Weingast B.R., Wittman D. The Oxford Handbook of Political Economy

Подождите немного. Документ загружается.

412 interpersonal comparisons of well-being

common to restrict the individual utilities by assuming, for instance, that consuming

more of all goods is always better than consuming less.

We assume that a social ranking is an ordering—a reﬂexive, transitive, and com-

plete social goodness relation. A social evaluation functional is welfarist if and only

if there exists a social evaluation ordering (referred to as a social welfare ordering

by Gevers 1979) on the set of all possible utility vectors such that, for any proﬁle

U =(U

,...,U

) and for any two alternatives x and y, x is socially at least as

good as y for the proﬁle U if and only if the utility vector u =(u

,...,u

(x),...,U

(x)) = U(x) is at least as good as the utility vector v =(v

,...,v

(y),...,U

(y)) = U(y) according to the social evaluation ordering. Welfarism is

a consequence of three axioms: unlimited domain, Pareto indiﬀerence, and binary

independence of irrelevant alternatives. We now provide formal deﬁnitions of these

axioms.

Unlimited domain requires the social evaluation functional to produce a social

ordering for every logically possible utility proﬁle. That is, no individual utility func-

tion and no combination of such functions is excluded as a possible way of assigning

individual well-being to the alternatives.

Unlimited domain: The social evaluation functional generates a social ordering for

every logically possible utility proﬁle.

Pareto indiﬀerence demands that if, according to a utility proﬁle U, the individual

utilities for two alternatives x and y are the same, then x and y must be equally good

according to the social ranking generated by U .

Pareto indiﬀerence: For all alternatives x and y and for all proﬁles U,ifU(x)=

U(y), then x and y are equally good for the proﬁle U.

Binary independence of irrelevant alternatives is a consistency condition that im-

poses restrictions across diﬀerent proﬁles. If the utilities for two alternatives x and y

are the same in two proﬁles U and V, then the social rankings of x and y resulting

from the two proﬁles should be the same.

Binary independence of irrelevant alternatives: For all alternatives x and y and for

all proﬁles U and V,ifU(x)=V(x)andU(y)=V(y), then the ranking of

x and

y for the proﬁle U is the same as the ranking of x and y for the proﬁle V.

For a social evaluation functional that satisﬁes unlimited domain, Pareto indif-

ference and binary independence of irrelevant alternatives together are equivalent

to welfarism. This result, which is implicit in d’Aspremont and Gevers (1977)and

explicit in Hammond (1979), is referred to as the welfarism theorem. It requires our

maintained assumption that there are at least three alternatives.

Theorem 1: Suppose a social evaluation functional satisﬁes unlimited domain. The

social evaluation functional satisﬁes Pareto indiﬀerence and binary independence of

irrelevant alternatives if and only if there exists a social evaluation ordering of all

utility vectors such that, for all alternatives x and y and for all proﬁles U, x is at least

charles blackorby & walter bossert 413

as good as y for the proﬁle U if and only if the utility vector U (x) is at least as good as

the utility vector U(y) according to the social evaluation ordering.

This result is quite remarkable. It states that, given the welfarism axioms, a single

ordering of utility vectors is suﬃcient to establish the social ranking for any pro-

ﬁle. Note that the social evaluation ordering does not depend on the proﬁle under

consideration—the same ordering must be used for any proﬁle. Thus, the only in-

formation required to rank two alternatives x and y for a proﬁle U consists of the

two utility vectors U(x)andU(y); knowledge of any other features of the individual

utility functions is not needed.

Blackorby, Bossert, and Donaldson (2005a) prove a generalized version of the

welfarism theorem where multiple proﬁles of individual and social non-welfare in-

formation are permitted.

3 Axioms and Examples

.............................................................................

We now formulate some basic axioms regarding the social evaluation ordering of

utility vectors. The ﬁrst of these is anonymity. It ensures that the social evaluation

ordering treats individuals impartially, paying no attention to their identities: if we

relabel the utilities in a utility vector u,theresultingvectorisasgoodasu.Suchare-

labeling is called a permutation of a utility vector. A permutation of u =(u

,...,u

)

is a utility vector v =(v

,...,v

)suchthateachindexi in u is matched to exactly

one index j in v such that u

= v

. For example, (u

, u

)isapermutationof

, u

). Anonymity requires that any permutation of a utility vector u is as

good as u itself. This is a strengthening of Arrow’s (1951) condition that prevents the

existence of a dictator.

Anonymity: For all utility vectors u and v,ifv is a permutation of u,thenu and v

are equally good.

As an example, consider the utility vectors u =(1, 0, −3) and v =(0, −3, 1). v is

a permutation of u because v

= u

, v

= u

,andv

= u

. Thus, anonymity requires

thetwovectorstobeequallygood.

Pareto principles impose monotonicity properties on the social evaluation order-

ing. They require that the social evaluation ordering respond positively to increases

in utility. The weak Pareto principle requires an increase in everyone’s utility to be

regarded as a social improvement.

Weak Pareto: For all utility vectors u and v,ifu

for all i,thenu is better

than v.

For example, weak Pareto requires that the utility vector u =(1, −2, 4) is better

than v =(−1, −3, 0) because everyone is better oﬀ in u than in v. In conjunction

414 interpersonal comparisons of well-being

with anonymity, weak Pareto also implies that if everyone is better oﬀ in a per-

mutation of a utility vector u than in v,thenu is better than v even if u and v

cannot be compared according to the weak Pareto principle alone. For example, let

u =(−2, 1, 4) and v =(−1, −3, 0). Weak Pareto alone is silent about the ranking of

u and v. However, anonymity implies that u and (1, −2, 4) are equally good and, as

mentioned earlier, (1, −2, 4) is better than v by weak Pareto. By transitivity, u must

be better than v.(SeealsoSuppes1966 for a discussion.)

The strong Pareto requirement extends weak Pareto to cases in which no one’s

utility decreases and at least one individual’s well-being increases.

Strong Pareto: For all utility vectors u and v,ifu

≥ v

for all i with at least one

strict inequality, then u is better than v.

To illustrate the diﬀerence between strong Pareto and weak Pareto, consider the

two utility vectors u =(2, −2, 0) and v =(2, −3, 0). Strong Pareto requires that u is

better than v but weak Pareto does not.

Continuity is a condition that prevents the goodness relation from exhibiting

“large” changes in response to “small” changes in the utility distribution.

Continuity: For all utility vectors u and v and for all sequences of utility vectors

u



j=1, 2,...

where u

=(u

,...,u

) for all j,

(a)ifthesequenceu



j=1, 2,...

approaches v and u

is at least as good as u for all

j,thenv is at least as good as u;

(b)ifthesequenceu



j=1, 2,...

approaches v and u is at least as good as u

for all

j,thenu is at least as good as v.

To illustrate the continuity axiom, consider the following example. Let u =

(1, −1, −1) and v =(0, 0, 0). Suppose that a sequence of utility vectors u



j=1, 2,...

given by u

=(0, 0, 1/j) for all j and that u is at least as good as u

for all j . Because

the sequence u



j=1, 2,...

approaches v,continuityrequiresthatu is at least as good

as v.

The next axiom is an equity requirement; see d’Aspremont and Gevers (1977)and

Deschamps and Gevers (1978). It is called minimal equity and, loosely speaking, it

requires that the social ordering does not always favor inequality. More precisely,

the axiom requires the following. Consider two utility vectors u and v such that the

utilities of all but two individuals i and j are the same in u and in v and the utilities of

i and j are closer together in u than in v in the sense that v

> u

;thatis,

in moving from v to u,thebetter-oﬀ individual j loses and the worse-oﬀ individual

i gains, without reversing their relative ranks. The pair given by u =(3, 4, 7) and

v =(3, 2, 12) is an example for such a situation. Individual 1 has the same utility in

both, individual 2 is worse oﬀ than individual 3 in both vectors, individual 2 gains and

individual 3 loses when moving from v to u. An extremely equality-averse ordering

would always declare v better than u in these circumstances, and the minimal equity

axiom requires that this is not the case: there must be at least one pair of utility vectors

u and v with these properties such that u is at least as good as v.

charles blackorby & walter bossert 415

Minimal equity: There exist utility vectors u and v and individuals i and j such

that v

> u

, u

= v

for all other individuals k and u is at least as good

as v.

Finally, we introduce a separability property. This independence condition limits

the inﬂuence of the well-being of unconcerned individuals on the social ordering.

Suppose that a social change aﬀects only the utilities of the members of a population

subgroup. Independence of the utilities of unconcerned individuals requires the social

assessment of the change to be independent of the utility levels of people outside the

sub-group.

Independence of the utilities of unconcerned individuals: For all utility vectors u,v,

and

v and for all sub-groups of individuals, if u

and v

for all individuals

i in this sub-group, and u

= v

and

for all individuals j not in this sub-

group, then u is at least as good as v if and only of

u is at least as good as

In this deﬁnition, the individuals not in the sub-group are the unconcerned—they

have the same utilities in u and v and in

u and

v. Independence of the utilities of

unconcerned individuals requires the ranking of u and v to depend on the utili-

ties of the concerned individuals, those in the sub-group, only. The corresponding

separability axiom for social evaluation functionals can be found in d’Aspremont

and Gevers (1977) where it is called separability with respect to unconcerned indi-

viduals. D’Aspremont and Gevers’s separability axiom is called elimination of (the

inﬂuence of) indiﬀerent individuals in Maskin (1978)andinRoberts(1980b). In

the case of two individuals, the independence axiom is implied by the strong Pareto

principle. Therefore it is usually applied to societies with at least three individuals.

To illustrate this independence condition, consider the utility vectors u =

(2, −1, 0), v =(1, 0, 0),

u =(2, −1, 5), and

v =(1, 0, 5). Individual 3 is unconcerned

when considering the ranking of u and v because her or his utility is the same in both

situations. The same is true for

u and

v—again, individual 3’s utility is the same in

both. The concerned individuals—individuals 1 and 2—have the same utilities in u

and in

u,andinv and in

v. Thus, the ranking of u and v should be the same as

the ranking of

u and

v: the utility level of the unconcerned individual 3 should not

inﬂuence the social ranking.

We conclude this section with some examples of social evaluation orderings,

restricting attention to those characterized in this chapter.

The strongly dictatorial social evaluation orderings pay attention to the utility of

a single individual only. That is, a social evaluation ordering is strongly dictatorial if

and only if there exists an individual k such that, for all utility vectors u and v, u is

at least as good as v if and only if individual k’s utility in u isgreaterthanorequal

to k’s utility in v. For example, suppose we have a strong dictatorship with individual

2 as dictator. Then the utility vector u =(−5, 2, −1) is better than the utility vector

v =(1, 1, 5) because u

= 2 is greater than v

= 1. Strong dictatorships satisfy weak

Pareto, continuity, minimal equity, and independence of the utilities of unconcerned

individuals. Anonymity and strong Pareto are violated.

416 interpersonal comparisons of well-being

A strong positional dictatorship assigns dictatorial power to a position in the

society rather than to a named individual. A social evaluation ordering is a strong

positional dictatorship if and only if there exists a position k such that, for all utility

vectors u and v, u is at least as good as v if and only if the utility of the kth-best-oﬀ

individual in u is greater than or equal to the utility of the kth-best-oﬀ individual in v.

An important special case is the maximin ordering which is obtained for k = n,thatis,

the social ranking is determined by the utility of the worst oﬀ.Ifk = 1, the maximax

ordering, which pays attention to the best oﬀ only, results. Note that strong positional

dictatorships are not strong dictatorships because the kth-best-oﬀ individual in one

utility vector is not necessarily the same as the kth-best-oﬀ individual in another. For

example, consider the maximin ordering (the strong positional dictatorship where

k = n)andtheutilityvectorsu =(2, 1, 3) and v =(0, 1, 4). According to maximin,

u is better than v because the worst-oﬀ individual in u (individual 2)hasahigher

utility than the worst-oﬀ individual in v (individual 1). Strong positional dictator-

ships satisfy anonymity, weak Pareto, and continuity. They violate strong Pareto

and independence of the utilities of unconcerned individuals. All strong positional

dictatorships except maximax satisfy minimal equity.

Utilitarianism ranks any two utility vectors by comparing their sums of utilities.

Thus, according to utilitarianism, for all utility vectors u and v, u is at least as

good as v if and only if the sum of the utilities in u is greater than or equal to

the sum of the utilities in v. For example, according to utilitarianism, the util-

ity vectors u =(1

, −1, 0) and (0, 0, 0) are equally good because they have the

same total utility. Utilitarianism satisﬁes all of the axioms introduced earlier in

this section.

A social evaluation ordering is a weakly utilitarian ordering if and only if it respects

the betterness relation of utilitarianism. That is, for all utility vectors u and v,if

total utility in u isgreaterthantotalutilityinv,thenu is better than v.Weak

utilitarianism is a class of orderings rather than a single ordering because the ranking

of two utility vectors with the same total utility is not speciﬁed. For example, if

u =(2, −1, 0) and v =(3, 0, −3), any weakly utilitarian ordering must declare u

better than v because total utility in u is higher. On the other hand, the ranking of

u =(1, −1, 0) and v =(0, 0, 0) is not determined: unlike utilitarianism, weak utili-

tarianism does not require the two vectors to be equally good.

Leximin is a modiﬁed version of maximin in which utility vector u is better

than utility vector v if the worst-oﬀ individual in u is better oﬀ than the worst-oﬀ

individual in v. If these individuals are equally well oﬀ, the utilities of the next-

worse-oﬀ individuals are used to determine the social ranking, and the procedure

continues until either there is a strict ranking or the two utility vectors are permuta-

tions of each other, in which case they are declared equally good. For example, accord-

ing to leximin, the utility vector u =(2, 1, −3) is better than the vector v =(−3, 1, 0).

This is the case because the worst-oﬀ individuals (individual 3 in

u, individual 1 in v)

are equally well oﬀ and the second-worst-oﬀ individual in u (individual 2)isbetter

oﬀ than the second-worst-oﬀ individual in v (individual 3). Leximin satisﬁes all of

our axioms except continuity.

charles blackorby & walter bossert 417

4 Information Invariance

.............................................................................

Information invariance conditions restrict the information regarding the measur-

ability and interpersonal comparability of individual utilities that can be used in

social evaluation. As mentioned in the introduction, allowing for interpersonal

comparisons of well-being is a promising way out of the negative conclusion of

Arrow’s theorem which we will discuss shortly. The most common way to rep-

resent informational environments is to deﬁne the set of invariance transforma-

tions that can be applied to utility vectors without changing their informational

contents. Information invariance with respect to the information assumption rep-

resented by the set of admissible transformations then requires that the ranking

of any two utility vectors is the same as the ranking of the transformed vectors.

This approach was developed in contributions such as d’Aspremont and Gevers

(1977), Roberts (1980a, 1980b), and Sen (1974) and we follow it in this chapter. We

present the information invariance assumptions that are used in the remainder of

the chapter and refer the reader to Blackorby and Donaldson (1982), Blackorby,

Donaldson, and Weymark (1984), Bossert and Weymark (2004), d’Aspremont (1985),

d’Aspremont and Gevers (1977, 2002), DeMeyer and Plott (1971), Dixit (1980), Gev-

ers (1979), Roberts (1980b), and Sen (1970, 1974, 1977, 1986), for example, for more

extensive discussions.

If the only information that can be used is ordinal utility information without

interpersonal comparability, we obtain Arrow’s (1951) informational environment

that requires information invariance with respect to ordinal non-comparability. The

set of admissible transformations consists of all vectors of independent increasing

transformations.

Information invariance with respect to ordinal non-comparability: For all util-

ity vectors u and v and for all increasing functions 

,...,

, u is at

least as good as v if and only if (

),...,

)) is at least as good as

(

),...,

)).

Ordinal non-comparability implies that intrapersonal comparisons of utility levels

are possible. This is the case because an inequality such as u

≥ v

is preserved

whenever an increasing transformation is applied to all utility values of individual

i:wehaveu

≥ v

if and only if 

) ≥ 

) for all increasing functions 

Interpersonal comparisons of utility levels are not possible in this informational

environment. For example, suppose that u

=1andv

=0.Clearly,u

.How-

ever, under ordinal non-comparability, this inequality does not permit us to say

that individual i is better oﬀ in utility vector u than individual j in v because

the inequality is not preserved by all vectors of admissible transformations. Letting



)=u

and 

)=v

+ 2, it follows that 

)=1< 2=

)andthe

inequality is reversed. Of the social evaluation orderings deﬁned in the previous sec-

tion, only strong dictatorships satisfy information invariance with respect to ordinal

non-comparability.

418 interpersonal comparisons of well-being

If utilities are cardinally measurable and no requirements regarding the interper-

sonal comparison of utilities are imposed, we obtain the information assumption

cardinal non-comparability.

Information invariance with respect to cardinal non-comparability: For all utility vec-

tors u and v, for all positive constants a

,...,a

and for all constants b

,...,b

u is at least as good as v if and only if (a

+ b

,...,a

+ b

) is at least as good

as (a

+ b

,...,a

+ b

Information invariance with respect to ordinal non-comparability and information

invariance with respect to cardinal non-comparability are equivalent (see Sen 1970

and, for a diagrammatic illustration, Blackorby, Donaldson, and Weymark 1984).

Intuitively, this is the case because, for any two utility vectors u and v and for any

vector of increasing transformations, the values of the transformations can be repli-

cated by the values of a vector of increasing aﬃne transformations because a straight

line is determined by two points on that line. As an immediate consequence of this

equivalence, it follows that, among the social evaluation orderings of the previous

section, strong dictatorships are the only ones satisfying information invariance with

respect to cardinal non-comparability.

If utility levels are comparable both intrapersonally and interpersonally but no

further information is available, we obtain ordinal full comparability. In that case,

only common increasing transformations can be applied to the utilities without

changing the information relevant for social evaluation.

Information invariance with respect to ordinal full comparability: For all utility

vectors u and v and for all increasing functions 

, u is at least as good as v if

and only if (

),...,

)) is at least as good as (

),...,

)).

Ordinal full comparability implies that utility levels can be compared interperson-

ally. The inequality u

≥ v

is preserved even for diﬀerent individuals i and j if

a common transformation 

is applied to all utility values: we have u

≥ v

and only if 

) ≥ 

) for all increasing functions 

. Comparisons of util-

ity diﬀerences are not possible in this informational environment, either intrap-

ersonally or interpersonally. For example, suppose that u

=2,v

=0,w

=3,and

=2.Wehaveu

−v

=2> 1=w

−t

. However, this inequality is not preserved

if an arbitrary increasing transformation is applied to the individual utilities. Let-

ting 

)=u

, it follows that 

) −

)=8< 19 = 

) −

)and

the inequality is reversed. Information invariance with respect to ordinal full com-

parability is satisﬁed by the strongly dictatorial social evaluation orderings, the

strong positional dictatorships, and leximin. Utilitarianism does not satisfy this

invariance property.

If utilities are cardinally measurable and fully interpersonally comparable, both

utility levels and utility diﬀerences can be compared interpersonally. In this case,

the only admissible transformations are increasing aﬃne transformations which are

identical across individuals.

charles blackorby & walter bossert 419

Information invariance with respect to cardinal full comparability: For all utility

vectors u and v, for all positive constants a

and for all constants b

, u is at

least as good as v if and only if (a

+ b

,...,a

+ b

) is at least as good as

+ b

,...,a

+ b

Utility levels and utility diﬀerences can be compared because inequalities both of

the form u

≥ v

and of the form u

−v

≥ w

−t



are preserved if a common

increasing aﬃne transformation is applied to all utilities. This information invariance

axiom is satisﬁed by the strong dictatorships, the strong positional dictatorships,

utilitarianism, and leximin.

5 Impossibilities and

Characterizations

.............................................................................

In the absence of interpersonal comparisons of well-being, there do not exist satisfac-

tory social evaluation principles. A variant of Arrow’s (1951) theorem (see also Blau

1957) formulated for social evaluation orderings states that only strong dictatorships

satisfy weak Pareto, continuity, and information invariance with respect to ordinal

non-comparability.

Theorem 2: A social evaluation ordering satisﬁes weak Pareto, continuity, and in-

formation invariance with respect to ordinal non-comparability if and only if it is a

strong dictatorship.

Sen (1970) proves that replacing information invariance with respect to ordi-

nal non-comparability by information invariance with respect to cardinal non-

comparability does not provide an escape from the negative conclusion of Arrow’s

theorem. This observation is an immediate consequence of the equivalence of the

two information invariance conditions mentioned in the previous section. Thus, we

obtain the following result.

Theorem 3: A social evaluation ordering satisﬁes weak Pareto, continuity, and in-

formation invariance with respect to cardinal non-comparability if and only if it is a

strong dictatorship.

Clearly, strongly dictatorial principles are not anonymous and, therefore, if this

impartiality requirement is added to the list of axioms in Theorem 2 or in Theorem 3,

an impossibility results. Continuity is not required in this impossibility theorem.

Theorem 4: There exists no social evaluation ordering satisfying anonymity, weak

Pareto, and information invariance with respect to ordinal or cardinal non-

comparability.

The impossibility result of Theorem 4 remains true if anonymity is weakened to the

requirement that rules out the existence of a dictator (see Arrow 1951).

420 interpersonal comparisons of well-being

The conclusion we draw from Theorem 4 is that interpersonal comparisons of

utility must be permitted to obtain reasonable principles for social evaluation. One

possibility is to replace information invariance with respect to ordinal or cardinal

non-comparability by information invariance with respect to ordinal full comparabil-

ity. If anonymity, weak Pareto, and continuity are added, we obtain a characterization

of the class of strong positional dictatorships.

Theorem 5: A social evaluation ordering satisﬁes anonymity, weak Pareto, continuity,

and information invariance with respect to ordinal full comparability if and only if it

is a strong positional dictatorship.

We now discuss some axiomatizations of the utilitarian and leximin principles.

In addition to having an information invariance assumption as one of the axioms,

another common feature of the remaining results is that they all employ the sepa-

rability condition independence of the utilities of unconcerned individuals. For that

reason, they all require the assumption that there are at least three individuals; for two

individuals, the axiom has no additional power given that the strong Pareto principle

is also imposed.

Deschamps and Gevers (1978) examine the class of social evaluation orderings

satisfying information invariance with respect to cardinal full comparability. If this

axiom is added to anonymity, strong Pareto, minimal equity, and independence of

the utilities of unconcerned individuals, only weakly utilitarian orderings and leximin

remainaspossibilities.

Theorem 6: Suppose there are at least three individuals. If a social evaluation order-

ing satisﬁes anonymity, strong Pareto, minimal equity, independence of the utilities

of unconcerned individuals, and information invariance with respect to cardinal full

comparability, then it is weakly utilitarian or leximin.

Some remarks regarding the role played by the various axioms in the theorem

statement are in order. Anonymity and strong Pareto ensure that the principle is

impartial and responds positively to increases in individual utilities. The indepen-

dence condition plays an important role because it brings out the separability prop-

erty that is shared by utilitarianism and leximin. Although other principles (such as

generalized utilitarianism which uses the sum of transformed utilities instead of the

sum of utilities; see, for instance, Blackorby, Bossert, and Donaldson 2002)satisfy

this property, they are ruled out by information invariance with respect to cardinal

full comparability. This assumption ensures that, in addition to utility levels, utility

diﬀerences are interpersonally comparable and the linearity of (weak) utilitarianism

guarantees that these principles satisfy the requirement whereas other generalized

utilitarian orderings do not. Minimal equity is required merely to rule out the ex-

tremely equality-averse leximax ordering. Leximax proceeds analogously to leximin

but it starts by comparing the utilities of the best oﬀ and works its way down to the

worst oﬀ in case of ties.

Theorem 6 is not an if-and-only-if result because not all weakly utilitarian order-

ings satisfy all axioms: the axioms also place restrictions on how a weakly utilitarian

charles blackorby & walter bossert 421

ordering ranks utility vectors which have the same sum. However, it shows that

there do not exist many orderings other than utilitarianism and leximin satisfying

the axioms of the theorem statement. The following two theorems illustrate how

utilitarianism or leximin can be obtained by modifying one or another of the axioms.

We begin with utilitarianism, which has received a considerable amount of atten-

tion in the literature on social choice. The following theorem is due to Maskin (1978;

see also Deschamps and Gevers 1978). It is obtained by replacing minimal equity with

continuity in Theorem 6.

Theorem 7: Suppose there are at least three individuals. A social evaluation

ordering satisﬁes anonymity, strong Pareto, continuity, independence of the utilities

of unconcerned individuals, and information invariance with respect to cardinal full

comparability if and only if it is utilitarian.

Theorem 7 illustrates the power of the continuity axiom. Leximin clearly is not

continuous and neither is leximax. Consequently, neither of these principles survives

if this property is added. Because minimal equity is required in Theorem 6 only

to exclude leximax, it is no longer needed in Theorem 7 because this principle is

already ruled out by continuity. Continuity also forces equal goodness according to

utilitarianism to be respected if the corresponding betterness relation is respected

and, therefore, utilitarianism rather than the weakly utilitarian rules are obtained.

D’Aspremont and Gevers (1977) provide an alternative characterization of utilitari-

anism that does not require the independence axiom (see also Blackwell and Girshick

1954;Milnor1954;andRoberts1980b).

A characterization of the leximin ordering due to d’Aspremont and Gevers (1977)

replaces information invariance with respect to cardinal full comparability by infor-

mation invariance with respect to ordinal full comparability in Theorem 6.

Theorem 8: Suppose there are at least three individuals. A social evaluation ordering

satisﬁes anonymity, strong Pareto, minimal equity, independence of the utilities of

unconcerned individuals, and information invariance with respect to ordinal full

comparability if and only if it is leximin.

Clearly, ordinal full comparability rules out all weakly utilitarian principles (including

utilitarianism itself) and, as a consequence, Theorem 8 follows immediately from

Theorem 6.

Hammond (1976) provides an alternative characterization of leximin. Because it

does not employ an information invariance condition, we do not state it here.

6 Concluding Remarks

.............................................................................

This chapter provides a brief introduction to welfarist social choice theory. It is argued

that the most promising route of escape from the negative conclusion of Arrow’s