appear regardless of electoral system. [[Michel Balinski|Balinski]] and [[Rida Laraki|Laraki]] discuss techniques of [[ballot design]] derived from [[psychometrics]] that minimize these psychological effects, such as asking voters to give each candidate a verbal grade (e.g. "bad", "neutral", "good", "excellent") and issuing instructions to voters that refer to their ballots as judgments of individual candidates.<ref name=":mj2" />{{Page needed|date=October 2024}} Similar techniques are often discussed in the context of [[contingent valuation]].<ref name="Arrow" />

=== Esoteric solutions ===
In addition to the above practical resolutions, there exist unusual (less-than-practical) situations where Arrow's requirement of IIA can be satisfied.

==== Supermajority rules ====
[[Supermajority]] rules can avoid Arrow's theorem at the cost of being poorly-decisive (i.e. frequently failing to return a result). In this case, a threshold that requires a <math>2/3</math> majority for ordering 3 outcomes, <math>3/4</math> for 4, etc. does not produce [[voting paradox]]es.<ref>{{Cite journal |last=Moulin |first=Hervé |date=1985-02-01 |title=From social welfare ordering to acyclic aggregation of preferences |url=https://dx.doi.org/10.1016/0165-4896%2885%2990002-2 |journal=Mathematical Social Sciences |volume=9 |issue=1 |pages=1–17 |doi=10.1016/0165-4896(85)90002-2 |issn=0165-4896}}</ref>

In [[Spatial model of voting|spatial (n-dimensional ideology) models of voting]], this can be relaxed to require only <math>1-e^{-1}</math> (roughly 64%) of the vote to prevent cycles, so long as the distribution of voters is well-behaved ([[quasiconcave]]).<ref name="Caplin-1988" /> These results provide some justification for the common requirement of a two-thirds majority for constitutional amendments, which is sufficient to prevent cyclic preferences in most situations.<ref name="Caplin-1988">{{Cite journal |last1=Caplin |first1=Andrew |last2=Nalebuff |first2=Barry |date=1988 |title=On 64%-Majority Rule |url=https://www.jstor.org/stable/1912699 |journal=Econometrica |volume=56 |issue=4 |pages=787–814 |doi=10.2307/1912699 |issn=0012-9682 |jstor=1912699}}</ref>

==== Infinite populations ====
[[Peter C. Fishburn|Fishburn]] shows all of Arrow's conditions can be satisfied for [[Uncountable set|uncountably infinite sets]] of voters given the [[axiom of choice]];<ref name="Fishburn197022">{{Cite journal |last=Fishburn |first=Peter Clingerman |year=1970 |title=Arrow's impossibility theorem: concise proof and infinite voters |journal=Journal of Economic Theory |volume=2 |issue=1 |pages=103–106 |doi=10.1016/0022-0531(70)90015-3}}</ref> however, Kirman and Sondermann demonstrated this requires disenfranchising [[Almost surely|almost all]] members of a society (eligible voters form a set of [[Measure (mathematics)|measure]] 0), leading them to refer to such societies as "invisible dictatorships".<ref>See Chapter 6 of {{cite book |last=Taylor |first=Alan D. |title=Social choice and the mathematics of manipulation |publisher=Cambridge University Press |year=2005 |isbn=978-0-521-00883-9 |location=New York |postscript=none}} for a concise discussion of social choice for infinite societies.</ref>