Editing Arrow's impossibility theorem (section)

=== 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" />