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EBR
FACULTY WORKING
PAPER NO.
741
Restricted Domains, Arrow Social Welfare Functions and
Noncorruptible and Nonmanipulable Social Choice
Correspondences: The Case of Private and
Public Alternatives
Zvi Ritz
Commerce and Business Administra
Bureau of Economic and Business Research
University of Illinois, Urbana-Champaign
College of
3o^
dy
BEiBM
FACULTY WORKING PAPER NO. 741
College of Commerce and Business Administration
University of Illinois at Urb ana- Champaign
February 1981
Restricted Domains, Arrow Social Welfare Functions
and Noncorruptible and Nonmanipulable Social Choice
Correspondences: The Case of Private and Public Alternatives
Zvi Ritz, Assistant Professor
Department of Business Administration
Acknowledgment
I wish to express my deep appreciation to
Ehud Kalai who introduced me to the field of social choice
and whose advice and support enabled me to complete this
paper.
Thanks are also due to Alvin Roth for lengthy discussions and valuable suggestions.
This research was supported
in part by NSF grants SOC 7620953 and SOC 7907542.
:
w
».
of
I.
gnu.
Abstract
An n-person social choice problem is considered in which the alternatives are (n+1) dimensional vectors, with the first component of such a
vector being that part of the alternative affecting all the individuals
together, while the (i+1) component is the part of the alternative
affecting individual
(individual
i
i
alone.
Assuming that individuals are selfish
must be indifferent between any two alternatives with the
same first and (i+1) components)
,
that they may be indifferent among
alternatives, and that each individual may choose his preferences out of
a different set of permissible preferences, we prove that any set of
restricted domains of preferences admits an n-person nondictatorial
Arrow-type social welfare function if and only if it admits an n-person
nondictatorial, nonmanipulable and noncorruptible social choice correspondence.
We also characterize all the sets of restricted domains of
preferences which admit two-person Arrow-type social welfare functions
(and therefore also admit two-person nonmanipulable and noncorruptible
social choice correspondences).
1.
Introduction
Given the impossibility results derived by Arrow in his seminal book
"Social Choice and Individual Values" on the possibility of constructing
nondictatorial social welfare functions, and by Gibbard
[7]
and by
Satterthwaite [18] on the possibility of constructing nondictatorial and
nonmanipulable (strategy-proof) social choice functions, much of the recent literature on social welfare and social choice functions falls into
two categories.
One category includes efforts to derive possibility re-
sults by relaxing some of the criteria introduced by Arrow [1], by
Gibbard
[7]
and by Satterthwaite [13] as desirable characteristics for
social welfare and for social choice functions, while the other Includes
attempts to obtain possibility results by restricting the domains of
preferences available to the individuals (for a comprehensive and detailed description of many of these efforts see Sen [23] and Kelly [12]).
The appeal of the "restricted domains" approach is in the observation
that in most standard economic models, a considerable structure is im-
posed both on the Individuals' preferences (e.g., assuming Individuals
with continuous and concave utility functions) and on the social alternatives (e.g., assuming economies with public goods only, with private
goods only, etc.), thus results derived in social choice theory following
this approach may be applicable to other economic models too.
The first to demonstrate the possibilities inherent in the "restricted
domains" approach were Black [2] and Arrow himself ([1], page 75) who
proved that majority rule is a nondictatorial social welfare function for
groups of individuals with "single-peaked preferences".
Arrow
[1,
II] also discusses the "difference between the ordering (done by an
Chap.
-2-
individual) of the social states according to direct consumption of the
individual and the ordering when the individual adds his general standards of equity."
He refers to the first type of ordering as reflecting
an individual's "tastes" while the second reflects his "values."
He
demonstrates that this distinction can be represented as restrictions on
the individuals' domains of preferences (when an individual orders by
"taste" he must be indifferent between any two social alternatives which
allocate to him the same (private) alternative) and he goes on to prove
that this distinction by itself is not enough to guarantee the existence
of a "desired" social welfare function ([1] theorem
3
and also Blau [3]).
In the model discussed here the "restricted domains" approach is
used to emphasize both differences between the way individuals choose
their preferences (according to "values" versus "tastes") and differences
among types of alternatives (e.g., public goods as opposed to private
goods).
For this purpose, when dealing with the possibility of aggre-
gating individuals' preferences into a social welfare function or a social
choice function, we distinguish among three types of "restricted domains."
Cases where individuals order (or choose among) the alternatives according to their "values" are cases of public alternatives only; cases where
individuals order according to their "tastes" are cases of private alternatives only, and cases where individuals order the social alternatives
according to their "tastes" and in addition the social alternatives are
such that part of each alternative affects all the individuals together,
while other parts affect each individual separately, are cases of mixed
alternatives.
-3-
This distinction may be demonstrated by the following example.
If a
group of individuals (rational, of course) must choose a cake from among
a variety of flavors,
it is a case of public alternatives.
If the same
group must decide how to divide a given cake among its members, and each
individual considers only the amount allocated to him when comparing different allocations (thus he is indifferent between any two allocations
that give him the same piece of cake), it is now a case of private al-
ternatives.
On the other hand, if individuals give importance not only
to the amount allocated to them but also to the amounts allocated to
other members of the group, then it is again a public alternatives case.
If the group must simultaneously decide on the flavor of the cake and
its apportionment and each individual considers only what Is allocated
to him when comparing the same flavor but different divisions, then it
is a case of mixed alternatives.
This work is an investigation of the possibility of constructing
Arrow-type social welfare functions and Gibbard-Satterthwaite-type social
choice correspondences for the case of mixed alternatives.
We assume
that every social alternative is a mix of public and private alternatives,
individuals may differ in their sets of private alternatives as well as
in their domains of preferences, and indifferences among alternatives
are allowed.
The principal result for this most general case Is that a
set of n (greater than one) restricted domains admits an n-person non-
dictatorial Arrow-type social welfare function if and only if it admits
an n-person nondictatorial, noncorruptible Gibbard-Satterthwaite-type
social choice correspondence, namely a nondictatorial, noncorruptible,
nonmanipulable, rational and efficient social choice correspondence.
_4-
Intuitively a noncorruptible social choice correspondence is
a
choice
correspondence for which no individual can, by misrepresenting his true
preferences, change the social outcome without changing the value of it
for himself.
This concept is a generalization of the Sattherwaite-
Sonnenschein [20] concept of nonbossiness.
In their model they discuss
strategy-proof allocation mechanisms in classical economic environments,
and such a mechanism is bossy if an individual can, by misrepresenting
his true preferences, change the bundles allocated to other individuals
while maintaining his unchanged.
Thus a mechanism can be bossy over
private goods only, while a social choice correspondence may be corruptible over public and mixed alternatives as well, as is demonstrated in
a later example (section 3).
Although both manipulable and corruptible
social choice correspondences include "strategic" behavior of individuals
who misrepresent their "true" preferences, there is a conceptual difference between the two.
A person manipulates a mechanism if, by behaving
"strategically", he improves his position directly ; he corrupts a social
mechanism if, by changing the value of the outcome to others (without
altering its value for himself), he creates the possibility of his indirectly improving his position by opening room for either "bribe taking"
or "blackmailing".
We also characterize all the sets of two or more re-
stricted domains of preferences which admit two-person Arrow-type social
welfare functions (and therefore two-person noncorruptible GibbardSatterthwaite-type social choice correspondences).
This work builds on, complements and generalizes a number of previous
contributions.
Muller
[8]
The most significant of them are results derived by Kalai-
and Maskin [13, 14],
They independently characterized all
-5-
the restricted domains of strict preferences (no indifferences among al-
ternatives are allowed) which admit Arrow-type social welfare functions
for the case of public alternatives only when all individuals have the
same restricted domain of preferences.
In addition they proved that
such domains admit Arrow-type social welfare functions if and only if
they admit rational, nonmanipulable, efficient and non-dictatorial social
choice functions.
(Since indifferences are not allowed, all such func-
tions are also noncorruptible.)
This further demonstrated the close
reciprocity between the Arrow set and the Gibbard-Satterthwaite sets of
criteria, a demonstration that started with the works of Gibbard
and Satterthwaite [19] combined.
[7]
(Pattanaik [15] proved one direction of
the equivalence for the cases in which individual preferences may be
restricted and a discussion of the possibility of full equivalence for
these cases appears in Blin-Satterthwaite
[4 J.)
Kalai-Ritz
[9]
char-
acterized all the restricted domains of strict preferences which admit
Arrow-type social welfare functions for the case of private alternatives
only and individuals which are symmetric in both their alternatives and
preferences set.
Ritz [17] generalized the Kalai-Ritz [9] results for
the case of private alternatives, nonsymmetric individuals and with in-
differences among alternatives allowed.
Kalai-Muller
[8]
He also proved, parallel to
and Maskin [14], that sets of restricted domains admit
Arrow-type social choice functions if and only if they admit nonmanipulable, noncorruptible, rational and non-dictatorial social choice cor-
respondences.
As all these models are special cases of the mixed alternatives
model discussed here (if all the private alternative sets contain
-6-
exactly one element and all the restricted domains are the same and contain only strict preferences then the mixed alternatives model is re-
duced to the models discussed by Kalai-Muller
[8]
and Maskin [13],
[14], while if the public alternatives set contains one element only,
then the general model is reduced to the cases discussed by Kalai-Ritz
[9]
and Ritz [17]), thus our results unify and generalize some of the
results derived by all the other authors.
The drawback of this model is
that we did not prove completely the independence between the existence
of Arrow-type social welfare functions for a group and its size, and
therefore we characterized only those sets of restricted domains which
admit two-person Arrow-type social welfare functions, while all the pre-
viously mentioned works characterize, for their respective cases, all
the restricted domains which admit Arrow-type social welfare functions
for any finite (and greater than one) number of individuals.
We also
proved the equivalence between the Arrow type social welfare functions
and the Gibbard-Sattherthwaite type social choice correspondences only
for Arrow social choice functions which obey positive association and
therefore appears to be less general.
([1], page 25),
However, since to quote Arrow
"We are trying to describe social welfare and not some
sort of illfare" we do not consider this as a major drawback.
The definitions and some preliminary results are introducted in
section
2.
In section 3 we prove the equivalence between Arrow-type
social welfare functions and noncorruptible Gibbard-Satterthwaite-type
social choice correspondences.
In section 4 we characterize all the
sets of restricted domains which admit two-person social choice functions, and in section
5
we convey some results on the dependency (or
-7-
independency) between the number of individuals and the existence of
Arrow-type social welfare functions.
Definition and Preliminary Results
2.
Consider a finite set N of n individuals (n
^
Let A
2).
(|A
q
|
^
1)
denote the set of all public alternatives available to all the individuals
For every individual
as a group.
e N and every a £
j
A«, let A (a) be
the nonempty set of all private alternatives which are available to indi-
vidual
when alternative a is available to the group.
j
x€
{(a;x)|a£ A~ and
(A ;A.) =
n
Let
A. (a)}, then E. denotes the set of all
reflexive, transitive and complete binary relations on (A n ;A.).
ment of
E
.
J
is called a jpreference relation.
(a;x),(b;y)£ (A.;A
and r 6 I
)
.
If
An ele-
for some
it is both (a;x)r(b;y) and (b;y)r(a;x)
then it is alb in r ((a;x) is indifferent to (b;x)) and if for some
G E. it
r'
in
r'
is (a;x)r'(b;y) and not
(b;y)r'(a;x) then it is (a;x)p(b;y)
((a;x) is preferred to (b;y)).
Let Q, be a nonempty subset of
E
the elements of Q
.;
missible preference relations for individual
j.
.
are the ad-
Throughout this work
we assume that any two alternatives are distinguishable, i.e., for every
j
e N, and every (a;x),(b;y) € (A ;A
Q
then there exists
(A,
Q
A
;A
.
<n)
V
;
)
- {(x
ment of (A ;A
Q
;
r'
G
ft
if for no r G
such that (b;y)p(a;x) in r
(x ;x.)
G
X = (x ;x.,...,x
n
)
X;L ,...,x n )|
),
),
Q
(A ;A
Q
)
1
P.
it is (a;x)p(b;y),
.
Let
.
for every
j
€
N}
.
An ele-
is called an n-person social al-
ternative (or n-person mixed alternatives allocation).
Let A
repre-
sent the set of all reflexive, transitive and complete binary relations
on (A ;A
).
An n-person social function (SF) over mixed alternatives
is a function h :Q
8
,
R = (r-,...,r
+A
)
where Q
= ft,xQ„*...xn
12
n
is called an n-person profile .
.
An element of
If for a profile
R and social alternatives X and Y it is both Xh n R )Y and Yh n
(R)X
(
then
-8n
it is XIY (X is indifferent to Y in h (R)),
n
and if it is Xh (R)Y but not
n
n
Yh (R)X then it is XPY (X is preferred to Y in h (R)).
n
h an n-person
SF is an n-person social welfare function (SWF) if it obeys the following conditions:
every k € N,
r,
k
)
in r
and there exists
j£
(n)
R£
Q
(x-.;x, )r, (y..;y, ),
(x ;x.)p(y ;y
in
(n)
For every X,Y e (A ;A
) and
Unanimity (U):
i)
if for
ft
N such that
then XPY, and if for every k S N,
(x ;x )i(y ;y )
Q
k
k
then XIY.
Independence of irrelevant alternatives (IIA):
ii)
For every
Cx^^y^)
(n)
(n)
X,Y e (A ;A
and R,S £ Q
if [for every k 6 N ,
)
iff
(x ;x )s (y ;y } and (y ;y )r (x ;x ) iff (y ;y )s (x ;x )] then
k k
k
k
k
k k
k k
n
n
Xh (R)Y iff Xh (S)Y.
A SF is dictatorial (D) if there exists
(n)
X,Y e (A ;A
) and R e
Q
a"
1
',
(x ;x )p(y ;y
Q
Q
j
G
)
in r
N,
is nondictatorial (ND) if it is not dictatorial.
s.t. for every
implies XPY.
A SF
An n-person Arrow
social welfare function (ASWF) is a nondictatorial SWF.
We say that 9
K = {i.,...,i
i
m
}
C
admits an m-person ASWF (m
(V)
N,
= 9.
such that Q
12
i,
* Q.
<^
x
i_
n) if there exists
m
(Throughout this work, if X = (x n ;x..
X.
= (XqJx.
,x ._. ,x
x
,...,x )).
n
j+1
U
,_j_,
, . . .
,x
)
admits an
i
m-person ASWF.
, . . .
a
x
...
,
..
1
and X_.|y = (x_;x.,
.
then
,x ),
. . .
n
,x.
.
,y,
Since in the definition of an ASWF we allow for a group indifference
between bundles of alternatives even if not every individual is indifferent between his respective alternatives, we first investigate the sig-
nificance of this, especially the possible existence of a "trivial"
ASWF, in the sense that the group is indifferent between every two
-9-
bundles of alternatives over which there is no unanimous agreement among
the group's members, (Kalai-Muller [8], Maskin [13] and Kalai-Ritz [9]
all assumed in their respective works that a social welfare function is
an antisymmetric, complete and transitive ordering of the social alternatives, thus eliminating the possibility of such an ASWF).
In lemma 1 we demonstrate that there are sets of restricted domains
for which no "trivial" ASWF exists, while in lemma
2
we prove that if a
set of restricted domains admits an ASWF which is not "trivial," then it
also admits an ASWF which allows indifferences among social alternatives
only if every individual in the group is indifferent to all of them.
^
For n
2,
g
is a Pareto unanimity collective choice rules
(Pareto CCR)
(Sen [21]) over mixed alternatives if for every X,Y £ (A~>A
R 6
(n)
.Q
n
it is xpy in g (R) iff for every
j
€ N it
is
^
anc*
(x ;x )r (y ;y
Q
Q
and for at least one individual k it is also (x ;x, )p(y ;y,
n
n
)
in
r,
;
)
in
any other case it is XIY in g (R).
Lemma 1 ;
is such that for at least one k
If Q
(A^A^
(x ;a),(x ;b),(x ;c) e
Q
for at least another % e N
on Q
n
,
€
N,
there exist r £
fl,
k
and
such that (x ;a)p(x ;b)p(x ;c) in r and
[A (x
Q)
>^
[
2,
then the Pareto CCR defined
is not an ASWF.
Proof .
Without loss of generality (w.l.o.g.) assume k =
(x ;x),(x ;y) S (A ;A
Q
Q
Also for
j
2
)
and r
2
£
Q.
2
=»
(x rt ;c,x,x,,
u
j
. . .
,x ),
n
% - 2
and let
such that (x ;x)p(y ;y) in r »
2
= 3,...,n let (x_;x.) e (A.;A.) and r.
mixed bundles X
1,
Y
S
fl
(x-.;a,y,x_,
U
J
.
. . .
Consider the
,x ),
n
-10-
Z = (x
u
;b,y,x_,
j
definition of g
. . .
,
,x
n
)
and the profile R = (r,r_,...,r
2.
n
then by the
),
the Pareto CCR, it is XIYPZIX in g (R), hence g
not transitive and therefore it is not an ASWF.
Lemma
2
If
R e
P.
Q.E.D.
:
n
admits
9.
f
an n-person ASWF which is not a Pareto CCR, then
an n-person ASWF such that for every X,Y £
admits h
P.
is
n
\
XIY in h (R) iff (x ;x.)i(y ;y.) in r. for every
Q
anc*
^'
^n'^
€ N.
j
Proof .
The ordered set of individuals (j, ,j~f*»»»j
)
is lexicographically
decisive over a set of bundles of alternatives, if for any two bundles
X,Y in the set, the group prefers X to Y iff for some k,
k
<_
<^
n,
in-
prefers his X alternative to his Y alternative, while every
dividual 1
individual
1
j
,
% =
l,...,k-l, is indifferent between his alternatives
(we also say that X is lexicographically preferred to Y by (j, ,»»»>j ))•
Since
R G Q
r
2
,
f
n
S uch
is not a Pareto CCR, there exist X,Y
that w.l.o.g.
n
and XPY in f (R).
(x^x^pCy^y
Define h
n
)
as follows.
n
n
and R e ?S \ XIY in h (R) iff (x ;x.)i(y ;y
Q
n
XPY in h (R) if XPY in
n
f (R)
cographically prefer X to Y.
in
.)
r^
(A ;A
W
Q
)
and
;y )p(x ;x ) in
Q
2
2
n)
For every X,Y G (A ;A^
Q
(y
in r. for every
or if XIY in f"(R) and (2,3,
. .
)
j^N;
.,n,l) lexi-
It can be easily demonstrated that h
is
reflexive, complete and obeys U, IIA and ND and the proof Is omitted.
To complete the proof that h
tive.
is an ASWF we now prove that it is transi-
(Throughout this work, to prove that a binary relation r is
transitive, we prove that for every a,b,c (i) apb and bpc imply ape and
(ii) aib and bic imply aic,
in r (e.g., see Sen [20] for analysis of
-11-
By the definition of h
binary relations).)
r
,
if for some X,Y,Z € (A ;A
and
^)
&
n
XIYIZ in h (R), then also
(n)
Q
n
n
This implies Xf (R)Yf (R)Z.
n
XPYPZ in h (R).
n
O
graphically preferred to
vidual in (2,3,
. . .
n
Z
by (2,3,
. . .
,n,l).
(x~;x )i(z_;z
. . .
in
)
r,
and let % be the
,n,l) such that (y n ;y )p(z n ;z
U
m
in r
)
for m = 2,. ...1-1.
m
in r
)
Since also XIZ in
n-person ASWF satisfies the positive association (PA) condi-
9.^
Q
£ N (x ;x
)r (y ;y
Q
Q
)
Observe that lemma
2
)
such that for every
and (x ;x )p(y ;y ) in
Q
Q
n
n
XPY in h (R) implies XPY in h (R').
implies (x ;x )r'(y ;y
Q
Q
implies (x ;x )p(y ;y
Q
Q
Lemma
in r»
)
can be slightly modified as to be:
3
If
admits
P.
f
an n-person ASWF which satisfies the PA condition
and is not a Pareto CCR, then
(2
'
n
admits h an n-person ASWF which
obeys the PA condition and such that for every X,Y S (A ;A
fi
R e
n
(n)
ft
and
Q.E.D.
n
tion, if for every X,Y e (A ;A^ ') and R,R' G
r
Let
(R) then
f
n
—an
.
in r
.)
by definition XPZ in h (R).
]
then
Let k be the first indi-
= min{k,Jt}, then by the transitivity of r. (x_;x )p(z ;z
h
n'
If also XPZ in f (R),
,n,l) such that (x.,;x, )p(y~;y,
first individual in (2,3,
m
ST
Hence X is lexicographically preferred to Y and Y is lexico-
in f (R).
(J
R£
and
n
So assume XIZ in f (R), which implies XIYIZ
by definition XPZ in h (R).
n
R£
(n
Thus suppose for some X,Y,Z e (A ;A
n
XIZ in h (R).
j
(n
Q
and the transitivity of all
,
XIY in h (R) iff (x ;x )i(y ;y
Q
Q
)
in r
for every
)
and
j£N.
Proof
Define h
as in the proof of lemma 2.
Suppose for some
(n)
(n)
X,Y e (A ;A
suc h that for every
) and R,R' 6 Q
Q
(x ;x )r (y ;y
Q
Q
)
implies (x ;x )r'(y ;y
Q
Q
)
j
e
N,
and (x ;x )p(y ;y
Q
Q
)
in r
-12-
implies (x ;x.)p(y ;y.) in
r*.,
Q
n
n
XPY in h (R) and Yh (R' )X.
Since
f
n
satisfies PA, it cannot be the case of XPY in f (R), thus it must be
XIY in
n
f
Let k be the first individual in (2,3,
(R).
.
..
,n,l) such
This implies (x ;x )p(y ;y ) in r , hence
Q
k
k
fc
n
n
Let m be the first
it must be both YPX in h (R') and XIY in f (R' ).
that (x ;x )p(y ;y
k
Q
C)
k
in r
)
.
fc
individual in (2,...,n,l) such that (y Q ;y )p(xQ ;x
)
in r', then it must
be both (y„;y )p(x rt ;x ) in r and m < k, which contradicts XPY in h (R).
m
m
m
Q.E.D.
be the set of all nonempty subsets of
Let n
(A.,; A
).
An
n-person social choice correspondence (SCC) over mixed alternatives is
a correspondence H
n
:
P.'
IT.
»
x IT
A SCC is a rational social
choice correspondence (RSCC) if for every B G H
R6
and
fi
it sat-
isfies:
n
H (R,B) C B.
1.
Feasibility (F):
2.
Independence of nonoptimal alternatives (INOA):
C
3.
C
B,
n
if H (R,B) n c £
(xn ;x ,)r .(y n ;y
4.
n
n
then H (R,C) = H (R,B)
For any X,Y
Unanimity (P):
.)
For every
^B,
if for every
j
<~i
e
C.
N,
and there exists k G N such that
n
(x ;x )p(y ;y ) in r , then Y f H (R,B).
k
Q k
fe
n
n
If X € H (R,B) then Y e H (R,B) iff
Uniqueness (UQ):
(x ;x )i(y ;y
Q
Q
)
in r
for every
j
€
N.
Since in the definition of a SCC we do not restrict the choice set
to include a single social alternative only, we thus avoid the need for
a tie-breaking function (needed to choose a unique social outcome among
the alternatives over which the group is indifferent (see Satterthwaite
[19])), however this gives rise to two different manipulability concepts.
,
-13-
H
R,R
G
?
r. =
an n-person SCC is point-manipulable if there exist B G
,
and k e N such that
£1
r.
J*
(x_;x, )p(y ;y,
)
in
r, .
H
,
j*
k)
an n-person SCC is set-manipulable if there
(n)
(n)
exist B e n
R,R» e n
and k e N such that r
,
t r* while for any
K.
j
(j
,
n
n
and there exist X G H (R',B) and Y £ H (R,B), such that
r'.,
other
j£N
for any other
r',
II
£ N
(j £ k),
Y G H (R,B) it is (x
Q
K.
n
r
- r', and for every X G H (R',B) and for every
;x,
)p(y Q ;y,
)
in
r,
If H
.
is not point-manipulable
or not set-manipulable it is said to be point-nonmanipulable , or
set-nonmanipulable respectively.
Fortunately, these two concepts are equivalent for rational social
choice correspondences, as is established by the following lemma.
Lemma
4
A rational social choice correspondence is point-manipulable iff
it is set-manipulable.
Proof
Part a .
Suppose H
,
an n-person RSCC is point-manipulable by individual
Hence there exist B €
k.
for every
j
€ N such that
II
j
R,R' e £T
,
4
j
k,
such that
r,
k
£ r'
k*
r,,
= r'
j
j
and for some X e H (R,B) and
YG
n
H (R',B) it is (y ;y )p(x ;x.) in r
Q
Y
n
n
H (R,B) and X £ H (R',B).
4.
n^
.
This implies, by DQ, that
Let C = {X,Y}; since C c B then by con-
n
n
n
dition INOA, it is {X} = H (R,C) and {Y} = H (R',C), hence H is also
set-manipulable.
Part b .
If H
,
an n-person RSCC is set-manipulable it is obviously also
point-manipulable.
Q.E.D.
-14-
Since in this work we are concerned with rational SCCs only, therefore from now on we use only the concept of manipulable (nonmanipulable
(NM)) SCC and do not distinguish between point and set manipulability.
H
R,R' e
B^II
an n-person SCC is corruptible if there exist
,
q"
1
and k e n such that r
'
f r£, for any other
j
€ N
,
(j
t k)
fc
r. =
both r
X^
for some
r'.,
and
n
and Y 9 H (R,B).
r'
corruptible.
n
H (R,B) and
Y^
H
n
H (R',B) it is
(x^x^!^^)
in
is noncorruptible (NC) if it is not
As we mentioned before,
the corruptibility notion is a
generalization of the bossiness concept of Satterthwaite and
Sonnenschein [20].
The following example is of a corruptible but not
bossy SCC.
Consider a group of three individuals with a set of three public alternatives A = {a,b,c} and the following restricted domains of preferences:
Q
= {r-}, Q- = {r-},
9.^
=
in r_, apbic in r- and bicpa in r,.
R e o
x n
x n
{r^r^r^} where apbpc
in r^, cpbpa
Define {a} = F(R,B) for every
and every B C A such that a G B,
{b} = F((r ,r ,r ),{b,c}) =
2
1
1
FCC^.r^), {b,c}
)
and {b} = F(R,{b}), {c} - F(R,{c}) for every R €
and {c} = F((r ,r ,r ),{b,c}
1
fi
x Q
x 9.y
2
4
It is
easy to verify that F is a nonmanipulable but corruptible (by individual
3) rational social choice correspondence.
The question of whether or
not there are restricted domains which admit nonmanipulable and cor-
ruptible social choice correspondences but do not admit nonmanipulable
and noncorruptible social choice correspondences is still an open one.
)
.
-15(n)
Fo r any B e rp
H
,
,
let B. = {(x ;x) e (A
;
A.) there exists X_ [x in B}
|
an n-person SCC is dictatorial (D) if there exists
(y ;y.)
(n)
BE
for every
II
€ B..
,
H
n
R6
(n)
ft
and
€
j
such that
N,
n
H (R,B) (x ;x .)r .(y ;y.) for every
Q
X€
A
is nondictatorial (ND) if it is not dictatorial.
SCC is Gibbard-Satterthwaite type social choice correspondence (GSSCC)
if it is a nonmanipulable , nondictatorial and rational social choice
If it is also noncorruptible then it is an extended
correspondence.
GSSCC .
We say that
there exists
admits an m-person GSSCC over mixed alternatives if
ft
RC—
n, K = {i, ,i„, ... ,i },
1
z
m
such that
=
'
ft
*
ft.
x
ft.
i_
i.
... x
1
admits an m— person GSSCC.
Equivalence Between Arrow Social Welfare Functions and Noncorruptible
Gibbard-Satterthwaite Type Social Choice Correspondences
2.
Theorem
n
>^
1
be a finite set, then for every n and m such that
Let (A_;A
)
m
admits an m-person noncorruptible Gibbard-Satterthwaite
>^
2,
Q
type SCC iff it admits an m-person Arrow SWF which satisfies the positive association condition and which is not a Pareto CCR.
Proof
Part a .
Assume that
admits a
9.
an m-person noncorruptible GSSCC, and
w.l.o.g. assume that K = {l,2,...,m}, i.e.,
H
h
,
thus
:
ft
m
(m)
ft
> A
and
(m)
ft
ft
=
*
ft.
1
will be used interchangeably.
as follows.
For every X,Y 6 (A ;A
x
...
ft
m
admits
Define
(m)
)
and
Re
(m)
ft
ft.
,
m
-16m
Xh (R)Y iff
m
h
1.
X£
m
We prove that h™ is an ASWF in three steps:
H (R,{X,Y}).
m
is a SWF, and 3.
h
is a SF, 2.
(m)
for every R e Q
an d every X,Y
«=
m
H (R,{X,Y}) or Y e
m
XIYIZ in h (R).
H^R.U.Y})
(A ;A
(m)
),
m
X e H (R,{X}) and either
Hence h™ is both reflexive
(or both).
^
An»^
)
R e ^
an(*
Then by the UQ condition it must be
(x ;x .)i(y_;y .)i(z-;z
n
H^R^X, Z} )
in r. for every
.)
= l,...,m, which implies
j
m
Suppose for some
and therefore XIZ in h (R).
(m)
and R 6
X,Y,Z 6 (A ;A
)
(m)
p.
m
by INOA Y £ H (R,B) and
Z
m
m
m
m
and Z ? H (R,£Y,Z}) 1 and
Therefore it must be
f H (R,B).
= H (R,B), by INOA {X} -
Let B = {X,Y,Z},
it is XPYPZ in h (R).
Y? H^R, {X,Y})
then XPY and YPZ imply both
{X}
is an ASWF.
This implies both {X,Y} = rf^R.U.Y}) and
{Y,Z} = E^(R,{X,Y>).
=
€E
Suppose for some X,Y,Z e
and complete.
{X, Z}
m
m
Since for no B G IT™' and R € a^ ' ± s H^R.B) empty, therefore
Step 1 .
X
h
H^R.U.Z}), and by definition
XPZ in
m
h (R).
Step 2 .
m
UQ and P for H
,
m
together imply that h satisfies unanimity.
doesn't satisfy IIA, i.e., (w.l.o.g.) there exist
Suppose h
(m)
(m)
suc h that for every
X,Y G (A ;A
), R,S € Q
iff (x ;x )s (y ;y ),
j
j
(y^y^r.^;^)
j£K,
m
j
:j
m
and YPX in h (S), which in turn implies {Y} = R^CS.IXjY}).
implies that not for every
j
G
K,
(x ;x .)i(y ;y
Q
Q
)
in
s
m
m
together with X € H (R,{X,Y}) imply {X} - h (R,{X,Y}.
method used by Schmeidler and Sonnenschein [20], let T~
j
= l,...,m let T. = (s .,... ,s .,r
Q
Q
Iff (y ;y )s (x ;x ), Xh (R)Y
()
j
j
(x ;x )r (y ;y
,..., r
);
By UQ this
and r
.
This
Applying the
=»
R and for
w.l.o.g. assume that for
)
-17-
= l,...,m
j
(y
;yjp(x
;x
j
for
)
and
in r
= l,...,(k-l).
j
)
s
and
in r
and for
j
s
,
for
h
3
Obviously
l<k<^m.
p/
m>
and every x,Y € (A ;A
Since H
H^T.^X.Y})
J
If k < m. then H
Hence m- < k
is
(m)
),
(x ;x )p(y ;y
k
k
such that for some X,Y £ B, (y_;y, )p(x n ;x,
9.
Y?
in r
)
implies XPY
k
)
in
and
and
r,
By INOA this implies X S H^R.iX.Y}) and, therefore by
m
m
H (R,{X,Y}), hence XPY in h (R), a contradiction.
is indeed an Arrow SWF.
therefore h
is not a Pareto CCR.
Suppose h
This com-
Since a
satisfies
doesn't obey PA, then
m)
o/ m ) such tha t for every
there exist X,Y 6 (A ;A^
) and R,R' €
(x ;xj)rj(y ;yj) implies
plies (x ;x.)p(y ;y
Q
)
(x^x^r'Cy^yj), (x^x^pCygjyj)
in r*
m
m
XPY in h (R) but Yh (R')X.
implies that for at least one individual k e K, it is (x
in
But
m.
_<
is not dictatorial then there exists B G H
pletes the proof that h
OQ,
in
.)
Suppose there exists individual k € K such that for every
«
m
X e H (R,B).
UQ,
2
is corruptible by individual k, again a contradiction, hence
in h (R).
R e
,m
. . .
obeys II A.
Step
R€
= ntj+1,
= m +l,...,m (x ;x .)i(y ;y
2
Q
Q
manipulable by individual k, a contradiction.
then a
j
Let k be such that {Y} = H^T. ,{X,Y}) and {X} k
r. and s..
J
(x ;x )p(y ;y
Q
Q
1
r,
(and therefore also in r,), thus {X,Y}
assume YPX in h (R T
).
Since h
fi
in
€
K,
im-
m
XPY in h (R)
;x,
)p(y„;y,
)
a (R',{X,Y}) and we may
obeys IIA there must be at least one
individual £ € N who changed his preferences between (x n ;x
(yA»y,)'
r..
j
)
and
Suppose exactly one individual % changed his preferences,
and w.l.o.g. assume £
1,
thus we may assume r
= r' for
j
2,...,m.
-18-
Case 1 .
(y ;y )p(x ;x
1
)
in
r.,
]L
In this case H™
and (x ;x )r (y ;y ).
1
1
1
is manipulable by individual 1 when his "true" preference is
r.
,
a
contradiction.
Case
(y ;y )i(x ;x )
()
1
1
2.
in r^ and (x ;x )p(y ;y
1
1
in rj.
)
Then H™ is
manipulable by individual 1 when his "true" preference is r', again a
contradiction.
Suppose more than one individual changed his aind.
W.l.o.g. assume
that these individuals are 1,2,... ,£ (notice that % < m).
Using the
Schmeidler-Sonnenscheim [21] method, let T_ = R and
T. = (r^,...,r'.,r.
H^T^
and {Y} {Y}
=
H^T
+1
,...,r
,{X,Y}
m
for
)
= !,...,<>.
j
Sir.ce {X}
let k be the first integer such that
),
m
,
- rf^Tg, {X, Y})
{X,Y}), then by cases 1 and 2 above, H
individual k, a contradiction.
Hence h
is manipulable by
obeys PA.
Part b .
Assume that
admits h
9.
an m-person Arrow SWF which obeys the
PA condition and which is not a Pareto CCR, and w.l.o.g. assume
K
{l,2,...,m}, i.e.,
admits h
9.
(m)
for every X,Y G (A ;A
) and
Q
i
G
B£
n
in r. for every
For every
K.
(m)
R6
Define H
and R G
9.^
m
.
By lemma 4 we may assume that
\
XIY in h (R) iff (x ;x )i(y ;y.)
m
Q
Q
as follows.
(ra)
fl
,
m
m
X G H (R,B) if f X € B and Xh (R)Y
for every Y G B.
We first prove that H™ is a RSCC and then that it is
also a NC GSSCC.
Since (A ;A
fl
B
G
n
m
m
H (R,B)
is a finite set,
so is also every
m
and therefore for every B there exists X G B such that Xh (R)Y
for every Y G B (h
bility.
O
is complete and transitive), hence H
obeys feasi-
m)
(m
such that C C B,
Suppose there exist C,B G n ' and R G p/
ncM
and rf^R.C) + H^R.B) n C.
-19m
m
There exists X' € H (R,C) such that X' ? H (R,B).
Case 1 .
m
m
there exists another Y e C such that Y 6 H (R,B).
nc^U
H (R,B)
Since
m
Since X' g H^R.B), then it must be YPX' in h (R), and since Y € C,
m
then by definition X £ H (R,C), a contradiction.
m
There exists X' 6 H (R,B) n c and X
Case 2 .
j*
with
m
m
? H (R,C) imply YPX' in h (R).
X'
H^R.B),
it implies X' f
By definition, since Y,X' £ B,
Hence H™ obeys INOA.
a contradiction.
obeys U, then by definition H
Since h
Since
m
there exists Y 6 c such that Y € H (R,C); this together
m
H (R,C)
0,
m
9 H (R,C).
1
The assumption
obeys P.
that XIY in h (R) if and only if every individual is indifferent be-
obeys UQ.
tween X and Y, guarantees that n
that H
R,R' £
m
(m)
fi
,
for every
in r
.
j
B
€
n^
m)
Then there exists
is manipulable.
Suppose H
is a RSCC.
This completes the proof
and X,Y G B such that w.l.o.g. r
- 2,...,m, X
(y ;y )p(x ;x
Q
m
m
€ H (R,B), Y € H (R\B) and
? rj, r. =
(y ;y )p(x ;x )
1
1
m
and the existence of k 6 K (k > 1) such that (xQ ;x, )p(y Q ;y,) in
(otherwise P is violated); since
r,
= r/,
then it implies X £
m
m
Together these imply XPY in h (R) and YPX in h (R'
tradicts the assumption that h
obeys PA, hence h
r
l
and
' r l'
r> " r
(x_;x..
f°r
i
,
j
B
G
IT
),
r.
H^R'
,B).
which either con-
obeys IIA or the assumption that it
is nonmanipulable.
there exist R,R* G $T
r'.
implies both Y £ H (R,B) (because of UQ)
in r
)
±
m'
Suppose H
is corruptible, then
and X,Y € B such that w.l.o.g.
= 2 » •••>*» X G Hm (R,B), Y € Hm (R',B), Y € Hm (R,B)
)i(y ;y.) in both
r.
and rj.
m
This implies both XPY in h (R)
m
and YPX in h (R'), which contradicts the assumption that h
satisfies
-20-
IIA, hence H
is noncorruptible;
which completes the proof that h
is
a noncorruptible GSSCC.
Q.E.D.
3.
Characterization of the Sets of Restricted Domains
Admitting Two-Person Arrow Social Welfare Functions
We first characterize all the sets of restricted domains of preferences which admit two-person Arrow SWFs and then, in light of theorem
we show that we also have characterized all the sets which admit twoperson noncorruptible Gibbard-Satterthwaite type SCC.
F =
{(X,Y)e (A l^ 2 h
(X-.JX.
)p(y n ;y.
)
x
(A ;A
Q
(2)
)
there exists
r^
12
Let
such that
in r.}, F contains all the ordered pairs of social
alternatives where the first individual can strictly prefer the alternative allocated to him in the first social alternative, to his second
mixed alternative.
F there exists r'
We refer to F as the feasible set .
e
Ji_
Let C = {(X,Y)
such that (y ;y~)p(x ;x«) in r'}.
n
n
C contains
all the ordered pairs in the feasible set, over which the second
individual can create conflict of interests with the first individual.
C is the conflict set .
We say that D
£
(a ;A
(2)
)
x
(A ;A
(2)
)
is
closed under decisive implications (CUDI) if the following conditions
hold for every X,Y,Z e (A ;A V ').
Q
1,
z
-21-
If for some r. S R
Gla.
and r
2
G
9^
(x ;x )p(y ;y )r (z ;z
(y ;y )r 2 (z ;z 2 )p(x ;x 2 ) and (X,Y)
2
If for some
Gib.
rG
1
1
G D then (X,Z) G
(x^x^pCygjy^pCzgjzp
R
1
1
),
D.
In r and
(X,Y),(Y,Z) G D, then (X,Z) G D.
If for some r.
G2a.
(z
;z
2
fi^
and r
2
G
fi
2
,
(x ;x )r (y ;y )p(z ;z ),
1
1
1
1
)p(x ;x )r (y ;y ) and (Y,Z) G D then (X,Z) G D.
2
2
2
If for some
G2b.
€
rG Q^
(x ;x )p(y ;y )p(z
2
Q
2
;
2
)
in r and (Z,X) G D,
then either (Z,Y) G D or (Y,X) G D.
We say that 9
(2)
is decomposable if there is a D which is closed under
decisive implications.
9
(?)
has a nontrivial decomposition if it is
decomposable with a D such that (F-C) ^ D
<^
p.
Intuitively D contains all the ordered pairs of social alternatives
over which the first individual is decisive.
he is not a dictator while (F-C)
vidual is not a dictator.
•j
D^
F guarantees that
D guarantees that the second indi-
We say that 9
if there exist j,k e N such that 9 .*9
Theorem
Thus
has a nontrivial decomposition
has a nontrivial decomposition.
2
admits a two-person Arrow SWF which is not a Pareto CCR, over
9
mixed alternatives, iff it has a nontrivial decomposition.
Proof
:
Part a .
Assume that 9
that
j
has a nontrivial decomposition.
= 1, k = 2 are such that 9
(2)
'
W.l.o.g. assume
= 9 xQ. has the nontrivial
-22-
Define h
decomposition.
2
For every X,Y G (A ;A
as follows.
(2)
')
n
and
R6Q< 2 \
2
XIY in h (R) iff
a.
(xgjx^KyQ^)
and (xg;x )i(y ;y ) in
2
2
r,
and r
2
respectively;
2
XPY in h (R) if
b.
(i)
(x ;x )r (y ;y ),
1
1
1
(x ;x )p(y ;y
Q
1
(ii)
(x ;x )p(y ;y
1
1
1
(x ;x )r (y ;y
Q
2
2
2
)
and either
)
or (x ;x )p(y ;y )
2
Q
2
)
in r^ ((y ;y )p(x ;x ) in r ) and
2
Q
2
2
)
in r
(or both), or
(X,Y) G D, or
(iii)
(x ;x 9 )p(y ;y
Q
(Y,X)
To prove that h
2
£
2
2
((y ;y )p(x ;x ) in r^) and
1
1
D.
is a two person Arrow SWF, first observe that part (a)
2
(2)
of the definition implies Xh (R)X for any X G (A ;A
)
Q
R S Q
(2)
,
thus h
2
is reflexive.
Then notice that if h
plete order, then there exist X,Y E (A ;A
(2)
Q
2
2
neither Xh (R)Y nor Yh (R)X.
)
and R 6
2
and every
is not a cora(2)
fl
'
such that
2
Since it is not XIY in h (R), then it must
either be the case of (x ;x. )p(y_;y.
n
)
and (y-;y„)p(x_;x„), or be the case
of (y ;y )p(x ;x ) and (x ;x )p(y ;y ) in r^ and r respectively.
2
1
Q
2
2
1
W.l.o.g. assume
(x^x^pCy^y^
in r. and (y ;y )p(x ;x ) in r (since
2
Q
2
2
these two cases are symmetric to each other).
This together with not
2
Xh (R)Y imply (X,Y) £ D, but by part (b.(iii)) in the definition, it
2
then must be YPX in h (R), a contradiction.
that h
2
Parts (a) and (b.(i)) imply
obeys the unanimity condition, and since h
2
was defined for
pairs of social alternatives only, it also obeys the IIA condition.
Thus to demonstrate that h
2
is at least a social welfare function, it
is only left to prove that it obeys transitivity.
Suppose it is not a
-23-
(2)
(2)
v
transitive relation, and there exist X,Y,Z G (A ;A V ') and R G Q '
2
We distinguish among the following three cases.
such XPYPZh (R)X.
((x^x^r^y^yp,
XPY because of unanimity
Case 1 .
(x ;x )r (y ;y )).
Q
2
2
2
If also YPZ because of unanimity, then (x ;x )r (z-;z ),
n
(xn ;x_)r 7 (z n ;z ? ) and either 1 or 2 (or both) strictly prefers X to Z.
2
By definition this implies XPZ in h (R), a contradiction.
(y ;y )p( 2
;
1
z
1
)
and (z ;z )p(y ;y ) in
2
2
YPZ implies (Y,Z) G D ;
(z
n
and r
2
respectively, then
2
(x^x^r^y^y^pCz^z^
and Zh (R)X imply
;z_)p(x„;x») in r~, and together with condition (G2a), this implies
(X,Z) e D.
(z
^
If
;z
1
Then by definition XPZ in h (R), a contradiction.
)p(y Q ;y
1
)
and (y ;y )p(z ;z ) in
Q
2
2
^
and r
2
respectively, then
YPZ implies both (Z,Y) 9 D and (x ;x )r (y ;y )p(z ;z )
2
2
2
Q
2
plies both (z
n
)p(xr) ;x1 )r 1 (y f.;y 7
;z
If
2
Zh (R)X im-
.
and (Z,X) G D, and by condition
)
(Gla), this implies (Z,Y) G D, a contradiction.
Case
2
XPY in h (R) because of (x ;x )p(y ;y
Q
Q
2.
)
and (X,Y) G D.
in r
2
If YPZ because of unanimity, then Zh (R)X and (x-jx^ )p(y ;y- )r (z ;z
n
1
f)
1
imply (y ;y )r (z ;z )p(x ;x ) and (X,Z)
2
2
2
2
Q
above
and r„ imply, by condition (Gla),
r.
If (y ;y )p(z ;z
1
1
)
(z
;x..
;z
Q
1
(X,Z) G D, a contradiction.
(X,Z) G D.
Then, by definition,
2
)p(z ;z
R
But (X,Y) G D and the
D.
and (z ;z )p(y ;y ) in el and r respectively, then
Q
2
2
2
(Y,Z) G D, and by condition (Gib),
(*
6"
)
implies XPZ in h (R), a contradiction.
)
and (y ;y )p(z ;z ) in r. and r respectively, in which
Q
2
2
2
Thus it must be
1
)p(y ;y
1
case YPZ implies (Z,Y)
G
D.
If also (x ;x.)p(z
2
;z
),
implies (y ;y )p(z ;z )p(x ;x ) in r and (X,Z) ^ D.
2
2
2
2
Q
(X,Y) G d,
)
(X,Z)
G
D and (Z,Y)
G
then Zh (R)X
But then r 2>
D together contradict condition (G2b).
So suppose (z ;z )r (x ;x )p(y ;y ).
1
1
1
1
If
(y ;y )p(z
2
;z
2
)r (x ;x
2
)
2
then
-24-
(X,Y) G d implies (Z,Y) G D, a contradiction.
by condition (G2a),
leaves as the only possible case, (y_;y„)p(x_;x„)p(z
2
Zh (R)X implies both (z
;z..
)p(x ;x-
n
and (Z,X) e D.
)
;z
)
This
Then
in r„.
Then, by condition
(Gib), it is (Z,Y) ^ D, again a contradiction.
XPY because of (y ;y )p(x ;x
Q
Case 3 .
respectively and (Y,X)
(z
Q
;z
2
D,
2
in r
)
and r^
unanimity then
(y^y^pU^z^r^Xg^)
2
and (X,Z)
9
2
If Zh"(R)X because of (x-jx. )p(z
tradiction.
(X,Z)
Q
and
and by condition (Gla), together they imply (Y,X)
D,
r9
(x ;x )p(y ;y
2
If Zh (R)X because of
D.
)r (x ;x )p(y ;y ) implies both
2
2
2
(Y,Z)
in
9
),
^
D,
;z.. )
in r,,
(z
a con-
D,
;z
2
)p(x ;x
2
then YPZ implies (Y,Z) ^ d, and together with r-,
and condition (G2b) imply (Y,X) e D, a contradiction.
2
This leaves Zh (R)X because of
in r_ and (Z,X) e D.
(z_;z..
)p(x ;xQ
in r-,
)
(x ;x )p(z ;z
Q
2
2
)
Then, either YPZ because of unanimity which to-
gether with condition (G2a) implies, (Y,X)
(y ;y )p(z ;z ) in r^,
1
1
(z
;z
2
<=
D,
YPZ because of
)p(y ;y 2 ) in r 2 and (Y,Z) e d which together
with condition (Gib), again implies (Y,X) e D, or YPZ because of
(z
;z
1
)p(y ;y ),
1
condition (G2b)).
Hence h
2
(y ;y )p(z
2
;z
)
2
and (Z,Y) ? D, and again (Y,X) € D (by
In all these three cases we reach a contradiction.
is transitive and therefore a social welfare function.
complete the proof that h
2
is an Arrow SWF, notice that since
To
(2)
ft
has a
(2)
such that
nontrivial decomposition, there exist X',Y' E (A ;A
)
Q
(X\Y') S
c and (X',Y')
(X",Y") e C n D.
function,
)
h
2
?
D and X",Y" ^ (A ;A
(2)
)
Q
These two cases imply that h
2
such that
is a ND social welfare
is not a Pareto CCR because of parts (b.ii) and (b.iii)
of the definition and the fact that
(2)
ft
has a nontrivial decomposition.
-25-
Part b.
Suppose
admits a two-person Arrow SWF which is not a Pareto CCR.
9.
(2)
W.l.o.g. assume
e F|(x
D - {(X,Y)
=
9.
9..
x H_
admits that 2-person ASWF.
Define
;x )p(y ;y ) in r implies XPY in h (R)}.
±
1
1
Since
net h
V
not a Pareto CCR (and by lemma 2), then there exist X',Y* € (A ;A
Q
and R' G JT
in
2
in h (R').
Then either by IIA or by U
2
2
(x^xppCy^yp
2
will always imply X'PY' in h (R), thus (X',Y') G D and (F-C) £ D.
in r
h
')
such that w.l.o.g. (x^;xpp(y^;yp in rj, (y^;ypp(x^;xp
'
and X'PY'
r'
is
is also ND,
then there exist R" G oS
2
and X",Y" G (A ;A
^
<"
2)
)
such that
()
(y'^yppW^p
in r^,
2
(x^xppCy^yp
in r^, and X"h (R")Y".
Hence (Y",X") G C and (Y",X") G D, therefore (F-C) C D C F.
the proof that
(2)
has a nontrivial decomposition, we show that D is
9.
Suppose there exist
closed under decisive implications.
r'
2
G
9.
2
To complete
G
r'
9.
,
(2)
and X,Y,Z G (A ;A V ') such that (x ;x )p(y ;y )r^(z ;z ),
()
1
1
1
Q
(y ;y ) r 2( z ;z 2 )p(xo ;x 2 ) and (X,Y) 6 D *
2
o
This imP lles both XPY in h2 ( R,
2
2
Then by transitivity XPZ in h (R*
and Yh (R')Z.
)
2
XPZ in h (R) for any R such that (x ;x )p(z ;z
Q
1
which proves that D obeys condition (Gla).
1
)
>
and by either IIA or U,
in r., hence (X,Z) G D
By using similar arguments
it is straightforward to show that D also obeys (Gib) and (G2a), thus
Suppose there
to complete the proof we show that D also obeys (G2b).
exist X,Y,Z
(
G(A ;A V
2)
')
Q
and r£ G
in r£, (Z,X) G D and (Y,X) G D.
fl
2
such that (x ;x )p(y ;y )p(z ;z )
2
2
2
2
(Y,X) G D implies XPY in h (R) for
(2)
every R G a
such that (x ;x )p(y ;y ) in r 2>
2
2
Q
(Z,X) G D implies ZPX
(2)
in h (R) for every R G n
such that (z.;z. )p(x ;x.
2
(Z,X) G D
£
f,
there exists such r
is r' and (z ;z )p(x ;x
Q
Q
)
in r
).
Choose R G
)
in
r-
(and since
such that r
9.
2
and then ZPXPY in h (R).
2
By transitivity
-26-
2
ZPY in h (R), and since (y ;y_)p(z ;z_) in r„,
n
(Z,Y)
G
then it must be
D.
Part c .
Suppose the only ASWFs
no j,k G N, such that 2. x
admits, are Pareto CCRs, then there are
ft
has a nontrivial decomposition, since
JL
otherwise it is possible, as was done in part (a) to define a two-person
Q.E.D.
ASWF which is not a Pareto CCR.
Corollary
1
admits a two-person noncorruptible Gibbard-Satterthwaite type
ft
SCC iff it is has a nontrivial decomposition.
Proof
Since any two-person Arrow SWF also obeys the PA condition, then by
theorem
1
and theorem
2
together, the proof is immediate.
Q.E.D.
The following corollary demonstrates that some of the previous char-
acterizations of domains admitting 2-person ASWFs, are special cases of
the characterization introduced in this paper.
Corollary
(i)
If
2
|A
n
|
=1, then conditions (Gla) to (G2b) are reduced to con-
ditions (Dla) to (D2c) of Ritz [17];
(ii)
if
|A
n
|
1,
A.
A,
ft
=
ft
for every
j
£ N and no indifferences
among alternatives are allowed, then conditions (Gla) to (G2b)
are reduced to conditions (DI1) and (DI2) of Kalai-Ritz [9];
-27-
if for every
(iii)
N and every a G A
j€
,
A. (a) = A,
o,
.
= n,
|
=1
a|
and no indifferences among alternatives are allowed, then con-
ditions (Gla) to (G2b) are reduced to conditions (Dlla) to
(DI2b) of Kalai-Muller [8].
Proof
Since
(i)
x,
1
|
aJ
= 1 (say A
n
= {a}),
then instead of (ajx..) we use
in
and instead of X = (a;x,,...,x
)
in
we use X = (x 1( ...,x
).
Conditions (Dla) to (D2c) are as follows.
ft.,
x py .rz and (X,Y) £ D then (X,(z.,,y )) e D.
2
1
1
rG
Q.
x ry pz
and (Y,Z) G d then ((x ,y ),Z) G d.
2
rG
o
x py pz
and (X,Y),(Y,Z) G D, then (X,Z)G
If for some r
G
Q.
D2b.
If for some r
G « 2> x.ry pz. and (Z,X) G d then (Z.Cx^y^) €
D2c.
If for some r
G Q
Dla.
If for some r
Dlb.
If for some
Die.
If for some
D2a.
G
t
,
x-py.rz. and (Z,X) G D then ((z ,y ),X) G D.
2
(X,Y) G D and (X,Y) £ C then x py rz
implies
(X.U^y^) G
if (X,Y) G C then condition (Gla) implies (X,(z ,y )) G D.
2
D;
In both
Similarly (G2a) implies (Dlb), con-
cases condition (Dla) is derived.
dition (Gib) implies (Die), condition (G2a) implies (D2a), condition
(Gla) Implies (D2b) and condition (G2b) implies condition (D2c).
(ii)
DI1.
Conditions (DI1) and (DI2) are as follows.
If for some p
6
Q, x py ,pz
1
,
(X,Y) G D and (Y,Z)
G
D,
then
;
(X,Z) G D.
DI2.
If for some p
(Y,X) G D.
G
ft,
x-py.pz
D.
and (Z,X) G d then either (Z,Y) ^ d
x py pz
2
2
or (Y,X) G D (or both).
If
D.
and (Z,X) G D, then (Z,Y)G
D or
-28-
Condition (Gib) implies condition (DI1) while condition (G2b) implies
(DI2).
(iii)
Since for every
A and
e N and every a e A_, A (a)
j
say A = {x}, then (A-.;A
)
= (A_;x) for every
instead of X = (x ;x,...,x) we just use
n
x..
j
e N.
=1
|A|
Thus
etc.
Conditions (Dlla) to (DI2b).
,p£
If there are p
DI1.
£D
such that xpypz and ypzpx then
P.
implies that (x,z) G
Dlla.
(x,y)
Dllb.
(z,x) £ D implies that (y,x) 6 D.
DI2.
there is a
If
pS
D,
and
S such that xpypz then
G d imply (x,z) £
DI2a.
(x,y) G D and (y,z)
DI2b.
(z,x) e D implies that either (y,x) 6 D or (z,y) 6 r
D,
and
(or both).
Condition (Gla) implies (Dlla), condition (G2a) implies condition (Dllb)
(when X - y, Y = z and Z = x).
Condition (Gib) implies condition (DI2a)
Q.E.D.
and (G2b) implies (DI2b).
To show the usefulness of theorem 2, we discuss two examples.
Example
For
Restricted domains containing inseparable
pairs of alternatives
1.
j
G N, (x ;x), (y_;y) G (A-;A.) is an inseparable pair of altern
natives in
P.
.
(denoted by ((x ;x),(y ;y)) if there exists r e
ft
.
that (x ;x)p(y ;y) in r, and for no (z~;z) e (A ;A.) and no r' G
Q
Q
Q
it is (x ;x)p(z ;z)p(y ;y) in r'.
Q
such
fi
,
(See Kalai-Ritz [9] and Ritz [17]
for discussion of inseparable pairs of private alternatives and Kalai-Ritz
-29-
Kim-
[10] for the case of inseparable pairs of public alternatives.
Roush [11] proved for the case of public alternatives, no indifferences
allowed and all individuals have the same restricted domain of preferences that the domains containing exactly one inseparable pair are the
largest restricted domains that still admit Arrow SWFs).
Assume that for every
P.
=
.
and
P.
2) and every a
£
A-.,
A (a)
A^
,
contains only antisymmetric, complete and transitive binary
fi
Suppose ((x„;x), (y„;y)) is an inseparable pair in Q.
relations.
Claim
£ N (n =
j
1
If JAqXAJ
> 2 or if
IA-xaJ =
and
2
=2, then
\p.\
P.xfl
has a non-
trivial decomposition.
Proof
{(X,Y) G F such that either (y ;y ) ^ (x ;x) or
Q
2
Q
Define D
(x ;x
Q
2
)
t (y ;y)}.
Q
Obviously D C F.
{A
X
M
=
;z)
e A xA
i
= fp
°
i'
s,t *
p2 }
- 2 then actually
(*o'* )p 1^0 ;
^
and
Hence (Y,X) S D and (X,Y) € D where X = (x ;x,x) and
Y - (7 ;y,7)» thus F-C
Q
Q
= 2 and |q|
|a xA.|
q
^o'^'^O'^'
(y ;y)P 2 (^ 5x).
(z
If
S
D
such that
(
to demonstrate that D ^ 0,
F«
S
x
»
x)
DC
If
I*
I
a xAtI >
( z q5
o
z)
F and F-C
t
2
then tnere exists
^o ,y ^
5 D«
and a S ain
it:
is easy
T<> prove that D is closed
under decisive implications is straightforward and the proof is omitted.
Q.E.D.
-30-
Example
2.
Let
|a
Decentralization .
q
|
^2.
strictions on
!1
(n = 2), the only re-
are that for every a,b e A- and every
(a;x)p(b;y) in r for some x e A (a) and
r for every x
j€H
Assume that for every
G A
(a) and y
£ A.(b).
y*
r68
,
if
e A.(b), then (a;x)p(b;y) in
We titled this example "decen-
tralization" since such restrictions allow separation of issues where
different individuals are endowed with the responsibility for different
areas and a coordunator is responsible for decisions pertaining to matters
involving more than one area.
Claim
2
The above Q^xfi
has a nontrivial decomposition.
Proof
Define D = {(X,Y) € f|x
q
^ y }
Q
.
The proof that F-C £ D £ F and
that D is closed under decisive implications is straightforward and
is omitted.
4.
Q.E.D.
The Group's Size and the Existence of Arrow
Social Welfare Functions
Kalai-Muller [8], Maskin [13],
[14], Kalai-Ritz
[9]
and Ritz [17]
all proved for their respective cases the independence of the existence
of an Arrow SWF from the group's size.
That is to say that if a set of
restricted domain of preferences admits a k-person Arrow SWF for a
finite k greater than one, then it admits an n-person Arrow SWF for any
given finite n greater than one (though these ASWFs need not be the same).
These results enabled the characterization of the domains admitting
Arrow SWFs in general, rather than characterizing only the domains
-31-
Unfortunately, we did not manage to
adraitting two-person Arrow SWF.
prove a parallel result for the general case analyzed here and the issue
of complete independence is still an open question, nevertheless the
following two theorems do contain some interesting results about the
relationship between the group's size and the possible existence of
Theorem 4 also displays some of the difficulties which
Arrow SWFs.
arise when dealing with a mix of private and public alternatives.
Theorem
3
For n >
2,
admits a k-person Arrow SWF then it admits an
if Q
m-person Arrow SWF for any
2
k
_<
m
<^
n.
_<
Proof
admits h
Suppose 8
(k
_>
2), a k-person Arrow SWF.
v
K = {l,2,...,k} C N be such that
Q,x...xn
1
k
xft
k+1
XIY in h
j
(ii)
'
admits h
Define h
.
on
as follows.
For any X,Y€ (A ;A
(i)
ft
W.l.o.g. let
k+1
(k+1)
)
and
R€
n
(k+1)
(R) iff (x ;x .)i(y ;y
Q
Q
.)
in r
for every
= l,...,k+l,
XPY in h
(1)
X
(2)
x
k+1
(R) if
-(k+l)
-(k+i)
in
r,
k
PY
IY
°r
-(k+l)
^^-(k+l))'
-(k+i)
and Cx JX
in hk(R
)pCy Jyk+l )
fc+l
-(k+i) )
.
k+1
The proof that h " is an Arrow SWF is straightforward and is omitted.
By repeating the above step as many times as needed we can prove that
il
admits an m-person Arrow SWF for any m such that k
_<
m
_<
n.
Q.E.D.
-32-
We now consider a special case of the general mixed alternatives
case, where the public and private alternatives are independent of each
other, every individual in the group has access to the same set of
private alternatives, indifferences among alternatives are not allowed
and all individuals have the same restricted domain of preferences.
Assume that for every
and
ft'
e N and every a G A
j
,
A. (a) = A^,
ft
ft',
.
contains only antisymmetric, transitive and complete binary
We say that
relations.
admits an n-person ASWF, if
admits an
ft'
Also assume that no group indifferences are allowed.
n-person ASWF.
Theorem
ft'
4
For n >
3,
ft'
admits an n-person ASWF iff it admits a 3-person ASWF.
Proof
Part a .
Assume that
ft'
admits a 3-person ASWF, then by theorem
3,
ft'
also
If
ft'
con-
admits an n-person ASWF.
Part b .
Assume that
ft'
an n-person ASWF, for n >
admits h
tains an inseparable pair of alternatives and [A-xA-l >
then by claim 1 of example
theorem
3
it also admits h
1,
3
ft'
admits h
2
|
a 3-person ASWF,
- 2,
|A
|
2
or
|ft'|
_>
2,
a 2-person ASWF, and then by
so let's assume that
doesn't contain any ISP (notice that if |a xAJ q
is either the case of |A
3.
2
and
|ft'|
= 1 and then for no n >
ft'
- 1 then it
2
is there
-33-
h
n
1
ft
an n-person ASWF, or it is the case of |A n
admits h
3
|
=1,
|A
|
=2
and then
a 3-person ASWF as was proved in Kalai-Ritz ([10],
theorem
3,
(ii))).
Individuals i,j ^ N are identical over X,Y € A xA? (notice that
now (A ;A
(n
Q
O
- (A xA^), denoted by (i,j), if r - r
Q
±
and (y n ;y.) = (y n >y.)-
(x^Xj) - (x ;x
,
Q
)
Individual k is dictator by (i,j) , denoted
k = d(i,j), if whenever (i,j) are identical, k is a dictator, i.e., for
every X,Y e A n x ^S and ever y R € &'
y
- y
h
are strict orders).
,
then (x n ;x,
)r. (y_;y,
)
such that
r.
= r,, x, = x. and
implies Xh (R)Y (notice that both
and
f.
Lemma 5
For n >
if h
4,
a SWF is such that whenever two individuals are
identical there exists a dictator by them, then this dictator is unique.
Proof
Suppose the claim is false.
Since
|
A_
|
^2,
(x ;x)r(x ;y).
Q
let (x ;x), (x ;y)
Q
Q
Case 1 .
For every (i,j), d(i,j) G {i,j}.
e A n xA and
i
r
€ n
'
be such tnat
Let X = (x ;x,x,y,y,... ,y), Y - (x ;y,y,x,x,y,
..
.
,y) and
n
n
R = (r,...,r), then Xh (R)Y because d(l,2) e {1,2} and also Yh (R)X be-
cause d(3,4) G {3,4}, a contradiction.
Case 2 .
There exists at least one identical pair (i,j) s.t. d(i,j) £ (i,j}.
W.l.o.g. assume that 1 = d(2,3).
Subcase 1 .
1
£ U,j}.
Suppose there exists (i,j) s.t. d(i,j) f
Then if k = d(i,j), let X = (x ;x,y,
Y = (x ;y,x,...,x) and R = (r,...,r).
Q
. . .
Then Xh (R)Y because of
and Yh (R)X because d(i,j) - k ^ 1, a contradiction.
,y),
1 =
d(2,3)
1.
-34-
Subcase
d(l,j)
2
^
1€
.
Since d(l,j) t
{i,j}.
1,
then also d(l,j)
s
5
j
or
^l»j}> and a contradiction can be easily constracted.
Q.E.D.
The following lemmas are proved for n =
This is done only in
4.
order to simplify the notations, the proofs for the cases of n > 4 are
identical.
Individual
i is
s.t. x_ = y n = b,
Lemma
b-dictator if for every R €
n
tt'
and X,Y £ A xA
x
(b;x,)r.(b;y.) implies Xh (R)Y.
6
Let h
4
If for every two identical Individuals there
be a SWF.
exists a dictator, and if for some
b^A.
there is a b-dictator then
these dictators are the same.
Proof
W.l.o.g. assume
pose for some b £
1
= d(i,j) for every I,j
A..,
I
fi
'
4
Sup-
Q.E.D.
is a public-dictator if for every R G
and y
j).
Then by the above assumptions it is
Xh (R)Yh (R)X, a contradiction.
that x- - X- « ... = x
j*
s,t * (b;x)r(b;y) and let X - (b;x,y,y,y),
(b;y,x,x,x) and R = (r,r,r,r).
j
(i
Then let
1 is the b-dictator.
(b;x),(b;y) £ A q*\ and r € n
Y
{1,2,3,4}
<=
,n
ft
= y_ = ... = y
,
and X,Y 6 A xA^ such
then (x ;x)r (y ;y)
Q
i
n
implies Xh (R)Y.
Lemma
7
Let h
be a SWF.
If for every two identical individuals there
exists a dictator k, and there is also
j
a public-dictator then k = j.
-35-
Proof
Suppose k ^
(a;x),(b;y) e
W.l.o.g. assume k = 1 and
j.
Aq*^ and
G a
r > rf
'
= 2.
j
Let
be sucn tnat (a;x)r(b;y) and
then
= 1 and h
(b;y)r'(a;x) (if there are no such r and
r'
2-person ASWF can be defined directly).
Let X = (a;x,x,x,x),
,
|ft*|
a
4
Y " (b;y,y,y,y) and R = (r,r , t r , ,r') then Xh (R)Y because of 1 = k(2,3)
4
and Yh (R)X because of
j
= 2,
a
contradiction.
Q.E.D.
k is an induced dictator if he is the dictator of lemmas 6 and 7
combined.
Preferences (r,r') G
there exists r" 6
over (a;x
r'
2
),
2
fl'
are ((a;x. ,Xg), (b;y ,y )) connected if
..
.
,,
which agrees with r over (a;x.
ft'
),
(b;y.
and with
)
(b;y >.
2
Lemma 8
Let h
4
be an ASWF for which there exists k, an induced dictator.
If there exist X,Y £ a xAi
4
and R G n
q
4
'
such that
^o ;yk^ r k^ x O ;xk^
4
Xh (R)Y then there are no i,j S N (i t j) such that (r ,r
((XqJx^x .),(y
and
are
)
;y ,y .)) connected.
1
Proof
W.l.o.g. assume that k -
1,
(r ,r
2
3
)
are ((x ;x ,x ),(y ;y 2> y ))
2
3
3
connected, (y.;y )r (x.jx.) and Xh (R)Y.
there exists
with
(7
r
;y~
3
2
r'
£
over (x ;x
)r'(y"
Q
;y" ).
2
which agrees with r» over (x n ;x 1
ft'
2
)
Since (r_,r-) are connected
,
(y_;y
2
)
) ,
(y n ;y,
) ,
and
w.l.o.g. assume (x ;x )r' (x_;x,) and
Let R' = (r^r'.r 1 ,^).
Q
2
4
By IIA it is Xh (R')Y.
-36-
_____
_____
Let Y - (y ;y ,y ,y ,7 ) and X - (x ;x ,x ,x ,x ).
4
2
2
1 2 2 4
1
4
by unanimity, Yh (R')X since
1'
—
_4
Then it Is Yh (R')Y
4
s—
is an induced dictator, and Xh (R' )X
—
4
Q.E.D.
by unanimity which leads to Yh (R')X, a contradiction.
Individual k is x-essential if there are X,Y € A xA
Q
such that x
Q
every
j^N,
Lemma
9
If h
4
\
= Y == *>
k
n
£ k and Xh (R)Y.
t y ,
Q
j
(x ;x)r (y ;x),
k Q
and
R6
(y Q ;y )r,(Xg;x
fl*
for
)
is an ASWF for which there exists k an induced dictator,
for no x e A
is there an individual
j
then
^ k who is x-essential.
Proof
W.l.o.g. assume k =
Suppose the claim is false.
x-essential, i.e., there exist X,Y e A n xAi
X = (x ;x ,x,x ,x
1
for
3
4
lemma
= 1,3,4 and Xh (R)Y.
8,
and R G n
4
'
j
= 2 is
such tnat
Y = (y ;y ,x,x ,x ), (x ;x)r (y ;x), (y ;y )r (x ;x..)
Q
4
Q
2
Q
()
3
1
),
4
j
4
and
1
Since 1 is an induced dictator, then by
no two preferences of r
2
,
r~ and r
4
There-
can be connected.
fore (x ;x)r (y ;x) implies (y ;x)r (x ;x) and (y ;x)r (x ;x),
4
Q
3
Q
2
Q
(y ;y )r (x ;x 3 ) implies (x ;x 3 )r 2 (y ;y 3 ) and (x Q ;x 3 )r 4 (y ;y 3 ) and
Q
3
3
(y ;y 4 )r 4 (x Q ;x 4 ) implies (x ;x 4 )r 2 (y Q ;y 4 ) and (xQ ;x 4 )r 3 (y ;y 4 ).
Case 1 .
Suppose (y ;y )r (y ;y ).
4
Q
3
3
If also (y ;y )r (y ;y ),
4
4
Q
3
then it
(x ;x )r (y ;y ), and therefore
4
3
4
4
4
(x ;x ,x,x ,x )h (R)(x ;x ,x,x ,x )h (R)(y ;y ,x,y ,y ) and
4
3
4
()
3
1
Q
3
3
Q
1
1
implies (x ;x )r (y ;y )
4
3
3
Q
,
(x ;x ,x,x ,x )h (R)(y ;y ,x,y ,y ).
4
4
4
1
3
1
and
r.
If
(x ;x )r (y ;y ) then r
2
4
3
2
Q
are connected over ((x Q ;x,x.,), (y ;x,y 4 ), a contradiction, and
if (y ;y )r (x ;x ) then (x ;x )r (y ;y ) and r 2> r are
3
4
Q
2
3
4
2
3
Q
-37-
((x ;x,x,
n
),
(y_;x,y_))-connected, again a contradiction.
(y ;y )r (y ;y ),
4
4
3
If
(x^x^x.x^x^h 4 (R)(y
then by unanimity
;y ,y,y ,y ),
1
3
3
but then either r 2' r 3 or r 2' r 4 are (( x x > x 4)> (yQ>x,y~))-connected,
o»
a contradiction.
Suppose (y ;y )r (y ;y ).
3
4
Q
3
Case 2 .
(x^XpX.x^.x^h
unanimity
either with r_ or with
3
(x ;x )r (y ;y ) and
4
3
3
Q
4
,
(R)(y ;y ,y,y ,y ) and r is connected
4
4
2
1
a contradiction.
3
(x^x^x.x-^x^h
4
n
;x..
(x-.:x_)r_(x_:x,
If
4
4
3
then
)
(R)(y ;y ,x,y ,y ); if also
4
4
1
then it must be (x ;x )r (y ;y )r (x ;x ),
4
4
4
3
4
Q
(y ;y )r (y ;y )
3
4
4
(x
r.
4
If also (x ;x )r (x ;x ) then by
4
3
Q
3
,x,x,,x-)h (R)(y_;y_ »x,y, ;y,
and r»,r, are connected, a contra-
)
diction; but if (y ;y4) r 4(y 5y ) then lc must be
3
^
x ;x 4^ r 4^ y ;y 4^ and
o
o
or with r, respectively, a contradiction.
r_ is connected either with r,
Q.E.D.
We say that k is (x,r)-dictatorial by
for every X,Y £
then
(
x ;x
An*^
r
;y
k^
k^ k^ y
4
and every R G
a4
im P lies
(
1
4
ft
j,
denoted k = d.(x,r), if
s.t. x. = y
= r,
= x and r
R)Y *
Lemma 10
Let h
4
be an ASWF for which there is k, an induced dictator.
there are x 6
A.. ,
j
6 N and
G
r
If
such that d.(x,r) exists then
ft'
k = d.(x,r).
Proof
W.l.o.g. assume k = 1,
2^1.
j
€
N,
x'
Also assume (x n ;x)r' (y n ;x).
for some y,z e A.
^ a
,
and
Let r G
ft'
r'
€
Q«
s .t.
d.Cx'.r') =
be such that (y n ;y)r(x_;z)
(notice that there must be such r).
Suppose
j
j*
1.
-38_
Then since
4
d.(x,p) it must be (x ;z,x,x,x)h ((r,r' ,r' ,r* ))(y„;y,x,x,x),
2
Q
which contradicts the assumption that
j
=
1,
then since
=
2
an induced dictator.
1 is
If
4
d^x.p)
it is (y ;x,y,y,y)h ((r' ,r,r,r))(x ;x,z,z,z)
Q
which again contradicts the assumption that 1 is an induced dictator.
Q.E.D.
Lemma 11
If h
4
is an ASWF and there exists k an induced dictator,
exist at least one individual
e
j
N,
alternative x G A
then there
and r e Q' for
which d.(x,r) does not exist.
J
Proof
Assume w.l.o.g. that k = 1.
Suppose this is not the case.
lemma 10, 1
d.(x,r) for every
4
an ASWF, there exist X,Y € A xA
4
Xh (R)Y.
N,
x
Q
;z
)
A-
€E
e
a nd R
-,
This implies that there exists
(x ;x )r (y ;y ).
2
2
2
exist (z
£
j
n
j
4
'
£
and r G Q'.
say
e A xA
x
and r e n
4
then (y ;y ,y ,y ,y )h (R' )(z
4
2
3
1
then
is
j
» 2, s.t.
such that Cy ;y )r(z ;z )r(x ;x ).
;L
1
;L
'
4
)
4
Since ((y ;y ),(x ;x )) is not an ISP, then there
1
1
Let R' = (r,r ,r,,r.) then by IIA also Xh (R' )Y.
1 = d (x ,r
3
3
3
Since h
(y ;y-)r., (x ;x-) and
s.t.
N,
Then by
;
Z;L
Since 1 = d.(y ,r
2
2
)
,y 2 ,x 3 ,x 4 ), and since also
(z^z^y^x^x^h 4 (R Xx^x-pX^x^x^,
f
a contra-
Q.E.D.
diction.
Let us now complete the proof of the theorem.
If h
(n
>^
4) is such
that there exists an Induced dictator for it, then by lemma 11 there
eA
exist x
assume
j
,
j
6 N and
r
= n and define h
G
&»
'
(n
for which there is no d.(x,r).
>^
4) as follows.
W.l.o.g.
For every X,Y & A xAi
q
1
-39-
andRE'Q
1
••
•
(y n »y-i
»
"" 1
Xh
,
n-1
n
(R)Y iff (x n ;x_,...,x . ,x)h ((r.
u
l
n— i
i
GN such that there
is no k
assume (i,j) to be (n-l,n).
1
X,Y e AqXA^" and
..
,r
.
It I s straightforward to show that h
»y _-i> x )«
If there is no induced dictator for h
i,j
,
fl'
Xh
,
€ N which
n_1
Case
then:
Define h
"
11 1
R£
,
n—
,,r))
is an ASWF.
There exist
1.
is dictator by (i,j).
W.l.o.g.
For every
as follows.
(R)Y iff
n
(x ;x ,...,x _ ,x _ )h ((r ,...,r _ ,r _ ))(y ;v
... v _ ,y _ ).
1
n 1
n 1
1
n 1 n 1
i5
n 1 ri 1
}
is an ASWF is straightforward.
proof that h
there exists k a dictator by (i,j).
there must exist b
there exist r G
Y = (b;y, ,...,y
1
Define
fl»
For every (i,j)
Since there is no induced dictator,
A n such that there is no b-dictator for h
€
R»
n— li»yn
)
a*
.
X - (b:x.,...,x
1
n-1,
such that (b;y. )r, (b;x,
i
J.
x
),
,x
n
)
i.e.,
,
and
Xh (R')Y when r
n
= r.
+ A^ as:
A-.XA.
g:
Case 2 .
The
in
g(X) - y if X - (b;? ,...,?)
n
= x
Define h
Xh
n_1
n_1
n
otherwise,
For every X,Y € A xa""
as follows:
n
1
and
(R)Y iff (X,g(X))h ((r ,...,r _ ,r))(Y,g(Y)).
n 1
1
is an ASWF is straightforward.
that h
theorem
4.
RG
a1
""1
,
Again, the proof
This completes the proof of
Q.E.D.
-40-
References
New
1.
Arrow, K. J., Social Choice and Individual Values , 2nd ed.
York: Wiley, 1963.
2.
Black, D.
3.
Blau, J. H., "The Existence of Social Welfare Functions,"
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4.
Blin, J. M. and M. A. Satterthwaite, "Individual Decisions and
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Dutta, B.„ "Existence of Stable Situations, Restricted Preferences
and Strategic Manipulation under Democratic Group Decision
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6.
Fishburn, P.
Princeton, N.J.:
7.
Gibbard, A., "Manipulation of Voting Schemes:
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Kalai, E. and E. Muller, "Characterizations of Domains Admitting
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Kalai, E. and Z. Ritz, "Characterization of the Private Alternatives Domains Admitting Arrow Social Welfare Functions,"
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Kim, K. H. and F. W. Roush,
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Kelly, J. S., Arrow Impossibility Theorems .
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Maskin, E., "Social Welfare Functions on Restricted Domains,"
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Maskin, E., "On Strategy Proofness and Social Welfare Functions
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16.
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North-
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Satterthwaite, M. A. and H. Sonnenschein, "Strategy-Proof Allocation Mechanisms," Discussion Paper No. 395, The Center for
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Schmeidler, D. and H. Sonnenschein, "The Possibility of a Cheat
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