RANKING SOCIAL DECISIONS WITHOUT INDIVIDUAL
PREFERENCES ON THE BASIS OF OPPORTUNITIES"
Carmen Herrero, Iñigo Iturbe-Ormaetxe & Jorge Nieto**
WP-AD 95-23
* Thanks are due to José Alcalde, Luis Corchón, Lany Kranich and Antonio Villar iux helpful comrnents.
Financia1 support From the DGICYT under project PB92-0342 and Protocolo Aquitania-Pals Vasco-Navarra is
gratefully acknowledged. The usual caveats applies.
+*
C. Herrero: University of Alicante and instituto Valenciano de investigaciones Económicas, 1. Iturhe:
University of Alicante, J. Nieto: Universidad Pública de Navarra.
Editor: Instituto Valenciano de
Investigaciones Económicas, S.A.
Primera Edición Diciembre 1995.
ISBN: 84-482-1174-X
Depósito Legal: V-5157-1995
Impreso por Copisteria Sanchis, S.L.,
mt,121-bajo, 46008-Valencia.
Printed in Spain.
RANKING SOCIAL DECISIONS WITHOUT INDIVIDUAL PREFERENCES
ON THE BASIS OF OPPORTUNITIES
Carmen Herrero, Iñigo Iturbe-Ormaetxe & Jorge Nieto
A B S T R A C T
In
this
paper
we
study
severa1
methods
of
opportunity sets by taking a s a primary notion
ranking
equality of
profiles
of
opportunity,
undestood a s equality of choice sets. Each of these social decision rules
looks f i r s t a t the size of
the common opportunity s e t available t o al1
members of
tbe society. If
this is not decisive, additional criteria a r e
used
lexicographic
in
a
procedure.
Axiomatic
characterizations
of
each
methos a r e also provided.
KEYWORDS: Equality of Opportunity, Ranking Profiles of Opportunity Sets.
1.- INTRODUCTION
The notion of
justice,
equality of
opportunity, a s an expression of
social
has played a major role in theoretical and applied economics and
indeed i t is almost ubiquitous in public economics.
The
most
equality
of
immediate
individual
interpretation
choice
sets
of
equality
[see Kolm
of
opportunity
is
(1973) (19951, Thomson
(1992)l. According t o this, equality of opportunity has mostly t o do with
equality of accessibility, and not with outcome equality [see also Arneson
(1989), Bossert
(1995), Cohen
Roemer
for
(1994)
justifiably
clear
restricts
conflict
different
the
between
practica1 relevance:
(1989), Dworkin
models
degree
the
of
ethical
in
which
outcome
value
(1981), Herrero
of
personal
equalityl.
previous
(1995) or
responsibiIity
Yet,
there
proposal
is
and
a
its
identical choice sets is a rather unlikely event even
f o r twins, and a theory focusing on this strict approach will leave a major
class of
rank
The relevant question is, therefore,
situations unsolved.
social
decisions
when
in
the
associated
profiles,
how t o
individual
opportunity sets a r e not identical.
This problem has been treated from a variety of viewpoints. We take
t h e less demanding approach in informational terms: the planner does not
take into account any information on individual preferences. To the best of
our knowledge the f i r s t attempt to rank distributions of
opportunity sets
in this setting is made in Kranich (1994).
Apart from the possible planner's lack of information about individual
preferences, two other difficulties appear related t o the use
of
private
information in evaluating opportunity sets. Tbe f i r s t one refers t o dynamic
considerations
and
changing
tastes
leading
to
the
preferente
for
flexibility model [see Kreps (197911. The second one is related t o adapting
preferences,
that
aspirations t o their
is,
to
the
possibilities,
idea
that
people
tend
to
adjust
a s argued by Elster by means of
s o u r g r a p e s paradox [see Elster (1982),pp.2191.
their
the
In Kolm's words [Kolm (199511,
"freedom
is
the
end-value
of
justice.
This
is
equivalent
to
considering individuals' preferences a s irrelevant for justice"
Kranich
(1994) suggests
a
way
of
evaluating
profiles
of
(finite)
opportunity sets faced by a group of agents. He proposses t o consider only
t h e difference in the number of elements faced by the agents a s a measure
of
the
degree
of
inequality
of
opportunity.
That
is,
equality
in
the
cardinality of the choice sets is al1 that matters.
There is a
source of discomfort in Kranich's proposal, namely t h a t
only the relative s i z e of the opportunity sets faced by the agents matters,
while neither the size of the common possibility s e t , nor the absolute s i z e
of the opportunity sets a r e taken into account. For instance, by means of
such a procedure, a profile A, in which both opportunity sets have a single
(and different) element is declared a s indifferent to a profile B, in which
both opportunity sets have many (and identical) elements.
The main proposal of this paper is to take a s a basic approach of
equality of opportunity the size of the cornrnon opportunity set, and declare
above in the ranking a situation having a bigger common set. A prohlem with
this
criterion
is
that
there
are
too
many
situations
which
are
not
distinguishable and i t seems sensible t o take into account also additional
characteristics of the opportunity profiles. Thus, a s a second step, and in
the case we face a couple of situations such that the size of the common
s e t s a r e identical, we may look a t those additional properties.
Since additional properties play a subsidiary role, we propose some
lexicographic method a s a proper way of ranking profiles of opportunity
sets: f i r s t we look a t the common opportunity set. If this is not decisive,
we
look
at
those
additional
properties.
With
this
in
mind,
three
alternative methods a r e studied in this paper, each stressing in different
ways the trade off between pure equality of
opportunity and efficiency.
For the sake of simplicity we only deal witb the two-person case.
In a f i r s t place we study the cornrnon opportunity relation, where only
equality of
Kranich's
opportunity issues a r e taking into account. In this case, t h e
cardinality
difference
relation
is
taken
as
a
subsidiary
criterion. In a second place, we propose the lexrnin opportunity relation,
where a t a second step we look a t the sizes of the individual opportunity
sets, starting by the smallest ones. Finally, we also study the utilitarian
opportunity relation. Now, the complementary criterion is the size of the
aggregate opportunity set.
Section 2 is devoted to notation and preliminaries.
Sections 3 t o 5
have a common structure. We present a particular social relation, then we
provide with an axiomatic characterizatíon of the relation, and we prove
the independence of the axioms used.
cornments, closes the paper.
Finally, Section 6, with some final
2.- PRELIMINARIES: NOTATION AND DEFINITIONS
Let N = {1,2, ...,n) denote the set of agents, and L be an infinite s e t
of opportunities. We denote the s e t of nonempty, finite subsets of L by
and
consider
lists
or
1
A = (A , . . . , A n ) , where A
i
profiles
of
f o r al1 i
E (e
opportunity
N. Let
E
sets
=.TI
(en
1
EN
of
the
(e,
form
denote the s e t of
(e
al1 prof iles.
A social relation is a complete, reflexive and transitive relation
defined on
t h a t ranks distributions of
(en
some fairnecs criteria. For A, B
E (e",
opportunities on the basis of
we write A
B meaning t h a t profile
2
A is socially considered a t least a s good a s profile B. A s usual, > and
a r e t h e asymmetric and reflexive parts, respectively, of
If
A,B
inclusion; A\B
For A
...
5
and
E
= { x
E
c
A
B,
Al x
C
B
#
A = (# A
denotes
i
2
strict
and
weak
B ); # A = number of elements in s e t A;
1
,...,#
b. f o r al1 i and a
i
# A ~ ( a
~ )
A"). Vector inequalities a r e denoted >>, >
with the usual interpretation í a >> b <=> a
2
-
2.
E", c(A) denotes a permutation of N such that # AC")
# A~'");
<=> a
A
2,
E
t
;c
b; a s b <=> a
> b f o r al1 i; a > b
i
i
z b. f o r al1 il.
i
I
The
lexicographic ordering L in R" is defined a s usually: for any a,b c IR",
a >. b <=> a. > b
and a
Consider
the
L
1'
i
1
2
now
A = (A ,A ), A'
E
(e.
k
= b
k
f o r al1 k
two-agents
< i.
case.
Thus,
Now, f o r a social relation
properties will be used in the sequel:
a
2
profile
on
Z
(e,
is
a
pair
the following
ANONYMITY (AN): F o r a n y A
E
1
2
g 2 , A = (A ,A 1, A
- (A ,A 1.
2
1
UNIFORMATION O F OPPORTUNITY S E T S (UNJ): F o r a n y A
X
Y S A'\A',
S A1\A2,
x
y
é
(A1\{x})u(A2\{x>),
6
é A2\{x2},
1
* y,
E
A1nA
2
,
y
L,
E
-
then
[ ( A ~ \ ~ X > ) U ~ ~ > , ( A ~ \ ~ X > A.
)U~~>I
E f2,
A
=
1
2
2
(A ,A 1, if x
E
1
2
A \A ,
t h e n [ A ~ , ( A ~ \ ~ X ~ > ) zU ~A.X > I
IRRELEVANT EXPANSIONS (IE): F o r a n y A
x
2
!E2, A = (A ,A 1, if x
UNCOMMON REPLACING (UR): F o r a n y A
x
2
t h e n I(A~V()U~~>,(A~\Y)U{X>)~
> A.
é A1n.4',
COMMON REPLACING (REP): F o r a n y A
1
A = (A ,A 1,
E f2,
E
1
2
2
gZ, A = (A ,A 1, if x, y é A1uA ,
-
t h e n [ A 1 u ~ x > , A 2 u ~ y > lA.
NEGATIVE MONOTONICITY (NM): F o r a n y A
1
E
f 2 , A = (A1,A2) if A' S A2 c B2,
2
t h e n A > (A ,B 1.
MONOTONICITY (MON): F o r any A
A1uAZ c B'UB'
E
x2,
1
A = (A ,A
2
if A'
2 B1,
A2
c
B2,
, t h e n B > A.
TRANSFER (SR): F o r a n y A E f2, A = (A1,A2), X
C
x
A2\A1,
é A'.
If
# (A1u{x>) 5 # (AZ\x), t h e n (A'U{X>,A~\X) > A.
IRRELEVANT TRANSFERS (IT): L e t A = (A1,A2)
(A'\{~),A~u{x>)
E
f Z , and x
E
A1\A2.
Then,
- A.
P r e v i o u s p r o p e r t i e s h a v e d i f f e r e n t s t a t u s . A f i r s t g r o u p of p r o p e r t i e s
have t o do either with the setting or with our basic principle in ranking
profiles
of
relations.
opportunity
A
alternative
second
sets,
group
complementary
and
of
are
fulfilled
properties
principies,
by
are
and
al1
reiated
they
the
proposed
with
possible
play
the
role
a
minirnal
of
differentiating among the relations we propose.
Within
principle.
the
It
first
says
group
that
of
agents'
AN
properties,
identities
play
is
no
role
on
fairness
the
social
ordering. UN1 is the expression of the priority of the common opportunity
set. It says that whenever opportunity sets becorne more similar, then t h e
new situation is ranked above the previous one. A s for REP and UR, they
reflect the planner's lack of inforrnation on individual preferences and/or
perceptions of the alternatives. So, REP has t o do with the way the planner
looks a t the cornmon set. Whenever sornething common is replaced by something
else, both situations a r e ranked equally. On the other hand, UR says t h a t
if we change a non comrnon element by a different one, the new situation is
not ranked beiow, reflecting not only the lack of information but also t h e
possibility
of
strict
irnprovements due t o a possible
enlargement of
the
common opportunity set.
In the second group of axioms, IE and NM follow the line of taking
only care of the pure concept of equality of opportunity. Adding different
opportunities t o both agents does not affect the ranking (IE), and if we
enlarge the opportunities of
degree
transfer
of
the agent facing the biggest set, then t h e
equality
of
opportunity
decreases
principle
as
a
decreasing
way
of
(NM).
TR
deals
inequalities:
with
the
whenever
we
transfer opportunities from those agents facing more opportunities t o those
facing less, the degree of inequality decreases. Both MON and IT deal with
different aspects of efficiency. MON means that whenever the opportunity
set of an agent expands, the situation becomes better. IT says that we rank
equally any situation having identical common opportunities irrespective of
who is the agent enjoying non common opportunities. Notice that IT implies
AN.
Consider now the following social relations:
DEFINITION O [Kranich (1994)l: For A, B
relation,
i
cd
x',
the cardinality difference
i s defined by:
DEFINITION 1: For A, B E
defined by: A
6
2
CO
the cornmon opportunity relation, z
(eZ,
co
is
B <=>
DEFINITION 2: For A, B
6 (e2,
the
utilitarian opportunity relation,
is
2
U
defined by:
DEFINITION 3: For A, B
defined by:
6
(ez,
the lexmin opportunity relation,
2
lo
is
Previous
social
relations
rank
profiles
of
opportunity
sets
in
different ways. For the cardinality difference relation al1 t h a t matters i s
whether opportunity sets have equal size.
consider
that
the
most
important
In the remaining relations we
criterion
deals
with
the
cornmon
opportunity set. With such a spirit, we s t a r t by considering the common
opportunity relation, which ranks profiles on the basis of the size of t h e
common
opportunity
differences in the
set,
size.
and,
at
a
The utilitarian
second
step,
by
looking
opportunity relation
at
looks
the
at
a
second step a t the size of the aggregate opportunities set. Finally, t h e
lexmin opportunity relation uses a s a complementary criterion the sizes of
both indiviual opportunity sets, starting by the smallest one.
3. AXIOMATIC CHARACTERIZATION OF THE COMMON OPPORTUNITY RELATION
THEOREM 1.- A weak preorder
on de2 satisfies AN, UNI, REP, UR, IE and NM
2
if and only if 2 coincides with 2
CO
.
PROOF:
Obviously,
satisfies
2
CO
al1 the
properties.
Let
us
see
the
sufficiency
part. By AN, we may assume, f o r any A E deZ that A = <r(A).
2' such that [ # (A1nA2),# Al = [ # (B1nB2),# B1. Then A
( i ) Let A, B
E
Choose x
A1nA2, y
E
E
- B.
-
B1nB2. Thus, by REP, [ ( ~ ~ \ ~ x ) ) u ~ y ~ , ( ~ ~ \A.
~x~)u~~~l
By repeating the procedure a suitable number of times, and by transitivity,
we obtain that C =
Now, take x
UR,
E
(c1,c2)- A,
A1\A2,
y
where
ci =
[A'\(A~~A~)IV(B~~B~).
B1\B2. Thus, by construction, x
E
1
[(C \ ~ X } ) U { ~ ~ ,2C ~
C.I By suitahly repetition
1
2
transitivity, we get that (B ,C )
2
of
1
1
2
C\C , and by
the procedure
and
C.
Apply a symrnetric procedure t o the previous one for x
1
E
E
2
1
A \A, y
E
B2\B1.
1
2
2
Thus, [B ,(C~\{X))U{~)I
2 (B ,C ). By repeating the procedure, B r (B ,C ).
Then, by transitivity, B r A.
By a similar procedure, starting by B, we also obtain A
therefore, A
1
-
# A
2
2
= # B'
-
# B
2
. TO
deZ such t h a t # (A'AA')
E
proof: A
application of IE, A
2
= # (B1nB2),
>
# B'.
Then, by repeated
- B.
A s a consequence, for any A
1
and
- B.
Assume, without loss of generality, t h a t # A'
B G B .
B,
- B.
(ii) Let A = (A ,A ), B = (B',B')
# A'
2
E
de
2
,
we may find B
-
1
2
A, B = (B ,B ), with
1
2
( i i i ) L e t A = (A ,A ), B = (B1,B2)
# A'
1
-
# A2
2
2
A S A c B
E
2
s u c h that # ( A1n A 2) = # ( B1n B 2),
2
> # B' - # B2. TO proof: A > B. Choose A, B s u c h that A1 = B1,
.
Then, b y NM, A > B.
1
2
(iv) L e t A = (A ,A ), B = (B',B')
E
1
1
2
2
s u c h t h a t # ( A n A ) > # ( B n B 1.
2'
T o p r o o f : A > B.
1
(iv.a) If m o r e o v e r , # A
-
2
# A
b
1
# B
1
-
2
# B2, c o n s t r ~ c tC = (B uX,B vX),
1
1
2
where X is chosen s u c h that # X = # ( B ' ~ B ~ -) # (A n ~ ' ) , X n (B UB ) =
Thus, #
(cinc2)
= # (A'~A'), # C' - #
c2 =
0.
# B1 - # B2. By UNI, C t- B, a n d
b y (iii), A > C . T h u s , b y t r a n s i t i v i t y , A > B.
(iv.b) T a k e n o w the c a s e # A'
S
ynA2 =
0.
# A2
2
< # B1 - # B . BY (i), we m a y a s s u m e
2
B2 c A' c A'.
that B1
-
NOW, t a k e X = A \B
1
,
a n d Y s u c h t h a t # Y = # X, a n d
By applying IE a s u i t a b l e n u m b e r of t i m e s ( # X), w e o b t a i n that
(B'UY,B~UX) = (B'UY,A~)
N o w t a k e Z = A1\B1.
- B.
Then, by UNI, A = [~(B'UY)\Y}UZ, A']
t-
( B 1 u ~ , A 2 ) ,a n d
b y t r a n s i t i v i t y , A t- B.
THEOREM 2.- AN, UNI, REP, UR, I E a n d NM a r e independent.
PROOF:
We p r o v i d e w i t h e x a m p l e s of w e a k p r e o r d e r s s a t i s f y i n g al1 the p r o p e r t i e s
b u t o n e at a n y time:
1
2
(AN) T a k e z1 defined b y (A , A )
[ # (B'~B'),
-
# B'
Z
2
(B1,B2) <=> [# (A'~A'),#
A'
- # A']
ZL
# B21. I t s a t i s f i e s REP, UNI, UR, IE a n d NM, b u t i t d o e s
n o t s a t i s f y AN: ((x,y,zj,(x,yj)
(UNI) Consider
Zl
=
2
cd'
n o t s a t i s f y UNI: (ta},{b})
>l({x,y},{x,y,z>).
I t s a t i s f i e s AN, REP, UR, I E a n d NM, b u t i t d o e s
- ({a},{aj).
(REP) Consider a fixed set K c L, Now, A > B <=> A >
3
B, and if A
CO
-
B,
CO
when # (A'nAZn~)> # (B1nB2n~).I t satisfies AN, UNI, UR, IE and NM, but
does not satisfy REP: Let A be such that x
E
1
A M',
e
x
K, y
K. Thus,
E
[(A'\(X>)U~~>,(A~\(X>)U{Y>)
>3 A.
(UR) Take a fixed set K c L. Now, A > B <=> A >
4
CO
B, and if A
-
B, when
CO
# [(A1uA2)n~1> # [ ( B ' U B ~ ) ~ K I .I t satisfies AN, UNI, REP, IE and NM, but
does not satisfy UR: Let A be such that x
y
st AZ\(x>. Thus, A
(IE) Take
2
>4
REP,
UR,
#
and
NM,
but
it
does
not
((x,z>,(x,y,t>)
(NM) Consider
[
#
2
6
1
-
#
AO(l)1
iL [
satisfies AN, UNI, REP, UR, IE but does
((x>,tx,y>lt6 ((x>,(x>).
B"(Z'
1
.
satisfy
y
e
K,
It
satisfies
AN,
UNI,
IE:
((x),(x,Y>).
2
defined a s follows: (A ,A )
(A1nAZ), # AO(')
K,
(B1,B2) <=>
5
# Bw(l)
(A1nAZ),
#
E
[A~,(A~\(X>)U(~>I.
defined a s follows: (A1,AZ)2
5
A2\A1, x
E
i6
#
not
1
2
(B ,B )
<=>
( B1n B2) , # Ae(')
fulfill
NM:
-
#
l. I t
AO(l)
4. THE UTILITARIAN OPPORTUNITY RELATION
THEOREM 3.- A weak preorder r on bez satisfies UNI, REP, UR, MON and IT
if and only if z coincides with
2
U
.
PROOF:
Since
2
U
obviously
satisfies al1 the properties,
we only deal with t h e
sufficiency part.
( i ) Let
B
A,
# (B'~B'), #
1
B
2
S B
.
repeated
A = (A1,A2), B
= (B1,BZ) such
that
1
1
2
(AnA )
#
= # (B'UB~). By IT, we may assume t h a t A
(A'vA')
1
=
2
A ,
'
Assume furthermore that A n BJ = 0 f o r al1 i , j = 1,2. Now, by
application
Then, by UR, B
A
be2,
E
2
of
REP,
-
I A ~ \ ( A ~ ~ A ~ ) ~ ( B ~ ~ B ~ ) , A ~ \ ( A ~ A.
~A~)~(B~~B~
A. By a similar argument, A
Z
B, and in consequence,
- B.
(ii) Let A, B
E
A = (A1,AZ), B = (B1,B2) be such t h a t # (Aln~') =
be2,
>
# (B1nB2), # (A'uA2)
B' S B',
# (B1uB2). By IT, we may consider that
> # B2. Consider C2
# A1 = # B1, # A'
A
c2 =
B2 such t h a t #
3
-
Then, by MON, (B1,C2) > B, and by (i), (B1,C2)
1
2
A ,
S
# A'.
A. Thus, by transitivity,
A > B.
(iii)
#
Let
B
A,
be2,
E
=
A
(A',A'),
B
=
(B1,BZ) be
1
( A ' ~ A ~ )> # (B'nB2). Again, by IT, assume t h a t A
2
moreover, # A
# A'.
2
2
# B
1
, take C
B',
3
2
t h a t # C2 > # A'
2
<
# B
2
.
1
G A , B
that
2
S B
1
.
If
2
1
C 2 B2 such that # C = # A , # C =
Then, by MON, C > B, and by (i), A
Consider now the cace # A
2
such
- C. Thus, by transitivity, A > B.
Let p = # A
1
-
# B'.
Take C2
+ p. Thus, by MON, (B1,C2) > B. Take now x
3
BZ such
E
cZ\Bi,
0
1
c 9 , and y
E
Thus, by UNI, l ~ ' u ( y ~ , ( ~ ~ \ ( x ~>) u(9',c2).
~y~l
L, y é 9'.
Applying UN1 p times, and choosing appropriately X c C\B'
1
2
2
1
step, we end up a t some D = (D ,D ) > (9 ,C ), such t h a t D'
# D',
- D,
Thus, by (i), A
# A2 = # D'.
in the last
C
2
D
,
# A
1
=
and by transitivity, A > B
..
THEOREM 4.- UNI, REP, UR, MON and IT a r e independent.
PROOF:
Again, we provide with examples of weak preorders such that satisfy al1 but
one of previous properties a t any time.
(UN11 Consider
r1 defined by A r
satisfies
UR,
REP,
and
MON
B iff
1
IT,
but
it
z # (B'uB~). I t
(A'uA2)
#
fails
to
satisfy
UNI:
((a,e,c),(a,f,d}) >; ((a,b,c),(a,b,d)l.
(REP) Consider
A >
2
i2
B <=> A >
defined
a s follows:
-
B, and if A
U
Take a
fixed set K
c L.
Now,
B, when # (A1nA2nK) > # ( 9 l n ~ ~ n It
~).
satisfies UNI, UR, MON and IT, but does not satisfy REP: Let A be such t h a t
x
A ' ~ A ~x, é K, Y
E
(UR) Consider
A >
3
E
K. T ~ U SI(A~\(X})U{~},(A~\(~})U{~})
,
>2 A.
defined
3
a s follows:
-
B <=> A > B, and if A
U
U
Take
a
fixed
s e t K c L,
Now,
B, when # l ( ~ ~ u A ~ ) n> K#l [ ( B ' U B ~ ) ~ KI tI .
satisfies UNI, REP, MON and IT, but does not satisfy UR: Let A be such t h a t
x
E
2
1
A \A, x
E
K, y é. K, y
(MON) Consider
Z
UNI, REP,
and
UR
4
e
AZ\(x). Thus,
defined by A
IT,
but
2
it
4
A
>3
[(A1,(A2\(~})u(y})l.
B iff # (A1nA2) i # (9'nBZ). I t satisfies
fails
to
fulfill
MON:
({a,b},(a,b,c>)-4 ((a,b),(a,b}).
(IT) Consider
2
5
defined by
2
5
=
2
lo
.
It satisfies UNI, REP, UR and MON,
but i t fails t o fulfill IT: (ta,b),(a,c,d)) >
((a},(a,b,c,d))..
5. T H E LEXIMIN OPPORTUNITY RELATION
THEOREM 5.- A w e a k p r e o r d e r z o n 2' s a t i s f i e s AN, UNI, REP, UR,MON, a n d
T R if a n d only if
coincides w i t h z
2
lo
.
PROOF:
By AN, Consider A = <r(A), B = o-(B). Without l o s s of g e n e r a l i t y , w e m a y
1
2
1
2
c o n s i d e r A = (A ,A ), B = (B ,B ) s u c h that AinBj = !a , f o r al1 i , j = 1,2.
(i) L e t
Then A
A,
-
B
2'
E
such
that
[#
(A'AA~), #
Al
=
[#
(B1nB2),
B1.
#
B. Notice t h a t t h i s is e x a c t l y case (i)
i n T h e o r e m 1. T h e s a m e
p r o o f applies.
( i i ) L e t A,
2'
E
= #(B'nBZ);
b e s u c h that # (A'nA2)
A'
= # B',
= # BZ a n d A1nA2 = A1nc2.
By MON,
#
< # B2. T h e n B > A.
# A2
c2
Choose
1
B
2
3
A'
s u c h that #
> A, a n d b y ( i ) , (A',c2)
(A ,C )
(iii) L e t A, B
E
c2
-
B. T h u s , b y t r a n s i t i v i t y , B > A.
f2 b e s u c h t h a t # (A'~A') = # (B'nB2);
# A'
< # B'. T h e n
B > A.
If
2
moreover # A
c'nc2,
# C' = # B',
T a k e C'
# B2,
5
#
c2 =
3
A',
2
C
2 A
2
s u c h t h a t A1nA2 =
# BZ. Then, b y MON, C > A, a n d b y (i), C
-
B.
T h u s , b y t r a n s i t i v i t y , B > A.
Consider
#
B2
2
now
#
the
case
(A2\x)
#
(A'U(X})~(A'\X)
t r a n s i t i v i t y , B > A.
#
A'
But
#
2
(A'u{x>,AZ\~) > A.
#
= #
A'
> # B'.
Take
X
S
A\A1
such
+ 1, a n d choose x B A1UA2. Then,
(A1u¿x})
5
#
B1,
(B1nB2). I n consequence,
(AZ\x)
#
B
2
#
that
by TR,
BZ,
and
(A'u¿x>,A2\~), a n d by
(iv) Let A, B
g2 be such that # (A1nA2) < # ( B ' ~ B ~ )Then
.
B > A.
E
1
Consider f i r s t the case A nA
1
UNI, ((x),{x)) 2- A. If B
2
1
2
= 0. Then, # (B nB )
2
= B
2
1. Take x
E
1
B nB
2
.
By
= {x), we a r e done. Otherwise, by MON,
B > ({x},{xj), and by transitivity, B > A.
Suppose now that A n A 2
that # (A'VO < # B',
?t D,
1
and # ( B ' ~ B ~=) # (A'nA2) + 1. Take X c A such
and x @ A'nA2. Then, by UNI, [ ( A ~ \ X ) U ~ X > , A ~>U ~ X ) I
A. Furthermore, # [(A1\X)utx)ln(A2u(x>) = # (B1nB2), # [ ( A 1 \ ~ ) u t x ~<l # B1.
Thus, by (iii), B > [(A1\~)utx),A2u(x>l,and by transitivity, B > A.
Finally, assume that A1nA2 s a, and # (B1nB2) = # (A1nA2) + p, p
Construct C' = A'UY,
c2=A'UY,
(cinc2) =
such t h a t #
# (B'~B')
2.
2
- 1.
By MON,
al1
but
C > A. Now, we a r e in the previous case and may act a s before..
THEOREM 6.- AN, UNI, REP, UR, MON, and TR a r e independent.
PROOF:
provide
We
with
examples
of
equality
relations
fulfilling
one
property a t any time:
(AN)
Define
2
1
in the following way: A
2
1
B <=> ( # (A1nA2), # A)
iL
(# (B'~B'), # B). I t satisfies UNI, REP, UR, MON and TR but i t fails t o
satisfy AN: (ta,b>,tc>)>l (tc>,ta,b)).
(UNI) Define r
2
in t h e following way: A
2
2
1
2
B <=> [# Aff"', # (A VA ) ]
tL
I# BW(",# [ B ~ U B ~ )ItI . satisfies AN, REP, UR, MON and TR but i t does not
satisfy UNI: ((a>,(b,c)) >; (td),(d)).
(REP) Consider z3 defined as follows:
A >
3
B <=> A >
lo
B, and if A
-
lo
Take a fixed s e t K c L,
Now,
B, when # (A'nA2n~) > # (B1nBZn~).I t
s a t i s f i e s AN, UNI, UR, MON and TR, but it does not satisfy REP: Let A be
such t h a t x E A1nAZ, x 6f K, y
E
K. Thus, [ ( A ' \ ~ X ~ ) U ~ ~ ~ , ( A ~ \>3~ XA.~ ) U ~ ~ ~ )
(UR) Consider
r
defined as follows: Take a fixed s e t K c L.
A t4 B <=> A
>,o
B, and if A
4
-
lo
B, when # (A'uA')~K
Now,
> # (B'uB')~K.
It
s a t i s f i e s AN, UNI, REP, MON and TR, but it does not satisfy UR: Let A be
such
A
x
that
E
x
A'\A',
E
K,
y
K,
6f
y
A\{X).
@
Thus,
t4 [ ( ~ ~ , ( A ~ \ t x ) ) u t y > ) l .
(MON) Take r5 defined as follows: (A1,A2) z5 (B',B2)
[ #
(A1nAZ), # ~ ~ " ' 1r
L
<=>
.
[ # (B1nBZ), # ~ ~ " ' 1 I t fulfills AN,
UNI, REP,
UR and TR, but it does n o t satisfy MON: ((a),(a,b>) -5 (ta),ta)).
(TR) Take
2
6
t o be t h e utilitarian opportunity relation.
UNI, REP, UR and MON, but
it
(ta},ta,b,c,d>) p6 ((a,b),(a,b}).m
does
not
satisfy
TR:
I t satisfies AN,
6 . FINAL REMARKS
We s t a r t by providing a summary of results in Table 1. The different
columns r e f e r to the different social relations studied in this paper. The
last column corresponds to the cardinality difference relation presented in
Kranich (1994). We summarize the behavior of the aforementioned relations
with
respect
properties
to
the
properties
correspond
to
the
presented
in
properties
used
Section
in
2.
the
The
starred
characterization
results offered in the paper.
AN
yes*
yes
yes*
UN I
yes*
yes*
yes*
no
REP
yes*
yes*
yes*
Y ~ S
UR
yes*
yes*
yes*
Y ~ S
1E
yes*
no
no
Y ~ S
NM
yes*
no
no
yes
Y ~ S
-
MON
-
-
RO
-
-
yes*
yes*
no
-
TR
ye S
no
yes*
IT
no
yes*
no
Yno
Table 1
Some remarks a r e in order:
(1) Among al1 the properties presented, AN and NM appear in Kranich
(1994).
Moreover,
IE
shares
Independence of Common Expansions.
the
same
spirit
than
Kranich's
( 2 ) The cardinality difference relation
satisfies TR, but i t does not
satisfy UNI. Nonetheless, UN1 is satisfied by al1 our relations.
(3) As f o r the remaining properties, REP and UR a r e also satisfied by
al1 the relations;
MON i s satisfied by al1 of
them but
and TR i s
2
CO
satisfied by al1 of them but
2
.
U
(4)
Relations
and
2
2
U
case, whereas
lo
extend straightforwardly
presents analogous difficulties t o
2
CO
to the
n-person
Zcd.
( 5 ) Throughout the paper we assumed that al1 information on individual
preferences
is
irrelevant
sornehow included
for
the
in the previous
social
ranking.
A
hidden
assurnption
one is that the possible diversity
or
similarity between the basic opportunities is also neglected. This can be a
matter of discussion. When we compare two individual choice sets, we do not
pay attention to the similarity between the alternatives they contain. In
the case we would like t o take these similarities into account, we should
deal,
at
the
same
time,
with
a
binary
relation
describing
such
similarities. When this similarity is conceived a s an equivalence relation,
a
simple strategy for
extending our results may be to reconstruct our
social relations over the quotient set. In the case the similarity relation
is
less
structured
additional research.
the
situation
becomes
more
complex
and
deserves
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