This note shows that results similar to Arrow's Impossibility Theorem can be proved by replacing the weak Pareto principle by a weaker condition called Pareto Neutrality and used by Xu (1990) to state another version of Sen''s liberal paradox. Our result strengthens Xu''s arguments for taking into account non-welfarist information into the social-choice-theoretic framework.Arrow's Impossibility Theorem

Herrade Igersheim

English

Research Papers in Economics

Extending Xu's results to Arrow's Impossibility Theorem 
Herrade Igersheim
CODE, Universitat Autònoma de Barcelona
Abstract
This note shows that results similar to Arrow's Impossibility Theorem can be proved by
replacing the weak Pareto principle by a weaker condition called Pareto Neutrality and used
by Xu (1990) to state another version of Sen's liberal paradox. Our result strengthens Xu's
arguments for taking into account non−welfarist information into the social−choice−theoretic
framework.
I am particularly grateful to Claude d'Aspremont and two anonymous referees for their helpful comments. The usual caveat, of
course, applies.
Citation: Igersheim, Herrade, (2005) "Extending Xu's results to Arrow's Impossibility Theorem." Economics Bulletin, Vol. 4,
No. 13 pp. 1−6
Submitted: October 4, 2005.  Accepted: November 29, 2005.
URL: http://www.economicsbulletin.com/2005/volume4/EB−05D70027A.pdf1. Introduction
The purpose of this note is to prove another version of Arrow’s famous
Impossibility Theorem (1963) by using a condition deﬁned by Xu (1990) in
a paper on Sen’s liberal paradox (1970a,b). Xu notes that the weak Pareto
principle consists of two parts: a neutrality factor called the Pareto Neutrality
on the one hand, and a unanimity factor called the Pareto Unanimity on the
other hand. Xu shows that the Pareto Neutrality is responsible for generating
the Impossibility of a Paretian liberal. We attempt here to prove that the
Pareto Neutrality is responsible for Arrow’s Impossibility Theorem as well.
Our study proceeds from a stream of research focusing on the under-
standing of analytical connections between Arrow’s and Sen’s impossibility
results, which the very recent work of Saari (1998, 2001) is part of. One of
the objectives of this study is to stress conceptual drawbacks of the social-
choice-theoretic framework and then to be able to circumvent these negative
results in a satisfying way.
In this note, we follow the line of reasoning of Wilson (1972) and Kelsey
(1985, 1988). Indeed, Kelsey (1985) proves another version of Sen’s lib-
eral paradox by replacing the weak Pareto principle by a condition of non-
imposition used by Wilson (1972) within the framework of Arrow’s Impos-
sibility Theorem. Beyond formulating a new analytical result, we aim at
strengthening Xu’s view on ﬂaws of the social choice theory, which tends to
challenge Wilson’s and Kelsey’s conclusions.
The remainder of the paper is organized as follows. In Section 2, we
present the basic concepts of our analysis. Section 3 deals with the proof of
our theorem and ﬁnally, Section 4 concludes the paper.
2. Basic concepts
Let N = f1;2;:::;ng be the ﬁnite set of individuals, which constitutes
society (n ¸ 2). X denotes the ﬁnite set of all conceivable social states and
contains at least three distinct social states. Ri is a preference relation of
the individual i 2 N on social states. We assume that Ri is a complete
pre-ordering on X (complete and transitive binary relation on X). Pi and
Ii are the asymmetric and symmetric parts of Ri, respectively. A n¡list of
individual preferences (R1;R2;:::;Rn) will be called a proﬁle and designed
by d. Let D be the set of all conceivable proﬁles. A collective choice rule f
speciﬁes a social preference relation for each proﬁle d of D: R = f (d): As for
Ri, P and I are the asymmetric and symmetric parts of R, respectively. If
R is always a complete pre-ordering, f is a “social welfare function” (SWF)
in the sense of Arrow (1963). If R is only complete and acyclic, f is a “social
decision function” (SDF) in the sense of Sen (1970b).
Some conditions required by Arrow’s and Sen’s impossibility theorems
must be deﬁned. We state these conditions below.
Condition 1 (U) Unrestricted domain The domain of f is the set D.
Condition 2 (P) Weak Pareto principle For any x;y 2 X, if xPiy for
every i 2 N, then xPy.
1Condition 3 (I) Independence of irrelevant alternatives 8x;y 2 X;
8d;d0 2 D, [8i; xRiy () xR0
iy] =) [xRy () xR0y]:
Before stating the conditions of nondictatorship and of minimal liber-
tarianism imposed by Arrow and Sen on the collective choice rule, another
deﬁnition, which clariﬁes the concept of decisiveness, must be introduced:
Deﬁnition 1 Decisiveness A set of individuals V of N is decisive for x
against y if xPy when xPiy for every i 2 V .
Accordingly:
Condition 4 (D) Nondictatorship There is no individual i such that:
8d 2 D;8x;y 2 X : xPiy =) xPy.
Condition 5 (L*) Minimal libertarianism There is at least one pair of
persons decisive both ways over at least one pair of alternatives each; i.e.
for each of them i, there is a pair of alternatives in X, x;y, such that xPiy
implies xPy and yPix implies yPx.
Theorem 1 (Arrow) There is no SWF satisfying Conditions U, P, I and
D.
Theorem 2 (Sen) There is no SDF satisfying Conditions U, P and L*.
Let us examine now how Xu (1990) proposes to decompose the weak
Pareto principle and which part of it gives rise to Sen’s liberal paradox.
Condition 6 (WPU) Weak Pareto Unanimity There exists x and y in
X, and a preference proﬁle d¤ such that xP ¤
i y for every i 2 N implies xP ¤y.
Condition 7 (PN) Pareto Neutrality For any x, y, a and b in X, and
for any preference proﬁles d and d0, if xPiy and aP 0
ib for every i 2 N, then
aP 0b () xPy.
Lemma 1 (Xu) Condition P is equivalent to Conditions PN and WPU.
Condition PN is highly demanding since it enables to “look around” to
get information from other pairs. This strong property is suﬃcient to bring
about the paradox of a Paretian liberal:
Theorem 3 (Xu) There is no SDF satisfying Conditions U, PN and L*.
The theorem that we shall prove in the next section is thus that Condition
PN triggers oﬀ another version of Arrow’s theorem.
23. Proof of the theorem
We begin by introducing some supplementary deﬁnitions and two condi-
tions.
Deﬁnition 2 Undecisiveness A set of individuals V of N is undecisive for
x against y if yRx when xPiy for every i 2 V .
Deﬁnition 3 Almost Undecisiveness A set of individuals V of N is al-
most undecisive for x against y if yRx when xPiy for every i 2 V and yPix
for every i 2 (N ¡ V ).
Notationally, let J = fjg denote a set which contains only one individual
j. Besides, UD(x;y) means that J is undecisive for x against y, and UD(x;y)
means that J is almost undecisive for x against y. Note that UD(x;y) =)
UD(x;y).
Condition 8 (AUD) Anti-Undecisiveness There is no individual i such
that: 8d 2 D;8x;y 2 X;xPiy =) yRx.
Please note that an individual that would satisfy Condition D could be
called a dictator. On the other hand, an individual that would satisfy Con-
dition AUD could be called an anti-dictator1.
Condition 9 (AP) Anti-weak Pareto principle For any x;y 2 X, if
xPiy for every i 2 N, then yRx.
We now turn to two lemmas from which our theorem will be derived.
Lemma 2 (Restatement of Lemma 1) Under Condition PN, if there ex-
ists a preference proﬁle d¤ such that uP ¤
i v for every i 2 N implies uP ¤v for
at least one pair of alternatives fu;vg in X, then Condition P holds.
Lemma 3 If there is no preference proﬁle d¤ such that uP ¤
i v for every i 2 N
implies uP ¤v for at least one pair of alternatives fu;vg in X, then Condition
AP holds.
The proof of Lemma 3, obvious, is omitted here.
We are now ready to prove our theorem.
1Actually, one could argue that an individual which satisﬁes Condition AUD could be
called a “weak” anti-dictator in the sense of Mas-Colell and Sonnenschein (1972), which
deﬁne on the other hand a “weak” dictator. Therefore, the case of collective impotence,
i.e. 8x;y 2 X : xIy, is taken into account in our theorem 4 thanks to Condition AUD.
3Theorem 4 There is no SWF satisfying Conditions U, PN, I, D and AUD.
Proof Two cases have to be distinguished: on the one hand, if there exists
a preference proﬁle d¤ such that uP ¤
i v for every i 2 N implies uP ¤v for at
least one pair of alternatives fu;vg in X, then Condition P holds according
to Lemma 2 and the proof is similar as Arrow’s. Hence, there is a dictator
in society, which leads to a contradiction with Condition D.
On the other hand, if there is no preference proﬁle d¤ such that uP ¤
i v
for every i 2 N implies uP ¤v for at least one pair of alternatives fu;vg in
X, then Condition AP holds according to Lemma 3 and we must prove that
there is no SWF satisfying Conditions U, AP, I and AUD. We follow the same
line of reasoning as Arrow’s proof (1963, pp. 97-100), but in an opposite way.
(1) Firstly, we prove that if there is some individual j such that J = fjg
is almost undecisive over any ordered pair of alternatives, then an SWF
satisfying Conditions U, AP and I implies that j must be an anti-dictator.
Suppose that J = fjg is almost undecisive for some x against some y,
i.e. 9 x;y 2 X such that UD(x;y). Let z be another alternative, and let
i refer to all individuals other than j. Assume that xPjy & yPjz and that
yPix & yPiz. By transitivity of the individual preferences, xPjz. Notice that
we have not speciﬁed the preferences of persons other than j between x and z.
Now, zRy from Condition AP. Further, yRx since J is almost undecisive for
x against y. Consequently, zRx by the transitivity of R. This result, zRx, is
obtained without any assumption about the preferences of individuals other
than j regarding x and z. If the preferences of i over fy;xg and fy;zg aﬀect
the ranking of the pair fx;zg in R, then Condition I is violated. Hence,
J = fjg is undecisive for x against z:
UD(x;y) =) UD(x;z):
The rest of the proof for (1) is as follows: the combinations of individual
orderings are the same as Arrow’s, while their consequences for R are oppo-
site. Finally, one obtains that UD(x;y) over any pair fx;yg in X implies
UD(a;b); 8a;b 2 X. Therefore, individual j is an anti-dictator and (1) is
proved.
(2) Secondly, we must prove that there exists an individual j such that
J = fjg is almost undecisive over at least one pair of alternatives. Hence,
according to (1), one can deduce that this individual j is an anti-dictator,
which contradicts Condition AUD. We make the contrary supposition, i.e.
that there is no such individual j, and show that it leads to an inconsistency.
If Condition AP holds, then for any x;y 2 X such that xPiy for every
i in N, we obtain yRx. Therefore, for any pair of alternatives, there is at
least one undecisive set, i.e. the set of all individuals. Thus, for any pair of
alternatives, there is also at least one almost undecisive set.
Here again, the rest of the proof for (2) uses the same arguments as
Arrow’s, but with opposite conclusions for R. It leads to a contradiction
since there exists at least one almost undecisive set over at least one pair of
alternatives, which contains only one individual. The theorem now follows
from (1) since such an individual must be an anti-dictator.
44. Conclusion
Our theorem 4 means that an SWF which satisﬁes Conditions U, PN
and I is either dictatorial (there is a dictator in society) or anti-dictatorial
(the social preference relation goes systematically against preferences of an
individual – even weakly: he is an anti-dictator). Please note that this result
is close to Wilson’s (1972), even though the set of invoked conditions is
diﬀerent.
Actually, Wilson (1972) and Kelsey (1985, 1988) both use a condition of
non-imposition to state their theorems, whereas Xu (1990) and Theorem 4
invoke Condition PN. Therefore, the implications one can draw from these
two groups of results are opposite. On the one hand, Wilson and Kelsey
tend to moderate the responsibility of the weak Pareto principle in the neg-
ative results developed into the social-choice-theoretic framework. On the
other hand, Xu and Theorem 4 agree with Sen’s argument against the weak
Pareto principle and plead for the integration of non-welfarist information
into individual preferences.
These observations must lead to further investigation by comparing, both
on analytical and conceptual levels, these two families of results. It will then
be possible to clarify conceptual issues they raise.
5References
Arrow, K. J. (1963) Social Choice and Individual Values (2nd ed.), Wiley:
New York.
Kelsey, D. (1985) “The liberal paradox – a generalisation” Social Choice and
Welfare 1, 245-250.
Kelsey, D. (1988) “What is Responsible for the ‘Paretian Epidemic’?” Social
Choice and Welfare 5, 303-306.
Mas-Colell, A. and Sonnenschein, H. F. (1972) “General Possibility Theorem
for Group Decision” Review of Economic Studies 39, 185-192.
Saari, D. G. (1998) “Connecting and resolving Sen’s and Arrow’s theorems”
Social Choice and Welfare 15, 239-261.
Saari, D. G. (2001) Decisions and elections. Explaining the unexpected,
Cambridge University Press: Cambridge.
Sen, A. K. (1970a) “The Impossibility of a Paretian Liberal” Journal of Po-
litical Economy 78, 152-157.
Sen, A. K. (1970b) Collective Choice and Social Welfare, Holden-Day: San
Francisco.
Wilson, R. (1972) “Social Choice Theory without the Pareto Principle” Jour-
nal of Economic Theory 5, 478-486.
Xu, Y. (1990) “The Libertarian paradox: some further observations” Social
Choice and Welfare 7, 343-351.
6

Extending Xu's results to Arrow''s Impossibility Theorem

http://www.accessecon.com/pubs/EB/2005/Volume4/EB-05D70027A.pdf

Extending Xu's results to Arrow''s Impossibility Theorem

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics