View source for Arrow's impossibility theorem

{{Short description|Mathematical result that there is no perfect voting system}}
'''Arrow's impossibility theorem''', the '''general possibility theorem''' or '''Arrow's paradox''' is an [[Proof of impossibility|impossibility theorem]] in [[Philosophy:Social choice theory|social choice theory]] that states that when voters have three or more distinct alternatives (options), no [[Social:Ranked voting|ranked voting]] [[Social:Electoral system|electoral system]] can convert the '''ranked preferences''' of individuals into a community-wide (complete and transitive) ranking while also meeting the specified set of criteria: ''[[Social:Unrestricted domain|unrestricted domain]]'', ''non-dictatorship'', ''[[Pareto efficiency]]'', and ''[[Social:Independence of irrelevant alternatives|independence of irrelevant alternatives]]''. The theorem is often cited in discussions of voting theory as it is further interpreted by the [[Gibbard–Satterthwaite theorem]]. The theorem is named after economist and Nobel laureate [[Biography:Kenneth Arrow|Kenneth Arrow]], who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book ''[[Philosophy:Social Choice and Individual Values|Social Choice and Individual Values]]''.  The original paper was titled "A Difficulty in the Concept of Social Welfare".<ref>{{cite journal |last1=Arrow |first1=Kenneth J. |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= [[Finance:Journal of Political Economy|Journal of Political Economy]] |volume=58 |issue=4 |pages=328–346 |archive-url= https://web.archive.org/web/20110720090207/http://gatton.uky.edu/Faculty/hoytw/751/articles/arrow.pdf |archive-date=2011-07-20| doi=10.1086/256963 |jstor=1828886|s2cid=13923619 }}</ref>

In short, the theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria:

* If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
* If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
* There is no "dictator": no single voter possesses the power to always determine the group's preference.

[[Social:Cardinal voting|Cardinal voting]] electoral systems are not covered by the theorem, as they convey more information than rank orders.<ref name=":0">{{cite web|url=https://electology.org/podcasts/2012-10-06_kenneth_arrow|title=Interview with Dr. Kenneth Arrow|date=October 6, 2012|website=The Center for Election Science|quote=''CES:'' you mention that your theorem applies to preferential systems or ranking systems. ... But ... Approval Voting, falls within a class called cardinal systems. ... ''Dr. Arrow:'' And as I said, that in effect implies more information. ... I'm a little inclined to think that score systems where you categorize in maybe three or four classes ... is probably the best.}}</ref><ref>{{Cite journal|last=Sen|first=Amartya|year=1999|title=The Possibility of Social Choice|journal=American Economic Review|language=en|volume=89|issue=3|pages=349–378|doi=10.1257/aer.89.3.349|jstor=117024|quote=Does Arrow's impossibility ... go away with the use of interpersonal comparisons in social welfare judgments? ... yes. The additional informational availability allows sufficient discrimination to escape impossibilities of this type. ... even weaker forms of comparability would still permit making consistent social welfare judgments, satisfying all of Arrow's requirements}}</ref> [[Gibbard's theorem]] and the [[Finance:Duggan–Schwartz theorem|Duggan–Schwartz theorem]] show that strategic voting remains a problem. The axiomatic approach Arrow adopted can treat all conceivable rules (that are based on preferences) within one unified framework. In that sense, the approach is qualitatively different from the earlier one in voting theory, in which rules were investigated one by one. One can therefore say that the contemporary paradigm of social choice theory started from this theorem.<ref>{{cite book|last1=Suzumura|first1=Kōtarō |editor-last1=Arrow|editor-first1=Kenneth J.|editor-last2=Sen |editor-first2=Amartya K.|editor-link2=Amartya Sen|editor-last3=Suzumura|editor-first3=Kōtarō|editor-link3=Kotaro Suzumura |title=Handbook of social choice and welfare|volume=1| year=2002 | publisher=Elsevier | location=Amsterdam, Netherlands | isbn=978-0-444-82914-6 | url=https://books.google.com/books?id=rh10cOpltLsC |chapter=Introduction |page=10 |ref=ArrowSenSuzumura2002 }}</ref>

The practical consequences of the theorem are debatable. Arrow has said: "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="ns12" /> When asked what he would change about US elections, he said, "The first thing that I'd certainly do is go to a system where people ranked all the candidates."<ref>{{cite web |last1=Aaron |first1=Hamlin |title=CES Podcast with Dr Arrow |url=https://electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/ |website=Center for Election Science |date=25 May 2015 |publisher=CES |access-date=9 March 2023}}</ref> Arrow's impossibility theorem does not apply to multi-winner voting such as [[Social:Proportional representation|proportional representation]].

== Statement ==
The need to aggregate [[Preference|preference]]s occurs in many disciplines: in [[Finance:Welfare economics|welfare economics]], where one attempts to find an economic outcome which would be acceptable and stable; in [[Decision theory|decision theory]], where a person has to make a rational choice based on several criteria; and most naturally in [[Social:Electoral system|electoral system]]s, which are mechanisms for extracting a governance-related decision from a multitude of voters' preferences.

The framework for Arrow's theorem assumes that we need to extract a preference order on a given set of options (outcomes). Each individual in the society (or equivalently, each decision criterion) gives a particular order of preferences on the set of outcomes. We are searching for a [[Social:Ranked voting|ranked voting]] electoral system, called a ''social welfare function'' (''preference aggregation rule''), which transforms the set of preferences (''profile'' of preferences) into a single global societal preference order. Arrow's theorem says that if the decision-making body has at least two members and at least three options to decide among, then it is impossible to design a social welfare function that satisfies all these conditions (assumed to be a reasonable requirement of a fair electoral system) at once:
; Non-dictatorship: The social welfare function should account for the wishes of multiple voters. It cannot simply mimic the preferences of a single voter.
; [[Social:Unrestricted domain|Unrestricted domain]], or universality: For any set of individual voter preferences, the social welfare function should yield a unique and complete ranking of societal choices. Thus:{{bulleted list
 | It must do so in a manner that results in a complete ranking of preferences for society.
 | It must deterministically provide the same ranking each time voters' preferences are presented the same way.
}}
; [[Social:Independence of irrelevant alternatives#In social choice|Independence of irrelevant alternatives]] (IIA): The social preference between x and y should depend only on the individual preferences between x and y (''[[Learning to rank#Pairwise approach|pairwise]] independence''). More generally, changes in individuals' rankings of ''irrelevant'' alternatives (ones outside a certain subset) should have no impact on the societal ranking of the subset. For example, if candidate ''x'' ranks socially before candidate ''y'', then ''x'' should rank socially before ''y'' even if a third candidate ''z'' is removed from participation. (See Remarks below.)
; Monotonicity, or positive association of social and individual values: If any individual modifies his or her preference order by promoting a certain option, then the societal preference order should respond only by promoting that same option or not changing, never by placing it lower than before. An individual should not be able to hurt an option by ranking it ''higher''.
; Non-imposition, or citizen sovereignty: Every possible societal preference order should be achievable by some set of individual preference orders. This means that the social welfare function is [[Surjective function|surjective]]: It has an unrestricted target space.

A later (1963)<ref name="Arrow 1963">{{Cite book|url=http://cowles.yale.edu/sites/default/files/files/pub/mon/m12-2-all.pdf |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|title=Social Choice and Individual Values|last=Arrow|first=Kenneth Joseph Arrow|date=1963|publisher=Yale University Press|isbn=978-0300013641|language=en}}</ref> version of Arrow's theorem replaced the monotonicity and non-imposition criteria with:
; [[Pareto efficiency]], or '''unanimity''': If every individual prefers a certain option to another, then so must the resulting societal preference order. This, again, is a demand that the social welfare function will be minimally sensitive to the preference profile.

This later version is more general, having weaker conditions. The axioms of monotonicity, non-imposition, and IIA together imply Pareto efficiency, whereas Pareto efficiency (itself implying non-imposition) and IIA together do not imply monotonicity.

===Independence of irrelevant alternatives (IIA)===
The IIA condition has three purposes (or effects):<ref>{{cite book|first1=Andreu|last1=Mas-Colell|first2=Michael Dennis|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|page=794|title=Microeconomic Theory|publisher=Oxford University Press|isbn=978-0-19-507340-9}}</ref>
;Normative: Irrelevant alternatives should not matter.
;Practical: Use of minimal information.
;Strategic: Providing the right incentives for the truthful revelation of individual preferences. Though the strategic property is conceptually different from IIA, it is closely related.

Arrow's death-of-a-candidate example (1963, page 26)<ref name="Arrow 1963"/> suggests that the agenda (the set of feasible alternatives) shrinks from, say, X = {a, b, c} to S = {a, b} because of the death of candidate c. This example is misleading since it can give the reader an impression that IIA is a condition involving ''two'' agenda and ''one'' profile. The fact is that IIA involves just ''one'' agendum ({x, y} in case of pairwise independence) but ''two'' profiles. If the condition is applied to this confusing example, it requires this: Suppose an aggregation rule satisfying IIA chooses b from the agenda {a, b} when the profile is given by (cab, cba), that is, individual 1 prefers c to a to b, 2 prefers c to b to a. Then, it must still choose b from {a, b} if the profile were, say: (abc, bac); (acb, bca); (acb, cba); or (abc, cba).

In different words, Arrow defines IIA as saying that the social preferences between alternatives x and y depend only on the individual preferences between x and y (not on those involving other candidates).

== Formal statement of the theorem ==
Let {{math|A}} be a set of '''outcomes''', {{math|''N''}} a number of '''voters''' or '''decision criteria'''. We shall denote the set of all full [[Total order|linear orderings]] of {{math|A}} by {{math|L(A)}}.

A (strict) '''social welfare function''' ('''preference aggregation rule''') is a function
:<math> F : \mathrm{L(A)}^N \to \mathrm{L(A)} </math>
which aggregates voters' preferences into a single preference order on {{math|A}}.<ref>Note that by definition, a ''social welfare function'' as defined here satisfies the Unrestricted domain condition.  Restricting the range to the social preferences that are never indifferent between distinct outcomes is probably a very restrictive assumption, but the goal here is to give a simple statement of the theorem.  Even if the restriction is relaxed, the impossibility result will persist.</ref>

An {{math|''N''}}-[[Tuple|tuple]] {{math|(''R''{{sub|1}}, …, ''R''{{sub|''N''}}) ∈ L(A){{sup|''N''}}}} of voters' preferences is called a ''preference profile''.  In its strongest and simplest form, Arrow's impossibility theorem states that whenever the set {{math|A}} of possible alternatives has more than 2 elements, then the following three conditions become incompatible:

; Unanimity, or weak Pareto efficiency: If alternative {{math|'''a'''}} is ranked strictly higher than {{math|'''b'''}} for all orderings {{math|''R''{{sub|1}} , …, ''R''{{sub|''N''}}}}, then {{math|'''a'''}} is ranked strictly higher than {{math|'''b'''}} by {{math|F(''R''{{sub|1}}, ''R''{{sub|2}}, …, ''R''{{sub|''N''}})}}. (Unanimity implies non-imposition.)
; Non-dictatorship: There is no individual, {{math|''i''}} whose strict preferences always prevail. That is, there is no {{math|''i'' ∈ {1, …, ''N''} }} such that for all {{math|(''R''{{sub|1}}, …, ''R''{{sub|''N''}}) ∈ L(A){{sup|''N''}}}} and all {{math|'''a'''}} and {{math|'''b'''}}, when {{math|'''a'''}} is ranked strictly higher than {{math|'''b'''}} by {{math|''R{{sub|i}}''}} then {{math|'''a'''}} is ranked strictly higher than {{math|'''b'''}} by {{math|F(''R''{{sub|1}}, ''R''{{sub|2}}, …, ''R''{{sub|''N''}})}}.
; [[Social:Independence of irrelevant alternatives|Independence of irrelevant alternatives]]: For two preference profiles {{math|(''R''{{sub|1}}, …, ''R''{{sub|''N''}})}} and {{math|(''S''{{sub|1}}, …, ''S''{{sub|''N''}})}} such that for all individuals {{math|''i''}}, alternatives {{math|'''a'''}} and {{math|'''b'''}} have the same order in {{math|''R{{sub|i}}''}} as in {{math|''S{{sub|i}}''}}, alternatives {{math|'''a'''}} and {{math|'''b'''}} have the same order in {{math|F(''R''{{sub|1}}, …, ''R''{{sub|''N''}})}} as in {{math|F(''S''{{sub|1}}, …, ''S''{{sub|''N''}})}}.

== Proof by decisive coalitions ==
Arrow's proof used the concept of ''decisive coalitions''.<ref name="Arrow 1963" />

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

The following proof is a simplification taken from (Sen, 2014)<ref>{{Cite book |last=Sen |first=Amartya |title=The Arrow Impossibility Theorem |chapter=Arrow and the Impossibility Theorem |url=https://www.degruyter.com/document/doi/10.7312/mask15328-003/html |date=2014-07-22 |pages=29–42 |publisher=Columbia University Press |isbn=978-0-231-52686-9 |language=en |doi=10.7312/mask15328-003}}</ref> and (Rubinstein, 2012).<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 |ol=29649010M |edition=2nd |at=Problem 9.5}}</ref> The simplified proof uses an addition 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
| name=Field expansion lemma
| note=
| 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.
}}
{{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>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
| name = Group contraction lemma
| note = 
| math_statement = If a coalition is decisive, and has size <math>\geq 2</math>, then it has a proper subset that is also decisive.
}}
{{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 group contraction lemma, there is a size-one decisive coalition — a dictator.

== Proof by the 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''.<ref>{{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 |volume=26 |issue=1 |pages=211–215 |citeseerx=10.1.1.193.6817 |doi=10.1007/s00199-004-0556-7 |jstor=25055941 |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 |s2cid=17101545}}</ref><ref>{{cite journal |last1=Yu |first1=Ning Neil |year=2012 |title=A one-shot proof of Arrow's theorem |journal=Economic Theory |volume=50 |issue=2 |pages=523–525 |doi=10.1007/s00199-012-0693-3 |jstor=41486021 |s2cid=121998270}}</ref>

We will prove that any social choice system respecting unrestricted domain, unanimity, and independence of irrelevant alternatives (IIA) is a dictatorship. The key idea is to identify a ''pivotal voter'' whose ballot swings the societal outcome.  We then prove that this voter is a partial dictator (in a specific technical sense, described below).  Finally we conclude by showing that all of the partial dictators are the same person, hence this voter is a dictator.

For simplicity we have presented all rankings as if there are no ties.  A complete proof taking possible ties into account is not essentially different from the one given here, except that one ought to say "not above" instead of "below" or "not below" instead of "above" in some cases. Full details are given in the original articles.

=== Part one: There is a "pivotal" voter for B over A ===
[[File:Diagram for part one of Arrow's Impossibility Theorem.svg|thumb|right|upright=1.35|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'''.]]
Say there are three choices for society, call them '''A''', '''B''', and '''C'''.  Suppose first that everyone prefers option '''B''' the least: everyone prefers  '''A''' to '''B''', and everyone prefers '''C''' to '''B'''.  By unanimity, society must also prefer both '''A''' and '''C''' to '''B'''.  Call this situation ''profile 0''.

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, 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, 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|thumb|right|upright=1.35|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|upright=1.35|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.

== Interpretations ==

Although Arrow's theorem is a mathematical result, it is often expressed in a non-mathematical way with a statement such as ''no voting method is fair'', ''every ranked voting method is flawed'', or ''the only voting method that is not flawed is a dictatorship''.<ref>{{cite web|url=http://www.chicagobooth.edu/capideas/blog/2016/march/what-economists-think-about-voting|url-status=dead|archive-url=https://web.archive.org/web/20160326065043/http://www.chicagobooth.edu/capideas/blog/2016/march/what-economists-think-about-voting|archive-date=2016-03-26|title=What economists think about voting|last=Cockrell|first=Jeff|date=2016-03-08|website=Capital Ideas|publisher=Chicago Booth|access-date=2016-09-05|quote=Is there such a thing as a perfect voting system? The respondents were unanimous in their insistence that there is not.}}</ref> These statements are simplifications of Arrow's result which are not universally considered to be true. What Arrow's theorem does state is that a deterministic preferential voting mechanism—that is, one where a preference order is the only information in a vote, and any possible set of votes gives a unique result—cannot comply with all of the conditions given above simultaneously.

Various theorists have suggested weakening the IIA criterion as a way out of the paradox. Proponents of ranked voting methods contend that the IIA is an unreasonably strong criterion.  It is the one breached in most useful [[Social:Electoral system|electoral system]]s. Advocates of this position point out that failure of the standard IIA criterion is trivially implied by the possibility of [[Condorcet paradox|cyclic preferences]]. If voters cast ballots as follows:
* 1 vote for A > B > C
* 1 vote for B > C > A
* 1 vote for C > A > B
then the pairwise majority preference of the group is that A wins over B, B wins over C, and C wins over A: these yield rock-paper-scissors preferences for any pairwise comparison.  In this circumstance, ''any'' aggregation rule that satisfies the very basic majoritarian requirement that a candidate who receives a majority of votes must win the election, will fail the IIA criterion, if social preference is required to be transitive (or acyclic).  To see this, suppose that such a rule satisfies IIA.  Since majority preferences are respected, the society prefers A to B (two votes for A > B and one for B > A), B to C, and C to A.  Thus a cycle is generated, which contradicts the assumption that social preference is transitive.

So, what Arrow's theorem really shows is that any majority-wins electoral system is a non-trivial game, and that [[Game theory|game theory]] should be used to predict the outcome of most voting mechanisms.<ref>This does not mean various normative criteria will be satisfied if we use equilibrium concepts in game theory. Indeed, the mapping from profiles to equilibrium outcomes defines a social choice rule, whose performance can be investigated by social choice theory. See {{harvp|Austen-Smith|Banks|1999}} Section 7.2.</ref> This could be seen as a discouraging result, because a game need not have efficient equilibria; e.g., a ballot could result in an alternative nobody really wanted in the first place, yet everybody voted for.

===Remark: Scalar rankings from a vector of attributes and the IIA property===
The IIA property might not be satisfied in human decision-making of realistic  complexity because the ''scalar'' preference ranking is effectively derived from the weighting—not usually explicit—of a ''vector'' of attributes (one book dealing with the Arrow theorem invites the reader to consider the related problem of creating a scalar measure for the track and field decathlon event—e.g. how does one make scoring 600 points in the discus event "commensurable" with scoring 600 points in the 1500&nbsp;m race) and this scalar ranking can depend sensitively on the weighting of different attributes, with the tacit weighting itself affected by the context and contrast created by apparently "irrelevant" choices.  Edward MacNeal discusses this sensitivity problem with respect to the ranking of "most livable city" in the chapter "Surveys" of his book ''MathSemantics: making numbers talk sense'' (1994).

== Alternatives based on functions of preference profiles ==
In an attempt to escape from the negative conclusion of Arrow's theorem, social choice theorists have investigated various possibilities ("ways out"). This section includes approaches that deal with
* aggregation rules (functions that map each preference profile into a social preference), and
* other functions, such as functions that map each preference profile into an alternative.
Since these two approaches often overlap, we discuss them at the same time. What is characteristic of these approaches is that they investigate various possibilities by eliminating or weakening or replacing one or more conditions (criteria) that Arrow imposed.

=== Infinitely many individuals ===
Several theorists (e.g., Fishburn<ref name = Fishburn1970>{{Cite journal| last = Fishburn | first = Peter Clingerman| title = Arrow's impossibility theorem: concise proof and infinite voters| journal = Journal of Economic Theory| volume = 2| issue = 1| pages = 103–106| year = 1970| doi = 10.1016/0022-0531(70)90015-3}}</ref> and Kirman and Sondermann<ref name = kirman-s72>{{Cite journal| last1 = Kirman | first1 = A.| last2 = Sondermann | first2 = D.| title = Arrow's theorem, many agents, and invisible dictators| journal = Journal of Economic Theory| volume = 5| issue = 2| pages = 267–277| year = 1972| doi = 10.1016/0022-0531(72)90106-8}}</ref>) point out that when one drops the assumption that there are only finitely many individuals, one can find aggregation rules that satisfy all of Arrow's other conditions.

However, such aggregation rules are practically of limited interest, since they are based on [[Ultrafilter|ultrafilter]]s, highly non-constructive mathematical objects. In particular, Kirman and Sondermann argue that there is an "invisible dictator" behind such a rule.<ref name = kirman-s72/> Mihara<ref name=mihara97>{{Cite journal|last1=Mihara |first1=H. R. |title=Arrow's Theorem and Turing computability |journal=Economic Theory |volume=10 |issue=2 |pages=257–276 |year=1997 |doi=10.1007/s001990050157 |jstor=25055038 |url=http://129.3.20.41/eps/pe/papers/9408/9408001.pdf |archive-url=https://web.archive.org/web/20110812013901/http://129.3.20.41/eps/pe/papers/9408/9408001.pdf |archive-date=2011-08-12 |url-status=dead |citeseerx=10.1.1.200.520 |s2cid=15398169 }}  Reprinted in {{cite book |editor1-first=K. V. |editor1-last=Velupillai |editor2-first=S. |editor2-last=Zambelli |editor3-first=S. |editor3-last=Kinsella |title=Computable Economics |series=International Library of Critical Writings in Economics |publisher=Edward Elgar |year=2011 |isbn=978-1-84376-239-3}}</ref><ref>{{Cite journal | last1 = Mihara | first1 = H. R.| title =  Arrow's theorem, countably many agents, and more visible invisible dictators | journal = Journal of Mathematical Economics | volume = 32 | issue = 3| pages = 267–277 | year = 1999 | doi = 10.1016/S0304-4068(98)00061-5 | url=http://econpapers.repec.org/paper/wpawuwppe/9705001.htm| citeseerx = 10.1.1.199.1970}}</ref> shows that such a rule violates algorithmic computability.<ref>Mihara's definition of a ''computable'' aggregation rule is based on computability of a simple game (see Rice's theorem).</ref> These results can be seen to establish the robustness of Arrow's theorem.<ref>See Chapter 6 of {{cite book | last=Taylor | first=Alan D. | title=Social choice and the mathematics of manipulation | year=2005 | publisher=Cambridge University Press | location=New York  | isbn=978-0-521-00883-9 |postscript=none}} for a concise discussion of social choice for infinite societies.</ref>

On the other hand, the ultrafilters (indeed, constructing them in an infinite model relies on the [[Axiom of choice|axiom of choice]]) are inherent in finite models as well (with no need of the axiom of choice).
They can be interpreted as decisive hierarchies, with the only difference that the hierarchy's top level - Arrow's dictator - always exists in a finite model but can be unattainable (= missing) in an infinite hierarchy. 
In the latter case, the "invisible dictator" is nothing else but the infinite decisive hierarchy itself.
If desired, it can be complemented with a limit point, which then becomes a "visible dictator".
Since dictators are inseparable from decisive hierarchies, the Dictatorship prohibition automatically prohibits decisive hierarchies, which is much less self-evident than the Dictatorship prohibition.<ref name="Tanguiane1994">{{Cite journal
|last=Tanguiane (Tangian)
|first=Andranick
|year=1994
|title=Arrow's paradox and mathematical theory of democracy
|journal=Social Choice and Welfare
|volume=11
|issue=1
|pages=1–82|doi=10.1007/BF00182898
|s2cid=154076212
}}</ref><ref name="Tangian2014">{{Cite book
|last=Tangian
|first=Andranik
|date=2014
|title=Mathematical theory of democracy
|publisher=Springer
|location=Berlin-Heidelberg}}</ref><ref name="Tangian2020">{{Cite book
|last=Tangian
|first=Andranik
|date=2020
|title=Analytical theory of democracy. Vols. 1 and 2
|publisher=Springer
|location=Cham, Switzerland}}</ref>
See also paragraph "Relaxing the Dictatorship prohibition".

=== Limiting the number of alternatives ===
When there are only two alternatives to choose from, [[May's theorem]] shows that only simple majority rule satisfies a certain set of criteria (e.g., equal treatment of individuals and of alternatives; increased support for a winning alternative should not make it into a losing one). On the other hand, when there are at least three alternatives, Arrow's theorem points out the difficulty of collective decision making. Why is there such a sharp difference between the case of less than three alternatives and that of at least three alternatives?

''Nakamura's theorem'' (about the core of simple games) gives an answer more generally. It establishes that if the number of alternatives is less than a certain integer called the '''[[Nakamura number]]''', then the rule in question will identify "best" alternatives without any problem; if the number of alternatives is greater or equal to the Nakamura number, then the rule will not always work, since for some profile a voting paradox (a cycle such as alternative A socially preferred to alternative B, B to C, and C to A) will arise. Since the Nakamura number of majority rule is 3 (except the case of four individuals), one can conclude from Nakamura's theorem that majority rule can deal with up to two alternatives rationally. Some super-majority rules (such as those requiring 2/3 of the votes) can have a Nakamura number greater than 3, but such rules violate other conditions given by Arrow.<ref>{{harvtxt|Austen-Smith|Banks|1999|loc=Chapter 3}} gives a detailed discussion of the approach trying to limit the number of alternatives.</ref>

=== Pairwise voting ===
A common way "around" Arrow's paradox is limiting the alternative set to two alternatives. Thus, whenever more than two alternatives should be put to the test, it seems very tempting to use a mechanism that pairs them and votes by pairs. As tempting as this mechanism seems at first glance, it is generally far from satisfying even [[Pareto efficiency]], not to mention IIA. The specific order by which the pairs are decided strongly influences the outcome. This is not necessarily a bad feature of the mechanism. Many sports use the tournament mechanism—essentially a pairing mechanism—to choose a winner.  This gives considerable opportunity for weaker teams to win, thus adding interest and tension throughout the tournament. This means that the person controlling the order by which the choices are paired (the agenda maker) has great control over the outcome. In any case, when viewing the entire voting process as one game, Arrow's theorem still applies.

=== Domain restrictions ===
Another approach is relaxing the universality condition, which means restricting the [[Domain of a function|domain]] of aggregation rules. The best-known result along this line assumes "single peaked" preferences.

Duncan Black has shown that if there is only one dimension on which every individual has a "single-peaked" preference, then all of Arrow's conditions are met by majority rule. Suppose that there is some predetermined linear ordering of the alternative set. An individual's preference is ''single-peaked'' with respect to this ordering if he has some special place that he likes best along that line, and his dislike for an alternative grows larger as the alternative goes further away from that spot (i.e., the graph of his utility function has a single peak if alternatives are placed according to the linear ordering on the horizontal axis).  For example, if voters were voting on where to set the volume for music, it would be reasonable to assume that each voter had their own ideal volume preference and that as the volume got progressively too loud or too quiet they would be increasingly dissatisfied.
If the domain is restricted to profiles in which every individual has a single peaked preference with respect to the linear ordering, then ''simple''<ref name="austensmith-b99">{{Cite book  | last1 = Austen-Smith | first1 = David | last2 = Banks | first2 = Jeffrey S. | title = Positive political theory I: Collective preference |url=https://books.google.com/books?id=nxXDn3nPxIAC&q=%22nakamura+number%22 | year = 1999 | publisher = University of Michigan Press | location = Ann Arbor | isbn = 978-0-472-08721-1 | access-date = 2016-02-16}}</ref> aggregation rules, which include majority rule, have an ''acyclic'' (defined below) social preference, hence "best" alternative.<ref>Indeed, many different social welfare functions can meet Arrow's conditions under such restrictions of the domain. It has been proved, however, that under any such restriction, if there exists any social welfare function that adheres to Arrow's criteria, then the majority rule will adhere to Arrow's criteria. See {{cite journal |last1=Campbell |first1=D. E. |last2=Kelly |first2=J. S. |title=A simple characterization of majority rule |journal=Economic Theory |volume=15 |issue=3 |year=2000 |pages=689–700 |doi=10.1007/s001990050318 |jstor=25055296|s2cid=122290254 }}</ref> In particular, when there are odd number of individuals, then the social preference becomes transitive, and the socially "best" alternative is equal to the median of all the peaks of the individuals (Black's [[Median voter theorem|median voter theorem]]<ref>{{Cite book  | last = Black | first = Duncan |  title = The theory of committees and elections | year = 1968 | publisher = University Press | location = Cambridge, Eng. | isbn = 978-0-89838-189-4 }}</ref>). Under single-peaked preferences, the majority rule is in some respects the most natural voting mechanism.

One can define the notion of "single-peaked" preferences on higher-dimensional sets of alternatives. However, one can identify the "median" of the peaks only in exceptional cases. Instead, we typically have the destructive situation suggested by McKelvey's Chaos Theorem:<ref>{{Cite journal | last1 = McKelvey | first1 = Richard D. |  title = Intransitivities in multidimensional voting models and some implications for agenda control | journal = Journal of Economic Theory | volume = 12 | issue = 3 | pages = 472–482 | year = 1976 | doi = 10.1016/0022-0531(76)90040-5}}</ref> for any ''x'' and ''y'', one can find a sequence of alternatives such that ''x'' is beaten by ''x<sub>1</sub>'' by a majority, ''x<sub>1</sub>'' by ''x<sub>2</sub>'', up to ''x<sub>k</sub>'' by ''y''.

In 1977, [[Biography:Ehud Kalai|Ehud Kalai]] and Eitan Muller gave a full characterization of domain restrictions admitting a nondictatorial Arrow Social Welfare function.<ref>{{Cite journal | last1 = Kalai | first1 = Ehud | first2 = Eitan  | last2 = Muller | title = Characterization of Domains Admitting Nondictatorial Social Welfare Functions and Nonmanipulable Voting Procedures | journal = Journal of Economic Theory | volume = 16 | issue = 2 | pages = 457–469 | year = 1977}}</ref>

=== Relaxing transitivity ===
By relaxing the transitivity of social preferences, we can find aggregation rules that satisfy Arrow's other conditions. If we impose ''neutrality'' (equal treatment of alternatives) on such rules, however, there exists an individual who has a "veto". So the possibility provided by this approach is also very limited.

First, suppose that a social preference is [[Quasitransitive relation|''quasi-transitive'']] (instead of transitive); this means that the strict preference <math>\succ</math> ("better than") is transitive: if <math>x \succ y</math> and <math>y \succ z</math>, then <math>x \succ z</math>. Then, there do exist non-dictatorial aggregation rules satisfying Arrow's conditions, but such rules are ''oligarchic''.<ref>{{cite journal|last=Gibbard|first=Allan F.|doi=10.1007/s10058-014-0158-1|title=Intransitive social indifference and the Arrow dilemma|year=2014|orig-year=1969|volume=18|issue=1|pages=3–10|journal=Review of Economic Design|s2cid=154682454}}</ref> This means that there exists a coalition L such that L is ''decisive'' (if every member in L prefers x to y, then the society prefers x to y), and each member in L ''has a veto'' (if she prefers x to y, then the society cannot prefer y to x).

Second, suppose that a social preference is ''acyclic'' (instead of transitive): there do not exist alternatives <math>x_1, \ldots, x_k</math> that form a ''cycle'' (<math>x_1 \succ x_2, \; x_2 \succ x_3, \; \ldots, \; x_{k-1} \succ x_k, \; x_k \succ x_1</math>). Then, provided that there are at least as many alternatives as individuals, an aggregation rule satisfying Arrow's other conditions is ''collegial''.<ref>{{cite journal |last=Brown |first=D. J. |year=1975 |title=Aggregation of Preferences |journal=[[Finance:Quarterly Journal of Economics|Quarterly Journal of Economics]] |volume=89 |issue=3 |pages=456–469 |jstor=1885263 | doi = 10.2307/1885263 }}</ref> This means that there are individuals who belong to the intersection ("collegium") of all decisive coalitions. If there is someone who has a veto, then he belongs to the collegium. If the rule is assumed to be neutral, then it does have someone who has a veto.

Finally, Brown's theorem left open the case of acyclic social preferences where the number of alternatives is less than the number of individuals. One can give a definite answer for that case using the ''Nakamura number''.  See limiting the number of alternatives.

=== Relaxing assumption IIA ===
There are numerous examples of aggregation rules satisfying Arrow's conditions except IIA. The Borda rule is one of them. These rules, however, are susceptible to ''strategic manipulation'' by individuals.<ref>{{Cite journal | last1 = Blair | first1 = Douglas | first2 = Eitan  | last2 = Muller | title = Essential aggregation procedures on restricted domains of preferences | journal = Journal of Economic Theory | volume = 30 | issue = 1 | pages = 34–53 | year = 1983 | doi = 10.1016/0022-0531(83)90092-3}}</ref>

See also [[Arrow's impossibility theorem#Interpretations|Interpretations of the theorem]] above.

=== Relaxing the Pareto criterion ===
Wilson (1972)<ref>{{cite journal | last = Wilson |first = Robert | year = 1972| title = Social Choice Theory without the Pareto principle | journal = [[Finance:Journal of Economic Theory|Journal of Economic Theory]] | volume =  5|issue = 3 | pages= 478–486| doi= 10.1016/0022-0531(72)90051-8}}</ref> shows that if an aggregation rule is non-imposed and non-null, then there is either a dictator or an inverse dictator, provided that Arrow's conditions other than Pareto are also satisfied. Here, an ''inverse dictator'' is an individual ''i'' such that whenever ''i'' prefers ''x'' to ''y'', then the society prefers ''y'' to&nbsp;''x''.

Amartya Sen offered both relaxation of transitivity and removal of the Pareto principle.<ref>{{cite journal |last=Sen |first=Amartya |date=September 1979a |title=Personal Utilities and Public Judgements: Or What's Wrong With Welfare Economics? |journal=[[Finance:The Economic Journal|The Economic Journal]] |volume=89 |issue=355 |pages=537–558 |jstor=2231867 |doi=10.2307/2231867}}</ref> He demonstrated another interesting impossibility result, known as the "impossibility of the Paretian Liberal" (see [[Philosophy:Liberal paradox|liberal paradox]] for details). Sen went on to argue that this demonstrates the futility of demanding Pareto optimality in relation to voting mechanisms.

=== Relaxing the dictatorship prohibition ===
[[Biography:Andranik Tangian|Andranik Tangian]] (2010) introduced measures of dictator's "representativeness", for instance, the "popularity index" defined as the average size of the social group whose pairwise preferences are shared (that is, represented) by the dictator, averaged over all pairs of alternatives and all preference profiles. It was shown that there always exist "good" Arrow's dictators who on the average represent a majority.<ref name="Tangian2010">{{Cite journal
 |last=Tangian |first=Andranik
 |year=2010
 |title= Application of the mathematical theory of democracy to Arrow's Impossibility Theorem (How dictatorial are Arrow's dictators?)
 |journal=Social Choice and Welfare
 |volume=35
 |issue=1
 |pages=135–167
 |doi=10.1007/s00355-009-0433-1
 |s2cid=206958453
}}</ref> Since they are rather representatives of the society - like democratically elected  presidents - there are no self-evident reasons to prohibit them. Restricting the notion of dictator to "bad" ones only, i.e. those who on the average represent a minority, Arrow's axioms were proven to be consistent.<ref name="Tangian2014" /><ref name="Tangian2020" />

=== Social choice instead of social preference ===
In social decision making, to rank all alternatives is not usually a goal.  It often suffices to find some alternative. The approach focusing on choosing an alternative investigates either ''social choice functions'' (functions that map each preference profile into an alternative) or ''social choice rules'' (functions that map each preference profile into a subset of alternatives).

As for social choice functions, the [[Gibbard–Satterthwaite theorem]] is well-known, which states that if a social choice function whose range contains at least three alternatives is strategy-proof, then it is dictatorial.

As for social choice rules, we should assume there is a social preference behind them. That is, we should regard a rule as choosing the maximal elements ("best" alternatives) of some social preference. The set of maximal elements of a social preference is called the ''core''. Conditions for existence of an alternative in the core have been investigated in two approaches. The first approach assumes that preferences are at least ''acyclic'' (which is necessary and sufficient for the preferences to have a maximal element on any ''finite'' subset).  For this reason, it is closely related to relaxing transitivity. The second approach drops the assumption of acyclic preferences. Kumabe and Mihara<ref name="kumabe-m10geb">{{Cite journal| last1 = Kumabe | first1 = M.| last2 = Mihara | first2 = H. R.| title = Preference aggregation theory without acyclicity: the core without majority dissatisfaction| journal = Games and Economic Behavior| volume = 72| pages = 187–201| year = 2011| doi = 10.1016/j.geb.2010.06.008| url=http://mpra.ub.uni-muenchen.de/23918/2/MPRA_paper_23918.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://mpra.ub.uni-muenchen.de/23918/2/MPRA_paper_23918.pdf |archive-date=2022-10-09 |url-status=live| arxiv = 1107.0431| bibcode = 2011arXiv1107.0431K| s2cid = 6685306}}</ref> adopt this approach. They make a more direct assumption that individual preferences have maximal elements, and examine conditions for the social preference to have a maximal element. See [[Nakamura number]] for details of these two approaches.

== Other alternatives ==

Arrow originally rejected cardinal utility as a meaningful tool for expressing social welfare,<ref>"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 [[Philosophy:Identity of indiscernibles|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|title=The Active Consumer: Novelty and Surprise in Consumer Choice|volume=20|series=Routledge Frontiers of Political Economy|editor-first=Marina|editor-last=Bianchi|publisher=Routledge|year=2002|contribution=Choice without utility? Some reflections on the loose foundations of standard consumer theory|pages=21–45|first=Fabio|last=Racnchetti}}</ref> and so focused his theorem on preference rankings, but later stated that a cardinal [[Philosophy:Score voting|score]] system with three or four classes "is probably the best".<ref name=":0" />

Arrow's framework assumes that individual and social preferences are "orderings" (i.e., satisfy completeness and transitivity) on the set of alternatives. This means that if the preferences are represented by a utility function, its value is an ''ordinal'' utility in the sense that it is meaningful so far as the greater value indicates the better alternative. For instance, having ordinal utilities of 4, 3, 2, 1 for alternatives a, b, c, d, respectively, is the same as having 1000, 100.01, 100, 0, which in turn is the same as having 99, 98, 1, .997. They all represent the ordering in which a is preferred to b to c to d. The assumption of ''ordinal'' preferences, which precludes ''interpersonal comparisons'' of utility, is an integral part of Arrow's theorem.

For various reasons, an approach based on cardinal utility, where the utility has a meaning beyond just giving a ranking of alternatives, is not common in contemporary economics. However, once one adopts that approach, one can take [[Social:Intensity of preference|intensities of preferences]] into consideration, or one can compare (i) gains and losses of utility or (ii) levels of utility, across different individuals. In particular, Harsanyi (1955)<ref>{{Cite journal|last=Harsanyi|first=John C.|date=1955|title=Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility|jstor=1827128|journal=Journal of Political Economy|volume=63|issue=4|pages=309–321|doi=10.1086/257678|s2cid=222434288}}</ref> gives a justification of [[Philosophy:Utilitarianism|utilitarianism]] (which evaluates alternatives in terms of the sum of individual utilities), originating from [[Biography:Jeremy Bentham|Jeremy Bentham]]. Hammond (1976)<ref>{{Cite journal|last=Hammond|first=Peter J.|date=1976|title=Equity, Arrow's Conditions, and Rawls' Difference Principle|jstor=1913445|journal=Econometrica|volume=44|issue=4|pages=793–804|doi=10.2307/1913445}}</ref> gives a justification of the maximin principle (which evaluates alternatives in terms of the utility of the worst-off individual), originating from [[Biography:John Rawls|John Rawls]].

Not all voting methods use, as input, only an ordering of all candidates.<ref>It is sometimes asserted that such methods may trivially fail the ''universality'' criterion.  However, it is more appropriate to consider that such methods fail Arrow's definition of an aggregation rule (or that of a function whose domain consists of preference profiles), if preference orderings cannot uniquely translate into a ballot.</ref> Methods which do not, often called "rated" or "cardinal" (as opposed to "ranked", "ordinal", or "preferential") electoral system, can be viewed as using information that only cardinal utility can convey. In that case, it is not surprising if some of them satisfy all of Arrow's conditions that are reformulated.{{refn| However, a modified version of Arrow's theorem may still apply to such methods (e.g., {{cite book|last1=Brams|last2=Fishburn |editor-last1=Arrow|editor-first1=Kenneth J.|editor-last2=Sen |editor-first2=Amartya K.|editor-link2=Amartya Sen|editor-last3=Suzumura|editor-first3=Kōtarō|editor-link3=Kotaro Suzumura |title=Handbook of social choice and welfare|volume=1| year=2002 | publisher=Elsevier | location=Amsterdam, Netherlands | isbn=978-0-444-82914-6 | url=https://books.google.com/books?id=rh10cOpltLsC |chapter=Chapter 4 |at=Theorem 4.2 framework |ref=ArrowSenSuzumura2002 }}}}
Range voting is such a method.<ref name="ns12">{{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><ref name =warrensmith>
{{cite web
 |first=Warren D.
 |last=Smith 
 |url=http://rangevoting.org/ArrowThm.html
 |title=How can range voting accomplish the impossible?
 |display-authors=0
 |access-date=2016-02-16
}}</ref>
Whether such a claim is correct depends on how each condition is reformulated.{{refn
| No voting method that nontrivially uses cardinal utility satisfies Arrow's IIA (in which preference profiles are replaced by lists of ballots or lists of utilities). For this reason, a weakened notion of IIA is proposed (e.g., {{harvtxt|Sen|1979|page=129}}). The notion requires that the social ranking of two alternatives depend only on the ''levels of utility'' attained by individuals at the two alternatives. (More formally, a ''social welfare functional'' <math>F</math> is a function that maps each list <math>u=(u_1, \ldots, u_n)</math> of utility functions into a social preference. <math>F</math> satisfies ''IIA (for social welfare functionals)'' if for all lists <math>u, u'</math> and for all alternatives <math>x, y</math>, if <math>u_i(x)=u'_i(x)</math> and <math>u_i(y)=u'_i(y)</math> for all <math>i</math>, then <math>xF(u)y \text{ iff } xF(u')y</math>.) Many cardinal voting methods (including range voting) satisfy the weakened version of IIA.
}} Other rated electoral system which pass certain generalizations of Arrow's criteria include [[Social:Approval voting|approval voting]] and majority judgment. Note that Arrow's theorem does not apply to single-winner methods such as these, but [[Gibbard's theorem]] still does: no non-defective electoral system is fully strategy-free, so the informal dictum that "no electoral system is perfect" still has a mathematical basis.<ref>{{Cite book|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)|last=Poundstone|first=William|date=2009-02-17|publisher=Macmillan|isbn=9780809048922|language=en}}</ref>

Finally, though not an approach investigating some kind of rules, there is a criticism by [[Biography:James M. Buchanan|James M. Buchanan]], Charles Plott, and others. It argues that it is silly to think that there might be ''social'' preferences that are analogous to ''individual'' preferences.<ref>{{Cite book|url=https://books.google.com/books?id=3TmW5ahsMLIC|title=Welfare Economics and Social Choice Theory|last1=Feldman|first1=Allan M.|last2=Serrano|first2=Roberto|year=2006|publisher=Springer Science & Business Media|isbn=9780387293684|language=en|quote=It is quite silly in the first place to think that there might be social preferences that are analogous to individual preferences. It is nonsense to talk about social preferences since society itself is nothing more than a collection of individuals, each with his own interests. ... this first reaction to Arrow's Theorem is logically attractive, but it can lead to varieties of nihilism that are unappealing to some people, including us.}}</ref> Arrow (1963, Chapter 8)<ref>{{Cite book|chapter-url=https://books.google.com/books?id=Lo2uCECV__8C|title=Social Choice and Individual Values|last=Arrow|first=Kenneth Joseph|year=1963|publisher=Yale University Press|isbn=978-0300013641|pages=103–109|chapter=Chapter VIII Notes on the Theory of Social Choice, Section III. What Is the Problem of Social Choice?|quote=these criticisms are based on misunderstandings of my position}}</ref> answers this sort of criticism seen in the early period, which come at least partly from misunderstanding.

== See also ==
* [[Condorcet paradox]]
* [[Gibbard's theorem]]
* [[Finance:Holmström's theorem|Holmström's theorem]]
* [[Finance:Market failure|Market failure]]
* Voting paradox
* [[Social:Comparison of electoral systems|Comparison of electoral systems]]

== References ==
{{reflist|30em}}

== Further reading ==
* {{cite book|last1=Campbell|first1=D. E. |editor-last1=Arrow|editor-first1=Kenneth J.|editor-last2=Sen |editor-first2=Amartya K.|editor-link2=Amartya Sen|editor-last3=Suzumura|editor-first3=Kōtarō|editor-link3=Kotaro Suzumura |title=Handbook of social choice and welfare|volume=1| year=2002 | publisher=Elsevier | location=Amsterdam, Netherlands | isbn=978-0-444-82914-6 | url=https://books.google.com/books?id=rh10cOpltLsC |chapter=Impossibility theorems in the Arrovian framework |pages=35–94 |ref=ArrowSenSuzumura2002 }} Surveys many of approaches discussed in #Alternatives based on functions of preference profiles.
* {{cite journal|first=Valentino|last=Dardanoni|title= A pedagogical proof of Arrow's Impossibility Theorem|journal=Social Choice and Welfare|volume=18|issue=1|year=2001|pages=107–112|jstor=41106398|doi=10.1007/s003550000062|s2cid=7589377|url=https://escholarship.org/content/qt96n108ts/qt96n108ts.pdf?t=li5b40}} [http://repositories.cdlib.org/ucsdecon/1999-25/ preprint].
* {{cite journal | first = Paul | last = Hansen | title = Another Graphical Proof of Arrow's Impossibility Theorem | doi=10.1080/00220480209595188 | year=2002 | volume=33 | issue=3 | journal= The Journal of Economic Education | pages= 217–235| s2cid = 145127710 }}
* {{cite book|title=The Mathematics of Behavior|first=Earl|last=Hunt|publisher=Cambridge University Press|year=2007|isbn=9780521850124|url=http://www.cambridge.org/9780521850124}}. The chapter "Defining Rationality: Personal and Group Decision Making" has a detailed discussion of the Arrow Theorem, with proof.
* {{cite book|first=Harold W.|last=Lewis|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 |title=Collective choice and social welfare |year=1979 |publisher=North-Holland |location=Amsterdam |isbn=978-0-444-85127-7}}
* {{cite book |last=Skala |first=Heinz J. |chapter=What Does Arrow's Impossibility Theorem Tell Us? |editor-first=G. |editor-last=Eberlein |editor2-first=H. A. |editor2-last=Berghel |title=Theory and Decision : Essays in Honor of Werner Leinfellner |publisher=Springer |year=2012 |isbn=978-94-009-3895-3 |pages=273–286 |chapter-url=https://books.google.com/books?id=Xrp9CAAAQBAJ&pg=PA273 }}
* {{cite journal | first1 = Pingzhong | last1 = Tang | first2 = Fangzhen | last2 = Lin | title = Computer-aided Proofs of Arrow's and Other Impossibility Theorems | doi=10.1016/j.artint.2009.02.005 | year=2009 | volume=173 | issue = 11 | journal=Artificial Intelligence | pages=1041–1053| doi-access = free }}

== External links ==
*{{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}}

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

{{Sourceattribution|Arrow's impossibility theorem}}