Editing Arrow's impossibility theorem (section)

=== Axioms of voting systems ===
==== Preferences ====
{{Further|Preference (economics)}}In the context of Arrow's theorem, citizens are assumed to have [[ordinal preferences]], i.e. [[Total order|orderings of candidates]]. If {{math|''A''}} and {{math|''B''}} are different candidates or alternatives, then <math>A \succ B</math> means {{math|''A''}} is preferred to {{math|''B''}}. Individual preferences (or ballots) are required to satisfy intuitive properties of orderings, e.g. they must be [[Transitive relation|transitive]]—if <math>A \succeq B</math> and <math>B \succeq C</math>, then <math>A \succeq C</math>. The social choice function is then a [[Function (mathematics)|mathematical function]] that maps the individual orderings to a new ordering that represents the preferences of all of society.

==== Basic assumptions ====
Arrow's theorem assumes as background that any [[Degeneracy (mathematics)|non-degenerate]] social choice rule will satisfy:<ref name="Gibbard1973">{{Cite journal |last=Gibbard |first=Allan |date=1973 |title=Manipulation of Voting Schemes: A General Result |url=https://www.jstor.org/stable/1914083 |journal=Econometrica |volume=41 |issue=4 |pages=587–601 |doi=10.2307/1914083 |jstor=1914083 |issn=0012-9682}}</ref>

* '''''[[Unrestricted domain]]''''' – the social choice function is a [[total function]] over the domain of all possible [[Ordinal utility|orderings of outcomes]], not just a [[partial function]].
** In other words, the system must always make ''some'' choice, and cannot simply "give up" when the voters have unusual opinions.
** Without this assumption, [[majority rule]] satisfies Arrow's axioms by "giving up" whenever there is a Condorcet cycle.<ref name="Campbell2000"/>
* ''[[Dictatorship mechanism|'''Non-dictatorship''']]'' – the system does not depend on only one voter's ballot.<ref name="Arrow 1963234"/>
** This weakens [[Anonymity (social choice)|''anonymity'']] ([[one vote, one value]]) to allow rules that treat voters unequally.
** It essentially defines ''social'' choices as those depending on more than one person's input.<ref name="Arrow 1963234"/>
* [[Surjective function|'''''Non-imposition''''']] – the system does not ignore the voters entirely when choosing between some pairs of candidates.<ref name="Wilson1972"/><ref name="Lagerspetz-2016">{{Citation |last=Lagerspetz |first=Eerik |title=Arrow's Theorem |date=2016 |work=Social Choice and Democratic Values |series=Studies in Choice and Welfare |pages=171–245 |url=https://doi.org/10.1007/978-3-319-23261-4_4 |access-date=2024-07-20 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-23261-4_4 |isbn=978-3-319-23261-4}}</ref>
** In other words, it is possible for any candidate to defeat any other candidate, given some combination of votes.<ref name="Wilson1972" /><ref name="Lagerspetz-2016" /><ref name="Quesada2002">{{Cite journal |last=Quesada |first=Antonio |date=2002 |title=From social choice functions to dictatorial social welfare functions |url=https://ideas.repec.org//a/ebl/ecbull/eb-02d70006.html |journal=Economics Bulletin |volume=4 |issue=16 |pages=1–7}}</ref>
** This is often replaced with the stronger '''[[Pareto efficiency]]''' axiom: if every voter prefers {{math|''A''}} over {{math|''B''}}, then {{math|''A''}} should defeat {{math|''B''}}. However, the weaker non-imposition condition is sufficient.<ref name="Wilson1972" />
Arrow's original statement of the theorem included [[Positive responsiveness|non-negative responsiveness]] as a condition, i.e., that ''increasing'' the rank of an outcome should not make them ''lose''—in other words, that a voting rule shouldn't penalize a candidate for being more popular.<ref name="Arrow1950" /> However, this assumption is not needed or used in his proof (except to derive the weaker condition of Pareto efficiency), and Arrow later corrected his statement of the theorem to remove the inclusion of this condition.<ref name="Arrow 1963234"/><ref>{{Cite journal |last1=Doron |first1=Gideon |last2=Kronick |first2=Richard |date=1977 |title=Single Transferrable Vote: An Example of a Perverse Social Choice Function |url=https://www.jstor.org/stable/2110496 |journal=American Journal of Political Science |volume=21 |issue=2 |pages=303–311 |doi=10.2307/2110496 |jstor=2110496 |issn=0092-5853}}</ref>

==== Independence ====
A commonly-considered axiom of [[Decision theory|rational choice]] is ''[[independence of irrelevant alternatives]]'' (IIA), which says that when deciding between {{math|''A''}} and {{math|''B''}}, one's opinion about a third option {{math|''C''}} should not affect their decision.<ref name="Arrow1950"/>
* '''''[[Independence of irrelevant alternatives]] (IIA)''''' – the social preference between candidate {{math|''A''}} and candidate {{math|''B''}} should only depend on the individual preferences between {{math|''A''}} and {{math|''B''}}.
** In other words, the social preference should not change from <math>A \succ B</math> to <math>B \succ A</math> if voters change their preference about whether <math>A \succ C</math>.<ref name="Arrow 1963234"/>
** This is equivalent to the claim about independence of [[Spoiler effect|spoiler candidates]] when using the [[Social welfare function#Constructing a social ordering|standard construction of a placement function]].<ref name="Quesada2002"/>
IIA is sometimes illustrated with a short joke by philosopher [[Sidney Morgenbesser]]:<ref name="Pearce">{{Cite journal |last=Pearce |first=David |title=Individual and social welfare: a Bayesian perspective |url=https://economia.uc.cl/wp-content/uploads/2022/12/Individual-and-Social-Welfare-A-Bayesian-Perspective-1-2.pdf |journal=Frisch Lecture Delivered to the World Congress of the Econometric Society}}</ref>
: Morgenbesser, ordering dessert, is told by a waitress that he can choose between blueberry or apple pie. He orders apple. Soon the waitress comes back and explains cherry pie is also an option. Morgenbesser replies "In that case, I'll have blueberry."
Arrow's theorem shows that if a society wishes to make decisions while always avoiding such self-contradictions, it cannot use ranked information alone.<ref name="Pearce" />