More on Social Choice Theory without the
Pareto Principle
(Very preliminary version)
Herrade Igersheim∗
Abstract
This note examines if the Pareto principle is responsible for the
negative results into the social-choice-theoretic framework, or, more
precisely, which aspect of the Pareto principle is responsible for them.
Two families of results have proved Arrow’s (1963) and Sen’s (1970a,b)
Impossibility Theorems by replacing the Pareto principle by weaker
conditions called Non-Imposition (Wilson, 1972; Kelsey, 1985) and
Pareto Neutrality (Xu, 1990; Igersheim, 2005). Our results strengthen
Kelsey’s arguments for the defense of the Pareto principle against Sen’s
criticisms.
Key Words: collective choice, Arrow’s Impossibility Theorem, Sen’s
liberal paradox, weak Pareto principle
Classification JEL: D6, D7
∗
CODE, Edifici B, Universitat Autònoma de Barcelona, 08193 Bellaterra, Cerdanyola
del Vallès, Barcelona, Spain (e-mail: [email protected]).
This is a very preliminary version. Proofs of Propositions 1 and 2 are available on request.
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1. Introduction
Our paper proceeds from a stream of research focusing on the understanding 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 socialchoice-theoretic framework and then to be able to circumvent these negative
results in a satisfying way.
More precisely, the purpose of this note is to examine the differences
between two families of results, which weaken the weak Pareto principle in
order to prover other versions of Arrow’s Impossibility Theorem (1963) and
Sen’s liberal paradox (1970a,b). Wilson (1972) and Kelsey (1985) both use a
condition of non-imposition to state their theorems, whereas Xu (1990) and
Igersheim (2005) invoke a condition called Pareto Neutrality. 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 negative results developed into the social-choicetheoretic framework. On the other hand, Xu and Igersheim 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 the conceptual issues they raise.
The remainder of the paper is organized as follows. In Section 2, we
present the basic concepts of our analysis. Section 3 presents our first proposition, which deals with Arrow’s Impossibility Theorem, whereas our second
proposition is exposed in Section 4 concerned with Sen’s liberal paradox.
Finally, Section 5 concludes the paper.
2. Basic concepts
Let N = {1, 2, ..., n} 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 ∈ 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).
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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 ∈ X, if xPi y for
every i ∈ N , then xP y.
Condition 3 (I) Independence of irrelevant alternatives ∀ x, y ∈ X,
∀ d, d0 ∈ D, [∀ i, xRi y ⇐⇒ xRi0 y] =⇒ [xRy ⇐⇒ xR0 y] .
Before stating the conditions of nondictatorship and of minimal libertarianism 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 xP y when xPi y for every i ∈ V .
Accordingly:
Condition 4 (D) Nondictatorship There is no individual i such that:
∀ d ∈ D, ∀ x, y ∈ X : xPi y =⇒ xP y.
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 xPi y
implies xP y and yPi x implies yP x.
Theorem 1 (Arrow, 1963) There is no SWF satisfying Conditions U, P,
I and D.
Theorem 2 (Sen, 1970a,b) There is no SDF satisfying Conditions U, P
and L*.
Let us examine now the conditions proposed respectively by Wilson (1972)
and Xu (1990) to decompose the weak Pareto principle.
Condition 6 (NI and SNI) Non-Imposition and Strict Non-Imposition For any x and y in X, there exists a preference profile d∗ such that xR∗ y
(respectively, xP ∗ y)1 .
Condition 7 (PN) Pareto Neutrality For any x, y, a and b in X, and
for any preference profiles d and d0 , if xPi y and aPi0 b for every i ∈ N , then
aP 0 b ⇐⇒ xP y.
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It should be noted that Condition SNI is in fact Arrow’s version of non-imposition.
3
Another condition must be introduced:
Condition 8 (AUD) Anti-Undecisiveness There is no individual i such
that: ∀ d ∈ D, ∀ x, y ∈ X, xPi y =⇒ 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 Condition AUD could be called an anti-dictator2 .
We then obtain two families of negative results, one with Conditions NI
or SNI, the other using Condition PN.
Theorem 3 (Wilson, 1972) There is no SWF satisfying Conditions U,
NI, I, D and AUD.
Theorem 4 (Kelsey, 1985) There is no SDF satisfying Conditions U, SNI
and L*.
Theorem 5 (Xu, 1990) There is no SDF satisfying Conditions U, PN and
L*.
Theorem 6 (Igersheim, 2005) There is no SWF satisfying Conditions U,
PN, I, D and AUD.
One must determine which condition enables to weaken Arrow’s and Sen’s
theorems without circumventing them, i.e., which condition is the most necessary to their emergence, in order to go against or in favor of the weak
Pareto principle.
3. Beyond Arrow’s result: a first proposition
On the NI’s side, a new result is established by Malawski and Zhou
(1994)3 .
Condition 9 (AP) Anti-weak Pareto principle For any x, y ∈ X, if
xPi y for every i ∈ N , then yRx.
Accordingly, a SWF is said Pareto optimal (PO) if Condition P is satisfied
or anti-Pareto optimal (APO) if Condition AP is met.
Theorem 7 (Malawski and Zhou, 1994) If a SWF satisfies Conditions
U, NI and I, then it is either Pareto-optimal or anti-Pareto optimal.
2
Actually, 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.
∀ x, y ∈ X : xIy, is taken into account in our restatements and results thanks to Condition
AUD.
3
For other extensions of Wilson’s result, see Tanaka (2003).
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This theorem can be restated with Condition PN: this is our Proposition
1.
Proposition 1 If a SWF satisfies Conditions U and PN, then it is either
Pareto-optimal or anti-Pareto optimal.
It should be realized that Condition I is not even required by Proposition
1. Would it mean that Condition PN involves more elements than Condition
NI and, thus, that Condition NI is the most necessary to establish negative
results in the social-choice-theoretic framework? A further investigation on
Sen’s liberal paradox could help us to conclude.
4. Beyond Sen’s paradox: a second proposition
On the PN’s side, Xu (1990) shows how Condition PN leads to Condition
P thanks to the notion of decisiveness.
Theorem 8 (Xu, 1990) For any SDF satisfying Condition U and PN, if a
person J is decisive over a pair {x,y}, then Condition P holds.
Condition NI can also bring about Condition P if an individual is decisive.
But it should be associated with Condition I in order to do so.
Proposition 2 For any SDF satisfying Condition U, NI and I, if a person
J is decisive over a pair {x,y}, then Condition P holds.
Note that Proposition 2 only uses Condition NI, not Condition SNI as
Kelsey (1985) does.
5. Concluding remarks
According to Kelsey (1985, p. 249), which comments his own results
based on Condition NI: “there are several ways to interpret the results in this
paper. It is possible to build a defense of the Pareto principle against Sen’s
criticisms based on these results (...). A second way of interpreting our result
is that it tells us which aspects of the Pareto principle cause the trouble in
Sen’s results”. According to Propositions 1 and 2, Condition NI seems to
be more fundamental to the emergence of Arrow’s and Sen’s theorems since
it does not involve some elements of independence such as Condition PN.
Indeed, Condition PN is highly demanding since it enables to “look around”
to get information from other pairs. But if the rejection of Condition PN can
be easily interpreted, i.e., one has to take into account non-welfarist information into the social-choice-theoretic framework, the meaning of Condition
NI remains ambiguous.
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References
Arrow, K. J. (1963) Social Choice and Individual Values (2nd ed.), Wiley:
New York.
Igersheim, H. (2005) “Extending Xu’s Results to Arrow’s Impossibility Theorem” Economics Bulletin 4 (13), 1-6.
Kelsey, D. (1985) “The liberal paradox – a generalisation” Social Choice and
Welfare 1, 245-250.
Malawski, M. and Zhou, L. (1994), “A note on social choice theory without
the Pareto principle” Social Choice and Welfare 11, 103-107.
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 Political Economy 78, 152-157.
Sen, A. K. (1970b) Collective Choice and Social Welfare, Holden-Day: San
Francisco.
Tanaka, Y. (2003) “A necessary and sufficient condition for Wilson’s impossibility theorem with strict non-imposition” Economics Bulletin 4 (17), 1-8.
Wilson, R. (1972) “Social Choice Theory without the Pareto Principle” Journal of Economic Theory 5, 478-486.
Xu, Y. (1990) “The Libertarian paradox: some further observations” Social
Choice and Welfare 7, 343-351.
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