Arrow's impossibility theorem: Difference between revisions
→Systems which evade Arrow's criteria
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Arrow's theorem only applies to [[Ordinal Voting|ordinal voting]] and not [[cardinal voting]]. It is, therefore, possible for several cardinal systems to pass all three fairness criteria.<ref>{{Cite journal| doi = 10.1086/259845| issn = 0022-3808| volume = 79| issue = 6| pages = 1397–1402| last = Ng| first = Y. K.| title = The Possibility of a Paretian Liberal: Impossibility Theorems and Cardinal Utility| journal = Journal of Political Economy| accessdate = 2020-03-20| date = 1971-11-01| url = https://www.journals.uchicago.edu/doi/10.1086/259845| 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|via=}}</ref><ref>{{Cite journal| doi = 10.2307/138144| issn = 0315-4890| volume = 18| issue = 2| pages = 195–200| last1 = Kemp| first1 = Murray| last2 = Asimakopulos| first2 = A.| title = A Note on “Social Welfare Functions” and Cardinal Utility*| journal = Canadian Journal of Economics and Political Science/Revue canadienne de economiques et science politique| accessdate = 2020-03-20| date = 1952-05-01| 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| 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=}}</ref><ref>{{Cite journal| doi = 10.1007/BF00126382| issn = 1573-7187| volume = 11| issue = 3| pages = 289–317| last = Harsanyi| first = John C.| title = Bayesian decision theory, rule utilitarianism, and Arrow's impossibility theorem| journal = Theory and Decision| accessdate = 2020-03-20| date = 1979-09-01| url = http://link.springer.com/10.1007/BF00126382| 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..|via=}}</ref><ref>{{Cite conference| publisher = Social Science Research Network| last = Vasiljev| first = Sergei| title = Cardinal Voting: The Way to Escape the Social Choice Impossibility| location = Rochester, NY| accessdate = 2020-03-20| date = 2008-04-01| url = https://papers.ssrn.com/abstract=1116545 | quote = We prove that cardinal voting can satisfy Pareto efficiency, independence of irrelevant alternative, unrestricted domain, and at the same time it can be nondictatorship in disproof of Arrow’s impossibility theorem.}}
</ref><ref>{{Cite web| title = Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow| work = The Center for Election Science| accessdate = 2020-03-20| date = 2015-05-25| url = https://www.electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/|quote=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.}}</ref> The typical example is [[score voting]] but there are also several [[Multi-Member System |multi-winner systems]]{{clarify}} which purport to pass all three of Arrow's original criteria. Additionally, there are cardinal systems which do not pass all criteria, but this is not due to Arrow's theorem; for example [[Ebert's Method]] fails [[Monotonicity]].
However, note that one of the assumptions used in Arrow's Theorem is that voters do not change their preferences on a given set of candidates regardless of whether candidates not in the set are running or not running. If even a single voter [[Normalization|normalizes]] their rated ballot or otherwise deviates from this assumption, it fails. Example:<blockquote>1: A:10 B:6 C:0
1: B:10 C:4 A:0
1: C:10 A:5 B:0</blockquote>Scores are A 15, B 16, and C 14, with B winning in [[Score voting]]. If C were to drop out, and all voters normalized in response, the election becomes:<blockquote>1: A:10 B:0
1: B:10 A:0
1: A:10 B:0</blockquote>Scores are A 20, B 10, and now A wins in Score voting. This example uses the standard [[Condorcet paradox]] but presented in rated form.
However, subsequent social choice theorists have expanded on Arrow's central insight, and applied his ideas more broadly. For example, the [[Gibbard-Satterthwaite theorem]] (published in 1973) holds that any deterministic process of collective decision making with multiple options will have some level of [[strategic voting]]. As a result of this much of the work of social choice theorists is to find out what types of [[strategic voting]] a system is susceptible to and the level of susceptibility for each. For example [[Single Member system | Single Member systems]] are not susceptible to [[Free riding]].
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