The topological equivalence of the pareto condition and the existence of a dictator

Journal

of Mathematical

Economics

9 (1982) 223-233.

North-Holland

THE TOPOLOGICAL EQUIVALENCE CONDITION AND THE EXISTENCE Graciela

October

Company

OF THE PARETO OF A DICTATOR

CHICHILNISKY*

Columbia Unioersity, New York, USA University of Essex, Colchester CO4 3X2, Received

Publishing

1979, final version

accepted

UK June 1981

The paper studies two standard properties of rules for aggregating individual into social preferences: non-dictatorship and the Pareto condition. Together with the condition of independence of irrelevant alternatives, these are the three basic axioms of Arrow’s social choice paradox. We prove the topological equivalence between the Pareto condition and the existence of a dictator for continuous rules. The axiom of independence of irrelevant alternatives is not required. The results use a topological framework for aggregation introduced in Chichilnisky (1980), but under different conditions. In Chichilnisky (1980) rules are anonymous and respect unanimity. Since anonymity is strictly stronger than the condition of non-dictatorship, while respect of unanimity is strictly weaker than the Pareto condition, the two sets of conditions are not comparable.

1. Introduction Arrow’s social choice paradox exhibits a contradiction between three conditions on the aggregation of individual preferences: non-dictatorship, the Pareto condition, and that of independence of irrelevant alternatives. Recent work has proven the existence of social choice paradoxes for continuous aggregation rules without requiring independence of irrelevant alternatives. In Chichilnisky (1980) the rules are required to be anonymous and respect unanimity, whereas in Chichilnisky (1981) decisive majority conditions are required. While these results exhibit the topological nature of social choice paradoxes, they are proven under rather different assumptions from Arrow’s For instance, the condition of respect of unanimity is strictly weaker than the Pareto condition, while that of anonymity is stronger than non-dictatorship, so that the results obtained under one set of conditions are not comparable to those obtained under the other. In addition, although the * This research was supported by the UNITAR Project on the Future and carried out at the Centre for Social Sciences, Columbia University, I thank K. Arrow and M. Hirsch for discussion and criticism; I am especially grateful to G. Heal, A. MasCollel and a referee for helpful suggestions.

0304-4068/82/0000-0000/$02.75

@ 1982 North-Holland

224

G. Chichilnisky, Non-dictatorship

and

thePareto condition

independence axiom is not assumed in this recent work, the aggregation rules are required to be continuous. The purpose of this paper is to show &at two of Arrow’s conditions, Pareto and non-dictatorship, are topologically inconsistent with each other, in the sense that any continuous social choice rule that satisfies the Pareto condition can be continuously deformed into a dictatorial rule. The independence axiom is not needed to prove this result. With more than two voters a weak form of positive association is required on the rule. The paper is organised as follows: Section 2 contains notations and definitions. Section 3 proves the result for a special case which admits a simple geometrical proof: when there are two voters and their preferences are linear. Section 4 proves the topological equivalence between the Pareto condition and the existence of a dictator, with any number of voters and unrestricted domains of preferences, when the rule satisfies a weak form of positive association. 2. Notation

and definitions

Let X denote the choice space. X will be assumed to be C* diffeomorphic’ to the closed unit ball B” in R”. A preference p is a C1 vector field* on X, p : X + R”, which is locally the gradient field of a utility function u on X. The vector p(x) at the choice x in X is therefore the normal to the tangent space of the indifference surface of u at x ; see fig. 1. Following a standard convention in social choice theory, preferences are ordinal, so that the vector field p(x) is normalized3 to be of unit length, i.e., IIp(x)ll= 1 for all x in X. The space of preferences on X denoted P is a subset of the Banach space of all C1 vector fields on X, V(X), and is characterized within V(X) by the normalization and the Frobenius integrability conditions; see e.g. Debreu (1972). V(X) is a Banach space when endowed with the C’ sup norm. Since both Frobenius conditions and the normalisation are closed conditions in the C’ sup norm, the space P endowed with the topology inherited from V(X) is a complete space. The space V(X) is infinite dimensional, and P contains infinite dimensional manifolds; see Chichilnisky (1976). We assume there are k agents (k 2~2). A profile pl, . . . , pk is an ordered k-tuple of preferences of the voters, so that the space of profiles is Pk. ’ Let Xc R”, Y c R” be smooth manifolds with boundary, f : X -+ Y is C2 if it admits an extension to a twice continuously differentiable map defined on a neighbourhood of X. X is Cz diffeomorphic to Y when there exists a C* one-to-one and onto map f : X + Y whose inverse is also C’. * A vector field u(x) on X is C’ if it admits an extension to a C’ vector field defined on a neighbourhood of X. 3 This implies that u is not satiated in the interior of X, X0. Related results can be obtained when preferences admit satiation in X0; see Chichilnisky (1979).

G. Chichilnisky,

Non-dictatorship

225

and the Pareto condition

l?(x) x

;-% x Fig. 1. p(x) is the normalized

gradient

of the preference

p at the choice

x.

A social aggregation rule is a map that assigns to each profile in Pk a social preference in P, 4 : Pk -+ P. A rule d, is said to satisfy a Pareto condition, or to be Pareto, when the following is always true: if for all voters 1, . . . , k, a choice x is preferred to another y (for instance if the utility functions that represent pl,. . . , pk give a higher value to x than to y) then the social preference $(pI, . . . , pk) prefers x to y. A rule C#I satisfies the weak positive association condition if 1 ,..., pi ,..., pk)=-pi for some i=l,..., k and some (PI ,..., pk)ePk, 4(P implies $(-Pi7

. . . T-Pi2 Pi9 -Pi2 . . . 9 Pi>

#

Pi*

A Pareto rule satisfying the weak positive association condition is denoted a W-Pareto rule. A rule C#Irespects unanimity when the restriction of the map b, on the set Pk):Pi=Pj, tli, j=l,..., k}, is the identity map on D, i.e., D=NPl,..*, 4/D = id,. Note that the Pareto condition implies respect of unanimity. A preference is called linear when it is induced by a linear utility function on X. An aggregation rule C#J~is dictatorial (with dictator d) if & is the projection onto the dth coordinate, &(pl, . . . , pk) = pd, V(p,, . . . , pk) E Pk. Let f and g be two continuous functions between two topological spaces Y and 2. f is said to be homotopic to g, or a continuous deformation of g, when there exists a continuous map II: Yx[O, satisfying

l]*Z,

II(y, 0) = f(y) and 17(y, 1) = g(y), for all y in Y,

3. An example

with two voters and linear preferences

We now study a particular case, where there are two voters and their preferences are linear. This case admits a simpler geometrical proof that the

226

G. Chichilnisky,

Non-dictatorship

and the Pareto condition

Fig. 2. The Pareto property for two voters and a two-dimensional choice space. The area shaded with circular lines is the dual cone of p1 and p2, the area doubly shaded is the cone determined by p1 and p2.

Pareto condition is topologically equivalent to the existence of a dictator. Let P denote the space of linear preferences on the choice space Xc R “+l, n 2 1. Each p E P can be identified by one vector, the gradient of the linear utility function corresponding to p. ‘Since gradients are normalised to be of unit length, the space of all linear preferences on X can therefore be identified with the space of their gradients, so that ii is the nth sphere in n+l . R , I.e., p =

S” ={TJ E R"+l : llull= l}.

Therefore with 2 voters the space of profiles p* is here (S”)*. When n = 1, Pz =S’ x S’ = T2, the two-dimensional torus. For any nz 1, a social choice rule is here a map $ : (S”)*+Y. Given two preferences pl, p2 in S”, the Pareto property implies that the function 4 maps the profile (pl, p2) into a preference p = c#J(P,, p2) which is contained in the cone determined by p1 and p2 in S”, as illustrated in fig. 2. This is because the Pareto condition implies that p must have a positive inner product with all vectors of the ‘dual cone’ of (pl, p2), i.e., with all vectors in the set {gES”:g.piZO

and

g*p,ZO}.

Thus, if 4 is Pareto, ~#~(pr,p2) must belong to the dual of this dual cone, i.e., p must belong to the cone determined by p1 and p2. This property is of course trivially satisfied by any q E S” when p1 and p2 are exactly opposed to each other, i.e., when p1 = -p2 in S”. Fig. 3 shows an example of a Pareto rule for two voters, and a. twodimensional choice space. The Pareto property implies respect of unanimity, i.e., ~$(p, p) = p for all p E S1. In particular, if a rule 4 is Pareto, for each p E S’, the inverse image of p under 4, 4-‘(p) c S’ X S’ intersects the set D =

{(PI,

PJ E w*

: Pl = Pd,

exactly at one point, 4-‘(p) tl D = {p, p}, Vp E S1. Fig. 4 represents a dictatorial rule: clearly, for each p E S’, 4-‘(p) {p,p} in this case as well. Note that the surfaces c#-‘(~)(V~ES~)

6 D = of the

G. Chichifnisky, Non-dictatorship and the Pareto condition

221

Fig. 3. The space of profiles of two voters’ preferences for two-dimensional choices and linear preferences is the toTus T’=(S’)‘. The curved lines indicate the surfaces b-‘(p) correstherefore its ponding to a fixed value p E p of the map 4 : p2 + p. The map 4 is Pareto; indifference surfaces intersect D exactly once.

Pareto rule in fig. 3, can be continuously deformed into those %of the dictatorial rule in fig. 4. This is an illustration of the following result. Fig. 5 shows an example of a rule that is clearly not Pareto, since each 4-‘(p) intersects D twice: the following theorem will prove that this rule is not topologically equivalent to a dictatorial rule either.

Fig. 4. A dictatorial

rule with two agents

and two commodities.

Theorem 1. Let 4 : P” + p be a continuous Pareto rule. Then 4 is homotopic to a dictatorial rule. Let p. be an arbitrary preference in p, and let p1 = pO, p2 = -pO. Continuity and the Pareto property imply that either

Proof.

4(PIT Pz)= POT or else 4(pl, p2)=-po.

Fig. 5. Each surface 4-‘(p) intersects the diagoqal D exactly two times: the map 4 is not Pareto. Therefore it is not topologically equivalent to a dictatorial rule either.

G. Chichilnisky,Non-dictatorship and the Pareto condition

228

Assume,

without

loss of generality,

that

PJ = PO.

&I,

Continuity then implies that for all other p in S”, 4(p, -p) = p. i.e., whenever agent one has a preference opposed to that of agent two, then agent one is a dictator. Therefore for any other pair of preferences q1 and q2, +(ql, q2)# -ql, because by the Pareto condition, either q1 # -q2 and &(ql, q2) is in the cone generated by q1 and q2 (and -ql does not satisfy this), or else q1 = -q2 in which case 4(q1, q2) = q1 as shown above. Define now the map H : (Sny x [O, l] + S”, by WP,,

This map denominator

P2> t>=

tPl+ (I-

tM(p1,

P2)

lItpI+ Cl-

tMPl>

P2)ll~

is always well defined because &pI,p2)# is never zero. Since for all (pl,pZ) E(S”)‘,

WP,,

p2, 0) = d4pl,

H

is the desired (Sn)2. 4. The topological a dictator

p2)

homotopy

equivalence

and

between

Npl,

the rule

-pl

so

that

the

p2, 1) = ply

4 and a dictatorial

rule

on

of the Pareto condition and the existence of

We now study the case of an unrestricted domain of smooth preferences P as defined on section 2, defined on an (n + 1)-dimensional choice space, n 2 2, and with k voters, k Z 2. For further references on algebraic topology, see e.g. Spanier (1963). Theorem 2. Let 4 : Pk -+ P be a continuous W-Pareto aggregation rule. Then C$ is homotopic to a dictatorial rule. Proof. Let x be a choice in X, and let S” be the unit sphere in R”+l. Given a chart for X at x, each preference p in P determines uniquely a point z in S”, the unit vector normal to the indifference surface of p at x in the orienting direction p(x). This determines a map r from the space of preferences P into S”, which is continuous by the choice of topology in P. The map r can be chosen to be continuous and onto S”, i.e., for all z in S” there exists some preference p on X that has z as a normal vector at x, i.e. T(p) = z. We can also define a continuous map h : S” -+ P assigning for each z in S” a preference on X with preferred direction equal to z at the choice

G. Chichilnisky,

Non-dictatorship

x. h(z) can be chosen as a linear orthogonal to the vector z.

The map I defined r 0 A, is the identity Let 4 : Pk + P be for each x in X we Pk

229

and the Pareto condition

preference

with

indifference

surfaces

by making the above diagram commutative, i.e., I= map on S”. a continuous rule satisfying the Pareto condition. Then, can define the map + by the diagram

m-P

I -5 I

nk

(syk

r

S”

(z,, . . . ,zJE(S”)~. all i.e., $(zr,. . ., zk)= f(#(A(z,), . . . , ,&q,))) for continuous, being the composition of continuous maps. For a given z,, E S”, define Gi ={(z,,

. . . ) zk) E (Sri))) such that

zi = zO, Vjf

tj

is

i}.

Let C(z,, zJ denote a shortest circular segment of a great circle on S” containing both zI and z2, which is uniquely defined when zr # z2. Since the rule 4 satisfies the Pareto condition, for any i = 1, . . . , k, the restriction map $/Gi must satisfy ICI/Gi(Z1,.

. . ZJ E

C(ziy ~0)

when

Zi#z,,

because rlrlGi(Z,, . . . , zk) must have positive inner product with all vectors in S” which have positive inner product with both z0 and zi. Since each Gi is homeomorphic to a sphere S”, we can define the degree of the map $/Gi : Gi + S,, and by the Pareto deg(+/Gi)

condition, is either

Now, since 4 is Pareto,

0 or 1 for all i = 1,. . . , k. 4 satisfies

the respect

of unanimity

t/j/D = id,.

condition,

i.e., (2)

Since D is homeomorphic to S”, we can define map $/cl/o : D -+ S”, and (2) implies deg($/D)

(1)

= 1.

the degree

of the restriction

(3)

230

Now,

Non-dictatorship and the Pareto condition

G. Chichilnisky,

let 17,((F~?‘)~) be the nth homotopy

group

of (Sn)k. Then

a%(w)k>= & n,(r);

(4)

i=l

see e.g. Spanier (1963, p. 419, B.5).4 Consider now the inclusion map in, : S” 3 (w)k>, defined

by inD(z>=(2,...,z)EDc(Sn)k

Similarly

for any

zES*,

for all i = 1,. . . , k define in,, : S” -+ (P)k

by ith

in,,(z)

= (zo,

place

. . . i.2;~. . . , zO) E Gi c (Sri)))

Let Pi :(S”)k + S” be the projection each i, the two composition maps:

PioinD:S”-+S”

and

for any

z E S”.

map onto the ith coordinate.

Then

for

Pi 0 in,, : S” + S”,

are both the identity map on S”. In particular these maps are that of id : S” + S”, i.e.,

the homotopy

classes of both

[Pi o inD] = [Pi o i&-J = [id].

(5)

Let [ino] denote the homotopy class of inGt : S” + (Sri))) in fl,,((Sn)k), let the map T:S”-+(S”)k be such that

and

(6) We can take Tto satisfy T(S”) c

(JGi c (Sn)ky

i=l

and

[Pi o T] = [Pi

’

inG,]

E

J!I, (S”),

for each i = 1,. . . , k. Therefore by (5) and (7) for each

(7) i = 1,. . . , k,

[Pi 0 T] = [Pi 0 inD] = [id], 4fl,,((S”)k) is written as @Fe1 &(S”) abelian group, since n 22.

(8) instead

of as

Xfcl

I&9”)

to indicate

that

it is an

G. Chichilnisky,

i.e., PT[T] =PT[in,], level, I’“:II,,((S”)k) implies

Non-dictatorship

and the Pareto condition

where Pi* is the map induced by Pi at the homotopy + II,,(F). Since (8) is true for all i = 1,. _ . , k, (4)

[T] = [in,].

(9)

[see also Chichilnisky cII/o Tl

231

(1981,

lemma l)]. Therefore

(10)

= c+ o w,

and by (6) this implies

(11) so that

deg($ oT) = i i=l Since deg($

0

deg($ 0 in,,) = deg(ll, 0 ini,).

in,) = deg($/D),

i$lde&

Q

and by (3) deg($/D) = 1, it follows that

in,,) = 1.

(12)

Since deg($o inG,) = deg($/Gi),

from (1) and (12) we obtain

deg($/Gd) = 1

for some

d E (1, . . . , k},

(13)

deg($/Gi) = 0

for all

d.

(14)

and

i=l

Now, the Pareto , ** 3 k

property

i#

and the continuity

of 4 imply that for all

i th place J,

0 iki(-zO)

=

44z,,

. . .

,%

..

. , zo)

E {zo,

-z,}.

(15)

This is because the Pareto condition implies if50 in&)

e C(z,, z)

for any

z E S”.

(16)

It follows also from (16) that if J/ 0 in&zo) = q,, then the value -zO is never assumed by the map +/Gi. Therefore for any i = 1, . . . , k,

J/ 0 inGi(-z,,) = z. By (12), therefore, Ic,o &,(-zo)

implies

1+5 0 in,,(-z,) = -zo,

= -z.

deg(4/Gi) = 0.

(17)

for some j = 1, . . . , k. Now by (16) if

232

G. Chichihisky,

Non-dictatorship

and the Pareto condition

then deg(+ 0 inci) > 0,

(18)

so that deg($ 0 in,,) = 1 by (1). It follows from (13) and (14) that for some d= 1,. . . , k, I/J0 inc,(-z,)

= -20,

(1%

and * 0 inc, (-zO) = z.

for all

i # d.

(20)

In particular, I+!J(z,,. . . , zk) # -z, for all profile (z,, . . . , zk) in Gi, all i=l,...,k. Note that in the argument given above the vector z0 is arbitrary chosen. Therefore, by continuity of 4, for any z: E S”, (21,.

. . , zk)#-zd>

(21)

(zi, . . . , zk) is in U i Gi(z$,

whenever

where

* . 7 Zk) E (Sri))) 1Zi =

Gi(zA)={(Zi,.

Without loss of generality, that for any zi E S”,

assume

I&(z,, -21, . . . ) -zJ

ZAP

Vi # d}.

d = 1. Then

(21) implies,

= 21.

(22)

Consider now any profile (z,, . . . , zk) E (Sn)k. Since positive association condition, if +(z,, it would

contradicting I/l(z,,

4 satisfies

the weak

. . . >zk)=-zl,

follow $(z,,

in particular,

that -z1,.

. . ,-~1>#~1,

(22). Therefore, . . . > zk)

#

we have proven

in particular

that

-zl,

for all (zi, . . . , zk) e (Sn)k. We now return to the map 4 : Pk + P. Since 4 satisfies both the Pareto and weak positive association conditions, then for each choice x E X, there exists one d = 1, . . . , k with 4(p,, . . . , p&t) # -pd(x). This iS because both the Pareto and the weak positive association condition hold locally, for any x in X; i.e., if y is preferred to x by pi, . . . , pk, and y is sufficiently close to x, then it follows that pi(x)(y-x)zo.* which is the Pareto

’ pk(x)(y-x)~o condition

for vectors

implies at the sphere

p(x)(y-x)zO, S” centred

at x.

G. Chichilnisky, Non-dictatorship

The first part of the proof implies exists some d = d(x) such that dth

4(-p&.

233

and the Pareto condition

therefore

that for any x in X there

place

. .mpd>

-pd,*.

(23)

. ~-Pd)b)=Pdb)*

By continuity of q5, d(x) must be the same for all x in X. In view of the Pareto and the weak positive association follows as proven above that for any pl, . . . , pk E pk, (PI,. . Therefore,

@d(X)+(1-t)~(Pl,.

We can then define

. ,pk)sPk,

. . , Pk)(X)#o*

the map H on Pk x [O, l] by ~pd(X)+(~-tbb(pb~~

pd(x)

.,Pk)(x)

.. . ' Pk' t”X’=IlrPd(x)+(l-f)~(p,,

Since

it

(24)

.,Pk)b)#-Pdb).

for any x EX, TV [0, l] and (pl,.

WP,,

conditions,

is the gradient

..

HP,,

. . . , Pk,

’

Pd(x) * x, it follows that for each at each x E X a field on X.

of the map

11, H(p,, (PI>. . . , Pk, t) E pk x[o, scalar multiple of a C1 gradient Since for all x E X,

H(p,,

. . . , P,c)(~>~/

. . . , pk, f) is in P, being

.,Pk, o)b)=~(p,,

. . ., pkb),

and

H is the desired the proof.

l>(x)

homotopy

=

pdb),

between

4 and a dictatorial

rule. This completes

References Arrow, K., 1963, Social choice and individual values, Cowles Foundation for research in economics monograph 12 (Yale University, New Haven, CT). Chichilnisky, G., 1976, Manifolds of preferences and equilibria, Report no. 27, Project on efficiency of decision making in economic systems (Harvard University, Cambridge, MA): Chichilnisky, G., 1980, Social choice and the topology of spaces of preferences, Advances in Mathematics. Chichilnisky, G., 1981, Structural instability of decisive majority rules, Journal of Mathematics Economics 9,207-221. Debreu, G., 1972, Smooth preferences, Econometrica. Spanier, E.H., 1963, Algebraic topology, McGraw-Hill series in higher mathematics (McGrawHill, New York).