## 1. The Will of the People?


Some of the trouble with social orderings is visible in a simple
but important example. Say there are three
alternatives \(A\), \(B\) and \(C\) to choose
among. There is a group of three people 1, 2 and 3 whose preferences
are to inform this choice, and they are asked to rank the alternatives
by their own lights from better to worse. Their individual preference
orderings turn out to be:


1. *ABC*
2. *BCA*
3. *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 so
on. Now, we might hope somehow to arrive at a single
"social" ordering of the alternatives that reflects the
preferences of all three. Then we could choose whichever alternative
is, socially, best--or, if there is a tie for first place, we
could choose some alternative that is as good as any other. Suppose,
taking the alternatives pair by pair, we put the matter to a vote: we
count one alternative as *socially preferred* to another if
there are more voters who prefer it than there are who prefer the
other one. We determine in this way that \(A\) is socially
preferred 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 \(A\) is
socially preferred to \(C\). By this reckoning, though, it is
just the other way around, since there are two voters who
prefer \(C\) to \(A\). We do not have a
social *ordering* of the alternatives at all. We have
a *cycle*. Starting from any alternative, moving to a socially
preferred one, and from there to the next, you soon find
yourself back where you 
started.[1]


This is the "paradox of voting". Discovered by the
Marquis de Condorcet (1785), it shows that possibilities for
choosing rationally can be lost when individual preferences are
aggregated into social preferences. Voter 1 has \(A\) at the top
of his individual ordering. This voter's preferences can
be *maximized*, by choosing \(A\). The preferences of 2 or
3 can also be maximized, by choosing instead their maxima, \(B\)
or \(C\). Pairwise majority decision doesn't result in a social
maximum, though. \(A\) isn't one because a majority prefers
something else, \(C\). Likewise, \(B\) and \(C\) are
not social maxima. The individual preferences lend themselves to
maximization; but, because they cycle, the social preferences do
not.


Are there other aggregation procedures that are better than
pairwise majority decision, or do the different ones have shortcomings
of 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 various
procedures and comparing their properties. Arrow broke new ground by
coming at it from the opposite direction. Starting with various
requirements that aggregation procedures might be expected to meet, he
asked which procedures fill the bill. Among his requirements
is *Social Ordering*, which insists that the result of
aggregation is always an ordering of the alternatives, never a
cycle. After the introduction in
 Section 2 of
the technical framework that Arrow set up in order to study social
choice,
 Section 3.1 sets out further conditions
that he imposed. Briefly, these are: *Unrestricted Domain*
which says that aggregation procedures must be able to handle any
individual preferences at all; *Weak Pareto*, which requires
them to respect unanimous individual
preferences; *Non-Dictatorship*, which rules out procedures by
which social preferences always agree with the strict preferences of
some one individual; and finally *Independence of Irrelevant
Alternatives*, which says that the social comparison among any two
given alternatives is to depend on individual preferences among only
that pair. Arrow's theorem, stated in
 Section 3.2, tells us that, except in
 the very simplest of cases, no
aggregation procedure whatsoever meets all the requirements.


The tenor of Arrow's theorem is deeply antithetical to the
political ideals of the Enlightenment. It turns out that Condorcet's
paradox is indeed not an isolated anomaly, the failure of one specific
voting method. Rather, it manifests a much wider problem with the very
idea of collecting many individual preferences into one. On the face
of it, anyway, there simply cannot be a common will of all the people
concerning collective decisions, that assimilates the tastes and
values of all the individual men and women who make up a society.


There are some who, following Riker (1982), take Arrow's theorem to
show that democracy, conceived as government by the will of the
people, is an incoherent illusion. Others argue that some conditions
of the theorem are unreasonable, and from their point of view the
prospects for collective choice look much brighter. After presenting
the theorem itself, this entry will take up some main points of
critical discussion. Section 4 considers the
meaning and scope of Arrow's conditions. Section 5
discusses aggregation procedures that are available when not all of
Arrows conditions need be satisfied, or when different underlying
assumptions are made about the nature of social
choice. Section 6 concludes with an overview of
proposals to study within Arrow's technical framework certain
aggregation problems other than the one that concerned him. 


Amartya Sen once expressed regret that the theory of social choice
does not share with poetry the amiable characteristic of communicating
before it is understood (Sen 1986). Arrow's theorem is not especially
difficult to understand and much about it is readily communicated, if
not in poetry, then at least in plain English. Informal presentations
go only so far, though, and where they stop sometimes
misunderstandings start. This exposition uses a minimum of technical
language for the sake of clarity.


## 2. Arrow's Framework


The problem of finding an aggregation procedure arises, as Arrow
framed it, in connection with some given alternatives between which
there is a choice is to be made. The nature of these alternatives
depends on the kind of choice problem that is being studied. In the
theory of elections, the alternatives are people who might stand as
candidates in an election. In welfare economics they are different
states of a society, such as distributions of income and labour
requirements. The alternatives conventionally are referred to using
lower case letters from the end of the alphabet
as \(x, y, z, \ldots\); the set of all
these alternatives is \(X\). The people whose tastes and values
will inform the choice are assumed to be finite in number, and they
are enumerated \(1, \ldots, n\).


Arrow's problem arises, then, only *after* some alternatives
and people have been fixed. It is for them that an aggregation
procedure is sought. Crucially, though, this problem
arises *before* relevant information about the people's
preferences among the alternatives has been gathered, whether that is
by polling or some other method for eliciting or determining
preferences. The question that Arrow's theorem answers is, more
precisely, this: Which procedures are there for arriving at a social
ordering of some given alternatives, on the basis of some given
people's preferences among them, no matter what these preferences turn
out to be?


In practice, meanwhile, we sometimes must select a procedure for
making social decisions without knowing for which alternatives and
people it will be used. In recurring elections for some public office,
for instance, there is a different slate of candidates each time, and
a different population of voters, and we must use the same voting
method to determine the winner, no matter who the candidates and
voters are and no matter how many of them there happen to be. Such
procedures are not directly available for study within Arrow's
framework, with its fixed set \(X\) of alternatives and people
\(1, \ldots, n\). Arrow's theorem is still relevant to them,
though. It tells us that *even* when the alternatives and
people are held fixed, then still there is no "good"
method for deriving social orderings. Now, if there is no good method
for voting even once, with the particular candidates and voters who
are involved on that occasion, then nor, presumably, is there a good
method that can be used repeatedly, with different candidates and
voters 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, in
his framework, merely *ordinal*. It is the kind of information
that is implicated in Condorcet's paradox of voting,
in Section 1, where each person ranks the
alternatives from better to worse but there is nothing beyond this
about how strong anybody's preferences are, or about how the
preferences of one person compare in strength to those of another. In
confining aggregation procedures to ordinal information, Arrow argued
that:




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


 His point was that even if people do have
stronger and weaker preferences, and even if the strengths of their
preferences can somehow be measured and made available as a basis for
social decisions, nevertheless ordinal information is all that
matters because preferences are "interpersonally
incomparable". Intuitively, what this means is that there is no
saying how much more strongly someone must prefer one thing to
another in order to make up for the fact that someone else's
preference is just the other way around. Arrow saw no reason to
provide aggregation procedures with information about the strength of preferences
because he thought that they cannot put such information to
meaningful use. 


Accordingly, the preferences of individual people are represented
in Arrow's framework by binary relations \(R\_{i}\) among the
alternatives: \(xR\_{i}y\) means that individual \(i\) *weakly
prefers* alternative \(x\) to alternative \(y\). That is, either
\(i\) *strictly prefers* \(x\) to \(y\), or else \(i\)
is *indifferent* between them, finding them equally good. Each
individual preference relation \(R\_{i}\) is assumed to
be *connected* (for all alternatives \(x\) and \(y\), either
\(xR\_{i}y\), or \(yR\_{i}x\), or both) and *transitive* (for all
\(x\), \(y\) and \(z\), if \(xR\_{i}y\) and \(yR\_{i}z\), then
\(xR\_{i}z\)). That these relations have these structural properties
was, for Arrow, a matter of the "rationality" of the
preferences they represent; for further discussion, see the entries on
Preferences and Philosophy of Economics. Connected, transitive
relations are called *weak orderings*. They are
"weak" in that they allow ties--in this connection,
indifference.



A *preference profile* is a list \(\langle R\_{1}, \ldots,
R\_{n}\rangle\) of weak orderings of the set \(X\) of alternatives, one
for each of the people \(1, \ldots, n\). The list of three
individual orderings in the paradox of voting is an example of a
preference profile for the alternatives \(A\), \(B\), and \(C\) and people
1, 2, and 3. A profile is a representation of the individual
preferences of everybody who will be consulted in the choice among the
alternatives. It is in the form of profiles that Arrow's aggregation
procedures receive information about individual preferences. Often it
is convenient to write \(\langle R\_{i}\rangle\) instead of \(\langle
R\_{1}, \ldots, R\_{n}\rangle\). Other profiles are written \(\langle
R^\*\_{i}\rangle\), and so on.


In restricting individual inputs to weak orderings of the
alternatives, Arrow overlooked the possibility that people could input
information about their preferences in the form of ordinal scores or
grades. Graded inputs enable an "escape" from Arrow's
impossibility that is explained in Section
5.3. Amartya Sen extended Arrow's framework to take into account
not only ordinal information about people's preferences among pairs of
alternatives, but also *cardinal* information about the utility
they derive from each one. In this way he was able to investigate the
consequences of other assumptions than Arrow's about the measurability
and interpersonal comparability of individual
preferences. See Section 5.4 and the
entry social choice theory for details
and references.


### 2.2 Multiple Profiles


Arrow required aggregation procedures to derive social orderings
from more than just a single profile, representing
everyone's *actual* preferences. In his framework they must
reckon with many profiles, representing preferences that the
people *could* have.


Variety among preferences is the result, in Arrow's account, of the
different standards by which we assess our options. Our preferences
depend on our "tastes" in personal consumption but
importantly, for social choice, they also depend on our socially
directed "values". Now, we are to some extent free to have
various tastes, values, and preferences; and we are free, also, to
have these independently of one another. Any individual can have a
range of preferences, then, and for any given sets of people and
alternatives there are many possible preference profiles. One profile
\(\langle R\_{i}\rangle\) represents the preferences of these people
among their alternatives in, if you will, one possible world. Another
profile \(\langle
R^\*\_{i}\rangle\) represents preferences of the same
people, and among the same alternatives, but in another possible world
where their tastes and values are different.


Arrow's rationale for requiring aggregation procedures to handle
many profiles was epistemic. As he framed the question of collective
choice, a procedure is sought for deriving a social ordering of some
given alternatives on the basis of some given people's tastes and
values. It is sought, though, before it is known just what these
tastes and values happen to be. The variety among profiles to be
reckoned with is a measure, in Arrow's account of the matter, of how
much is known or assumed about everybody's preferences *a
priori*, which is to say before these have been elicited. When
less is known, there are more profiles from which a social ordering
might have to be derived. When more is known, there are fewer of
them.


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


With many profiles in play there can be "interprofile"
conditions on aggregation procedures. These coordinate the results of
aggregation at several profiles at once. One such condition that plays
a crucial role in Arrow's theorem is *Independence of Irrelevant
Alternatives*. It requires that whenever everybody's preferences
among two alternatives are in one profile the same as in another, the
collective ordering must also be the same at the two profiles, as far
as these alternatives are concerned. There is to be this much
similarity among social orderings even as people's tastes and values
change. Sections 3.1
and 4.5 discuss in more detail the meaning of
this controversial requirement, and the extent to which it is
reasonable to impose it on aggregation procedures.


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



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


Little apparently agreed with Arrow that there might be a different
social ordering were people's tastes different, but unlike Arrow he
thought that it wouldn't have to be similar to the actual or current
ordering in any special way. Little's objection was taken to support
the "single profile" approach to social welfare judgments
of Abram Bergson (1938) and Paul Samuelson (1947), and there was a
debate about which approach was best, theirs or Arrow's. Arguably,
what was at issue in this debate was not--or should not have
been--whether aggregation procedures must handle more than a
single preference profile, but instead whether there should be any
coordination of the output at different profiles. Among others Sen
(1977) and Fleurbaey and Mongin (2005) have made this point. If they
are right then the substance of Little's objection can be accommodated
within Arrow's multi-profile framework simply by not imposing any
interprofile constraints. Be this as it may, Arrow's framework is
nowadays the dominant one. 


### 2.3 Social Welfare Functions


Sometimes a certain amount is known about everybody's preferences
before these have been elicited. Profiles that are compatible with
what is known represent preferences that the people could have, and
might turn out actually to have, and it is from these
"admissible" profiles that we may hope to derive social
orderings. Technically, a *domain*, in Arrow's framework, is a
set of admissible profiles, each concerning the same alternatives \(X\)
and people \(1, \ldots, n\). A *social welfare function* \(f\)
assigns to each profile \(\langle R\_{i}\rangle\) in some domain a binary
relation \(f\langle R\_{i}\rangle\) on \(X\). Intuitively, \(f\) is an
aggregation procedure and \(f\langle R\_{i}\rangle\) represents the
social 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 the
further requirement that \(f\langle R\_{i}\rangle\) is always a weak
ordering of the set \(X\) of alternatives. Informally speaking, this
means that the output of the social welfare function must always be a
ranking of the alternatives from better to worse, perhaps with
ties. It may never be a cycle. This requirement will appear here, as
it does in other contemporary presentations of Arrow's theorem, as a
separate condition of *Social Ordering* that social welfare
functions might be required to meet. See
 Section 3.1. This way, we can consider the
 consequences of dropping this
condition without changing any basic parts of the framework.
See Section 4.2.


Arrow established a convention that is still widely observed of
using '\(R\)' to denote the social preference derived from
"\(\langle R\_{i}\rangle\)". The social welfare function used to derive it
is, in his notation, left implicit. One advantage of writing
'\(f\langle R\_{i}\rangle\)' instead of '\(R\)' is
that when we state the conditions of the impossibility theorem, in the
next section, the social welfare function will figure explicitly in
them. This makes it quite clear that what these conditions constrain
is the *functional* relationship between individual and social
preferences. Focusing attention on this was an important innovation of
Arrow's approach.


## 3. Impossibility


With the conceptual framework now in
place, Section 3.1 sets out the
"conditions" or constraints that Arrow imposed on social
welfare functions, and Section 3.2 states the
theorem itself. Section 4 explains the
conditions more fully, discusses reasons that Arrow gave for imposing
them, and considers whether it is proper to do so. 


Arrow's conditions often are called axioms, and his approach is
said to be axiomatic. This might be found misleading. Unlike axioms of
logic or geometry, Arrow's conditions are not supposed to express more
or less indubitable truths, or to constitute an implicit definition of
the object of study. Arrow himself took them to be questionable
"value judgments" that "express the doctrines of
citizens' sovereignty and rationality in a very general form"
(Arrow 1951 [1963]: 31). Indeed, as we will see
in Section 4, and as Arrow himself recognized,
sometimes it is not even desirable that social welfare functions
should satisfy all conditions of the impossibility theorem.


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


### 3.1 These Conditions...


A first requirement is that the social welfare function \(f\) can
handle 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 \(X\).
> 
> 
> 


Condition *U* requires that \(f\) is defined for each
"logically possible" profile of individual preferences. A
second requirement is that, in each case, \(f\) produces an ordering of
the alternatives, perhaps with ties:



> 
> *Social Ordering* (*SO*): For any profile \(\langle
> R\_{i}\rangle\) in the domain, \(f\langle R\_{i}\rangle\) is a weak
> ordering of \(X\).
> 
> 
> 


Notice that, as the paradox of voting in
 Section 1 shows, these two conditions
 *U* and *SO* by
themselves already rule out aggregating preferences by pairwise
majority decision, if there are at least three alternatives to choose
between, and three people whose preferences are to be taken into
account.


To state the next requirements it is convenient to use some
shorthand. For any given individual ordering \(R\_{i}\), let \(P\_{i}\) be
the *strict* or *asymmetrical* part of \(R\_{i}: xP\_{i}y\)
if \(xR\_{i}y\) but not \(yR\_{i}x\). Intuitively, \(xP\_{i}y\) means
that \(i\) really does prefer \(x\) to \(y\), in that \(i\) is not indifferent
between them. Similarly, let \(P\) be the strict part of \(f\langle
R\_{i}\rangle\). The next condition of Arrow's theorem is: 



> 
> *Weak Pareto* (*WP*): For any profile \(\langle
> R\_{i}\rangle\) 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 strict
preferences. That is, whenever everyone strictly prefers one
alternative to another, the social ordering that \(f\) derives must
agree. Pairwise majority decision
satisfies 
*WP*.[2]
 Many other well-known voting methods such
as Borda counting satisfy it as well (see
 Section 5.2). So *WP* requires
 that \(f\) is to this extent like
them.


The next condition ensures that social preferences are not based
entirely on the preferences of any one person. Person \(d\) is
a *dictator* of \(f\) if for any alternatives \(x\) and \(y\), and
for any profile \(\langle\ldots, R\_{d},\ldots\rangle\) in the domain of
\(f\): if \(xP\_{d}y\), then \(xPy\). When a dictator strictly prefers one
thing to another, the society always does as well. Other people's
preferences can still influence social preferences. So can
"non-welfare" features of the alternatives such as, in
the case of social states, the extent to which people are equal,
their rights are respected, and so on. But all these can make a
difference only when the dictator is indifferent between two
alternatives, having no strict preference one way or the other. The
condition is now simply:



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


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


To state the last condition of Arrow's theorem, another piece of
shorthand is handy. For any given relation \(R\), and any set \(S\), let
\(R|S\) be the *restriction of* \(R\) *to* \(S\). It is that part of \(R\)
concerning just the elements of 
\(S\).[3]
 The *restriction of* \(\langle
R\_{1}, \ldots, R\_{n}\rangle\) *to* \(S\), written \(\langle
R\_{1}, \ldots, R\_{n}\rangle|S\), is just \(\langle R\_{1}|S,
\ldots, R\_{n}|S\rangle\). Take for instance the profile from the
paradox of voting in Section 1:


1. *ABC*
2. *BCA*
3. *CAB*


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


1. *AC*
2. *CA*
3. *CA*


Now the remaining condition can be stated: 



> 
> *Independence of Irrelevant Alternatives* (*I*): For
> all alternatives \(x\) and \(y\) in \(X\), and all profiles \(\langle
> R\_{i}\rangle\) 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 some
alternatives \(x\) and \(y\) are concerned, so too must the social
preference relations \(f\langle R\_{i}\rangle\) and \(f\langle
R^\*\_{i}\rangle\) be identical, as far as \(x\) and \(y\) are concerned. For
example, consider the profile:


1. *BAC*
2. *CAB*
3. *BCA*


Its restriction to the pair {\(A\),\(C\)} is identical to that of the
profile of the paradox of voting. Suppose the domain of a social
welfare function includes both of these profiles. Then, to
satisfy *I*, it must derive from each one the same social
preference among \(A\) and \(C\). The social preference among \(A\) and \(C\)
is, in this sense, to be "independent" of anybody's
preferences among either of them and the remaining
"irrelevant" alternative \(B\). The same is to hold for any
two profiles in the domain, and for any other pair taken from the set
\(X\) = {\(A\), \(B\), \(C\)} of all alternatives. Some voting methods do not
satisfy *I* (see Section 5.2), but
pairwise majority decision does. To see whether \(x\) is socially
preferred to \(y\), by this method, you need look no further than the
individual preferences among \(x\) and \(y\). 


### 3.2 ...are Incompatible


Arrow discovered that, except in the very simplest of cases, the
five conditions of Section 3.1 are
incompatible.



> 
> *Arrow's Theorem*: Suppose there are more than two
> alternatives. Then no social welfare function \(f\)
> satisfies *U*, *SO*, *WP*, *D*,
> and *I*.
> 
> 
> 


Arrow (1951) has the original proof of this
"impossibility" theorem. See among many other works Kelly
1978, Campbell and Kelly 2002, Geanakoplos 2005 and Gaertner 2009 for
variants and different proofs.


## 4. The Conditions, again


Taken separately, the conditions of Arrow's theorem do not seem
severe. Apparently, they ask of an aggregation procedure only that it
will come up with a social preference ordering no matter what
everybody prefers (*U* and *SO*), that it will resemble
certain democratic arrangements in some ways (*WP*
and *I*), and that it will not resemble certain undemocratic
arrangements in another way (*D*). Taken together, though,
these conditions exclude all possibility of deriving social
preferences. It is time to consider them more closely. 


### 4.1 Unrestricted Domain


Arrow's domain condition *U* says that the domain of the
social welfare function includes every list of \(n\) weak orderings of
\(X\). For example, suppose the alternatives are \(A\), \(B\), and \(C\), and
that the people are 1, 2, and 3. There are 13 weak orderings of three
alternatives, so the unrestricted domain contains 2197 (that is,
\(13^{3}\)) lists of weak orderings of \(A\), \(B\), and \(C\). A social
welfare function \(f\) for these alternatives and people, if it
satisfies *U*, maps each one of these "logically
possible" preference profiles onto a collective preference among
\(A\), \(B\), and \(C\).


In Arrow's account, the different profiles in a domain represent
preferences that the people might turn out to have. To
impose *U*, on his epistemic rationale, amounts to assuming
that they might have any preferences at all: it is only when their
preferences could be anything that it makes sense to require the
social welfare function to be ready for everything. Arrow wrote in
support of *U*:




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


There have been misunderstandings. Some think *U* requires
of social welfare functions that they can handle "any old"
alternatives. It does nothing of the sort. What it requires is that the
social welfare function can handle the widest possible range
of *preferences* among whichever alternatives there are to
choose among, and whether there happen to be many of these or only a
few of them is beside the point: the domain of a social welfare
function can be completely unrestricted even if there are in \(X\) just
two alternatives. One way to sustain this unorthodox understanding
of *U* is, perhaps, to think of Arrow's \(x, y, z, \ldots\) not as alternatives properly
speaking--not as candidates in elections, social states, or what
have you--but as names or labels that *represent* these on
different occasions for choosing. Then, it might be thought, variety
among the alternatives to which the labels can be attached will generate
variety among the profiles that an aggregation procedure might be
expected to handle. Blackorby et al. (2006) toy with this idea at one
point, but they quickly set it aside. It does not seem to have been
explored in the literature.


Of course, there is nothing to keep anyone from reinterpreting
Arrow's basic notions, including the set \(X\) of alternatives, in any
way they like; a theorem is a theorem no matter what interpretation
it is given. It is important to realize, though, that to interpret
\(x, y, z, \ldots\) as labels is not standard, and can
only make nonsense of much of the theory of social choice to which
Arrow's theorem has given 
rise.[4]


Arrow already knew that *U* is a stronger domain condition
than is needed for an impossibility result. The *free triple
property* and the *chain property* are weaker conditions
that replace *U* in some versions of Arrow's theorem (Campbell
and Kelly 2002). These versions, being more informative, are, from a
logical point of view, better. *U* is simpler to state than
their domain conditions, though, and might be found more
intuitive. Notice that the weaker domain conditions still require a
lot of variety among profiles. A typical proof of an Arrow-style
impossibility theorem requires that the domain is unrestricted with
respect to some three alternatives. In this case there is always a
preference profile like the one implicated in the paradox of voting
in Section 1, from which pairwise majority
decision derives a cycle.


Whether it is sensible to impose *U* or any other domain
condition on a social welfare function depends very much on the
particulars of the choice problem being studied. Sometimes, in the
nature of the alternatives under consideration, and the way in which
individual preferences among them are determined, imposing *U*
certainly is not appropriate. If for instance the alternatives are
different ways of dividing up a pie among some people, and it is known
prior to selecting a social welfare function that these people are
selfish, each caring only about the size of his own piece, then it
makes no obvious sense to require of a suitable function that it can
handle cases in which some people prefer to have less for themselves
than to have more. The social welfare function will never be called on
to 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 economics
> that each individual values different social states solely according
> to his consumption under them. If this be the case, we should only
> require that our social welfare function be defined for those sets of
> individual orderings which are of the type described; only such should
> be admissible (Arrow 1951 [1963]: 24).
> 


Section 5.1 considers some of the
possibilities that open up when there is no need to reckon with all
"logically possible" individual preferences.


### 4.2 Social Ordering


Condition *SO* requires that the result of aggregating
individual 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 a *ranking* of the
alternatives from better to worse, perhaps with ties. There is never
to be a cycle of social preferences, like the one derived by pairwise
majority decision in the paradox of voting,
in Section 1.


Arrow did not state *SO* as a separate
condition. He built it into the very notion of a social
welfare function, arguing that the result of aggregating
preferences will have to be an ordering if it is to "reflect
rational choice-making" (Arrow 1951 [1963]: 19). Criticized by
Buchanan (1954) for transferring properties of individual
choice to collective choice, Arrow in the second edition of *Social
Choice and Individual Values* gave a different rationale. There he
argued that transitivity is important because it ensures that
collective choices are independent of the path taken to them (Arrow
1951 [1963]: 120). He did not develop this idea further.


Charles Plott (1973) elaborated a suitable notion of path
independence. Suppose we arrive at our choice by what he
called *divide and conquer*: first we divide the alternatives
into some smaller sets--say, because these are more
manageable--and we choose from each one. Then we gather together
all the alternatives that we have chosen from the smaller sets, and we
choose again from among these. There are many ways of making the
initial division, and a choice procedure is said to be *path
independent* if the choice we arrive at in the end is independent
of which division we start with (Plott 1973: 1080). In Arrow's
account, social choices are made from some given
"environment" \(S\) of feasible alternatives by maximizing a
social ordering \(R\): the choice \(C(S)\) from among \(S\) is the set of
those \(x\) within \(S\) such that for any \(y\) within \(S, xRy\). It is not
difficult to see how intransitivity of \(R\) can result in path
dependence. Consider again the paradox of voting
of Section 1. \(A\) is strictly favoured above
\(B\), and \(B\) above \(C\); but, contrary to transitivity, \(C\) is strictly
favoured above \(A\). Starting with the division \(\{\{A,B\}, \{B,C\}\}\),
our choice from among \(\{A,B,C\}\) will be \(\{A\}\); but starting
instead with \(\{\{A,C\}, \{B,C\}\}\) we will arrive in the end at
\(\{B\}\).


Plott's analysis reveals a subtlety. The full strength
of *SO* is not needed to secure path independence of
choice. It is sufficient that social preference is a (complete
and) *quasi-transitive* relation, having a strict component
that is transitive but an indifference component that, perhaps, is
not transitive. Sen (1969: Theorem V) demonstrated the compatibility
of this weaker requirement with all of Arrow's other conditions, but
noted that the aggregation function he came up with would not
generally be found attractive. Its unattractiveness was no
accident. Allan Gibbard showed that the only social welfare functions
made available by allowing intransitivity of social indifference,
while keeping Arrow's other requirements in place, are what he
called *liberum veto oligarchies* (Gibbard 1969, 2014). There
has in every case to be some group of individuals, the oligarchs,
such that the society always strictly prefers one alternative to
another if all of the oligarchs strictly prefer it, but never does so
if that would go against the strict preference of any
oligarch.[5] A
dictatorship, in Arrow's sense, is a liberum veto oligarchy of
one. Relaxing *SO* by limiting the transitivity requirement to
strict social preferences therefore does not seem a promising way of
securing, in spite of Arrow's theorem, the existence of acceptable
social welfare functions.



### 4.3 Weak Pareto


Condition *WP* requires that whenever everybody ranks one
alternative strictly above another the social ordering agrees. This
has long been a basic assumption in welfare economics and might seem
completely uncontroversial. That the community should prefer one
social state to another whenever each individual does, Arrow argued in
connection with compensation, is "not debatable except perhaps
on a philosophy of systematically denying people whatever they
want" (Arrow 1951 [1963]: 34).


But *WP* is not as harmless as it might seem, and in
combination with *U* it tightly constrains the possibilities
for social choice. This is evident from Sen's (1970) demonstration
that these two conditions conflict with the idea that for each person
there is a personal domain of states of affairs, within which his
preferences must prevail in case of conflict with others'. This
important problem of the "Paretian libertarian" meanwhile
has its own extensive literature. For further discussion, see the
entry social choice theory.


We may think of *WP* as a vestige of what Sen called:



> 
> *Welfarism*: The judgement of the relative goodness of
> alternative states of affairs must be based exclusively on, and taken
> as an increasing function of, the respective collections of
> individual utilities in these states (Sen 1979: 468).
> 
> 
> 


In Arrow's ordinal framework, welfarism insists that individual
preference orderings are the *only* basis for deriving social
preferences. Non-welfare factors--physical characteristics of
social states, people's motives in having the preferences they do,
respect for rights, equality--none of these are to make any
difference except indirectly, through their reflections in individual
preferences. *WP* asserts the demands of welfarism in the
special case in which everybody's strict preferences coincide. Sen
argued that even these limited demands might be found excessive on
moral 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 strictly
prefers one alternative to another the society always prefers it as
well. The preferences of people other than the dictator can still make
a difference, and so can non-welfare factors, but only when the
dictator is indifferent between two alternatives, having no strict
preference one way or the other. Arrow's non-dictatorship
condition *D* says that there is to be no dictator. Plainly it
rules out many undemocratic arrangements, such as identifying social
preferences in every case with the individual preferences of some one
person. This apparently straightforward condition has attracted very
little attention in the literature. 


In fact there is more to the non-dictatorship condition than meets
the eye. An Arrovian dictator is just someone whose strict preferences
invariably are a subset of the society's strict preferences, and that
by itself doesn't mean that his preferences form a basis for social
preferences, or that the dictator has any power or control over
these. Aanund Hylland once made a related point while objecting to the
unreflective imposition of *D* in single profile analyses of
social choice:



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


The non-dictatorship condition for this reason sometimes goes too
far. Even pairwise majority voting, that paradigm of a democratic
procedure, is in Arrow's sense sometimes a dictatorship. Consider
Zelig. He has no tastes, values or preferences of his own but
temporarily takes on those of another, whoever is close at hand. He is
a human chameleon, the ultimate
conformist.[6]
Zelig one day finds himself on a committee of three that will choose
among several options using the method of pairwise majority voting
and, given his peculiar character, the range of individual orderings
that can arise is somewhat restricted. In each admissible profile, two
of the three individual orderings are identical: Zelig's and that of
whoever is seated closest to him at the committee
meeting.[7] Now
suppose it so happens that Zelig strictly prefers one option \(x\) to
another, \(y\). Then someone else does too; that makes two of the
three and so, when they vote, the result is a strict collective
preference for \(x\) above \(y\). The committee's decision procedure
is, in Arrow's sense, a dictatorship, and Zelig is the dictator. But
of course really Zelig is a follower, not a leader, and majority
voting is as democratic as can be. It's just that this one mad little
fellow has a way of always ending up on the winning side.


Arrow imposed *D* in conjunction with the
requirement *U* that the domain is completely
unrestricted. Perhaps this condition expresses something closer to its
intended meaning then. With an unrestricted domain, a dictator, unlike
Zelig, is someone whose preferences conflict with everybody else's in
a range of cases, and it is in each instance his preferences that
agree with social preferences, not theirs. However this may be, the
example of Zelig shows that whether it is appropriate to
impose *D* on social welfare functions depends on the details
of the choice problem at hand. The name of this condition is
misleading. 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 all
individual preferences among a pair of alternatives are the same in
one profile as they are in another, the social preference among these
alternatives must also be the same for the two profiles. Speaking
figuratively, what this means is that when the social welfare function
goes about the work of aggregating individual orderings, it has to
take each pair of alternatives separately, paying no attention to
preferences for alternatives other than them. Some aggregation
procedures work this way. Pairwise majority decision does: it counts
\(x\) as weakly preferred to \(y\), socially, if as many people weakly
prefer \(x\) to \(y\) as the other way around, and plainly there is no
need to look beyond \(x\) and \(y\) to find this out.


Condition *I* is not Arrow's formulation. It is a simpler
one that has since become the standard in expositions of the
impossibility theorem. Arrow's formulation concerns choices made from
within various "environments" \(S\) of feasible options by
maximizing social orderings:



> 
> *Independence of Irrelevant Alternatives* (*choice
> version*): 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, in
the sense of the social ordering \(f\langle R\_{i}\rangle\), as good as
any other; and \(C^\*(S)\) stands for the maxima by \(f\langle
R^\*\_{i}\rangle\). This is Arrow's Condition 3 (Arrow 1951 [1963]:
27).[8] 


Iain McLean (2003) finds a first statement
of *Independence*, and appreciation of its significance,
already in (Condorcet 1785). Meanwhile much controversy has surrounded
this condition, and not a little confusion. Some of each can be traced
to an example with which Arrow sought to
motivate it. When one candidate in an election dies
after polling, he wrote,



> 
> [...] the choice to be made among the set \(S\) of surviving
> candidates should be independent of the preferences of individuals for
> candidates not in \(S\). [...] Therefore, we may require of our
> social welfare function that the choice made by society from a given
> environment depend only on the orderings of individuals among the
> alternatives in that environment (Arrow 1951 [1963]: 26).
> 


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



> 
> Alternatively stated, if we consider two sets of individual
> orderings such that, for each individual, his ordering of those
> particular alternatives in a given environment is the same each time,
> then we require that the choice made by society from that environment
> be the same when individual values are given by the first set of
> orderings 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 candidate
involves different values and preference profiles. As he set the
example up, it is natural to imagine that everybody's values and
preferences stay the same while one candidate becomes unfeasible
("we'd all still prefer \(A\), but sadly he's not with us any
more"). Apparently, then, Arrow's example misses its mark. There
has been much discussion of this point in the literature. Hansson
(1973) argues that Arrow confused his independence condition for
another; compare Bordes and Tideman (1991) for a contrary view. For
discussion of several notions of independence whose differences have
not always been appreciated, see Ray (1973).


The following condition has also been called *Independence
of 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 and
> only if \(x f\langle R\_{i}^\*\rangle y\).
> 
> 
> 


If the intention is to express Arrow's independence condition this
is a mistake because \(I^\*\), though similar in appearance
to *I*, has a different content. *I* says that whenever
everybody's preferences concerning a pair of options are the same in
one profile as they are in another, the social preference must also be
the same at the two profiles, as far as this pair is concerned. This
is not what \(I^\*\) says because the embedded antecedent 'for
all \(i: xR\_{i}y\) if and only if \(xR^\*\_{i}y\)' is satisfied
not only when everybody's preferences among \(x\) and \(y\) are the
same in \(\langle R\_{i}\rangle\) as they are in \(\langle
R\_{i}^\*\rangle\), but in other instances as well. For example,
suppose that in \(\langle R\_{i}\rangle\) everybody is indifferent
between some social state \(T\) and another state \(S\) (in which case
for 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 the
antecedent '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. \(I^\*\) sometimes
constrains \(f\) though \(I\) does not and it is a more demanding
condition.


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


*Independence of Irrelevant Alternatives* might be said to
require that the social comparison among any given pair of
alternatives, say social states, depends only on individual
preferences among this pair. This is correct but it leaves some room
for misunderstanding. *I* says that the
only *preferences* that count are those concerning just these
two social states. That doesn't mean that preferences are the only
thing that counts, though. And, indeed, as far as *I* is
concerned, non-welfare features of the two states may also make a
difference. 


The doctrine that individual preferences are the *only*
basis for comparing the goodness of social states is welfarism
(mentioned already in Section 4.3). An example
illustrates how nasty it can be:


In the status quo \(S\), Peter is filthy rich and Paul is abjectly
poor. 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's
vast wealth to Paul. Paul prefers \(T\) to \(S\) ("I need to
eat") and Peter prefers \(S\) to \(T\) ("not my
problem"). This is one case. Compare it to another. Social state
\(T^\*\) arises from a different status quo \(S^\*\), also by taking from
Peter and giving to Paul. This time, though, their fortunes are
reversed. 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. Even
so, we may assume, the pattern of Peter's and Paul's preferences is
the same in the second case as it is in the first, because each of
them prefers to have more for himself than to have less. Paul prefers
\(T^\*\) to \(S^\*\) ("I need another Bugatti") and Peter
prefers \(S^\*\) to \(T^\*\) ("wish it were my problem"). Since
everybody's preferences are the same in the two cases, welfarism
requires that the relative social goodness is the same as well. In
particular, it allows us to count \(T\) socially better than \(S\) only if
we also count \(T^\*\) better than \(S^\*\). Whatever we think about taking
from the rich to give to the poor, though, taking from the poor to
give to the rich is quite another thing. As Samuelson said of a
similar case, "[o]ne need not be a doctrinaire egalitarian to be
speechless at this requirement" (Samuelson 1977: 83).


Condition *I* does not express welfarism. Applied to this
example, *I* states that there is to be no change in the social
comparison among the status quo \(S\) and the result \(T\) of
redistribution unless Peter's preferences among these states change,
or Paul's do (assume they are the only people involved). In this
sense, the social preference among these states may be said to depend
"only" on individual preferences among them. *I*
says the same about \(S^\*\) and \(T^\*\) or about any other pair of
alternatives. But *I* is silent about any relationship between
the social comparison among \(S\) and \(T\), on the one hand, and the
social comparison among \(S^\*\) and \(T^\*\), on the other. In particular,
it leaves a social welfare function free to count \(T\) socially better
than \(S\) (for *in*creasing equality), while also counting \(T^\*\)
worse than \(S^\*\) (for *de*creasing equality). Intuitively
speaking, *I* allows a social welfare function to "shift
gears" 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 \(\langle
> R\_{i}^\*\rangle\): 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 \(x f\langle
> R\_{i}\rangle y\) if and only if \(z f\langle R\_{i}^\*\rangle w\), and \(y
> f\langle R\_{i}\rangle x\) if and only if \(w f\langle R\_{i}^\*\rangle
> z\).
> 
> 
> 


 *SN* is more demanding than 
*I*.[9]
*I* requires consistency for each
pair of alternatives separately, as we go from one profile in the
domain to the next. *SN* also requires this, but in addition it requires consistency as
we go from one *pair* to the next, whether that is within a
single profile or among several different ones. This is
how *SN* keeps non-welfare features from making any difference:
by compelling the social welfare function to treat any two pairs of
alternatives the same way, if the pattern of individual preference is
the same for both.


Since \(I^\*\) and *SN* are logically stronger than *I*,
obviously a version of Arrow's theorem can be had using either one of
them instead of *I*. 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 of *Independence of Irrelevant Alternatives* is
not easily grasped, and its ramifications are not immediately obvious.
It is therefore surprising to see just how little has been said, over
the many decades that have passed since Arrow published his famous
theorem, to justify imposing this condition on social welfare
functions. Let us turn, now, to some arguments for and against.


We have discussed Arrow's attempt to motivate *I* using the
example of the dead candidate in an election. In the second edition
of *Social Choice and Individual Values* he offered another
rationale. *Independence*, he argued, embodies the principle
that welfare judgments are to be based on observable behavior. Having
expressed approval for Bergson's use of indifference maps, Arrow
continued:



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


Arrow seems to be saying that social decisions have to be made on
the basis of preferences for feasible alternatives because these are
the only ones that are observable. Arguably, though, this is
insufficient support. Arrow's choice version of *Independence*,
as we have seen, concerns *all* environments \(S\). The
observability argument, though, apparently just concerns some
"given" feasible alternatives. See Hansson (1973: 38) on
this point.


Gerry Mackie (2003) argues that there has been equivocation on the
notion of irrelevance. It is true that we often take nonfeasible
alternatives to be irrelevant. That presumably is why, in elections,
we do not ordinarily put the names of dead people on ballots, along
with those of the live candidates. But *I* also excludes from
consideration information on preferences for alternatives that, in an
ordinary sense, *are* relevant. An example illustrates Mackie's
point. George W. Bush, Al Gore, and Ralph Nader ran in the United
States presidential election of 2000. Say we want to know whether
there was a social preference for Gore above Bush. *I*
requires that this question be answerable independently of whether the
people preferred either of them to, say, Abraham Lincoln, or preferred
George Washington to Lincoln. This seems right. Neither Lincoln nor
Washington ran for President that year. They were, intuitively,
irrelevant alternatives. But *I* also requires that the
ranking of Gore with respect to Bush should be independent of voters'
preferences for Nader, and this does not seem right because
he *was* on the ballot and, in the ordinary sense, he was a
relevant alternative to them. Certainly Arrow's observability
criterion does not rule out using information on preferences for
Nader. They were as observable as any in that election.


A different rationale has been suggested for
imposing *I* specifically
in the case of voting. Many voting procedures are known to present
opportunities for voters to manipulate outcomes by misrepresenting
their preferences. Section 5.2 discusses the
example of Borda counting, which allows voters to promote their own
favorite candidates by strategically putting others' favorites at the
bottom of their lists. Borda counting, it will be seen,
violates *I*. Proofs of the Gibbard-Sattherthwaite theorem
(Gibbard 1973, Sattherthwaite 1975) associate vulnerability to
strategic voting systematically with violation of *I*, and Iain
McLean argues on this ground that voting methods ought to satisfy this
condition: "Take out [*I*] and you have gross
manipulability" (McLean 2003: 16). This matter of strategic
voting did not play a part in Arrow's presentation of the
impossibility theorem, though, and was not dealt with seriously in
the literature until after its publication. See the entry
on social choice theory for discussion
of this important theme in contemporary theory of social choice.


## 5. Possibilities


Arrow's theorem, it has been said, is about the impossibility of
trying to do too much with too little information. This remark directs
attention towards two main avenues leading from Arrow-inspired gloom
toward a sunnier view of the possibilities for collective decision
making: not trying to do so much, and using more information. One way
of not trying to do so much is to relax the requirement, it is a part
of *SO*, that all social preferences are
transitive. Section 4.2 briefly considered this
idea but found it unpromising. Another way is to soften the demand
of *U* that there be a social ordering for each
"logically possible" preference profile. That is, we can
restrict the domains of social welfare
functions. Section 5.1 discusses this important
"escape route" from Arrow's theorem in some detail. One
way of using more information is to loosen the independence
constraint *I*. This allows social welfare functions to make
use of more of the information that is carried by individual
preference orderings. See Section 5.2. Another
way is to extend Arrow's framework so as to allow individuals to
contribute richer information than is carried by preference
orderings. This information can come in the form of scores or grades,
as discussed in Section 5.3, or in the form of
cardinal measurements of individual utilities,
in Section 5.4.


### 5.1 Domain Restrictions


Sometimes, in the nature of the alternatives under consideration
and how individual preferences among them are determined, not all
individual preferences can arise. When studying such a case within
Arrow's framework there is no need for a social welfare function that
can handle each and every \(n\)-tuple of individual orderings. Some but
not all profiles are admissible, and the domain is said to
be *restricted*. In fortunate cases, it is then possible to find a
social welfare function that meets all assumptions and conditions of
Arrow's theorem--apart, of course, from *U*. Such domains
are said to be *Arrow consistent*. This Section considers some
important examples of Arrow consistent domains.


For a simple illustration, consider the following profile: 


1. *ABC*
2. *ABC*
3. *CBA*


Here, two of the three people have the same strict preference
ordering. Reckoning the collective preference by pairwise majority
decision, it is easy to see that the result is the ordering of this
majority: *ABC*. Consider now a domain made up entirely of
such profiles, in which most of the three voters share the same
strict preferences. On such a domain, pairwise majority decision
always derives an ordering and so it satisfies *SO*. This
social welfare function is nondictatorial as well provided the
domain, though restricted, still retains a certain variety. In the
above profile, voter \(3\) strictly prefers \(B\) to \(A\). Both of the
others strictly prefer \(A\) to \(B\), though, and that is the
social preference: with this profile in the domain, \(3\) is no
dictator. Pairwise majority
decision satisfies 
*D* if each of the voters disagrees in this way with both of
the others, in some or other profile.[10]
It always satisfies *WP*
and *I*. On such a domain, we have now seen, this aggregation
procedure satisfies all of Arrow's non-domain conditions. Such a
domain is Arrow consistent.


Full *identity* of preferences is not needed for Arrow
consistency. It can be enough that everybody's preferences are
similar, even if they never entirely agree. An example illustrates the
case of *single peaked* domains.


Suppose three bears get together to decide how hot their common pot
of porridge will be. Papa bear likes hot porridge, the hotter the
better. Mama bear likes cold porridge, the colder the better. Baby
bear most likes warm porridge; hot porridge is next best as far as he
is concerned ("it will always cool off"), and he doesn't
like cold porridge at all. These preferences among hot, warm and cold
porridge 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 ]](fig1.png)

Figure 1




This preference profile is *single peaked*. Each bear has a
"bliss point" somewhere along the ordering of the options
by their temperature, and each bear likes options less and less as we
move along this common ordering away from the bliss point, on either
side. Single peaked *preferences* arise with respect to the
left-right orientation of political candidates, the cost of
alternative public projects, and other salient attributes of
options. Single peaked *profiles*, in which everybody's
preferences are single peaked with respect to a common ordering, arise
naturally when everybody cares about the same thing in the options
under consideration--temperature, left-right orientation, cost,
or what have you--even if, as with the bears, there is no further
consensus about which options are better than which.


Duncan Black (1948) showed that if the number of
voters is odd, and their preference profile is single peaked,
pairwise majority decision always delivers up an
ordering.[11]
Furthermore, he showed, the maximum of this ordering is the bliss
point of the *median* voter--the voter whose bliss point
has, on the common ordering, as many voters' bliss points to one side
as it has to the other. In the example this is Baby bear, and warm
porridge is the collective maximum. This example illustrates the way
in which single peakedness can facilitate compromise.


Say the number of voters is odd. Now consider a
single-peaked *domain*--one that is made up entirely of
single peaked profiles. Black's result tells us that pairwise
majority decision on this domain satisfies *SO*. Provided the
domain is sufficiently inclusive (so that for each \(i\) there is
within the domain some profile in which \(i\) is not the median voter)
it also satisfies *D*. Pairwise majority decision always
satisfies *WP* and *I*, so such a domain is Arrow
consistent.

 With an even number of voters, single peakedness
does not ensure satisfaction of *SO*. For example, suppose
there are just two voters and that their individual orderings are:


1. *CAB*
2. *BCA*


Pairwise majority decision derives from this profile a weak social
preference for \(A\) to \(B\), since there is one who weakly prefers \(A\)
to \(B\), and one who weakly prefers \(B\) to \(A\). Similarly, it derives
a weak social preference for \(B\) to \(C\). Transitivity requires a weak
social preference for \(A\) to \(C\), but there is none. On the contrary,
there is a strict social preference for \(C\) above \(A\), since that is
the unanimous preference of the voters. Still, this profile is
single 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.]](fig2.png)

Figure 2




Majority decision with "phantom" voters can be used to
establish Arrow consistency when the number of people is even. Let
there be \(2n\) people, and let each profile in the domain be single
peaked with respect to one and the same ordering of the
alternatives. Let \(R\_{2n+1}\) be an ordering that also is single peaked
with respect to this common ordering. \(R\_{2n+1}\) represents the
preferences of a "phantom" voter. Now take each profile
\(\langle R\_{1},\ldots, R\_{2n}\rangle\) in the domain and expand it
into \(\langle R\_{1},\ldots, R\_{2n}, R\_{2n+1}\rangle\), by adding
\(R\_{2n+1}\). The set of all the expanded profiles is a single peaked
domain and, because the real voters together with the phantom are odd
in number, Black's result applies to it. Let \(g\) be pairwise majority
decision for the expanded domain. We obtain a social welfare function
\(f\) for the original domain by assigning to each profile the ordering
that \(g\) 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 \(f\) satisfies *SO* because \(g\) does. It
satisfies *WP* because there are more real voters than
phantoms (\(2n\) to \(1\); we could have used any odd number of phantoms
smaller than \(2n\)). \(f\) satisfies *I* because the phantom
ordering is the same in all profiles of the expanded domain. If the
domain includes sufficient variety among profiles then \(f\) also
satisfies *D* and is Arrow consistent. This nice idea of
phantom voters was introduced by Moulin (1980), who used it to
characterize a class of voting schemes that are non-manipulable, in
that they do not provide opportunities for strategic
voting. 


Domain restrictions have been the focus of much
research in recent decades. Gaertner (2001) provides a general
overview. Le Breton and Weymark (2006) survey work on domain
restrictions that arise naturally when analyzing economic problems in
Arrow's framework. Miller (1992) suggests that deliberation can
facilitate rational social choice by transforming initial preferences
into single peaked preferences. List and Dryzek (2003) argue that
deliberation can bring about a "structuration" of
individual preferences that facilitates democratic decision making
even without achieving full single-peakedness. List et al. (2013)
present empirical evidence that deliberation sometimes does have this
effect.


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



> 
> 
> [O]ne and only one of the [...] possible
> patterns of individuals' orderings is needed. [...]
> From *it* (not from each of them all) comes a social
> ordering. (Samuelson 1967: 48-49)
> 


 According to Sen (1977), though, the
Bergson-Samuelson social welfare function has more than a single
profile in its domain. It has in fact a completely unrestricted
domain, for while according to Samuelson only one profile is needed
"it could be *any* one" (Samuelson 1967: 49). What
distinguishes the single profile approach, on Sen's way of
understanding it, is that there is to be no coordinating the behavior
of the social welfare function at several different profiles, by
imposing on it interprofile conditions such as *I*
and *SN* (see Section 4.5). Either
way, though, and just as Samuelson insisted, Arrow's theorem does not
limit the single profile approach because one of its conditions is
inappropriate in connection with it. Either *U* is
inappropriate (if there is a single profile in the domain) or
else *I* is inappropriate (if there are no interprofile
constraints).


Certain impossibility theorems that are closely related to Arrow's
have been thought relevant to single-profile choice even so. These
theorems do not use Arrow's *inter*profile condition \(I\) but
use instead an *intra*profile neutrality condition. This
condition says that whenever within any *single* profile the
pattern of individual preferences for one pair \(x\),\(y\) of options is
the same as for another pair \(z,w\), the social ordering derived
from this profile must also be the same for \(x\), \(y\) as it is for \(z\),
\(w\):



> 
> *Single-Profile Neutrality* (*SPN*): For any \(\langle
> R\_{i}\rangle\), 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 \(z f\langle
> R\_{i}\rangle w\), and \(y f\langle R\_{i}\rangle x\) if and only if \(w
> f\langle R\_{i}\rangle z\).
> 
> 
> 


*SPN* follows from the strong neutrality (*SN*)
condition of Section 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 theorem
using *SPN* instead of *I*. These theorems were
supposed to block the Bergson-Samuelson approach. In fact,
condition *SPN* is just as easily set aside as *SN*,
and for the same reason: both exclude non-welfare information that is
relevant to the comparison of social states from an ethical
standpoint. Samuelson (1977) ridiculed *SPN* using an example
about redistributing chocolate. It is similar in structure to the
example of Peter and Paul, in
 Section 4.5.


### 5.2 More Ordinal Information


*Independence of Irrelevant Alternatives* severely limits
which information about individual preferences may be used for
what. It requires a social welfare function, when assembling the
social preference among a pair of alternatives, to take into account
only those of people's preferences that concern just this pair. This
Section discusses two kinds of information that is implicit in
preferences for other alternatives, and illustrates their use in social
decision making: information about the positions of alternatives in
individual orderings, and information about the fairness of social
states. 


*Positional* voting methods take into account where the
candidates come in the different individual orderings--whether it
is first, or second, ... or last. *Borda counting* is an
important example. Named after Jean-Charles de Borda, a contemporary
of Condorcet, it had already been proposed in the 13th
Century by the pioneering writer and social theorist Ramon
Lull. Nicholas of Cusa in the 15th Century recommended it
for electing Holy Roman Emperors. Borda counting is used in some
political elections and on many other occasions for voting, in clubs
and other organizations. Consider the profile:


1. *ABCD*
2. *BACD*
3. *BACD*


Let each candidate receive four points for coming first in some
voter's ordering, three for coming second, two for a third place and a
single point for coming last; the alternatives then are ordered by the
total number of points they receive, from all the voters. The Borda
count 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 method
applies with the obvious adaptation to any election with a finite
number of candidates.


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


1. *ACDB*
2. *BACD*
3. *BACD*


Then \(B\) will receive just 9 points \((1+4+4)\). \(A\) receives
the same 10 as before, though, and now outranks \(B\). This example
illustrates two important points. First, Borda counting does not
satisfy Arrow's condition *I*, since while each voter's ranking
of \(A\) with respect to \(B\) is the same in the two profiles, the
social ordering of this pair is different. Second, Borda counting
provides opportunities for voters to manipulate the outcome of an
election by strategic voting. If everybody's preferences are as in the
first profile, voter \(1\) might do well to misrepresent his
preferences by putting \(B\) at the bottom of his list. In this way,
he can promote his own favorite, \(A\), to the top of the social
ordering (he will get away with this, of course, only if the other
voters do not see what he is up to and adjust their own rankings
accordingly, by putting his favorite \(A\) at the bottom). The
susceptibility of Borda counting to strategic voting has long been
known. When this was raised as an objection, Borda's indignant
response is said to have been that his scheme was intended for honest
people. Lull and Nicholas of Cusa recommended, before voting by this
method, earnest oaths to tell the truth and stripping oneself of all
sins.



For further discussion of positionalist voting methods, see the
entries voting methods
and social choice theory; for an
analytical overview, see Pattanaik's (2002) handbook article. Barbera
(2010) reviews what is known about strategic voting. 


Mark Fleurbaey (2007) has shown that social welfare functions need
more ordinal information than *I* allows them if they are to
respond appropriately to a certain fairness of social states. He gives
the example of Ann, who has ten apples and two oranges, and Bob, with
three apples and eleven oranges. This allocation is said to be
"envy free" if, intuitively, she is at least as happy with
her own basket of fruit as she would be with his and, similarly, he is
as happy with his own basket as he would be with hers. Let the
distribution of fruit in one social state \(S\) be as described, and
consider the state \(S^\*\) in which the allocations are reversed. That
is, in \(S^\*\) it is Ann that has three apples and eleven oranges,
while Bob has ten apples and two oranges. More technically, \(S\)
is *envy free* if Ann weakly prefers \(S\) to \(S^\*\), and Bob
does too. We might expect that, other things being equal, the envy
freeness of a social state will promote it in the social ordering
above an alternative state that is not envy free. But *I* does
not 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 orange
from Bob and give both of them to Ann. Let \(T\) be the state arising
from this transfer. We assume for the sake of the example that Ann
always strictly prefers having more of everything for herself to
having 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 preferences
among these states are opposite and do not by themselves entail a
social preference one way or the other. Now, consider first a profile
in which the status quo \(S\) is envy free but \(T\) is not. (Let both
Ann 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 worse
off.) Relative to this profile, a social welfare function promoting
envy freeness comes out against the transfer from Bob to Ann by
ranking \(S\) strictly above \(T\). But consider another profile, in
which it is \(T\) that is envy free (now Ann and Bob are indifferent
between the baskets in \(T\), and agree that in \(S\) Ann is worse off
than Bob). Relative to this second profile, \(T\) outranks \(S\) in
the social ordering. In direct conflict with *I*, the social
preference among \(S\) and \(T\) switches as we go from one profile to
the other, although the individual preferences among \(S\) and \(T\)
stay the same. The social ranking of \(S\) and \(T\) turns on
preferences 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 options
> should only depend on the population's preferences on these two
> options and on what options are indifferent to each of these options
> for each individual (Fleurbaey 2007: 23).
> 
> 
> 


Fleurbaey (2007) discusses social welfare functions satisfying weak
independence together with Arrow's conditions apart, of course,
from *I*. The approach to social welfare that is sketched there
is 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 to
allow people to contribute more ordinal information about their
preferences than is admissible within Arrow's framework. Arrow's
purpose in limiting individual inputs to weak orderings of the
alternatives was to exclude cardinal information about utilities. This
limitation, though, is too tight for its purpose. One way for people
to communicate their preferences is to *grade* their
alternatives. Now, although in general grades do not carry cardinal
information, Arrow's framework has no provision for people to input
their preferences using them. This might seeem a small oversight on
Arrow's part but it is critical, because there are ways of aggregating
graded inputs that are not avaliable for preferance orderings, and
with them comes another "escape" from Arrow's
impossibility. Showing how it goes requires a slight extension of
Arrow's framework.



Let a *grade language* \(L\) be a strictly ordered collection
of expressions, the grades. The grades could be adjectival expressions
of a natural language, such as *excellent*, *good*, *acceptable*, *poor* and *terrible*,
or they could be characters such as the familiar letter grades of
academic evaluation. They could be numerals (then often
called *scores*) or the strings of stars commonly used to
evaluate hotels and restaurants. They could be any symbols that come
in some fixed order (from "top" to
"bottom").



A *grade function* \(G\_{i}\) maps the set \(X\) of alternatives
into some given grade language \(L\). Intuitively, \(G\_{i}(x)\) is
\(i\)'s grade for alternative \(x\). A grade function \(G\_{i}\)
contains *as 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 \(G\_{i}\) contains *more* information than
the corresponding ordering because we cannot always go in reverse:
different grade functions correspond to the same ordering. For
instance, one way to convey that \(A\) is preferable to \(B\) is to
say that whereas \(A\) is *good*, \(B\) is
merely *acceptable*; another is to say that whereas \(A\)
is *good*, \(B\) is *terrible*, but this doesn't come to
the same thing because *acceptable* and *terrible*
don't. The additional information in grades is in general not cardinal
information because for instance the expressions *good*, *acceptable* and *terrible*, ordinary
adjectives of English, do not carry cardinal information about how
good \(A\) and \(B\) are.



A *grade 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 welfare
function \(f\) are defined as in Arrow's framework, putting grade
profiles instead of Arrow's profiles of weak orders (the output of a
social welfare function is not a social grading of the alternatives
but a social ordering of them, just in Arrow's original
framework). The conditions of Arrow's theorem are reformulated
accordingly; 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*): For
> all alternatives \(x\) and \(y\) in \(X\), and all profiles \(\langle
> G\_{i}\rangle\) 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 that
whether one alternative ranks higher than another in the social
ordering depends only on everybody's grades for these two
alternatives. Other alternatives are in this matter
"irrelevant."



There are social welfare functions that satisfy all conditions of
Arrow's theorem, reformulated for graded inputs. With an odd number
\(n\) of people, for instance, the social welfare function *median
grading* fills the bill. Given a profile \(\langle G\_{i}\rangle\)
of grade functions, the *median* 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 to
bottom. The social ordering associated with this profile goes by the
alternatives' median grades: \(x f\langle G\_{i}\rangle y\) means that
the 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 \(d\) is
a *dictator* of \(f\) if for any alternatives \(x\) and \(y\),
and for any profile \(\langle\ldots, G\_{d},\ldots\rangle\) in the
domain of \(f\): if \(xP\_{d}y\), then \(xPy\). When \(f\) is median
grading, this means that whenever a dictator expresses a strict
preference for one alternative over another, by giving it a higher
grade, this alternative also ranks strictly higher in the social
ordering, having a higher median grade. Except in the trivial case of
a language with only a single grade (which makes it impossible to
express strict preferences, giving everybody vacuous dictatorial
powers), and provided there are at least two alternatives and three
people, median grading on an unrestricted domain has no dictator. To
see that it is so, consider any given person \(p\), and any two
alternatives \(A\) and \(B\). In an unrestricted domain there is some
profile in which \(p\)'s grade for \(A\) is higher than \(p\)'s grade
for \(B\), but in which everybody else's grades for these alternatives
are just the other way around: everybody other than \(p\) gives to
\(B\) the same grade that \(p\) gives to \(A\), and gives to \(A\) the
same grade that \(p\) gives to \(B\). In this profile, the median
grade 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. Repeating
this demonstration for each person, no one else is a dictator
either. Satisfaction of the other reformulated conditions on the
unrestricted domain of grade profiles is similarly
straightforward.



Michel Balinski and Rida Laraki (2007) showed that scoring and grading
enable an "escape" from Arrow's impossibility while
staying within an ordinal framework. Their theory of *majority
judgment* is a generalization of median grading that makes more
use than it does of the ordinal information in grade profiles
(Balinski and Laraki 2010). Balinski and Laraki insist that people
must share what they call a "common language" of grades,
without giving this notion a precise sense. Indeed it can be shown
that, when different people might have very different thresholds for
awarding grades, there is in grade profiles no more information than
in the corresponding profiles of weak orderings. Then a close relative
of Arrow's impossibility returns (Morreau 2016).



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



> 
> Now there's another possible way of thinking about it, which is not
> included in my theorem \(\ldots\) [E]ach voter does not just give a
> ranking. But says, this is good. And this is not good. Or this is very
> good. And this is bad. So I have three or four classes \(\ldots\) This
> changes 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 the
preferences of individuals \(i\) as utility functions \(U\_{i}\) that
map the alternatives onto real numbers: \(U\_{i}(x)\) is the utility
that \(i\) obtains from \(x\). A preference profile in Sen's
framework is a list \(\langle U\_{1}, \ldots, U\_{n}\rangle\) of utility
functions, and a domain is a set of these. An aggregation function,
now a social welfare *functional*, maps each profile in some
domain onto a weak ordering of the alternatives.


Sen showed how to study various assumptions concerning the
measurability and interpersonal comparability of utilities by
coordinating the social orderings derived from profiles that,
depending on these assumptions, carry the same information. For
instance, ordinal measurement with interpersonal
noncomparability--built by Arrow right into his technical
framework--amounts, in Sen's more flexible set up, to a
requirement that the same social ordering is to be derived from any
utility profiles that reduce to the same list of orderings. At the
other extreme, utilities are measured on a ratio scale with full
interpersonal comparability if those profiles yield the same social
ordering that can be obtained from each other by rescaling, or
multiplying 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 by
itself enough to avoid an impossibility result. In addition, utilities
have to be interpersonally comparable. Intuitively speaking, to put
information about preference strengths to good use it has to be
possible to compare the strengths of different individuals'
preferences. See Sen (1970: Theorem 8\*2). Interpersonal comparability
opens up many possibilities for aggregating utilities and preferences.
Two important ones can be read off from classical utilitarianism and
Rawls's difference principle. For details, see the entry
on social choice theory.


## 6. Reinterpretations



The Arrow-Sen framework lends itself to the study of a range of
aggregation problems other than those for which it was originally
developed. 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 relevant
to public decisions (see Estlund 2008, but compare Peter 2011). This
idea receives some support from Condorcet's jury theorem. It tells us,
simply put, that if individual people are more likely than not to
judge correctly in some matter of fact, independently of one another,
then the collective judgment of a sufficiently large group, arrived at
by majority voting, is almost certain to be correct (Condorcet
1785). The phenomenon of the "wisdom of crowds",
facilitated by cognitive diversity among individuals, provides further
and arguably better support for the epistemic conception (Page 2007,
Landemore 2012). But there are theoretical limits to the possibilities
for collective judgment on matters of fact. Starting with Kornhauser
and Sager's (1986) discussion of group deliberation in legal settings,
work on the theory of judgment aggregation has explored paradoxes and
impossibility theorems closely related to those that Condorcet and
Arrow discovered in connection with preference aggregation. See List
(2012) and the entry social choice
theory for overviews.


### 6.2 Multi-Criterial Decision


In many decision problems there are several criteria by which to
compare 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 and
conditions are appropriate, then tells us that there is no procedure
for arriving at an "overall" ordering that assimilates
different criterial comparisons.


The Arrow-Sen framework has been used to study multi-criterial
evaluation in industrial decision making (Arrow and Raynaud 1986) and
in engineering design (Scott and Antonsson 2000; compare Franssen
2005). Anandi Hattiangadi (forthcoming) argues that Arrow's theorem
limits the possibilities for an interpretationist metasemantics, in
which the correct interpretation of linguistic expressions is the one
that is, judging by relevant criteria, overall best.


Kenneth May (1954) used Arrow's framework to study the
determination of individual preferences. It had been found
experimentally that people's preferences, elicited separately for
different pairs of options, often are cyclical. May explained this by
analogy with the paradox of voting as the result of preferring one
alternative to another when it is better by more criteria than not.
More generally, he reinterpreted Arrow's theorem as an argument that
intransitivity of individual preferences is to be expected when
different criteria "pull in different directions". Susan
Hurley (1985, 1989) considered a similar problem in practical
deliberation when the criteria are moral values. She argued that
Arrow's theorem does not apply in this case. One strand of her
argument is that, unlike a person, a moral criterion can rank any
given alternatives just one way. It cannot "change its
mind" about them (Hurley 1985: 511), and this makes it
inappropriate to impose the analogue of the domain
condition *U* on procedures for weighing moral reasons. 


There are multicriterial problems in theoretical deliberation as
well. Okasha (2011) uses the Arrow-Sen framework to study the problem
of choosing among rival scientific theories by criteria including fit
to data, simplicity, and scope. He argues that the impossibility
theorem threatens the rationality of theory choice. See Morreau (2015)
for a reason to think that it does not apply to this problem, and
Morreau (2014) for a demonstration that impossibility theorems
relevant to single profile choice (see 
Sections 2.2 and 5.1) 
might sometimes apply even so. In related work, Jacob Stegenga (2013)
argues that Arrow's theorem limits the possibilities for combining
different kinds of evidence. Eleonora Cresto and Diego Tajer
(forthcoming) counter that confirmational holism requires a
restriction on the domains of evidence-aggregation functions that
blocks Stegenga's argument: they cast the Duhem problem in a positive
light as one aspect of a phenomenon that makes evidence aggregation
possible.


### 6.3 Overall Similarity


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


Overall similarity lies at the foundation of David Lewis's
metaphysics (Lewis 1968, 1973a, 1973b). He wrote little about how
similarities and differences in various respects might go together to
yield overall similarities, though (Lewis 1979) gives some idea of
what he had in mind. The Arrow-Sen framework lends itself to studying
this aggregation problem as well; and an impossibility theorem, if it
applies, limits the possibilities for arriving at overall similarities
of the sort that Lewis presupposes. Morreau (2010) presents the case
that 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 notion
of *verisimilitude* has distinguished "likeness"
and "content" dimensions, and the question arises whether
these can be combined into a single ordering of theories by their
overall verisimilitude. Zwart and Franssen (2007) argue that Arrow's
theorem does not apply to this problem but, using a theorem inspired
by it, they argue that there is no good way to combine the different
dimensions even so. See Schurz and Weingartner (2010) and Oddie (2013)
for constructive criticism of their views.