Arrow’s Theorem

First published Mon Oct 13, 2014; substantive revision Tue Nov 26, 2019

Kenneth Arrow’s “impossibility” theorem—or“general possibility” theorem, as he calledit—answers a very basic question in the theory of collectivedecision-making. Say there are some alternatives to choose among. Theycould be policies, public projects, candidates in an election,distributions of income and labour requirements among the members of asociety, or just about anything else. There are some people whosepreferences will inform this choice, and the question is: whichprocedures are there for deriving, from what is known or can be foundout about their preferences, a collective or “social”ordering of the alternatives from better to worse? The answer isstartling. Arrow’s theorem says there are no such procedureswhatsoever—none, anyway, that satisfy certain apparently quitereasonable assumptions concerning the autonomy of the people and therationality of their preferences. The technical framework in whichArrow gave the question of social orderings a precise sense and itsrigorous answer is now widely used for studying problems in welfareeconomics. The impossibility theorem itself set much of the agenda forcontemporary social choice theory. Arrow accomplished this while stilla graduate student. In 1972, he received the Nobel Prize in economicsfor his contributions.

1. The Will of the People?

Some of the trouble with social orderings is visible in a simplebut important example. Say there are threealternatives \(A\), \(B\) and \(C\) to chooseamong. There is a group of three people 1, 2 and 3 whose preferencesare to inform this choice, and they are asked to rank the alternativesby their own lights from better to worse. Their individual preferenceorderings turn out to be:

ABC
BCA
CAB

That is, person 1 prefers \(A\) to \(B\),prefers \(B\) to \(C\), and prefers \(A\)to \(C\); person 2 prefers \(B\) to \(C\), and soon. Now, we might hope somehow to arrive at a single“social” ordering of the alternatives that reflects thepreferences of all three. Then we could choose whichever alternativeis, socially, best—or, if there is a tie for first place, wecould choose some alternative that is as good as any other. Suppose,taking the alternatives pair by pair, we put the matter to a vote: wecount one alternative associally preferred to another ifthere are more voters who prefer it than there are who prefer theother one. We determine in this way that \(A\) is sociallypreferred to \(B\), since two voters (1 and 3) prefer \(A\)to \(B\), but only one (voter 2) prefers \(B\)to \(A\). Similarly, there is a social preference for \(B\)to \(C\). We might therefore expect to find that \(A\) issocially preferred to \(C\). By this reckoning, though, it isjust the other way around, since there are two voters whoprefer \(C\) to \(A\). We do not have asocialordering of the alternatives at all. We haveacycle. Starting from any alternative, moving to a sociallypreferred one, and from there to the next, you soon findyourself back where you started.^[1]

This is the “paradox of voting”. Discovered by theMarquis de Condorcet (1785), it shows that possibilities forchoosing rationally can be lost when individual preferences areaggregated into social preferences. Voter 1 has \(A\) at the topof his individual ordering. This voter’s preferences canbemaximized, by choosing \(A\). The preferences of 2 or3 can also be maximized, by choosing instead their maxima, \(B\)or \(C\). Pairwise majority decision doesn’t result in a socialmaximum, though. \(A\) isn’t one because a majority preferssomething else, \(C\). Likewise, \(B\) and \(C\) arenot social maxima. The individual preferences lend themselves tomaximization; but, because they cycle, the social preferences donot.

Are there other aggregation procedures that are better thanpairwise majority decision, or do the different ones have shortcomingsof their own? Condorcet, his contemporary Jean Charles de Borda(1781), and later Charles Dodgson (1844) and Duncan Black (1948),among others, all addressed this question by studying variousprocedures and comparing their properties. Arrow broke new ground bycoming at it from the opposite direction. Starting with variousrequirements that aggregation procedures might be expected to meet, heasked which procedures fill the bill. Among his requirementsisSocial Ordering, which insists that the result ofaggregation is always an ordering of the alternatives, never acycle. After the introduction inSection 2 ofthe technical framework that Arrow set up in order to study socialchoice,Section 3.1 sets out further conditionsthat he imposed. Briefly, these are:Unrestricted Domainwhich says that aggregation procedures must be able to handle anyindividual preferences at all;Weak Pareto, which requiresthem to respect unanimous individualpreferences;Non-Dictatorship, which rules out procedures bywhich social preferences always agree with the strict preferences ofsome one individual; and finallyIndependence of IrrelevantAlternatives, which says that the social comparison among any twogiven alternatives is to depend on individual preferences among onlythat pair. Arrow’s theorem, stated inSection 3.2, tells us that, except in the very simplest of cases, noaggregation procedure whatsoever meets all the requirements.

The tenor of Arrow’s theorem is deeply antithetical to thepolitical ideals of the Enlightenment. It turns out that Condorcet’sparadox is indeed not an isolated anomaly, the failure of one specificvoting method. Rather, it manifests a much wider problem with the veryidea of collecting many individual preferences into one. On the faceof it, anyway, there simply cannot be a common will of all the peopleconcerning collective decisions, that assimilates the tastes andvalues of all the individual men and women who make up a society.

There are some who, following Riker (1982), take Arrow’s theorem toshow that democracy, conceived as government by the will of thepeople, is an incoherent illusion. Others argue that some conditionsof the theorem are unreasonable, and from their point of view theprospects for collective choice look much brighter. After presentingthe theorem itself, this entry will take up some main points ofcritical discussion.Section 4 considers themeaning and scope of Arrow’s conditions.Section 5discusses aggregation procedures that are available when not all ofArrows conditions need be satisfied, or when different underlyingassumptions are made about the nature of socialchoice.Section 6 concludes with an overview ofproposals to study within Arrow’s technical framework certainaggregation problems other than the one that concerned him.

Amartya Sen once expressed regret that the theory of social choicedoes not share with poetry the amiable characteristic of communicatingbefore it is understood (Sen 1986). Arrow’s theorem is not especiallydifficult to understand and much about it is readily communicated, ifnot in poetry, then at least in plain English. Informal presentationsgo only so far, though, and where they stop sometimesmisunderstandings start. This exposition uses a minimum of technicallanguage for the sake of clarity.

2. Arrow’s Framework

The problem of finding an aggregation procedure arises, as Arrowframed it, in connection with some given alternatives between whichthere is a choice is to be made. The nature of these alternativesdepends on the kind of choice problem that is being studied. In thetheory of elections, the alternatives are people who might stand ascandidates in an election. In welfare economics they are differentstates of a society, such as distributions of income and labourrequirements. The alternatives conventionally are referred to usinglower case letters from the end of the alphabetas \(x, y, z, \ldots\); the set of allthese alternatives is \(X\). The people whose tastes and valueswill inform the choice are assumed to be finite in number, and theyare enumerated \(1, \ldots, n\).

Arrow’s problem arises, then, onlyafter some alternativesand people have been fixed. It is for them that an aggregationprocedure is sought. Crucially, though, this problemarisesbefore relevant information about the people’spreferences among the alternatives has been gathered, whether that isby polling or some other method for eliciting or determiningpreferences. The question that Arrow’s theorem answers is, moreprecisely, this: Which procedures are there for arriving at a socialordering of some given alternatives, on the basis of some givenpeople’s preferences among them, no matter what these preferences turnout to be?

In practice, meanwhile, we sometimes must select a procedure formaking social decisions without knowing for which alternatives andpeople it will be used. In recurring elections for some public office,for instance, there is a different slate of candidates each time, anda different population of voters, and we must use the same votingmethod to determine the winner, no matter who the candidates andvoters are and no matter how many of them there happen to be. Suchprocedures are not directly available for study within Arrow’sframework, with its fixed set \(X\) of alternatives and people\(1, \ldots, n\). Arrow’s theorem is still relevant to them,though. It tells us thateven when the alternatives andpeople are held fixed, then still there is no “good”method for deriving social orderings. Now, if there is no good methodfor voting even once, with the particular candidates and voters whoare involved on that occasion, then nor, presumably, is there a goodmethod that can be used repeatedly, with different candidates andvoters each time.

2.1 Individual Preferences

Arrow assumed that social orderings will be derived, if at all,from information about people’s preferences. This information is, inhis framework, merelyordinal. It is the kind of informationthat is implicated in Condorcet’s paradox of voting,inSection 1, where each person ranks thealternatives from better to worse but there is nothing beyond thisabout how strong anybody’s preferences are, or about how thepreferences of one person compare in strength to those of another. Inconfining aggregation procedures to ordinal information, Arrow arguedthat:

[I]t seems to make no sense to add the utility ofone individual, a psychic magnitude in his mind, with the utility ofanother individual. (Arrow 1951 [1963]: 11)

His point was that even if people do havestronger and weaker preferences, and even if the strengths of theirpreferences can somehow be measured and made available as a basis forsocial decisions, nevertheless ordinal information is all thatmatters because preferences are “interpersonallyincomparable”. Intuitively, what this means is that there is nosaying how much more strongly someone must prefer one thing toanother in order to make up for the fact that someone else’spreference is just the other way around. Arrow saw no reason toprovide aggregation procedures with information about the strength of preferencesbecause he thought that they cannot put such information tomeaningful use.

Accordingly, the preferences of individual people are representedin Arrow’s framework by binary relations \(R_{i}\) among thealternatives: \(xR_{i}y\) means that individual \(i\)weaklyprefers alternative \(x\) to alternative \(y\). That is, either\(i\)strictly prefers \(x\) to \(y\), or else \(i\)isindifferent between them, finding them equally good. Eachindividual preference relation \(R_{i}\) is assumed tobeconnected (for all alternatives \(x\) and \(y\), either\(xR_{i}y\), or \(yR_{i}x\), or both) andtransitive (for all\(x\), \(y\) and \(z\), if \(xR_{i}y\) and \(yR_{i}z\), then\(xR_{i}z\)). That these relations have these structural propertieswas, for Arrow, a matter of the “rationality” of thepreferences they represent; for further discussion, see the entries onPreferences and Philosophy of Economics. Connected, transitiverelations are calledweak orderings. They are“weak” in that they allow ties—in this connection,indifference.

Apreference profile is a list \(\langle R_{1}, \ldots,R_{n}\rangle\) of weak orderings of the set \(X\) of alternatives, onefor each of the people \(1, \ldots, n\). The list of threeindividual orderings in the paradox of voting is an example of apreference profile for the alternatives \(A\), \(B\), and \(C\) and people1, 2, and 3. A profile is a representation of the individualpreferences of everybody who will be consulted in the choice among thealternatives. It is in the form of profiles that Arrow’s aggregationprocedures receive information about individual preferences. Often itis convenient to write \(\langle R_{i}\rangle\) instead of \(\langleR_{1}, \ldots, R_{n}\rangle\). Other profiles are written \(\langleR^*_{i}\rangle\), and so on.

In restricting individual inputs to weak orderings of thealternatives, Arrow overlooked the possibility that people could inputinformation about their preferences in the form of ordinal scores orgrades. Graded inputs enable an “escape” from Arrow’simpossibility that is explained inSection5.3. Amartya Sen extended Arrow’s framework to take into accountnot only ordinal information about people’s preferences among pairs ofalternatives, but alsocardinal information about the utilitythey derive from each one. In this way he was able to investigate theconsequences of other assumptions than Arrow’s about the measurabilityand interpersonal comparability of individualpreferences. SeeSection 5.4 and theentrysocial choice theory for detailsand references.

2.2 Multiple Profiles

Arrow required aggregation procedures to derive social orderingsfrom more than just a single profile, representingeveryone’sactual preferences. In his framework they mustreckon with many profiles, representing preferences that thepeoplecould have.

Variety among preferences is the result, in Arrow’s account, of thedifferent standards by which we assess our options. Our preferencesdepend on our “tastes” in personal consumption butimportantly, for social choice, they also depend on our sociallydirected “values”. Now, we are to some extent free to havevarious tastes, values, and preferences; and we are free, also, tohave these independently of one another. Any individual can have arange of preferences, then, and for any given sets of people andalternatives there are many possible preference profiles. One profile\(\langle R_{i}\rangle\) represents the preferences of these peopleamong their alternatives in, if you will, one possible world. Anotherprofile \(\langleR^*_{i}\rangle\) represents preferences of the samepeople, and among the same alternatives, but in another possible worldwhere their tastes and values are different.

Arrow’s rationale for requiring aggregation procedures to handlemany profiles was epistemic. As he framed the question of collectivechoice, a procedure is sought for deriving a social ordering of somegiven alternatives on the basis of some given people’s tastes andvalues. It is sought, though, before it is known just what thesetastes and values happen to be. The variety among profiles to bereckoned with is a measure, in Arrow’s account of the matter, of howmuch is known or assumed about everybody’s preferencesapriori, which is to say before these have been elicited. Whenless is known, there are more profiles from which a social orderingmight have to be derived. When more is known, there are fewer ofthem.

There are other reasons for working with many profiles, even whenpeople’s actual preferences are known fully in advance. Serge Kolm (1996)suggested that counterfactual preferences are relevant when wecome to justify the use of some given procedure.Sensitivity analysis, used to manage uncertainty about errors in theinput, and to determine which information is critical in the sensethat the output turns on it, also requires that procedures handle arange of inputs.

With many profiles in play there can be “interprofile”conditions on aggregation procedures. These coordinate the results ofaggregation at several profiles at once. One such condition that playsa crucial role in Arrow’s theorem isIndependence of IrrelevantAlternatives. It requires that whenever everybody’s preferencesamong two alternatives are in one profile the same as in another, thecollective ordering must also be the same at the two profiles, as faras these alternatives are concerned. There is to be this muchsimilarity among social orderings even as people’s tastes and valueschange.Sections 3.1and4.5 discuss in more detail the meaning ofthis controversial requirement, and the extent to which it isreasonable to impose it on aggregation procedures.

Ian Little raised the following objection in an early discussionof (Arrow 1951):

If tastes change, we may expect a new ordering of all theconceivable states; but we do not require that the difference betweenthe new and the old ordering should bear any particular relation tothe changes of taste which have occurred. We have, so to speak, a newworld and a new order; and we do not demand correspondence between thechange in the world and the change in the order (Little 1952:423–424).

Little apparently agreed with Arrow that there might be a differentsocial ordering were people’s tastes different, but unlike Arrow hethought that it wouldn’t have to be similar to the actual or currentordering in any special way. Little’s objection was taken to supportthe “single profile” approach to social welfare judgmentsof Abram Bergson (1938) and Paul Samuelson (1947), and there was adebate about which approach was best, theirs or Arrow’s. Arguably,what was at issue in this debate was not—or should not havebeen—whether aggregation procedures must handle more than asingle preference profile, but instead whether there should be anycoordination of the output at different profiles. Among others Sen(1977) and Fleurbaey and Mongin (2005) have made this point. If theyare right then the substance of Little’s objection can be accommodatedwithin Arrow’s multi-profile framework simply by not imposing anyinterprofile constraints. Be this as it may, Arrow’s framework isnowadays the dominant one.

2.3 Social Welfare Functions

Sometimes a certain amount is known about everybody’s preferencesbefore these have been elicited. Profiles that are compatible withwhat is known represent preferences that the people could have, andmight turn out actually to have, and it is from these“admissible” profiles that we may hope to derive socialorderings. Technically, adomain, in Arrow’s framework, is aset of admissible profiles, each concerning the same alternatives \(X\)and people \(1, \ldots, n\). Asocial welfare function \(f\)assigns to each profile \(\langle R_{i}\rangle\) in some domain a binaryrelation \(f\langle R_{i}\rangle\) on \(X\). Intuitively, \(f\) is anaggregation procedure and \(f\langle R_{i}\rangle\) represents thesocial preferences that it derives from \(\langle R_{i}\rangle\).Arrow’s social welfare functions are sometimes called“constitutions”.

Arrow incorporated into the notion of a social welfare function thefurther requirement that \(f\langle R_{i}\rangle\) is always a weakordering of the set \(X\) of alternatives. Informally speaking, thismeans that the output of the social welfare function must always be aranking of the alternatives from better to worse, perhaps withties. It may never be a cycle. This requirement will appear here, asit does in other contemporary presentations of Arrow’s theorem, as aseparate condition ofSocial Ordering that social welfarefunctions might be required to meet. SeeSection 3.1. This way, we can consider the consequences of dropping thiscondition without changing any basic parts of the framework.SeeSection 4.2.

Arrow established a convention that is still widely observed ofusing ‘\(R\)’ to denote the social preference derived from“\(\langle R_{i}\rangle\)”. The social welfare function used to derive itis, in his notation, left implicit. One advantage of writing‘\(f\langle R_{i}\rangle\)’ instead of ‘\(R\)’ isthat when we state the conditions of the impossibility theorem, in thenext section, the social welfare function will figure explicitly inthem. This makes it quite clear that what these conditions constrainis thefunctional relationship between individual and socialpreferences. Focusing attention on this was an important innovation ofArrow’s approach.

3. Impossibility

With the conceptual framework now inplace,Section 3.1 sets out the“conditions” or constraints that Arrow imposed on socialwelfare functions, andSection 3.2 states thetheorem itself.Section 4 explains theconditions more fully, discusses reasons that Arrow gave for imposingthem, and considers whether it is proper to do so.

Arrow’s conditions often are called axioms, and his approach issaid to be axiomatic. This might be found misleading. Unlike axioms oflogic or geometry, Arrow’s conditions are not supposed to express moreor less indubitable truths, or to constitute an implicit definition ofthe object of study. Arrow himself took them to be questionable“value judgments” that “express the doctrines ofcitizens’ sovereignty and rationality in a very general form”(Arrow 1951 [1963]: 31). Indeed, as we will seeinSection 4, and as Arrow himself recognized,sometimes it is not even desirable that social welfare functionsshould satisfy all conditions of the impossibility theorem.

Arrow restated the conditions in the second edition ofSocialChoice and Individual Values (Arrow 1963). They appear here inthe canonical form into which they have settled since then.

3.1 These Conditions…

A first requirement is that the social welfare function \(f\) canhandle any combination of any individual preferences at all:

Unrestricted Domain (U): The domain of \(f\)includes every list \(\langle R_{1}, \ldots, R_{n}\rangle\) of \(n\)weak orderings of \(X\).

ConditionU requires that \(f\) is defined for each“logically possible” profile of individual preferences. Asecond requirement is that, in each case, \(f\) produces an ordering ofthe alternatives, perhaps with ties:

Social Ordering (SO): For any profile \(\langleR_{i}\rangle\) in the domain, \(f\langle R_{i}\rangle\) is a weakordering of \(X\).

Notice that, as the paradox of voting inSection 1 shows, these two conditionsU andSO bythemselves already rule out aggregating preferences by pairwisemajority decision, if there are at least three alternatives to choosebetween, and three people whose preferences are to be taken intoaccount.

To state the next requirements it is convenient to use someshorthand. For any given individual ordering \(R_{i}\), let \(P_{i}\) bethestrict orasymmetrical part of \(R_{i}: xP_{i}y\)if \(xR_{i}y\) but not \(yR_{i}x\). Intuitively, \(xP_{i}y\) meansthat \(i\) really does prefer \(x\) to \(y\), in that \(i\) is not indifferentbetween them. Similarly, let \(P\) be the strict part of \(f\langleR_{i}\rangle\). The next condition of Arrow’s theorem is:

Weak Pareto (WP): For any profile \(\langleR_{i}\rangle\) in the domain of \(f\), and any alternatives \(x\)and \(y\), if for all \(i\), \(xP_{i}y\),then \(xPy\).

WP requires \(f\) to respect unanimous strictpreferences. That is, whenever everyone strictly prefers onealternative to another, the social ordering that \(f\) derives mustagree. Pairwise majority decisionsatisfiesWP.^[2] Many other well-known voting methods suchas Borda counting satisfy it as well (seeSection 5.2). SoWP requires that \(f\) is to this extent likethem.

The next condition ensures that social preferences are not basedentirely on the preferences of any one person. Person \(d\) isadictator of \(f\) if for any alternatives \(x\) and \(y\), andfor any profile \(\langle\ldots, R_{d},\ldots\rangle\) in the domain of\(f\): if \(xP_{d}y\), then \(xPy\). When a dictator strictly prefers onething to another, the society always does as well. Other people’spreferences can still influence social preferences. So can“non-welfare” features of the alternatives such as, inthe case of social states, the extent to which people are equal,their rights are respected, and so on. But all these can make adifference only when the dictator is indifferent between twoalternatives, having no strict preference one way or the other. Thecondition is now simply:

Nondictatorship (D): \(f\) has no dictator.

To illustrate, pick some person \(d\), any one at all, and from eachprofile \(\langle R_{i}\rangle\) in the domain take the ordering \(R_{d}\)representing the preferences of \(d\). Now, in each case, let the socialpreference be that. In other words, for each profile \(\langleR_{i}\rangle\), let \(f\langle R_{i}\rangle\) be \(R_{d}\). This socialwelfare function \(f\) bases the social ordering entirely on thepreferences of \(d\), its dictator. It is intuitively undemocraticandD rules it out.

To state the last condition of Arrow’s theorem, another piece ofshorthand is handy. For any given relation \(R\), and any set \(S\), let\(R|S\) be therestriction of \(R\)to \(S\). It is that part of \(R\)concerning just the elements of \(S\).^[3] Therestriction of \(\langleR_{1}, \ldots, R_{n}\rangle\)to \(S\), written \(\langleR_{1}, \ldots, R_{n}\rangle|S\), is just \(\langle R_{1}|S,\ldots, R_{n}|S\rangle\). Take for instance the profile from theparadox of voting inSection 1:

ABC
BCA
CAB

Its restriction to the set \(\{A,C\}\) of alternatives is:

Now the remaining condition can be stated:

Independence of Irrelevant Alternatives (I): Forall alternatives \(x\) and \(y\) in \(X\), and all profiles \(\langleR_{i}\rangle\) and \(\langle R^*_{i}\rangle\) in the domain of \(f\), if\(\langle R_{i}\rangle|\{x,y\} = \langle R^*_{i}\rangle|\{x,y\}\), then\(f\langle R_{i}\rangle |\{x,y\} = f\langle R^*_{i}\rangle|\{x,y\}\).

I says that whenever two profiles \(\langle R_{i}\rangle\)and \(\langle R^*_{i}\rangle\) are identical, as far as somealternatives \(x\) and \(y\) are concerned, so too must the socialpreference relations \(f\langle R_{i}\rangle\) and \(f\langleR^*_{i}\rangle\) be identical, as far as \(x\) and \(y\) are concerned. Forexample, consider the profile:

BAC
CAB
BCA

Its restriction to the pair {\(A\),\(C\)} is identical to that of theprofile of the paradox of voting. Suppose the domain of a socialwelfare function includes both of these profiles. Then, tosatisfyI, it must derive from each one the same socialpreference among \(A\) and \(C\). The social preference among \(A\) and \(C\)is, in this sense, to be “independent” of anybody’spreferences among either of them and the remaining“irrelevant” alternative \(B\). The same is to hold for anytwo profiles in the domain, and for any other pair taken from the set\(X\) = {\(A\), \(B\), \(C\)} of all alternatives. Some voting methods do notsatisfyI (seeSection 5.2), butpairwise majority decision does. To see whether \(x\) is sociallypreferred to \(y\), by this method, you need look no further than theindividual preferences among \(x\) and \(y\).

3.2 …are Incompatible

Arrow discovered that, except in the very simplest of cases, thefive conditions ofSection 3.1 areincompatible.

Arrow’s Theorem: Suppose there are more than twoalternatives. Then no social welfare function \(f\)satisfiesU,SO,WP,D,andI.

Arrow (1951) has the original proof of this“impossibility” theorem. See among many other works Kelly1978, Campbell and Kelly 2002, Geanakoplos 2005 and Gaertner 2009 forvariants and different proofs.

4. The Conditions, again

Taken separately, the conditions of Arrow’s theorem do not seemsevere. Apparently, they ask of an aggregation procedure only that itwill come up with a social preference ordering no matter whateverybody prefers (U andSO), that it will resemblecertain democratic arrangements in some ways (WPandI), and that it will not resemble certain undemocraticarrangements in another way (D). Taken together, though,these conditions exclude all possibility of deriving socialpreferences. It is time to consider them more closely.

4.1 Unrestricted Domain

Arrow’s domain conditionU says that the domain of thesocial welfare function includes every list of \(n\) weak orderings of\(X\). For example, suppose the alternatives are \(A\), \(B\), and \(C\), andthat the people are 1, 2, and 3. There are 13 weak orderings of threealternatives, so the unrestricted domain contains 2197 (that is,\(13^{3}\)) lists of weak orderings of \(A\), \(B\), and \(C\). A socialwelfare function \(f\) for these alternatives and people, if itsatisfiesU, maps each one of these “logicallypossible” preference profiles onto a collective preference among\(A\), \(B\), and \(C\).

In Arrow’s account, the different profiles in a domain representpreferences that the people might turn out to have. ToimposeU, on his epistemic rationale, amounts to assumingthat they might have any preferences at all: it is only when theirpreferences could be anything that it makes sense to require thesocial welfare function to be ready for everything. Arrow wrote insupport ofU:

If we do not wish to require any prior knowledgeof the tastes of individuals before specifying our social welfarefunction, that function will have to be defined for every logicallypossible set of individual orderings. (Arrow 1951 [1963]: 24)

There have been misunderstandings. Some thinkU requiresof social welfare functions that they can handle “any old”alternatives. It does nothing of the sort. What it requires is that thesocial welfare function can handle the widest possible rangeofpreferences among whichever alternatives there are tochoose among, and whether there happen to be many of these or only afew of them is beside the point: the domain of a social welfarefunction can be completely unrestricted even if there are in \(X\) justtwo alternatives. One way to sustain this unorthodox understandingofU is, perhaps, to think of Arrow’s \(x, y, z, \ldots\) not as alternatives properlyspeaking—not as candidates in elections, social states, or whathave you—but as names or labels thatrepresent these ondifferent occasions for choosing. Then, it might be thought, varietyamong the alternatives to which the labels can be attached will generatevariety among the profiles that an aggregation procedure might beexpected to handle. Blackorby et al. (2006) toy with this idea at onepoint, but they quickly set it aside. It does not seem to have beenexplored in the literature.

Of course, there is nothing to keep anyone from reinterpretingArrow’s basic notions, including the set \(X\) of alternatives, in anyway they like; a theorem is a theorem no matter what interpretationit is given. It is important to realize, though, that to interpret\(x, y, z, \ldots\) as labels is not standard, and canonly make nonsense of much of the theory of social choice to whichArrow’s theorem has given rise.^[4]

Arrow already knew thatU is a stronger domain conditionthan is needed for an impossibility result. Thefree tripleproperty and thechain property are weaker conditionsthat replaceU in some versions of Arrow’s theorem (Campbelland Kelly 2002). These versions, being more informative, are, from alogical point of view, better.U is simpler to state thantheir domain conditions, though, and might be found moreintuitive. Notice that the weaker domain conditions still require alot of variety among profiles. A typical proof of an Arrow-styleimpossibility theorem requires that the domain is unrestricted withrespect to some three alternatives. In this case there is always apreference profile like the one implicated in the paradox of votinginSection 1, from which pairwise majoritydecision derives a cycle.

Whether it is sensible to imposeU or any other domaincondition on a social welfare function depends very much on theparticulars of the choice problem being studied. Sometimes, in thenature of the alternatives under consideration, and the way in whichindividual preferences among them are determined, imposingUcertainly is not appropriate. If for instance the alternatives aredifferent ways of dividing up a pie among some people, and it is knownprior to selecting a social welfare function that these people areselfish, each caring only about the size of his own piece, then itmakes no obvious sense to require of a suitable function that it canhandle cases in which some people prefer to have less for themselvesthan to have more. The social welfare function will never be called onto handle such cases for the simple reason that they will never arise.Arrow made this point as follows:

[I]t has frequently been assumed or implied in welfare economicsthat each individual values different social states solely accordingto his consumption under them. If this be the case, we should onlyrequire that our social welfare function be defined for those sets ofindividual orderings which are of the type described; only such shouldbe admissible (Arrow 1951 [1963]: 24).

Section 5.1 considers some of thepossibilities that open up when there is no need to reckon with all“logically possible” individual preferences.

4.2 Social Ordering

ConditionSO requires that the result of aggregatingindividual preferences is always a weak ordering of the alternatives,a binary relation among them that is both transitive and connected.Intuitively, the result has to be aranking of thealternatives from better to worse, perhaps with ties. There is neverto be a cycle of social preferences, like the one derived by pairwisemajority decision in the paradox of voting,inSection 1.

Arrow did not stateSO as a separatecondition. He built it into the very notion of a socialwelfare function, arguing that the result of aggregatingpreferences will have to be an ordering if it is to “reflectrational choice-making” (Arrow 1951 [1963]: 19). Criticized byBuchanan (1954) for transferring properties of individualchoice to collective choice, Arrow in the second edition ofSocialChoice and Individual Values gave a different rationale. There heargued that transitivity is important because it ensures thatcollective choices are independent of the path taken to them (Arrow1951 [1963]: 120). He did not develop this idea further.

Charles Plott (1973) elaborated a suitable notion of pathindependence. Suppose we arrive at our choice by what hecalleddivide and conquer: first we divide the alternativesinto some smaller sets—say, because these are moremanageable—and we choose from each one. Then we gather togetherall the alternatives that we have chosen from the smaller sets, and wechoose again from among these. There are many ways of making theinitial division, and a choice procedure is said to bepathindependent if the choice we arrive at in the end is independentof which division we start with (Plott 1973: 1080). In Arrow’saccount, social choices are made from some given“environment” \(S\) of feasible alternatives by maximizing asocial ordering \(R\): the choice \(C(S)\) from among \(S\) is the set ofthose \(x\) within \(S\) such that for any \(y\) within \(S, xRy\). It is notdifficult to see how intransitivity of \(R\) can result in pathdependence. Consider again the paradox of votingofSection 1. \(A\) is strictly favoured above\(B\), and \(B\) above \(C\); but, contrary to transitivity, \(C\) is strictlyfavoured above \(A\). Starting with the division \(\{\{A,B\}, \{B,C\}\}\),our choice from among \(\{A,B,C\}\) will be \(\{A\}\); but startinginstead with \(\{\{A,C\}, \{B,C\}\}\) we will arrive in the end at\(\{B\}\).

Plott’s analysis reveals a subtlety. The full strengthofSO is not needed to secure path independence ofchoice. It is sufficient that social preference is a (completeand)quasi-transitive relation, having a strict componentthat is transitive but an indifference component that, perhaps, isnot transitive. Sen (1969: Theorem V) demonstrated the compatibilityof this weaker requirement with all of Arrow’s other conditions, butnoted that the aggregation function he came up with would notgenerally be found attractive. Its unattractiveness was noaccident. Allan Gibbard showed that the only social welfare functionsmade available by allowing intransitivity of social indifference,while keeping Arrow’s other requirements in place, are what hecalledliberum veto oligarchies (Gibbard 1969, 2014). Therehas in every case to be some group of individuals, the oligarchs,such that the society always strictly prefers one alternative toanother if all of the oligarchs strictly prefer it, but never does soif that would go against the strict preference of anyoligarch.^[5] Adictatorship, in Arrow’s sense, is a liberum veto oligarchy ofone. RelaxingSO by limiting the transitivity requirement tostrict social preferences therefore does not seem a promising way ofsecuring, in spite of Arrow’s theorem, the existence of acceptablesocial welfare functions.

4.3 Weak Pareto

ConditionWP requires that whenever everybody ranks onealternative strictly above another the social ordering agrees. Thishas long been a basic assumption in welfare economics and might seemcompletely uncontroversial. That the community should prefer onesocial state to another whenever each individual does, Arrow argued inconnection with compensation, is “not debatable except perhapson a philosophy of systematically denying people whatever theywant” (Arrow 1951 [1963]: 34).

ButWP is not as harmless as it might seem, and incombination withU it tightly constrains the possibilitiesfor social choice. This is evident from Sen’s (1970) demonstrationthat these two conditions conflict with the idea that for each personthere is a personal domain of states of affairs, within which hispreferences must prevail in case of conflict with others’. Thisimportant problem of the “Paretian libertarian” meanwhilehas its own extensive literature. For further discussion, see theentrysocial choice theory.

We may think ofWP as a vestige of what Sen called:

Welfarism: The judgement of the relative goodness ofalternative states of affairs must be based exclusively on, and takenas an increasing function of, the respective collections ofindividual utilities in these states (Sen 1979: 468).

In Arrow’s ordinal framework, welfarism insists that individualpreference orderings are theonly basis for deriving socialpreferences. Non-welfare factors—physical characteristics ofsocial states, people’s motives in having the preferences they do,respect for rights, equality—none of these are to make anydifference except indirectly, through their reflections in individualpreferences.WP asserts the demands of welfarism in thespecial case in which everybody’s strict preferences coincide. Senargued that even these limited demands might be found excessive onmoral grounds (Sen 1979: Section IV).Section 4.5 has further discussion of welfarism.

4.4 Non-Dictatorship

Someone is a dictator, in Arrow’s sense, if whenever he strictlyprefers one alternative to another the society always prefers it aswell. The preferences of people other than the dictator can still makea difference, and so can non-welfare factors, but only when thedictator is indifferent between two alternatives, having no strictpreference one way or the other. Arrow’s non-dictatorshipconditionD says that there is to be no dictator. Plainly itrules out many undemocratic arrangements, such as identifying socialpreferences in every case with the individual preferences of some oneperson. This apparently straightforward condition has attracted verylittle attention in the literature.

In fact there is more to the non-dictatorship condition than meetsthe eye. An Arrovian dictator is just someone whose strict preferencesinvariably are a subset of the society’s strict preferences, and thatby itself doesn’t mean that his preferences form a basis for socialpreferences, or that the dictator has any power or control overthese. Aanund Hylland once made a related point while objecting to theunreflective imposition ofD in single profile analyses ofsocial choice:

In the single-profile model, a dictator is aperson whose individual preferences coincide with the social ones inthe one and only profile under consideration. Nothing is necessarilywrong with that; the decision process can be perfectly democratic, andone person simply turns out to be on the winning side on allissues. (Hylland 1986: 51, footnote 10)

The non-dictatorship condition for this reason sometimes goes toofar. Even pairwise majority voting, that paradigm of a democraticprocedure, is in Arrow’s sense sometimes a dictatorship. ConsiderZelig. He has no tastes, values or preferences of his own buttemporarily takes on those of another, whoever is close at hand. He isa human chameleon, the ultimateconformist.^[6]Zelig one day finds himself on a committee of three that will chooseamong several options using the method of pairwise majority votingand, given his peculiar character, the range of individual orderingsthat can arise is somewhat restricted. In each admissible profile, twoof the three individual orderings are identical: Zelig’s and that ofwhoever is seated closest to him at the committeemeeting.^[7] Nowsuppose it so happens that Zelig strictly prefers one option \(x\) toanother, \(y\). Then someone else does too; that makes two of thethree and so, when they vote, the result is a strict collectivepreference for \(x\) above \(y\). The committee’s decision procedureis, in Arrow’s sense, a dictatorship, and Zelig is the dictator. Butof course really Zelig is a follower, not a leader, and majorityvoting is as democratic as can be. It’s just that this one mad littlefellow has a way of always ending up on the winning side.

Arrow imposedD in conjunction with therequirementU that the domain is completelyunrestricted. Perhaps this condition expresses something closer to itsintended meaning then. With an unrestricted domain, a dictator, unlikeZelig, is someone whose preferences conflict with everybody else’s ina range of cases, and it is in each instance his preferences thatagree with social preferences, not theirs. However this may be, theexample of Zelig shows that whether it is appropriate toimposeD on social welfare functions depends on the detailsof the choice problem at hand. The name of this condition ismisleading. Sometimes there is nothing undemocratic about having a“dictator”, in Arrow’s technical sense.

4.5 Independence of Irrelevant Alternatives

Arrow’s independence condition requires that whenever allindividual preferences among a pair of alternatives are the same inone profile as they are in another, the social preference among thesealternatives must also be the same for the two profiles. Speakingfiguratively, what this means is that when the social welfare functiongoes about the work of aggregating individual orderings, it has totake each pair of alternatives separately, paying no attention topreferences for alternatives other than them. Some aggregationprocedures work this way. Pairwise majority decision does: it counts\(x\) as weakly preferred to \(y\), socially, if as many people weaklyprefer \(x\) to \(y\) as the other way around, and plainly there is noneed to look beyond \(x\) and \(y\) to find this out.

ConditionI is not Arrow’s formulation. It is a simplerone that has since become the standard in expositions of theimpossibility theorem. Arrow’s formulation concerns choices made fromwithin various “environments” \(S\) of feasible options bymaximizing social orderings:

Independence of Irrelevant Alternatives (choiceversion): For all environments \(S\) within \(X\), and all profiles\(\langle R_{i}\rangle\) and \(\langle R^*_{i}\rangle\) in the domain of\(f\), if \(\langle R_{i}\rangle|S = \langle R^*_{i}\rangle|S\), then\(C(S)= C^*(S)\).

Here \(C(S)\) is the set of those options from \(S\) that are, inthe sense of the social ordering \(f\langle R_{i}\rangle\), as good asany other; and \(C^*(S)\) stands for the maxima by \(f\langleR^*_{i}\rangle\). This is Arrow’s Condition 3 (Arrow 1951 [1963]:27).^[8]

Iain McLean (2003) finds a first statementofIndependence, and appreciation of its significance,already in (Condorcet 1785). Meanwhile much controversy has surroundedthis condition, and not a little confusion. Some of each can be tracedto an example with which Arrow sought tomotivate it. When one candidate in an election diesafter polling, he wrote,

[…] the choice to be made among the set \(S\) of survivingcandidates should be independent of the preferences of individuals forcandidates not in \(S\). […] Therefore, we may require of oursocial welfare function that the choice made by society from a givenenvironment depend only on the orderings of individuals among thealternatives in that environment (Arrow 1951 [1963]: 26).

Evidently Arrow took this for his choice versionof the independence condition. He continued:

Alternatively stated, if we consider two sets of individualorderings such that, for each individual, his ordering of thoseparticular alternatives in a given environment is the same each time,then we require that the choice made by society from that environmentbe the same when individual values are given by the first set oforderings as they are when given by the second (Arrow 1951 [1963]:26–27).

It is not clear why Arrow thought the case of the dead candidateinvolves different values and preference profiles. As he set theexample up, it is natural to imagine that everybody’s values andpreferences stay the same while one candidate becomes unfeasible(“we’d all still prefer \(A\), but sadly he’s not with us anymore”). Apparently, then, Arrow’s example misses its mark. Therehas been much discussion of this point in the literature. Hansson(1973) argues that Arrow confused his independence condition foranother; compare Bordes and Tideman (1991) for a contrary view. Fordiscussion of several notions of independence whose differences havenot always been appreciated, see Ray (1973).

The following condition has also been calledIndependenceof Irrelevant Alternatives:

(\(I^*\)) For all \(x\) and \(y\), and all \(\langle R_{i}\rangle\) and\(\langle R_{i}^*\rangle\) in the domain of \(f\), if for all \(i: xR_{i}y\)if and only if \(xR^*_{i}y\), then \(x f\langle R_{i}\rangle y\) if andonly if \(x f\langle R_{i}^*\rangle y\).

If the intention is to express Arrow’s independence condition thisis a mistake because \(I^*\), though similar in appearancetoI, has a different content.I says that whenevereverybody’s preferences concerning a pair of options are the same inone profile as they are in another, the social preference must also bethe same at the two profiles, as far as this pair is concerned. Thisis not what \(I^*\) says because the embedded antecedent ‘forall \(i: xR_{i}y\) if and only if \(xR^*_{i}y\)’ is satisfiednot only when everybody’s preferences among \(x\) and \(y\) are thesame in \(\langle R_{i}\rangle\) as they are in \(\langleR_{i}^*\rangle\), but in other instances as well. For example,suppose that in \(\langle R_{i}\rangle\) everybody is indifferentbetween some social state \(T\) and another state \(S\) (in which casefor all \(i\), both \(T R_{i} S\) and \(S R_{i} T\)), while in\(\langle R_{i}^*\rangle\) everybody strictly prefers \(T\) to \(S\)(for all \(i\), \(T R^*_{i} S\) but not \(S R^*_{i} T\)). Then theantecedent ‘for all \(i: T R_{i} S\) if and only if \(T R^*_{i}S\)’ is satisfied, although individual preferences among \(T\)and \(S\) are not the same in the two profiles. \(I^*\) sometimesconstrains \(f\) though \(I\) does not and it is a more demandingcondition.

The additional demands of \(I^*\) are sometimes excessive. Let \(T\) bethe result of reforming some tried and true status quo, \(S\). Nowsuppose we favor reform if it is generally thought that change will befor the better, but not otherwise. Then we will be on the lookout fora social welfare function \(f\) that derives a strict social preferencefor \(T\) above \(S\) when everybody strictly prefers \(T\) to \(S\), but astrict preference for \(S\) above \(T\) when everyone is indifferentbetween these states. \(I^*\) rules outevery \(f\) that conforms to this desideratum because it requires a weaksocial preference for \(T\) to \(S\) in both cases or in neither.

Independence of Irrelevant Alternatives might be said torequire that the social comparison among any given pair ofalternatives, say social states, depends only on individualpreferences among this pair. This is correct but it leaves some roomfor misunderstanding.I says that theonlypreferences that count are those concerning just thesetwo social states. That doesn’t mean that preferences are the onlything that counts, though. And, indeed, as far asI isconcerned, non-welfare features of the two states may also make adifference.

The doctrine that individual preferences are theonlybasis for comparing the goodness of social states is welfarism(mentioned already inSection 4.3). An exampleillustrates how nasty it can be:

In the status quo \(S\), Peter is filthy rich and Paul is abjectlypoor. Would it be better to take from Peter and give to Paul? Let \(T\)be the social state resulting from transferring a little of Peter’svast wealth to Paul. Paul prefers \(T\) to \(S\) (“I need toeat”) and Peter prefers \(S\) to \(T\) (“not myproblem”). This is one case. Compare it to another. Social state\(T^*\) arises from a different status quo \(S^*\), also by taking fromPeter and giving to Paul. This time, though, their fortunes arereversed. In \(S^*\) it is Peter who is poor and Paul is the rich one,so this is a matter of taking from the poor to give to the rich. Evenso, we may assume, the pattern of Peter’s and Paul’s preferences isthe same in the second case as it is in the first, because each ofthem prefers to have more for himself than to have less. Paul prefers\(T^*\) to \(S^*\) (“I need another Bugatti”) and Peterprefers \(S^*\) to \(T^*\) (“wish it were my problem”). Sinceeverybody’s preferences are the same in the two cases, welfarismrequires that the relative social goodness is the same as well. Inparticular, it allows us to count \(T\) socially better than \(S\) only ifwe also count \(T^*\) better than \(S^*\). Whatever we think about takingfrom the rich to give to the poor, though, taking from the poor togive to the rich is quite another thing. As Samuelson said of asimilar case, “[o]ne need not be a doctrinaire egalitarian to bespeechless at this requirement” (Samuelson 1977: 83).

ConditionI does not express welfarism. Applied to thisexample,I states that there is to be no change in the socialcomparison among the status quo \(S\) and the result \(T\) ofredistribution unless Peter’s preferences among these states change,or Paul’s do (assume they are the only people involved). In thissense, the social preference among these states may be said to depend“only” on individual preferences among them.Isays the same about \(S^*\) and \(T^*\) or about any other pair ofalternatives. ButI is silent about any relationship betweenthe social comparison among \(S\) and \(T\), on the one hand, and thesocial comparison among \(S^*\) and \(T^*\), on the other. In particular,it leaves a social welfare function free to count \(T\) socially betterthan \(S\) (forincreasing equality), while also counting \(T^*\)worse than \(S^*\) (fordecreasing equality). Intuitivelyspeaking,I allows a social welfare function to “shiftgears” as we go from one pair of social states to the next,depending on the non-welfare features encountered there.

The condition that expresses welfarism is:

Strong Neutrality (SN): For all alternatives \(x\),\(y\), \(z\) and \(w\), and all profiles \(\langle R_{i}\rangle\) and \(\langleR_{i}^*\rangle\): IF for all \(i\): \(xR_{i}y\) if and only if \(zR^*_{i}w\),and \(yR_{i}x\) if and only if \(wR^*_{i}z\), THEN \(x f\langleR_{i}\rangle y\) if and only if \(z f\langle R_{i}^*\rangle w\), and \(yf\langle R_{i}\rangle x\) if and only if \(w f\langle R_{i}^*\ranglez\).

SN is more demanding thanI.^[9]I requires consistency for eachpair of alternatives separately, as we go from one profile in thedomain to the next.SN also requires this, but in addition it requires consistency aswe go from onepair to the next, whether that is within asingle profile or among several different ones. This ishowSN keeps non-welfare features from making any difference:by compelling the social welfare function to treat any two pairs ofalternatives the same way, if the pattern of individual preference isthe same for both.

Since \(I^*\) andSN are logically stronger thanI,obviously a version of Arrow’s theorem can be had using either one ofthem instead ofI. Such a theorem will be less interesting,though—not only because it is logically weaker but also because,as we have seen, these more demanding conditions often are unreasonable.

The meaning ofIndependence of Irrelevant Alternatives isnot easily grasped, and its ramifications are not immediately obvious.It is therefore surprising to see just how little has been said, overthe many decades that have passed since Arrow published his famoustheorem, to justify imposing this condition on social welfarefunctions. Let us turn, now, to some arguments for and against.

We have discussed Arrow’s attempt to motivateI using theexample of the dead candidate in an election. In the second editionofSocial Choice and Individual Values he offered anotherrationale.Independence, he argued, embodies the principlethat welfare judgments are to be based on observable behavior. Havingexpressed approval for Bergson’s use of indifference maps, Arrowcontinued:

The Condition of Independence of Irrelevant Alternatives extendsthe requirement of observability one step farther. Given the set ofalternatives available for society to choose among, it could beexpected that, ideally, one could observe all preferences among theavailable alternatives, but there would be no way to observepreferences among alternatives not feasible for society. (Arrow 1963:110)

Arrow seems to be saying that social decisions have to be made onthe basis of preferences for feasible alternatives because these arethe only ones that are observable. Arguably, though, this isinsufficient support. Arrow’s choice version ofIndependence,as we have seen, concernsall environments \(S\). Theobservability argument, though, apparently just concerns some“given” feasible alternatives. See Hansson (1973: 38) onthis point.

Gerry Mackie (2003) argues that there has been equivocation on thenotion of irrelevance. It is true that we often take nonfeasiblealternatives to be irrelevant. That presumably is why, in elections,we do not ordinarily put the names of dead people on ballots, alongwith those of the live candidates. ButI also excludes fromconsideration information on preferences for alternatives that, in anordinary sense,are relevant. An example illustrates Mackie’spoint. George W. Bush, Al Gore, and Ralph Nader ran in the UnitedStates presidential election of 2000. Say we want to know whetherthere was a social preference for Gore above Bush.Irequires that this question be answerable independently of whether thepeople preferred either of them to, say, Abraham Lincoln, or preferredGeorge Washington to Lincoln. This seems right. Neither Lincoln norWashington ran for President that year. They were, intuitively,irrelevant alternatives. ButI also requires that theranking of Gore with respect to Bush should be independent of voters’preferences for Nader, and this does not seem right becausehewas on the ballot and, in the ordinary sense, he was arelevant alternative to them. Certainly Arrow’s observabilitycriterion does not rule out using information on preferences forNader. They were as observable as any in that election.

A different rationale has been suggested forimposingI specificallyin the case of voting. Many voting procedures are known to presentopportunities for voters to manipulate outcomes by misrepresentingtheir preferences.Section 5.2 discusses theexample of Borda counting, which allows voters to promote their ownfavorite candidates by strategically putting others’ favorites at thebottom of their lists. Borda counting, it will be seen,violatesI. Proofs of the Gibbard-Sattherthwaite theorem(Gibbard 1973, Sattherthwaite 1975) associate vulnerability tostrategic voting systematically with violation ofI, and IainMcLean argues on this ground that voting methods ought to satisfy thiscondition: “Take out [I] and you have grossmanipulability” (McLean 2003: 16). This matter of strategicvoting did not play a part in Arrow’s presentation of theimpossibility theorem, though, and was not dealt with seriously inthe literature until after its publication. See the entryonsocial choice theory for discussionof this important theme in contemporary theory of social choice.

5. Possibilities

Arrow’s theorem, it has been said, is about the impossibility oftrying to do too much with too little information. This remark directsattention towards two main avenues leading from Arrow-inspired gloomtoward a sunnier view of the possibilities for collective decisionmaking: not trying to do so much, and using more information. One wayof not trying to do so much is to relax the requirement, it is a partofSO, that all social preferences aretransitive.Section 4.2 briefly considered thisidea but found it unpromising. Another way is to soften the demandofU that there be a social ordering for each“logically possible” preference profile. That is, we canrestrict the domains of social welfarefunctions.Section 5.1 discusses this important“escape route” from Arrow’s theorem in some detail. Oneway of using more information is to loosen the independenceconstraintI. This allows social welfare functions to makeuse of more of the information that is carried by individualpreference orderings. SeeSection 5.2. Anotherway is to extend Arrow’s framework so as to allow individuals tocontribute richer information than is carried by preferenceorderings. This information can come in the form of scores or grades,as discussed inSection 5.3, or in the form ofcardinal measurements of individual utilities,inSection 5.4.

5.1 Domain Restrictions

Sometimes, in the nature of the alternatives under considerationand how individual preferences among them are determined, not allindividual preferences can arise. When studying such a case withinArrow’s framework there is no need for a social welfare function thatcan handle each and every \(n\)-tuple of individual orderings. Some butnot all profiles are admissible, and the domain is said toberestricted. In fortunate cases, it is then possible to find asocial welfare function that meets all assumptions and conditions ofArrow’s theorem—apart, of course, fromU. Such domainsare said to beArrow consistent. This Section considers someimportant examples of Arrow consistent domains.

For a simple illustration, consider the following profile:

ABC
ABC
CBA

Here, two of the three people have the same strict preferenceordering. Reckoning the collective preference by pairwise majoritydecision, it is easy to see that the result is the ordering of thismajority:ABC. Consider now a domain made up entirely ofsuch profiles, in which most of the three voters share the samestrict preferences. On such a domain, pairwise majority decisionalways derives an ordering and so it satisfiesSO. Thissocial welfare function is nondictatorial as well provided thedomain, though restricted, still retains a certain variety. In theabove profile, voter \(3\) strictly prefers \(B\) to \(A\). Both of theothers strictly prefer \(A\) to \(B\), though, and that is thesocial preference: with this profile in the domain, \(3\) is nodictator. Pairwise majoritydecision satisfiesD if each of the voters disagrees in this way with both ofthe others, in some or other profile.^[10]It always satisfiesWPandI. On such a domain, we have now seen, this aggregationprocedure satisfies all of Arrow’s non-domain conditions. Such adomain is Arrow consistent.

Fullidentity of preferences is not needed for Arrowconsistency. It can be enough that everybody’s preferences aresimilar, even if they never entirely agree. An example illustrates thecase ofsingle peaked domains.

Suppose three bears get together to decide how hot their common potof porridge will be. Papa bear likes hot porridge, the hotter thebetter. Mama bear likes cold porridge, the colder the better. Babybear most likes warm porridge; hot porridge is next best as far as heis concerned (“it will always cool off”), and he doesn’tlike cold porridge at all. These preferences among hot, warm and coldporridge can be represented as a preference profile:

Papa: Hot Warm Cold
Mama: Cold Warm Hot
Baby: Warm Hot Cold

Or else they can be pictured like this:

[A graph, y-axis labeled 'preference' and x-axis having 'Cold', 'Warm', and 'Hot' in that order. First line, labeled 'Mama', goes from Cold/preference high to Hot/preference low. Second line, labeled 'Papa', goes from Cold/preference low to Hot/preference high. Third line, labeled 'Baby', goes from Cold/preference low to Warm/preference high to Hot/preference medium ]

Figure 1

This preference profile issingle peaked. Each bear has a“bliss point” somewhere along the ordering of the optionsby their temperature, and each bear likes options less and less as wemove along this common ordering away from the bliss point, on eitherside. Single peakedpreferences arise with respect to theleft-right orientation of political candidates, the cost ofalternative public projects, and other salient attributes ofoptions. Single peakedprofiles, in which everybody’spreferences are single peaked with respect to a common ordering, arisenaturally when everybody cares about the same thing in the optionsunder consideration—temperature, left-right orientation, cost,or what have you—even if, as with the bears, there is no furtherconsensus about which options are better than which.

Duncan Black (1948) showed that if the number ofvoters is odd, and their preference profile is single peaked,pairwise majority decision always delivers up anordering.^[11]Furthermore, he showed, the maximum of this ordering is the blisspoint of themedian voter—the voter whose bliss pointhas, on the common ordering, as many voters’ bliss points to one sideas it has to the other. In the example this is Baby bear, and warmporridge is the collective maximum. This example illustrates the wayin which single peakedness can facilitate compromise.

Say the number of voters is odd. Now consider asingle-peakeddomain—one that is made up entirely ofsingle peaked profiles. Black’s result tells us that pairwisemajority decision on this domain satisfiesSO. Provided thedomain is sufficiently inclusive (so that for each \(i\) there iswithin the domain some profile in which \(i\) is not the median voter)it also satisfiesD. Pairwise majority decision alwayssatisfiesWP andI, so such a domain is Arrowconsistent.

With an even number of voters, single peakednessdoes not ensure satisfaction ofSO. For example, supposethere are just two voters and that their individual orderings are:

CAB
BCA

Pairwise majority decision derives from this profile a weak socialpreference for \(A\) to \(B\), since there is one who weakly prefers \(A\)to \(B\), and one who weakly prefers \(B\) to \(A\). Similarly, it derivesa weak social preference for \(B\) to \(C\). Transitivity requires a weaksocial preference for \(A\) to \(C\), but there is none. On the contrary,there is a strict social preference for \(C\) above \(A\), since that isthe unanimous preference of the voters. Still, this profile issingle peaked with respect to the common ordering \(BCA\):

[A graph, y-axis labeled 'preference' and x-axis having 'B', 'C', and 'A' in that order. First line, labeled '1' goes from B/low preference to C/high preference to A/medium preference. Second line, labeled '2', goes from B/high preference to A/low preference.]

Figure 2

Majority decision with “phantom” voters can be used toestablish Arrow consistency when the number of people is even. Letthere be \(2n\) people, and let each profile in the domain be singlepeaked with respect to one and the same ordering of thealternatives. Let \(R_{2n+1}\) be an ordering that also is single peakedwith respect to this common ordering. \(R_{2n+1}\) represents thepreferences of a “phantom” voter. Now take each profile\(\langle R_{1},\ldots, R_{2n}\rangle\) in the domain and expand itinto \(\langle R_{1},\ldots, R_{2n}, R_{2n+1}\rangle\), by adding\(R_{2n+1}\). The set of all the expanded profiles is a single peakeddomain and, because the real voters together with the phantom are oddin number, Black’s result applies to it. Let \(g\) be pairwise majoritydecision for the expanded domain. We obtain a social welfare function\(f\) for the original domain by assigning to each profile the orderingthat \(g\) assigns to its expansion. That is, we put:

\[f\langle R_{1},\ldots, R_{2n}\rangle = g\langle R_{1},\ldots, R_{2n}, R_{2n+1}\rangle.\]

This \(f\) satisfiesSO because \(g\) does. ItsatisfiesWP because there are more real voters thanphantoms (\(2n\) to \(1\); we could have used any odd number of phantomssmaller than \(2n\)). \(f\) satisfiesI because the phantomordering is the same in all profiles of the expanded domain. If thedomain includes sufficient variety among profiles then \(f\) alsosatisfiesD and is Arrow consistent. This nice idea ofphantom voters was introduced by Moulin (1980), who used it tocharacterize a class of voting schemes that are non-manipulable, inthat they do not provide opportunities for strategicvoting.

Domain restrictions have been the focus of muchresearch in recent decades. Gaertner (2001) provides a generaloverview. Le Breton and Weymark (2006) survey work on domainrestrictions that arise naturally when analyzing economic problems inArrow’s framework. Miller (1992) suggests that deliberation canfacilitate rational social choice by transforming initial preferencesinto single peaked preferences. List and Dryzek (2003) argue thatdeliberation can bring about a “structuration” ofindividual preferences that facilitates democratic decision makingeven without achieving full single-peakedness. List et al. (2013)present empirical evidence that deliberation sometimes does have thiseffect.

As Samuelson described it, the single profile approach might seemto amount to the most severe of domain restrictions:

[O]ne and only one of the […] possiblepatterns of individuals’ orderings is needed. […]Fromit (not from each of them all) comes a socialordering. (Samuelson 1967: 48–49)

According to Sen (1977), though, theBergson-Samuelson social welfare function has more than a singleprofile in its domain. It has in fact a completely unrestricteddomain, for while according to Samuelson only one profile is needed“it could beany one” (Samuelson 1967: 49). Whatdistinguishes the single profile approach, on Sen’s way ofunderstanding it, is that there is to be no coordinating the behaviorof the social welfare function at several different profiles, byimposing on it interprofile conditions such asIandSN (seeSection 4.5). Eitherway, though, and just as Samuelson insisted, Arrow’s theorem does notlimit the single profile approach because one of its conditions isinappropriate in connection with it. EitherU isinappropriate (if there is a single profile in the domain) orelseI is inappropriate (if there are no interprofileconstraints).

Certain impossibility theorems that are closely related to Arrow’shave been thought relevant to single-profile choice even so. Thesetheorems do not use Arrow’sinterprofile condition \(I\) butuse instead anintraprofile neutrality condition. Thiscondition says that whenever within anysingle profile thepattern of individual preferences for one pair \(x\),\(y\) of options isthe same as for another pair \(z,w\), the social ordering derivedfrom this profile must also be the same for \(x\), \(y\) as it is for \(z\),\(w\):

Single-Profile Neutrality (SPN): For any \(\langleR_{i}\rangle\), and any alternatives \(x\), \(y\), \(z\) and \(w\): IF for all\(i: xR_{i}y\) if and only if \(zR_{i}w\), and \(yR_{i}x\) if and only if\(wR_{i}z\), THEN \(x f\langle R_{i}\rangle y\) if and only if \(z f\langleR_{i}\rangle w\), and \(y f\langle R_{i}\rangle x\) if and only if \(wf\langle R_{i}\rangle z\).

SPN follows from the strong neutrality (SN)condition ofSection 4.5, on identifying\(\langle R_{i}\rangle\) with \(\langle R_{i}^*\rangle\). Parks (1976),and independently Kemp and Ng (1976), showed that there are“single profile” versions of Arrow’s theoremusingSPN instead ofI. These theorems weresupposed to block the Bergson-Samuelson approach. In fact,conditionSPN is just as easily set aside asSN,and for the same reason: both exclude non-welfare information that isrelevant to the comparison of social states from an ethicalstandpoint. Samuelson (1977) ridiculedSPN using an exampleabout redistributing chocolate. It is similar in structure to theexample of Peter and Paul, inSection 4.5.

5.2 More Ordinal Information

Independence of Irrelevant Alternatives severely limitswhich information about individual preferences may be used forwhat. It requires a social welfare function, when assembling thesocial preference among a pair of alternatives, to take into accountonly those of people’s preferences that concern just this pair. ThisSection discusses two kinds of information that is implicit inpreferences for other alternatives, and illustrates their use in socialdecision making: information about the positions of alternatives inindividual orderings, and information about the fairness of socialstates.

Positional voting methods take into account where thecandidates come in the different individual orderings—whether itis first, or second, … or last.Borda counting is animportant example. Named after Jean-Charles de Borda, a contemporaryof Condorcet, it had already been proposed in the 13^thCentury by the pioneering writer and social theorist RamonLull. Nicholas of Cusa in the 15^th Century recommended itfor electing Holy Roman Emperors. Borda counting is used in somepolitical elections and on many other occasions for voting, in clubsand other organizations. Consider the profile:

ABCD
BACD
BACD

Let each candidate receive four points for coming first in somevoter’s ordering, three for coming second, two for a third place and asingle point for coming last; the alternatives then are ordered by thetotal number of points they receive, from all the voters. The Bordacount of \(A\) is then 10 (or \(4+3+3\)) and that of \(B\) is 11 \((3+4+4)\),so \(B\) outranks \(A\) in the social ordering. This methodapplies with the obvious adaptation to any election with a finitenumber of candidates.

Now suppose voter \(1\) moves \(B\) from second place to last on hisown list, and we have the profile:

ACDB
BACD
BACD

Then \(B\) will receive just 9 points \((1+4+4)\). \(A\) receivesthe same 10 as before, though, and now outranks \(B\). This exampleillustrates two important points. First, Borda counting does notsatisfy Arrow’s conditionI, since while each voter’s rankingof \(A\) with respect to \(B\) is the same in the two profiles, thesocial ordering of this pair is different. Second, Borda countingprovides opportunities for voters to manipulate the outcome of anelection by strategic voting. If everybody’s preferences are as in thefirst profile, voter \(1\) might do well to misrepresent hispreferences by putting \(B\) at the bottom of his list. In this way,he can promote his own favorite, \(A\), to the top of the socialordering (he will get away with this, of course, only if the othervoters do not see what he is up to and adjust their own rankingsaccordingly, by putting his favorite \(A\) at the bottom). Thesusceptibility of Borda counting to strategic voting has long beenknown. When this was raised as an objection, Borda’s indignantresponse is said to have been that his scheme was intended for honestpeople. Lull and Nicholas of Cusa recommended, before voting by thismethod, earnest oaths to tell the truth and stripping oneself of allsins.

For further discussion of positionalist voting methods, see theentriesvoting methodsandsocial choice theory; for ananalytical overview, see Pattanaik’s (2002) handbook article. Barberà(2010) reviews what is known about strategic voting.

Mark Fleurbaey (2007) has shown that social welfare functions needmore ordinal information thanI allows them if they are torespond appropriately to a certain fairness of social states. He givesthe example of Ann, who has ten apples and two oranges, and Bob, withthree apples and eleven oranges. This allocation is said to be“envy free” if, intuitively, she is at least as happy withher own basket of fruit as she would be with his and, similarly, he isas happy with his own basket as he would be with hers. Let thedistribution of fruit in one social state \(S\) be as described, andconsider the state \(S^*\) in which the allocations are reversed. Thatis, in \(S^*\) it is Ann that has three apples and eleven oranges,while Bob has ten apples and two oranges. More technically, \(S\)isenvy free if Ann weakly prefers \(S\) to \(S^*\), and Bobdoes too. We might expect that, other things being equal, the envyfreeness of a social state will promote it in the social orderingabove an alternative state that is not envy free. ButI doesnot allow this.

To see why not, consider whether it would be socially preferable,starting from the status quo \(S\), to take one apple and one orangefrom Bob and give both of them to Ann. Let \(T\) be the state arisingfrom this transfer. We assume for the sake of the example that Annalways strictly prefers having more of everything for herself tohaving less, and that Bob’s preferences are similarly self-interested,so that in all admissible profiles Ann strictly prefers \(T\) to\(S\), while Bob strictly prefers \(S\) to \(T\). Their preferencesamong these states are opposite and do not by themselves entail asocial preference one way or the other. Now, consider first a profilein which the status quo \(S\) is envy free but \(T\) is not. (Let bothAnn and Bob be indifferent between the baskets each one has in \(S\)but not between those in \(T\), in which both agree that Bob is worseoff.) Relative to this profile, a social welfare function promotingenvy freeness comes out against the transfer from Bob to Ann byranking \(S\) strictly above \(T\). But consider another profile, inwhich it is \(T\) that is envy free (now Ann and Bob are indifferentbetween the baskets in \(T\), and agree that in \(S\) Ann is worse offthan Bob). Relative to this second profile, \(T\) outranks \(S\) inthe social ordering. In direct conflict withI, the socialpreference among \(S\) and \(T\) switches as we go from one profile tothe other, although the individual preferences among \(S\) and \(T\)stay the same. The social ranking of \(S\) and \(T\) turns onpreferences among these states and the “irrelevant”\(S^*\) and similarly defined \(T^*\), because the fairness of \(S\)and \(T\) does.

Fleurbaey recommends a weaker condition, attributing it to Hansson(1973) and to Pazner (1979):

Weak Independence: Social preferences on a pair of optionsshould only depend on the population’s preferences on these twooptions and on what options are indifferent to each of these optionsfor each individual (Fleurbaey 2007: 23).

Fleurbaey (2007) discusses social welfare functions satisfying weakindependence together with Arrow’s conditions apart, of course,fromI. The approach to social welfare that is sketched thereis developed at length in (Fleurbaey and Maniquet 2011).

5.3 Even More Ordinal Information: Scores and Grades

Another way to have social orderings in spite of Arrow’s theorem is toallow people to contribute more ordinal information about theirpreferences than is admissible within Arrow’s framework. Arrow’spurpose in limiting individual inputs to weak orderings of thealternatives was to exclude cardinal information about utilities. Thislimitation, though, is too tight for its purpose. One way for peopleto communicate their preferences is tograde theiralternatives. Now, although in general grades do not carry cardinalinformation, Arrow’s framework has no provision for people to inputtheir preferences using them. This might seeem a small oversight onArrow’s part but it is critical, because there are ways of aggregatinggraded inputs that are not avaliable for preferance orderings, andwith them comes another “escape” from Arrow’simpossibility. Showing how it goes requires a slight extension ofArrow’s framework.

Let agrade language \(L\) be a strictly ordered collectionof expressions, the grades. The grades could be adjectival expressionsof a natural language, such asexcellent,good,acceptable,poor andterrible,or they could be characters such as the familiar letter grades ofacademic evaluation. They could be numerals (then oftencalledscores) or the strings of stars commonly used toevaluate hotels and restaurants. They could be any symbols that comein some fixed order (from “top” to“bottom”).

Agrade function \(G_{i}\) maps the set \(X\) of alternativesinto some given grade language \(L\). Intuitively, \(G_{i}(x)\) is\(i\)’s grade for alternative \(x\). A grade function \(G_{i}\)containsas much information as some weak ordering \(R_{i}\)of the alternatives. This can be seen by putting \(xR_{i}y\) if\(G_{i}(x) \ge G_{i}(y)\): \(i\) weakly prefers \(x\) to \(y\) if\(i\)’s grade for \(x\) is at least as high as \(i\)’s grade for\(y\). In general \(G_{i}\) containsmore information thanthe corresponding ordering because we cannot always go in reverse:different grade functions correspond to the same ordering. Forinstance, one way to convey that \(A\) is preferable to \(B\) is tosay that whereas \(A\) isgood, \(B\) ismerelyacceptable; another is to say that whereas \(A\)isgood, \(B\) isterrible, but this doesn’t come tothe same thing becauseacceptable andterribledon’t. The additional information in grades is in general not cardinalinformation because for instance the expressionsgood,acceptable andterrible, ordinaryadjectives of English, do not carry cardinal information about howgood \(A\) and \(B\) are.

Agrade profile is a list \(\langle G_{1}, \ldots,G_{n}\rangle\) of grade functions, one for each of the people \(1,\ldots, n\). Notions of a multiprofile domain and a social welfarefunction \(f\) are defined as in Arrow’s framework, putting gradeprofiles instead of Arrow’s profiles of weak orders (the output of asocial welfare function is not a social grading of the alternativesbut a social ordering of them, just in Arrow’s originalframework). The conditions of Arrow’s theorem are reformulatedaccordingly; for instance, the reformulated domain assumption is:

Unrestricted Domain (U): The domain of \(f\)includes every list \(\langle G_{1}, \ldots, G_{n}\rangle\) of \(n\)grade functions defined on \(X\).

Arrow’s independence condition becomes:

Independence of Irrelevant Alternatives (I): Forall alternatives \(x\) and \(y\) in \(X\), and all profiles \(\langleG_{i}\rangle\) and \(\langle G^*_{i}\rangle\) in the domain of \(f\), if\(\langle G_{i}\rangle|\{x,y\} = \langle G^*_{i}\rangle|\{x,y\}\), then\(f\langle G_{i}\rangle |\{x,y\} = f\langle G^*_{i}\rangle|\{x,y\}\).

Intuitively, the reformulated independence assumption says thatwhether one alternative ranks higher than another in the socialordering depends only on everybody’s grades for these twoalternatives. Other alternatives are in this matter“irrelevant.”

There are social welfare functions that satisfy all conditions ofArrow’s theorem, reformulated for graded inputs. With an odd number\(n\) of people, for instance, the social welfare functionmediangrading fills the bill. Given a profile \(\langle G_{i}\rangle\)of grade functions, themedian grade for an alternative \(x\)is the grade that is in the middle, when we list all the grades\(G_{1}(x), \ldots, G_{n}(x)\) for \(x\) in order, from top tobottom. The social ordering associated with this profile goes by thealternatives’ median grades: \(x f\langle G_{i}\rangle y\) means thatthe median grade for \(x\) is either the same as the median grade for\(y\), or else it is higher.

To verify that median grading satisfies the reformulated conditions,take for instance the nondictatorship condition. A person \(d\) isadictator of \(f\) if for any alternatives \(x\) and \(y\),and for any profile \(\langle\ldots, G_{d},\ldots\rangle\) in thedomain of \(f\): if \(xP_{d}y\), then \(xPy\). When \(f\) is mediangrading, this means that whenever a dictator expresses a strictpreference for one alternative over another, by giving it a highergrade, this alternative also ranks strictly higher in the socialordering, having a higher median grade. Except in the trivial case ofa language with only a single grade (which makes it impossible toexpress strict preferences, giving everybody vacuous dictatorialpowers), and provided there are at least two alternatives and threepeople, median grading on an unrestricted domain has no dictator. Tosee that it is so, consider any given person \(p\), and any twoalternatives \(A\) and \(B\). In an unrestricted domain there is someprofile in which \(p\)’s grade for \(A\) is higher than \(p\)’s gradefor \(B\), but in which everybody else’s grades for these alternativesare just the other way around: everybody other than \(p\) gives to\(B\) the same grade that \(p\) gives to \(A\), and gives to \(A\) thesame grade that \(p\) gives to \(B\). In this profile, the mediangrade for \(A\) is \(p\)’s grade for \(B\), and the median grade for\(B\) is \(p\)’s grade for \(A\), which is higher. So although \(p\)strictly prefers \(A\) to \(B\), \(B\) ranks strictly higher than\(A\) in the social ordering, and \(p\) is not a dictator. Repeatingthis demonstration for each person, no one else is a dictatoreither. Satisfaction of the other reformulated conditions on theunrestricted domain of grade profiles is similarlystraightforward.

Michel Balinski and Rida Laraki (2007) showed that scoring and gradingenable an “escape” from Arrow’s impossibility whilestaying within an ordinal framework. Their theory ofmajorityjudgment is a generalization of median grading that makes moreuse than it does of the ordinal information in grade profiles(Balinski and Laraki 2010). Balinski and Laraki insist that peoplemust share what they call a “common language” of grades,without giving this notion a precise sense. Indeed it can be shownthat, when different people might have very different thresholds forawarding grades, there is in grade profiles no more information thanin the corresponding profiles of weak orderings. Then a close relativeof Arrow’s impossibility returns (Morreau 2016).

Arrow came to realize, late in his life, that scoring and gradingcreate possibilities for democracy that his framework unnecessarilyrules out of consideration. In a 2012 interview he noted:

Now there’s another possible way of thinking about it, which is notincluded in my theorem \(\ldots\) [E]ach voter does not just give aranking. But says, this is good. And this is not good. Or this is verygood. And this is bad. So I have three or four classes \(\ldots\) Thischanges the nature of voting.

For a transcript of this interview, see the Other Internet Resources.

5.4 Cardinal Information

Amartya Sen (1970) extended Arrow’s framework by representing thepreferences of individuals \(i\) as utility functions \(U_{i}\) thatmap the alternatives onto real numbers: \(U_{i}(x)\) is the utilitythat \(i\) obtains from \(x\). A preference profile in Sen’sframework is a list \(\langle U_{1}, \ldots, U_{n}\rangle\) of utilityfunctions, and a domain is a set of these. An aggregation function,now a social welfarefunctional, maps each profile in somedomain onto a weak ordering of the alternatives.

Sen showed how to study various assumptions concerning themeasurability and interpersonal comparability of utilities bycoordinating the social orderings derived from profiles that,depending on these assumptions, carry the same information. Forinstance, ordinal measurement with interpersonalnoncomparability—built by Arrow right into his technicalframework—amounts, in Sen’s more flexible set up, to arequirement that the same social ordering is to be derived from anyutility profiles that reduce to the same list of orderings. At theother extreme, utilities are measured on a ratio scale with fullinterpersonal comparability if those profiles yield the same socialordering that can be obtained from each other by rescaling, ormultiplying all utility functions by the same positive real number.Sen explored different combinations of such assumptions.

One important finding was that having cardinal utilities is not byitself enough to avoid an impossibility result. In addition, utilitieshave to be interpersonally comparable. Intuitively speaking, to putinformation about preference strengths to good use it has to bepossible to compare the strengths of different individuals’preferences. See Sen (1970: Theorem 8*2). Interpersonal comparabilityopens up many possibilities for aggregating utilities and preferences.Two important ones can be read off from classical utilitarianism andRawls’s difference principle. For details, see the entryonsocial choice theory.

6. Reinterpretations

The Arrow-Sen framework lends itself to the study of a range ofaggregation problems other than those for which it was originallydeveloped. This Section briefly discusses some of them.

6.1 Judgment Aggregation

On an epistemic conception, the value of democratic institutions lies,in part, in their tendency to arrive at the truth in matters relevantto public decisions (see Estlund 2008, but compare Peter 2011). Thisidea receives some support from Condorcet’s jury theorem. It tells us,simply put, that if individual people are more likely than not tojudge correctly in some matter of fact, independently of one another,then the collective judgment of a sufficiently large group, arrived atby majority voting, is almost certain to be correct (Condorcet1785). The phenomenon of the “wisdom of crowds”,facilitated by cognitive diversity among individuals, provides furtherand arguably better support for the epistemic conception (Page 2007,Landemore 2012). But there are theoretical limits to the possibilitiesfor collective judgment on matters of fact. Starting with Kornhauserand Sager’s (1986) discussion of group deliberation in legal settings,work on the theory of judgment aggregation has explored paradoxes andimpossibility theorems closely related to those that Condorcet andArrow discovered in connection with preference aggregation. See List(2012) and the entrysocial choicetheory for overviews.

6.2 Multi-Criterial Decision

In many decision problems there are several criteria by which tocompare alternatives and, putting these criteria in place of people,it is natural to study such problems within the Arrow-Sen framework.Arrow’s theorem, if analogues of its various assumptions andconditions are appropriate, then tells us that there is no procedurefor arriving at an “overall” ordering that assimilatesdifferent criterial comparisons.

The Arrow-Sen framework has been used to study multi-criterialevaluation in industrial decision making (Arrow and Raynaud 1986) andin engineering design (Scott and Antonsson 2000; compare Franssen2005). Anandi Hattiangadi (forthcoming) argues that Arrow’s theoremlimits the possibilities for an interpretationist metasemantics, inwhich the correct interpretation of linguistic expressions is the onethat is, judging by relevant criteria, overall best.

Kenneth May (1954) used Arrow’s framework to study thedetermination of individual preferences. It had been foundexperimentally that people’s preferences, elicited separately fordifferent pairs of options, often are cyclical. May explained this byanalogy with the paradox of voting as the result of preferring onealternative to another when it is better by more criteria than not.More generally, he reinterpreted Arrow’s theorem as an argument thatintransitivity of individual preferences is to be expected whendifferent criteria “pull in different directions”. SusanHurley (1985, 1989) considered a similar problem in practicaldeliberation when the criteria are moral values. She argued thatArrow’s theorem does not apply in this case. One strand of herargument is that, unlike a person, a moral criterion can rank anygiven alternatives just one way. It cannot “change itsmind” about them (Hurley 1985: 511), and this makes itinappropriate to impose the analogue of the domainconditionU on procedures for weighing moral reasons.

There are multicriterial problems in theoretical deliberation aswell. Okasha (2011) uses the Arrow-Sen framework to study the problemof choosing among rival scientific theories by criteria including fitto data, simplicity, and scope. He argues that the impossibilitytheorem threatens the rationality of theory choice. See Morreau (2015)for a reason to think that it does not apply to this problem, andMorreau (2014) for a demonstration that impossibility theoremsrelevant to single profile choice (seeSections 2.2 and5.1) might sometimes apply even so. In related work, Jacob Stegenga (2013)argues that Arrow’s theorem limits the possibilities for combiningdifferent kinds of evidence. Eleonora Cresto and Diego Tajer(forthcoming) counter that confirmational holism requires arestriction on the domains of evidence-aggregation functions thatblocks Stegenga’s argument: they cast the Duhem problem in a positivelight as one aspect of a phenomenon that makes evidence aggregationpossible.

6.3 Overall Similarity

Things are more similar to each other in one respect, less similarin another. Much philosophy relies on notions of aggregate or“overall” similarity and Arrow’s framework has also been usedto study these.

Overall similarity lies at the foundation of David Lewis’smetaphysics (Lewis 1968, 1973a, 1973b). He wrote little about howsimilarities and differences in various respects might go together toyield overall similarities, though (Lewis 1979) gives some idea ofwhat he had in mind. The Arrow-Sen framework lends itself to studyingthis aggregation problem as well; and an impossibility theorem, if itapplies, limits the possibilities for arriving at overall similaritiesof the sort that Lewis presupposes. Morreau (2010) presents the casethat a variant of Arrow’s theorem does apply. Kroedel and Huber (2013)take a more optimistic view of overall similarity.

According to Popper (1963), some scientific theories, though false,are closer to the truth than others. Work on his notionofverisimilitude has distinguished “likeness”and “content” dimensions, and the question arises whetherthese can be combined into a single ordering of theories by theiroverall verisimilitude. Zwart and Franssen (2007) argue that Arrow’stheorem does not apply to this problem but, using a theorem inspiredby it, they argue that there is no good way to combine the differentdimensions even so. See Schurz and Weingartner (2010) and Oddie (2013)for constructive criticism of their views.

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Acknowledgments

I thank Mark Fleurbaey, Christian List, Gerry Mackie and JohnWeymark for their comments and suggestions.

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