View source for Arrow's impossibility theorem

{{Short description|Proof all ranked voting rules have spoilers}}
{{off topic|date=October 2024}}
{{Electoral systems|expanded=Social and collective choice}}

'''Arrow's impossibility theorem''' is a key result in [[social choice theory]], showing that no [[Ranked voting|ranking]]-based [[Social choice function|decision rule]] can satisfy the requirements of [[Rational choice models|rational choice theory]].<ref name="plato.stanford.edu"/> Most notably, [[Kenneth Arrow|Arrow]] showed that no such rule can satisfy all of a certain set of seemingly simple and reasonable conditions that include [[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 most 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> However, Arrow's theorem is substantially broader, and can be applied to methods of social decision-making other than voting. It therefore generalizes [[Nicolas de Condorcet|Condorcet]]'s [[voting paradox]], and shows similar problems exist for every [[Social choice function|collective decision-making procedure]] based on [[Ordinal utility|relative comparisons]].<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>

[[Plurality voting|Plurality-rule]] methods like [[First-past-the-post voting|first-past-the-post]] 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> particularly in [[Center squeeze|situations]] where they are not [[Condorcet cycle|forced]].<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 rare<ref name=":532322">{{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> situations called [[cyclic tie]]s.<ref name="Holliday23222"/> Under some idealized models of voter behavior (e.g. [[Black's median voter theorem|Black's left-right spectrum]]), spoiler effects can disappear entirely for these methods.<ref name=":2">{{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=":422">{{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>

Arrow's theorem does not cover [[rated voting]] rules, and thus cannot be used to inform their susceptibility to the [[spoiler effect]]. However, [[Gibbard's theorem]] shows these methods' susceptibility to [[strategic voting]], and [[Spoiler effect#Rated voting|generalizations of Arrow's theorem]] describe cases where rated methods are susceptible to the spoiler effect.

== Background ==
{{Main|Social welfare function|Voting systems|Social choice theory}}
When [[Kenneth Arrow]] proved his theorem in 1950, it inaugurated the modern field of [[social choice theory]], a branch of [[welfare economics]] studying mechanisms to aggregate [[Preference (economics)|preferences]] and [[Belief aggregation|beliefs]] across a society.<ref name=":1332">{{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> Such a mechanism of study can be a [[Market (economics)|market]], [[voting system]], [[constitution]], or even a [[Morality|moral]] or [[Ethics|ethical]] framework.<ref name="plato.stanford.edu" />

=== 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=":13">{{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=":13" /><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 name=":11">{{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=":14">{{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=":14" />

== Theorem ==

=== Intuitive argument ===
[[Condorcet paradox|Condorcet's example]] is already enough to see the impossibility of a fair [[Ranked voting|ranked voting system]], given stronger conditions for fairness than Arrow's theorem assumes.<ref name=":10">{{Cite journal |last=McLean |first=Iain |date=1995-10-01 |title=Independence of irrelevant alternatives before Arrow |url=https://dx.doi.org/10.1016/0165-4896%2895%2900784-J |journal=Mathematical Social Sciences |volume=30 |issue=2 |pages=107–126 |doi=10.1016/0165-4896(95)00784-J |issn=0165-4896}}</ref> Suppose we have three candidates (<math>A</math>, <math>B</math>, and <math>C</math>) and three voters whose preferences are as follows:

{| class="wikitable" style="text-align: center;"
! Voter !! First preference !! Second preference !! Third preference
|- 
! Voter 1 
| A || B || C
|- 
! Voter 2 
| B || C || A
|- 
! Voter 3 
| C || A || B
|}

If <math>C</math> is chosen as the winner, it can be argued any fair voting system would say <math>B</math> should win instead, since two voters (1 and 2) prefer <math>B</math> to <math>C</math> and only one voter (3) prefers <math>C</math> to <math>B</math>.  However, by the same argument <math>A</math> is preferred to <math>B</math>, and <math>C</math> is preferred to <math>A</math>, by a margin of two to one on each occasion. Thus, even though each individual voter has consistent preferences, the preferences of society are contradictory: <math>A</math> is preferred over <math>B</math> which is preferred over <math>C</math> which is preferred over <math>A</math>.

Because of this example, some authors credit [[Condorcet]] with having given an intuitive argument that presents the core of Arrow's theorem.<ref name=":10" /> However, Arrow's theorem is substantially more general; it applies to methods of making decisions other than one-man-one-vote elections, such as [[Market (economics)|markets]] or [[weighted voting]], based on [[Ranked voting|ranked ballots]].

=== Formal statement ===
Let <math>A</math> be a set of ''alternatives''. A voter's [[preference (economics)|preferences]] over <math>A</math> are a [[Connected relation|complete]] and [[Transitive relation|transitive]] [[binary relation]] on <math>A</math> (sometimes called a [[total preorder]]), that is, a subset <math>R</math> of <math>A \times A</math> satisfying:
# (Transitivity) If <math>(\mathbf{a}, \mathbf{b})</math> is in <math>R</math> and <math>(\mathbf{b}, \mathbf{c})</math> is in <math>R</math>, then <math>(\mathbf{a}, \mathbf{c})</math> is in <math>R</math>,
# (Completeness) At least one of <math>(\mathbf{a}, \mathbf{b})</math> or <math>(\mathbf{b}, \mathbf{a})</math> must be in <math>R</math>.
The element <math>(\mathbf{a}, \mathbf{b})</math> being in <math>R</math> is interpreted to mean that alternative <math>\mathbf{a}</math> is preferred to alternative <math>\mathbf{b}</math>. This situation is often denoted <math>\mathbf{a} \succ \mathbf{b}</math> or <math>\mathbf{a}R\mathbf{b}</math>. Denote the set of all preferences on <math>A</math> by <math>\Pi(A)</math>. Let <math>N</math> be a positive integer. An [[Ranked voting|''ordinal (ranked)'']] ''social welfare function'' is a function<ref name="Arrow1950"/>
: <math> \mathrm{F} : \Pi(A)^N \to \Pi(A) </math>
which aggregates voters' preferences into a single preference on <math>A</math>. An <math>N</math>-[[tuple]] <math>(R_1, \ldots, R_N) \in \Pi(A)^N</math> of voters' preferences is called a ''preference profile''.

'''Arrow's impossibility theorem''': If there are at least three alternatives, then there is no social welfare function satisfying all three of the conditions listed below:<ref name="Gean">{{cite journal |last=Geanakoplos |first=John |year=2005 |title=Three Brief Proofs of Arrow's Impossibility Theorem |url=https://cowles.yale.edu/sites/default/files/files/pub/d11/d1123-r4.pdf |url-status=live |journal=[[Economic Theory (journal)|Economic Theory]] |volume=26 |issue=1 |pages=211–215 |citeseerx=10.1.1.193.6817 |doi=10.1007/s00199-004-0556-7 |jstor=25055941 |s2cid=17101545 |archive-url=https://ghostarchive.org/archive/20221009/https://cowles.yale.edu/sites/default/files/files/pub/d11/d1123-r4.pdf |archive-date=2022-10-09}}</ref>
; [[Pareto efficiency]]
: If alternative <math>\mathbf{a}</math> is preferred to <math>\mathbf{b}</math> for all orderings <math>R_1, \ldots, R_N</math>, then <math>\mathbf{a}</math> is preferred to <math>\mathbf{b}</math> by <math>F(R_1, R_2, \ldots, R_N)</math>.<ref name="Arrow1950" />
; [[Dictatorship mechanism|Non-dictatorship]]
: There is no individual <math>i</math> whose preferences always prevail. That is, there is no <math>i \in \{1, \ldots, N\}</math> such that for all <math>(R_1, \ldots, R_N) \in \Pi(A)^N</math> and all <math>\mathbf{a}</math> and <math>\mathbf{b}</math>, when <math>\mathbf{a}</math> is preferred to <math>\mathbf{b}</math> by <math>R_i</math> then <math>\mathbf{a}</math> is preferred to <math>\mathbf{b}</math> by <math>F(R_1, R_2, \ldots, R_N)</math>.<ref name="Arrow1950" />
; [[Independence of irrelevant alternatives]]
: For two preference profiles <math>(R_1, \ldots, R_N)</math> and <math>(S_1, \ldots, S_N)</math> such that for all individuals <math>i</math>, alternatives <math>\mathbf{a}</math> and <math>\mathbf{b}</math> have the same order in <math>R_i</math> as in <math>S_i</math>, alternatives <math>\mathbf{a}</math> and <math>\mathbf{b}</math> have the same order in <math>F(R_1, \ldots, R_N)</math> as in <math>F(S_1, \ldots, S_N)</math>.<ref name="Arrow1950" />

=== Formal proof ===
{{Collapse top|title=Proof by decisive coalition}}

Arrow's proof used the concept of ''decisive coalitions''.<ref name="Arrow 1963234"/>

Definition:

* A subset of voters is a '''coalition'''.
* A coalition is '''decisive over an ordered pair <math>(x, y)</math>''' if, when everyone in the coalition ranks <math>x \succ_i y</math>, society overall will always rank <math>x \succ y</math>.
* A coalition is '''decisive''' if and only if it is decisive over all ordered pairs.

Our goal is to prove that the '''decisive coalition''' contains only one voter, who controls the outcome—in other words, a [[Dictatorship mechanism|dictator]].

The following proof is a simplification taken from [[Amartya Sen]]<ref>{{Cite book |last=Sen |first=Amartya |url=https://www.degruyter.com/document/doi/10.7312/mask15328-003/html |title=The Arrow Impossibility Theorem |date=2014-07-22 |publisher=Columbia University Press |isbn=978-0-231-52686-9 |pages=29–42 |chapter=Arrow and the Impossibility Theorem |doi=10.7312/mask15328-003}}</ref> and [[Ariel Rubinstein]].<ref>{{Cite book |last=Rubinstein |first=Ariel |url=https://openlibrary.org/books/OL29649010M/Lecture_Notes_in_Microeconomic_Theory |title=Lecture Notes in Microeconomic Theory: The Economic Agent |publisher=Princeton University Press |year=2012 |isbn=978-1-4008-4246-9 |edition=2nd |at=Problem 9.5 |ol=29649010M}}</ref> The simplified proof uses an additional concept:

* A coalition is '''weakly decisive''' over <math>(x, y)</math> if and only if when every voter <math>i</math> in the coalition ranks <math>x \succ_i y</math>, ''and'' every voter <math>j</math> outside the coalition ranks <math>y \succ_j x</math>, then <math>x \succ y</math>.

Thenceforth assume that the social choice system satisfies unrestricted domain, Pareto efficiency, and IIA. Also assume that there are at least 3 distinct outcomes.
{{Math theorem
| math_statement = if a coalition <math>G</math> is weakly decisive over <math>(x, y)</math> for some <math>x \neq y</math>, then it is decisive.
| name = Field expansion lemma
| note = 
}} {{Math proof|proof=Let <math>z</math> be an outcome distinct from <math>x, y</math>.

Claim: <math>G</math> is decisive over <math>(x, z)</math>.

Let everyone in <math>G</math> vote <math>x</math> over <math>z</math>. By IIA, changing the votes on <math>y</math> does not matter for <math>x, z</math>. So change the votes such that <math>x \succ_i y \succ_i z</math> in <math>G</math> and <math>y \succ_i x</math> and <math>y \succ_i z</math> outside of <math>G</math>.

By Pareto, <math>y \succ z</math>. By coalition weak-decisiveness over <math>(x, y)</math>, <math>x \succ y</math>. Thus <math>x \succ z</math>.  <math>\square</math>

Similarly, <math>G</math> is decisive over <math>(z, y)</math>.

By iterating the above two claims (note that decisiveness implies weak-decisiveness), we find that <math>G</math> is decisive over all ordered pairs in <math>\{x, y, z\}</math>. Then iterating that, we find that <math>G</math> is decisive over all ordered pairs in <math>X</math>.}} {{Math theorem
| math_statement = If a coalition is decisive, and has size <math>\geq 2</math>, then it has a proper subset that is also decisive.
| name = Group contraction lemma
| note = 
}} {{Math proof|proof=Let <math>G</math> be a coalition with size <math>\geq 2</math>. Partition the coalition into nonempty subsets <math>G_1, G_2</math>.

Fix distinct <math>x, y, z</math>. Design the following voting pattern (notice that it is the cyclic voting pattern which causes the Condorcet paradox):

<math>\begin{align}
  \text{voters in } G_1&: x \succ_i y \succ_i z \\
  \text{voters in } G_2&: z \succ_i x \succ_i y \\
  \text{voters outside } G&: y \succ_i z \succ_i x
\end{align}</math>

(Items other than <math>x, y, z</math> are not relevant.)

Since <math>G</math> is decisive, we have <math>x \succ y</math>. So at least one is true: <math>x \succ z</math> or <math>z \succ y</math>.

If <math>x \succ z</math>, then <math>G_1</math> is weakly decisive over <math>(x, z)</math>. If <math>z \succ y</math>, then <math>G_2</math> is weakly decisive over <math>(z, y)</math>. Now apply the field expansion lemma.}}

By Pareto, the entire set of voters is decisive. Thus by the group contraction lemma, there is a size-one decisive coalition—a dictator.
{{Collapse bottom}}

{{Collapse top|title=Proof by showing there is only one pivotal voter}}
Proofs using the concept of the '''pivotal voter''' originated from Salvador Barberá in 1980.<ref>{{Cite journal |last=Barberá |first=Salvador |date=January 1980 |title=Pivotal voters: A new proof of arrow's theorem |journal=Economics Letters |volume=6 |issue=1 |pages=13–16 |doi=10.1016/0165-1765(80)90050-6 |issn=0165-1765}}</ref> The proof given here is a simplified version based on two proofs published in ''[[Economic Theory (journal)|Economic Theory]]''.<ref name="Gean"/><ref>{{cite journal |last1=Yu |first1=Ning Neil |year=2012 |title=A one-shot proof of Arrow's theorem |journal=[[Economic Theory (journal)|Economic Theory]] |volume=50 |issue=2 |pages=523–525 |doi=10.1007/s00199-012-0693-3 |jstor=41486021 |s2cid=121998270}}</ref>

==== Setup ====

Assume there are ''n'' voters. We assign all of these voters an arbitrary ID number, ranging from ''1'' through ''n'', which we can use to keep track of each voter's identity as we consider what happens when they change their votes. [[Without loss of generality]], we can say there are three candidates who we call '''A''', '''B''', and '''C'''. (Because of IIA, including more than 3 candidates does not affect the proof.)

We will prove that any social choice rule respecting unanimity and independence of irrelevant alternatives (IIA) is a dictatorship. The proof is in three parts:
# We identify a ''pivotal voter'' for each individual contest ('''A''' vs. '''B''', '''B''' vs. '''C''', and '''A''' vs. '''C'''). Their ballot swings the societal outcome.
# We prove this voter is a ''partial'' dictator. In other words, they get to decide whether A or B is ranked higher in the outcome.
# We prove this voter is the same person, hence this voter is a [[Dictatorship mechanism|dictator]].

==== Part one: There is a pivotal voter for A vs. B ====
[[File:Diagram_for_part_one_of_Arrow's_Impossibility_Theorem.svg|right|thumb|Part one: Successively move '''B''' from the bottom to the top of voters' ballots. The voter whose change results in '''B''' being ranked over '''A''' is the ''pivotal voter for'' '''B''' ''over'' '''A'''.]]
Consider the situation where everyone prefers '''A''' to '''B''', and everyone also prefers '''C''' to '''B'''. By unanimity, society must also prefer both '''A''' and '''C''' to '''B'''. Call this situation ''profile[0, x]''.

On the other hand, if everyone preferred '''B''' to everything else, then society would have to prefer '''B''' to everything else by unanimity. Now arrange all the voters in some arbitrary but fixed order, and for each ''i'' let ''profile i'' be the same as ''profile 0'', but move '''B''' to the top of the ballots for voters 1 through ''i''. So ''profile 1'' has '''B''' at the top of the ballot for voter 1, but not for any of the others. ''Profile 2'' has '''B''' at the top for voters 1 and 2, but no others, and so on.

Since '''B''' eventually moves to the top of the societal preference as the profile number increases, there must be some profile, number ''k'', for which '''B''' ''first'' moves ''above'' '''A''' in the societal rank. We call the voter ''k'' whose ballot change causes this to happen the ''pivotal voter for '''B''' over '''A'''''. Note that the pivotal voter for '''B''' over '''A''' is not, [[A priori knowledge|a priori]], the same as the pivotal voter for '''A''' over '''B'''. In part three of the proof we will show that these do turn out to be the same.

Also note that by IIA the same argument applies if ''profile 0'' is any profile in which '''A''' is ranked above '''B''' by every voter, and the pivotal voter for '''B''' over '''A''' will still be voter ''k''. We will use this observation below.

==== Part two: The pivotal voter for B over A is a dictator for B over C ====
In this part of the argument we refer to voter ''k'', the pivotal voter for '''B''' over '''A''', as the ''pivotal voter'' for simplicity. We will show that the pivotal voter dictates society's decision for '''B''' over '''C'''. That is, we show that no matter how the rest of society votes, if ''pivotal voter'' ranks '''B''' over '''C''', then that is the societal outcome. Note again that the dictator for '''B''' over '''C''' is not a priori the same as that for '''C''' over '''B'''. In part three of the proof we will see that these turn out to be the same too.
[[File:Diagram_for_part_two_of_Arrow's_Impossibility_Theorem.svg|right|thumb|Part two: Switching '''A''' and '''B''' on the ballot of voter ''k'' causes the same switch to the societal outcome, by part one of the argument. Making any or all of the indicated switches to the other ballots has no effect on the outcome.]]
In the following, we call voters 1 through ''k − 1'', ''segment one'', and voters ''k + 1'' through ''N'', ''segment two''. To begin, suppose that the ballots are as follows:

* Every voter in segment one ranks '''B''' above '''C''' and '''C''' above '''A'''.
* Pivotal voter ranks '''A''' above '''B''' and '''B''' above '''C'''.
* Every voter in segment two ranks '''A''' above '''B''' and '''B''' above '''C'''.

Then by the argument in part one (and the last observation in that part), the societal outcome must rank '''A''' above '''B'''. This is because, except for a repositioning of '''C''', this profile is the same as ''profile k − 1'' from part one. Furthermore, by unanimity the societal outcome must rank '''B''' above '''C'''. Therefore, we know the outcome in this case completely.

Now suppose that pivotal voter moves '''B''' above '''A''', but keeps '''C''' in the same position and imagine that any number (even all!) of the other voters change their ballots to move '''B''' below '''C''', without changing the position of '''A'''. Then aside from a repositioning of '''C''' this is the same as ''profile k'' from part one and hence the societal outcome ranks '''B''' above '''A'''. Furthermore, by IIA the societal outcome must rank '''A''' above '''C''', as in the previous case. In particular, the societal outcome ranks '''B''' above '''C''', even though Pivotal Voter may have been the ''only'' voter to rank '''B''' above '''C'''. [[Condorcet paradox|By]] IIA, this conclusion holds independently of how '''A''' is positioned on the ballots, so pivotal voter is a dictator for '''B''' over '''C'''.

==== Part three: There exists a dictator ====
[[File:Diagram_for_part_three_of_Arrow's_Impossibility_Theorem.svg|thumb|Part three: Since voter ''k'' is the dictator for '''B''' over '''C''', the pivotal voter for '''B''' over '''C''' must appear among the first ''k'' voters. That is, ''outside'' of segment two. Likewise, the pivotal voter for '''C''' over '''B''' must appear among voters ''k'' through ''N''. That is, outside of Segment One.]]
In this part of the argument we refer back to the original ordering of voters, and compare the positions of the different pivotal voters (identified by applying parts one and two to the other pairs of candidates). First, the pivotal voter for '''B''' over '''C''' must appear earlier (or at the same position) in the line than the dictator for '''B''' over '''C''': As we consider the argument of part one applied to '''B''' and '''C''', successively moving '''B''' to the top of voters' ballots, the pivot point where society ranks '''B''' above '''C''' must come at or before we reach the dictator for '''B''' over '''C'''. Likewise, reversing the roles of '''B''' and '''C''', the pivotal voter for '''C''' over '''B''' must be at or later in line than the dictator for '''B''' over '''C'''. In short, if ''k''<sub>X/Y</sub> denotes the position of the pivotal voter for '''X''' over '''Y''' (for any two candidates '''X''' and '''Y'''), then we have shown

: ''k''<sub>B/C</sub> ≤ k<sub>B/A</sub> ≤ ''k''<sub>C/B</sub>.

Now repeating the entire argument above with '''B''' and '''C''' switched, we also have

: ''k''<sub>C/B</sub> ≤ ''k''<sub>B/C</sub>.

Therefore, we have

: ''k''<sub>B/C</sub> = k<sub>B/A</sub> = ''k''<sub>C/B</sub>

and the same argument for other pairs shows that all the pivotal voters (and hence all the dictators) occur at the same position in the list of voters. This voter is the dictator for the whole election.
{{Collapse bottom}}

=== Stronger versions ===

Arrow's impossibility theorem still holds if Pareto efficiency is weakened to the following condition:<ref name="Wilson1972"/>

; Non-imposition
: For any two alternatives '''a''' and '''b''', there exists some preference profile {{math|''R''{{sub|1}} , …, ''R''{{sub|''N''}}}} such that {{math|'''a'''}} is preferred to {{math|'''b'''}} by {{math|F(''R''{{sub|1}}, ''R''{{sub|2}}, …, ''R''{{sub|''N''}})}}.

== 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=":14"/>

=== {{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=":10"/>

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=":10" />

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=":72">{{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=":532322"/> or even non-existent.<ref name=":2" />

==== {{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=":2" /><ref name=":422"/>

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=":2" /><ref name=":422"/><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=":2" /><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=":72" /><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 name=":032">{{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=":4">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 name=":16">{{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=":52">"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=":14" /> 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=":52" /><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 name=":62">{{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 name=":152">{{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=":4" />

=== 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=":0" /> 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=":0">{{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>

== Common misconceptions ==
Arrow's theorem is not related to [[strategic voting]], which does not appear in his framework,<ref name="Arrow 1963234"/><ref name="plato.stanford.edu"/> though the theorem does have important implications for strategic voting (being used as a lemma to prove [[Gibbard's theorem]]<ref name="Gibbard1973"/>). The Arrovian framework of [[Social welfare function|social welfare]] assumes all voter preferences are known and the only issue is in aggregating them.<ref name="plato.stanford.edu" />

[[Monotonicity criterion|Monotonicity]] (called [[Positive response|positive association]] by Arrow) is not a condition of Arrow's theorem.<ref name="Arrow 1963234" /> This misconception is caused by a mistake by Arrow himself, who included the axiom in his original statement of the theorem but did not use it.<ref name="Arrow1950" /> Dropping the assumption does not allow for constructing a social welfare function that meets his other conditions.<ref name="Arrow 1963234" />

Contrary to a common misconception, Arrow's theorem deals with the limited class of [[Ranked voting|ranked-choice voting systems]], rather than voting systems as a whole.<ref name="plato.stanford.edu" /><ref>{{cite web |last1=Hamlin |first1=Aaron |date=March 2017 |title=Remembering Kenneth Arrow and His Impossibility Theorem |work=The Center for Election Science |url=https://electionscience.org/commentary-analysis/voting-theory-remembering-kenneth-arrow-and-his-impossibility-theorem/ |access-date=5 May 2024 |publisher=Center for Election Science}}</ref>

== See also ==
{{Portal|Economics
}}

* [[Comparison of electoral systems]]
* [[Condorcet paradox]]
* [[Doctrinal paradox]]
* [[Gibbard–Satterthwaite theorem]]
* [[Gibbard's theorem]]
* [[Holmström's theorem]]
* [[May's theorem]]
* [[Market failure]]

== References ==
{{Reflist|2}}

== Further reading ==

* {{cite book |last1=Campbell |first1=D. E. |url=https://books.google.com/books?id=rh10cOpltLsC |title=Handbook of social choice and welfare |publisher=Elsevier |year=2002 |isbn=978-0-444-82914-6 |editor-last1=Arrow |editor-first1=Kenneth J. |editor-link1=Kenneth Arrow |volume=1 |location=Amsterdam, Netherlands |pages=35–94 |chapter=Impossibility theorems in the Arrovian framework |ref=ArrowSenSuzumura2002 |editor-last2=Sen |editor-first2=Amartya K. |editor-link2=Amartya Sen |editor-last3=Suzumura |editor-first3=Kōtarō |editor-link3=Kotaro Suzumura}} Surveys many of approaches discussed in [[#Alternatives based on functions of preference profiles]]{{Broken anchor|date=2024-07-19|bot=User:Cewbot/log/20201008/configuration|target_link=#Alternatives based on functions of preference profiles|reason= The anchor (Alternatives based on functions of preference profiles) [[Special:Diff/1215537415|has been deleted]].}}.
* {{cite journal |last=Dardanoni |first=Valentino |year=2001 |title=A pedagogical proof of Arrow's Impossibility Theorem |url=https://escholarship.org/content/qt96n108ts/qt96n108ts.pdf?t=li5b40 |journal=Social Choice and Welfare |volume=18 |issue=1 |pages=107–112 |doi=10.1007/s003550000062 |jstor=41106398 |s2cid=7589377}} [http://repositories.cdlib.org/ucsdecon/1999-25/ preprint].
* {{cite journal |last=Hansen |first=Paul |year=2002 |title=Another Graphical Proof of Arrow's Impossibility Theorem |journal=The Journal of Economic Education |volume=33 |issue=3 |pages=217–235 |doi=10.1080/00220480209595188 |s2cid=145127710}}
* {{cite book |last=Hunt |first=Earl |author-link=Earl B. Hunt |url=http://www.cambridge.org/9780521850124 |title=The Mathematics of Behavior |publisher=Cambridge University Press |year=2007 |isbn=9780521850124}}. The chapter "Defining Rationality: Personal and Group Decision Making" has a detailed discussion of the Arrow Theorem, with proof.
* {{cite book |last=Lewis |first=Harold W. |title=Why flip a coin? : The art and science of good decisions |publisher=John Wiley |year=1997 |isbn=0-471-29645-7}} Gives explicit examples of preference rankings and apparently anomalous results under different electoral system. States but does not prove Arrow's theorem.
* {{Cite book |last1=Sen |first1=Amartya Kumar |author-link1=Amartya Sen |title=Collective choice and social welfare |publisher=North-Holland |year=1979 |isbn=978-0-444-85127-7 |location=Amsterdam}}
* {{cite book |last=Skala |first=Heinz J. |title=Theory and Decision : Essays in Honor of Werner Leinfellner |publisher=Springer |year=2012 |isbn=978-94-009-3895-3 |editor-last=Eberlein |editor-first=G. |pages=273–286 |chapter=What Does Arrow's Impossibility Theorem Tell Us? |editor2-last=Berghel |editor2-first=H. A. |chapter-url=https://books.google.com/books?id=Xrp9CAAAQBAJ&pg=PA273}}
* {{cite journal |last1=Tang |first1=Pingzhong |last2=Lin |first2=Fangzhen |year=2009 |title=Computer-aided Proofs of Arrow's and Other Impossibility Theorems |journal=Artificial Intelligence |volume=173 |issue=11 |pages=1041–1053 |doi=10.1016/j.artint.2009.02.005 |doi-access=free}}

== External links ==
{{subject bar|auto=y|d=y|Economics}}
* {{SEP|arrows-theorem}}
* [https://www.math.ucla.edu/~tao/arrow.pdf A proof by Terence Tao, assuming a much stronger version of non-dictatorship]
{{Decision theory paradoxes}}
{{Game theory}}
<references group="note" />

{{DEFAULTSORT:Arrow's General Possibility Theorem}}
[[Category:Voting theory]]
[[Category:Economics theorems]]
[[Category:Eponymous paradoxes]]
[[Category:Paradoxes in economics]]
[[Category:Theorems in discrete mathematics]]
[[Category:Decision-making paradoxes]]