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| {{For|Abel's impossibility theorem|Abel–Ruffini theorem}}
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| In [[social choice theory]], '''Arrow’s impossibility theorem''', the '''General Possibility Theorem''', or '''Arrow’s paradox''', states that, when voters have three or more distinct alternatives (options), no [[rank order]] [[voting system]] can convert the '''ranked preferences''' of individuals into a community-wide (complete and transitive) ranking while also meeting a specific set of criteria. These criteria are called ''[[unrestricted domain]]'', ''[[non-dictatorship]]'', ''[[Pareto efficiency]]'', and ''[[independence of irrelevant alternatives]]''. The theorem is often cited in discussions of election theory as it is further interpreted by the [[Gibbard–Satterthwaite theorem]].
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| The theorem is named after economist [[Kenneth Arrow]], who demonstrated the theorem in his Ph.D. thesis and popularized it in his 1951 book ''[[Social Choice and Individual Values]]''. The original paper was titled "A Difficulty in the Concept of Social Welfare".<ref>Arrow, K.J., "[http://gatton.uky.edu/Faculty/hoytw/751/articles/arrow.pdf A Difficulty in the Concept of Social Welfare]", ''[[Journal of Political Economy]]'' '''58'''(4) (August, 1950), pp. 328–346.</ref>
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| In short, the theorem states that no rank-order voting system can be designed that satisfies these three "fairness" criteria:
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| * If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
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| * 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).
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| * There is no "dictator": no single voter possesses the power to always determine the group's preference.
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| Voting systems that use [[cardinal utility]] (which conveys more information than rank orders; see the [[#Rated_voting_systems_and_other_approaches|subsection]] discussing the cardinal utility approach to overcoming the negative conclusion) are not covered by the theorem.<ref>[http://electology.org/interview-with-dr-kenneth-arrow/ Interview with Dr. Kenneth Arrow]: "''CES:'' Now, you mention that your theorem applies to preferential systems or ranking systems. ''Dr. Arrow:'' Yes ''CES:'' But the system that you’re just referring to, Approval Voting, falls within a class called cardinal systems. So not within ranking systems. ''Dr. Arrow:'' And as I said, that in effect implies more information.</ref> The theorem can also be sidestepped by weakening the notion of independence. Arrow 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 Identity of the Indiscernables demanded then the excision of cardinal utility from our thought patterns." Arrow (1967), as quoted on [http://books.google.com/books?id=7ECXDjlCpB0C&pg=PA33 p. 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.
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| 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.{{#tag:ref | Suzumura, 2002,<ref name = handbook02>{{Cite book | author=Suzumura, Kōtarō; Arrow, Kenneth Joseph; Sen, Amartya Kumar | authorlink= | coauthors= | title=Handbook of social choice and welfare, vol 1 | year=2002 | publisher=Elsevier | location=Amsterdam, Netherlands | isbn=978-0-444-82914-6 | pages=}}</ref> Introduction, page 10.}}
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| == Statement of the theorem ==
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| The need to aggregate [[preference]]s occurs in many disciplines: in [[welfare economics]], where one attempts to find an economic outcome which would be acceptable and stable; in [[decision theory]], where a person has to make a rational choice based on several criteria; and most naturally in [[voting systems]], which are mechanisms for extracting a decision from a multitude of voters' preferences.
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| 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 [[Ranked voting systems|ranked voting 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. The theorem considers the following properties, assumed to be reasonable requirements of a fair voting method:
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| ; [[Non-dictatorship]]: The social welfare function should account for the wishes of multiple voters. It cannot simply mimic the preferences of a single voter.
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| ; [[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:
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| :* It must do so in a manner that results in a complete ranking of preferences for society.
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| :* It must [[Deterministic system (mathematics)|deterministically]] provide the same ranking each time voters' preferences are presented the same way.
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| ; [[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 (''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. (See Remarks below.)
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| ; Positive association of social and individual values: (or '''[[monotonicity criterion|monotonicity]]''') 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''.
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| ; 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.
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| 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 at once.
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| A later (1963) version of Arrow's theorem can be obtained by replacing the monotonicity and non-imposition criteria with:
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| ; [[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.
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| The later version of this theorem is stronger—has weaker conditions—since monotonicity, non-imposition, and independence of irrelevant alternatives together imply Pareto efficiency, whereas Pareto efficiency and independence of irrelevant alternatives together do not imply monotonicity. (Incidentally, Pareto efficiency on its own implies non-imposition.)
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| '''Remarks on IIA'''
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| * The IIA condition can be justified for three reasons ([[Andreu Mas-Colell|Mas-Colell]], Whinston, and Green, 1995, page 794): (i) normative (irrelevant alternatives should not matter), (ii) practical (use of minimal information), and (iii) 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.
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| * Arrow's death-of-a-candidate example (1963, page 26) 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) or (acb, bca) or (acb, cba) or (abc, cba).
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| == Formal statement of the theorem ==
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| Let <math> \mathrm{A} </math> be a set of '''outcomes''', <math> \mathrm{N} </math> a number of '''voters''' or '''decision criteria'''. We shall denote the set of all full [[total order|linear orderings]] of <math> \mathrm{A} </math> by <math> \mathrm{L(A)} </math>.
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| A (strict) '''social welfare function''' ('''preference aggregation rule''') is a function
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| <math> F : \mathrm{L(A)}^N \to \mathrm{L(A)} </math>
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| which aggregates voters' preferences into a single preference order on <math> \mathrm{A} </math>.<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>
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| The <math> \mathrm{N} </math>-[[tuple]] <math> (R_1, \ldots, R_N) </math> of voters' preferences is called a ''preference profile''. In its strongest and simplest form, Arrow's impossibility theorem states that whenever the set <math> \mathrm{A} </math> of possible alternatives has more than 2 elements, then the following three conditions become incompatible:
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| ; '''unanimity''', or '''Pareto efficiency''': If alternative '''a''' is ranked above '''b''' for all orderings <math> R_1 , \ldots, R_N </math>, then '''a''' is ranked higher than '''b''' by <math> F(R_1, R_2, \ldots, R_N) </math>. (Note that unanimity implies non-imposition).
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| ; non-dictatorship: There is no individual ''i'' whose preferences always prevail. That is, there is no <math> i \in \{1, \ldots,N\} </math> such that {{nobreak|<math> \forall (R_1, \ldots, R_N) \in \mathrm{L(A)}^N, \quad F(R_1,R_2, \ldots, R_N) = R_i</math>.}}
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| ; '''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 ''i'', alternatives '''a''' and '''b''' have the same order in <math>R_i</math> as in <math>S_i</math>, alternatives '''a''' and '''b''' have the same order in <math> F(R_1,R_2, \ldots, R_N)</math> as in <math> F(S_1,S_2, \ldots, S_N) </math>.
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| == Informal proof ==
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| Based on two proofs<ref>[http://ideas.repec.org/p/cwl/cwldpp/1123r3.html Three Brief Proofs of Arrow’s Impossibility Theorem]</ref><ref>Yu, Ning Neil (2012) [https://sites.google.com/site/neilningyu/publications/doc/preprint%203%20-%20economic%20theory.pdf?attredirects=0 ''A One-shot Proof of Arrow’s Impossibility Theorem'']</ref> appearing in [[Economic Theory (journal)|Economic Theory]]. For simplicity we have presented all rankings as if ties are impossible. A complete proof taking possible ties into account is not essentially different from the one below, 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.
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| We will prove that any social choice system respecting unrestricted domain, unanimity, and independence of irrelevant alternatives (<tt>IIA</tt>) 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 [[Non-dictatorship|dictator]].
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| === Part One: There is a "pivotal" voter for B over A ===
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| [[File:Diagram for part one of Arrow's Impossibility Theorem.svg|thumb|right|300px|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'''.]]
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| Say there are three choices for society, call them '''A''', '''B''', and '''C'''. Suppose first that everyone prefers option '''B''' the least. That is, everyone prefers every other option to '''B'''. By unanimity, society must prefer every option to '''B'''. Specifically, society prefers '''A''' and '''C''' to '''B'''. Call this situation ''Profile 0''.
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| 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.
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| Since '''B''' eventually moves to the top of the societal preference, there must be some profile, number ''k'', for which '''B''' moves '''above A''' in the societal rank. We call the voter 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.
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| Also note that by <tt>IIA</tt> 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.
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| === Part Two: The pivotal voter for B over A is a dictator for B over C ===
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| In this part of the argument we refer to voter ''k'', the pivotal voter for '''B''' over '''A''', as ''Pivotal Voter'' for simplicity. We will show that 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 knowledge|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.
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| 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:
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| [[File:Diagram for part two of Arrow's Impossibility Theorem.svg|thumb|right|300px|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.|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.]]
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| * Every voter in Segment One ranks '''B''' above '''C''' and '''C''' above '''A'''.
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| * Pivotal Voter ranks '''A''' above '''B''' and '''B''' above '''C'''.
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| * Every voter in Segment Two ranks '''A''' above '''B''' and '''B''' above '''C'''.
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| 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.
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| Now suppose that Pivotal Voter moves '''B''' above '''A''', but keeps '''C''' in the same position and imagine that any number (or all!) of the other voters change their ballots to move '''C''' above '''B''', 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 <tt>IIA</tt> 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'''. By <tt>IIA</tt> this conclusion holds independently of how '''A''' is positioned on the ballots, so Pivotal Voter is a dictator for '''B''' over '''C'''.
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| === Part Three: There can be at most one dictator ===
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| [[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.]]
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| 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 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
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| ''k''<sub>B/C</sub> ≤ k<sub>B/A</sub> ≤ ''k''<sub>C/B</sub>.
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| Now repeating the entire argument above with '''B''' and '''C''' switched, we also have
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| ''k''<sub>C/B</sub> ≤ ''k''<sub>B/C</sub>.
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| Therefore we have
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| ''k''<sub>B/C</sub> = k<sub>B/A</sub> = ''k''<sub>C/B</sub>
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| 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.
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| == Interpretations of the theorem ==
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| 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 isn't flawed is a dictatorship"''. 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.
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| Arrow did use the term "fair" to refer to his criteria. Indeed, [[Pareto efficiency]], as well as the demand for non-imposition, seems acceptable to most people.{{Cn|date=January 2014}}
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| Various theorists have suggested [[Independence of irrelevant alternatives#Local independence|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 [[voting system]]s.
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| Advocates of this position point out that failure of the standard IIA criterion is trivially implied by the possibility of cyclic preferences.
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| If voters cast ballots as follows:
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| *1 vote for A > B > C
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| *1 vote for B > C > A
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| *1 vote for C > A > B
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| 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.
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| So, what Arrow's theorem really shows is that any majority-wins voting system is a non-trivial game, and that [[game theory]] should be used to predict the outcome of most voting mechanisms.{{#tag:ref |This does not mean various normative criteria will be satisfied if we use equilibrium concepts in game theory.
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| Indeed, the mapping from profiles to equilibrium outcomes defines a social choice rule, whose performance can be investigated by social choice theory.
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| See Austen-Smith and Banks (1999<ref name=austensmith-b99>{{Cite book | author = Austen-Smith, David; Banks, Jeffrey S. | title = [http://books.google.com/books?id=nxXDn3nPxIAC&q=%22nakamura+number%22 Positive political theory I: Collective preference] | year = 1999 | publisher = University of Michigan Press | location = Ann Arbor | isbn = 978-0-472-08721-1 | pages = }}</ref>), Section 7.2.}}
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| 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.
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| '''Remark: Scalar rankings from a vector of attributes and the IIA property'''.
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| 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 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).
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| == Other possibilities ==
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| In an attempt to escape from the negative conclusion of Arrow's theorem, social choice theorists have investigated various possibilities ("ways out").
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| These investigations can be divided into the following two:
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| *those investigating functions whose domain, like that of Arrow's social welfare functions, consists of profiles of preferences;
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| *those investigating other kinds of rules.
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| === Approaches investigating functions of preference profiles ===
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| This section includes approaches that deal with
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| *aggregation rules (functions that map each preference profile into a social preference), and
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| *other functions, such as functions that map each preference profile into an alternative.
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| Since these two approaches often overlap, we discuss them at the same time.
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| What is characteristic of these approaches is that they investigate various possibilities by eliminating or weakening or replacing
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| one or more conditions (criteria) that Arrow imposed.
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| ==== Infinitely many individuals ====
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| Several theorists (e.g., Kirman and Sondermann, 1972<ref name = kirman-s72>{{cite doi | 10.1016/0022-0531(72)90106-8}}</ref>) point out that when one drops the assumption that there are only finitely many individuals,
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| one can find aggregation rules that satisfy all of Arrow's other conditions.
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| However, such aggregation rules are practically of limited interest, since they are based on [[ultrafilter]]s, highly nonconstructive mathematical objects.
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| In particular, Kirman and Sondermann argue that there is an "invisible dictator" behind such a rule.
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| Mihara (1997,<ref name = mihara97>{{cite doi |10.1007/s001990050157}}</ref> 1999<ref>{{cite doi |10.1016/S0304-4068(98)00061-5}}</ref>)
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| 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
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| (see [[Rice_theorem#An_analogue_of_Rice.27s_theorem_for_recursive_sets|Rice's theorem]]).</ref>
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| These results can be seen to establish the robustness of Arrow's theorem.{{#tag:ref |
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| See Taylor (2005,<ref>{{Cite book | last=Taylor | first=Alan D. | coauthors= | title=Social choice and the mathematics of manipulation | year=2005 | publisher=Cambridge University Press | location=New York | isbn=0-521-00883-2 | pages=}}</ref> Chapter 6)
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| for a concise discussion of social choice for infinite societies.}}
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| ==== Limiting the number of alternatives ====
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| When there are only two alternatives to choose from, [[May's theorem]] shows that only simple majority rule satisfies a certain set of criteria
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| (e.g., equal treatment of individuals and of alternatives; increased support for a winning alternative should not make it into a losing one).
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| On the other hand, when there are at least three alternatives, Arrow's theorem points out the difficulty of collective decision making.
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| Why is there such a sharp difference between the case of less than three alternatives and that of at least three alternatives?
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| ''Nakamura's theorem'' (about the core of simple games) gives an answer more generally.
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| It establishes that if the number of alternatives is less than a certain integer called the '''[[Nakamura number]]''',
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| then the rule in question will identify "best" alternatives without any problem;
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| if the number of alternatives is greater or equal to the Nakamura number, then the rule will not always work,
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| 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.
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| Since the Nakamura number of majority rule is 3 (except the case of four individuals), one can conclude from Nakamura's theorem
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| that majority rule can deal with up to two alternatives rationally.
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| Some super-majority rules (such as those requiring 2/3 of the votes) can have a Nakamura number greater than 3,
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| but such rules violate other conditions given by Arrow.{{#tag:ref |
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| Austen-Smith and Banks (1999,<ref name=austensmith-b99 /> Chapter 3) gives a detailed discussion of the approach trying to limit the number of alternatives.}}
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| '''Remark'''. 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.
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| ==== Domain restrictions ====
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| Another approach is relaxing the universality condition, which means restricting the [[domain (mathematics)|domain]] of aggregation rules.
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| The best-known result along this line assumes "single peaked" preferences.
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| [[Duncan Black]] has shown that if there is only one dimension on which every individual has a "single-peaked" preference,
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| then all of Arrow's conditions are met by [[majority rule]].
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| Suppose that there is some predetermined linear ordering of the alternative set.
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| 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.
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| If the domain is restricted to profiles in which every individual has a single peaked preference with respect to the linear ordering,
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| then ''simple'' (<ref name=austensmith-b99/>) aggregation rules, which includes majority rule, have an ''acyclic'' (defined below) social preference,
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| hence "best" alternatives.{{#tag:ref |
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| 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 |last=Campbell |first=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 }} }}
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| In particular, when there are odd number of individuals, then the social preference becomes transitive, and the socially "best" alternative is equal to
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| the median of all the peaks of the individuals (Black's [[median voter theorem]]<ref>{{Cite book | author = Black, Duncan | title = The theory of committees and elections | year = 1968 | publisher = University Press | location = Cambridge, Eng. | isbn = 0-89838-189-4 | pages = }}</ref>).
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| Under single-peaked preferences, the majority rule is in some respects the most natural voting mechanism.
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| One can define the notion of "single-peaked" preferences on higher dimensional sets of alternatives.
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| However, one can identify the "median" of the peaks only in exceptional cases.
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| Instead, we typically have the destructive situation suggested by McKelvey's Chaos Theorem
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| (1976):<ref>{{cite doi |10.1016/0022-0531(76)90040-5}}</ref>
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| for any ''x'' and ''y'', one can find a sequence of alternatives such that
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| ''x'' is beaten by <math>x_1</math> by a majority, <math>x_1</math> by <math>x_2</math>, <math>\ldots</math>,
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| <math>x_k</math> by ''y''.
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| | |
| ==== Relaxing transitivity ====
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| By relaxing the transitivity of social preferences, we can find aggregation rules that satisfy Arrow's other conditions.
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| If we impose ''neutrality'' (equal treatment of alternatives) on such rules, however, there exists an individual who has a "veto".
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| So the possibility provided by this approach is also very limited.
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| First, suppose that a social preference is ''quasi-transitive'' (instead of transitive);
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| this means that the strict preference <math>\succ</math> ("better than") is transitive:
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| if <math>x \succ y</math> and <math>y \succ z</math>, then <math>x \succ z</math>.
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| Then, there do exist non-dictatorial aggregation rules satisfying Arrow's conditions, but such rules are ''oligarchic'' (Gibbard, 1969).
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| This means that there exists a coalition L such that
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| L is ''decisive'' (if every member in L prefers x to y, then the society prefers x to y), and
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| each member in L ''has a veto'' (if she prefers x to y, then the society cannot prefer y to x).
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| Second, suppose that a social preference is ''acyclic'' (instead of transitive):
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| there does not exist alternatives <math>x_1, \ldots, x_k</math> that form a ''cycle'' (<math>x_1 \succ x_2</math>, <math>x_2 \succ x_3</math>, <math>\ldots</math>, <math>x_{k-1} \succ x_k</math>, <math>x_k \succ x_1</math>).
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| Then, provided that there are at least as many alternatives as individuals, an aggregation rule satisfying Arrow's other conditions
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| is ''collegial'' (Brown, 1975<ref>{{cite journal |last=Brown |first=D. J. |year=1975 |title=Aggregation of Preferences |journal=[[Quarterly Journal of Economics]] |volume=89 |issue=3 |pages=456–469 |jstor=1885263 }}</ref>).
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| This means that there are individuals who belong to the intersection ("collegium") of all decisive coalitions.
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| If there is someone who has a veto, then he belongs to the collegium.
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| If the rule is assumed to be neutral, then it does have someone who has a veto.<ref name=austensmith-b99/>
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| | |
| Finally, Brown's theorem left open the case of acyclic social preferences where the number of alternatives is less than the number of individuals.
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| One can give a definite answer for that case using the ''Nakamura number''. See [[#Limiting the number of alternatives]].
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| ==== Relaxing IIA ====
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| There are numerous examples of aggregation rules satisfying Arrow's conditions except IIA.
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| The [[Borda count|Borda rule]] is one of them.
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| These rules, however, are susceptible to ''strategic manipulation'' by individuals
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| (Blair and Muller, 1983<ref>{{cite doi |10.1016/0022-0531(83)90092-3}}</ref>).
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| | |
| See also [[#Interpretations of the theorem|Interpretations of the theorem]] above.
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| | |
| ==== Relaxing the Pareto criterion ====
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| Wilson (1972) shows that if an aggregation rule is non-imposed and non-null, then there is either a dictator or an inverse dictator,
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| provided that Arrow's conditions other than Pareto are also satisfied.
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| Here, an ''inverse dictator'' is an individual i such that whenever i prefers x to y, then the society prefers y to x.
| |
| | |
| '''Remark'''. [[Amartya Sen]] offered both relaxation of transitivity and removal of the Pareto principle.<ref>{{cite journal |last=Sen |first=Amartya |year=1979 |title=Personal Utilities and Public Judgements: Or What's Wrong With Welfare Economics |journal=[[Economic Journal]] |volume=89 |issue=355 |pages=537–588 |jstor=2231867 }}</ref>
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| He demonstrated another interesting impossibility result, known as the "impossibility of the Paretian Liberal". (See [[liberal paradox]] for details). Sen went on to argue that this demonstrates the futility of demanding Pareto optimality in relation to voting mechanisms.
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| | |
| ==== Social choice instead of social preference ====
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| In social decision making, to rank all alternatives is not usually a goal. It often suffices to find some alternative.
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| The approach focusing on choosing an alternative investigates either ''social choice functions'' (functions that map each preference profile into an alternative)
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| 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.
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| | |
| As for social choice rules, we should assume there is a social preference behind them.
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| That is, we should regard a rule as choosing the maximal elements ("best" alternatives) of some social preference.
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| The set of maximal elements of a social preference is called the ''core''.
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| Conditions for existence of an alternative in the core have been investigated in two approaches.
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| The first approach assumes that preferences are at least ''acyclic'' (which is necessary and sufficient for the preferences to have a maximal element
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| on any ''finite'' subset). For this reason, it is closely related to [[#Relaxing transitivity]].
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| The second approach drops the assumption of acyclic preferences.
| |
| Kumabe and Mihara (2011<ref name=kumabe-m10geb>{{cite doi | 10.1016/j.geb.2010.06.008}}</ref>) adopt this approach.
| |
| They make a more direct assumption that individual preferences have maximal elements,
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| and examine conditions for the social preference to have a maximal element.
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| See [[Nakamura number]] for details of these two approaches.
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| | |
| === Rated voting systems and other approaches ===
| |
| | |
| Arrow's framework assumes that individual and social preferences are "orderings" (i.e., satisfy completeness and transitivity) on the set of alternatives.
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| 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.
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| For instance, having ordinal utilities of 4, 3, 2, 1 for alternatives a, b, c, d, respectively, is the same as
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| having 1000, 100.01, 100, 0, which in turn is the same as having 99, 98, 1, .997.
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| They all represent the ordering in which a is preferred to b to c to d.
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| 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,
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| is not common in contemporary economics.
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| However, once one adopts that approach, one can take [[intensity of preference|intensities of preferences]] into consideration, or
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| one can compare (i) gains and losses of utility or (ii) levels of utility,
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| across different individuals.
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| In particular, Harsanyi (1955) gives a justification of [[utilitarianism]] (which evaluates alternatives in terms of the sum of individual utilities), originating from [[Jeremy Bentham]].
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| Hammond (1976) gives a justification of the [[Maximin_principle#Maximin_in_philosophy|maximin principle]] (which evaluates alternatives in terms of the utility of the worst-off individual), originating from [[John Rawls]].
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| 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
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| methods fail Arrow's definition of an aggregation rule (or that of a function whose domain consists of preference profiles),
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| if preference orderings cannot uniquely translate into a ballot.</ref>
| |
| Methods which don't, often called "rated" or "cardinal" (as opposed to "ranked", "ordinal", or "preferential") voting systems, 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.{{#tag:ref | However, a modified version of
| |
| Arrow's theorem may still apply to such methods (e.g., Brams and Fishburn, 2002,<ref name = handbook02 /> Chapter 4, Theorem 4.2).}}
| |
| [[Range voting]] is such a method.<ref name =warrensmith>{{cite web
| |
| |author=Warren D. Smith, et al.
| |
| |url=http://rangevoting.org/ArrowThm.html
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| |title=How can range voting accomplish the impossible?
| |
| }}</ref><ref name =ns12>New Scientist 12 April 2008 pages 30-33</ref>
| |
| Whether such a claim is correct depends on how each condition is reformulated.
| |
| {{#tag:ref |
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| 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., Sen, 1979,<ref name= sen79>{{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 }}</ref> page 129).
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| The notion requires that the social ranking of two alternatives depend only on the ''levels of utility'' attained by individuals at the two alternatives.
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| (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.
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| <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>.)
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| Many cardinal voting methods (including Range voting) satisfy the weakened version of IIA.
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| }}
| |
| Other rated voting systems which pass certain generalizations of Arrow's criteria include [[Approval voting]] and [[Majority Judgment]]. Note that although Arrow's theorem does not apply to such methods, the [[Gibbard–Satterthwaite theorem]] still does: no system is fully strategy-free, so the informal dictum that "no voting system is perfect" still has a mathematical basis.
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| Finally, though not an approach investigating some kind of rules, there is a criticism by [[James M. Buchanan]] and others.
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| It argues that it is silly to think that there might be ''social'' preferences that are analogous to ''individual'' preferences.
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| Arrow (1963, Chapter 8) answers this sort of criticism seen in the early period, which come at least partly from misunderstanding.
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| == See also ==
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| *[[Holmström's theorem]]
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| *[[Voting paradox]]
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| *[[Market failure]]
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| == Notes ==
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| {{reflist|30em}}
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| == References ==
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| *Campbell, D.E. and Kelly, J.S. (2002) Impossibility theorems in the Arrovian framework, in Handbook of social choice and welfare (ed. by Kenneth J. Arrow, Amartya K. Sen and Kotaro Suzumura), volume 1, pages 35–94, Elsevier. Surveys many of approaches discussed in [[#Approaches investigating functions of preference profiles]].
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| * ''The Mathematics of Behavior'' by Earl Hunt, Cambridge University Press, 2007. The chapter "Defining Rationality: Personal and Group Decision Making" has a detailed discussion of the Arrow Theorem, with proof. [http://www.cambridge.org/9780521850124 URL to CUP information on this book]
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| * ''Why flip a coin? : the art and science of good decisions'' by Harold W. Lewis, John Wiley, 1997. Gives explicit examples of preference rankings and apparently anomalous results under different voting systems. States but does not prove Arrow's theorem. ISBN 0-471-29645-7
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| *[[Amartya Sen|Sen, A. K.]] (1979) “Personal utilities and public judgements: or what's wrong with welfare economics?” [[The Economic Journal]], 89, 537-558, arguing that Arrow's theorem was wrong because it did not incorporate non-utility information and the utility information it did allow was impoverished http://www.jstor.org/stable/2231867
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| * Yu, Ning Neil (2012) ''A one-shot proof of Arrow's theorem. ''[[Economic Theory (journal)|Economic Theory]], volume 50, issue 2, pages 523-525, Springer. http://link.springer.com/article/10.1007%2Fs00199-012-0693-3
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| == External links ==
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| * [http://ideas.repec.org/p/cwl/cwldpp/1123r3.html Three Brief Proofs of Arrow’s Impossibility Theorem]
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| * [http://repositories.cdlib.org/ucsdecon/1999-25/ A Pedagogical Proof of Arrow’s Impossibility Theorem]
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| * [http://www.jstor.org/stable/1183438 Another Graphical Proof of Arrow’s Impossibility Theorem]
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| * [http://www.springerlink.com/content/v00202437u066604/ A One-Shot Proof of Arrow’s Impossibility Theorem]
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| * [http://www.sciencedirect.com/science/article/pii/S0004370209000320/ Computer-aided Proofs of Arrow's and Other Impossibility Theorems]
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| {{Game theory}}
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| {{DEFAULTSORT:Arrow's Impossibility Theorem}}
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| [[Category:Voting systems]]
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| [[Category:Economics theorems]]
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