Maximal elements for non-transitive binary relations

Economics Letters North-Holland

MAXIMAL

163

14 (1984) 163-165

ELEMENTS

FOR NON-TRANSITIVE

BINARY

RELATIONS

Ghanshyam

MEHTA

U~roers~i~of Quernslund, St. Received

9 December

Lucirr, Queensland 4067, Ausrrdrrr

1982

Using a fixed-point theorem of Browder. we prove the existence of a maximal element non-transitive relation on a set X which is not assumed to be compact or convex. theorem is a generalization of the results available in the existing literature.

for a This

1. Introduction

The object of this paper is to prove a result about the existence of maximal elements for non-transitive relations when the underlying set X is assumed to be neither compact nor convex. In the literature the existence of maximal elements, in the absence of transitivity, has been proved under the assumption that the underlying set X is compact and convex. See Bergstrom (1975) Sonnenschein (1971). Gale and Mas-Cole11 (1975), Aliprantis and Brown (1982), and Ky Fan (1961). In this paper, Browder’s fixed-point theorem is used to prove a more general theorem.

2. Existence

of maximal

elements

The existence of maximal elements theorem of Browder (1968, p. 285). Theorem topological (i) (ii)

I.

Let space.

K be a compact, Let

will be deduced

convex

T be a multivalued

for each x E K, T(x) is a non-empty foreachxEK,T-‘(x)={yEK/xET(y))isopeninK.

0165.1765/84/$3.00

subset

mapping convex

@ 1984, Elsevier Science Publishers

from the following

of a Hausdorff

linear

of K into 2h SUL.~that subset of K;

B.V. (North-Holland)

G. Mehtu / Maximal elements for non - transrtwr relutiom

164

Then there is a point x,, such that x,) E T( x,) ). We now prove

elements

the following for subsets of R”.

theorem

about

the existence

of maximal

Theorem 2. Let X be u subset of R”. Let P be u binar>a relutron on X suti.sjjYng- the following conditions: (a) for each x, x G P(x) and P(x) is conuex; (b) P~‘(x)={yIx~P(y)} isopenin Xforallx; (c) there exists rtn x,) such that [Pm ‘(.~,))I’ is compact. Then there exists a muximal P( x’) n x = +.

element for P, i.e., there is an x’ .suc,h thut

Proof. Let F(x) = [P ‘(x)]‘. F(x) is then closed for each x in X. In view of (b) and (c) it suffices to prove that f-l:‘_, F(x,) # + for each finite subset {x,, x2,. ., x,,} of X, since one can easily verify that there is a maximal element x’ for P if fl, ExF( x) is non-empty. Suppose, per absurdum, that f-I:, , F(x,) = C#Ifor some { x,. x2.. .x,, ). Then

for each

xES=convex hull of {x,,x~....,x,,}, is non-empty, since at least one { .V E s/x e F(Y)) i = 1, 2,. . . .H must be in A(x), because otherwise

be non-empty. We prove B(x)=

next

that

B(x)

= { _r:E X,/x

{YEX/XEF(_):)}

of

the set A(x)= the points x,.

f-l:‘_, F(x,)

E F( .v)} is convex.

= {_),EX/xE

would

Now

[F()‘)jC)

which is convex for each x by condition (a). It follows that ,4(x) = S n B(x) is convex

for each x. Since A(x) f C#I

for each x we define a mapping T: S+2’ so that T-‘(X)=

K=

by

T(x)=A(x)

forallx,

condition

(i) of Browder’s

T satisfies {~ES/XE

{y~S/~&4(y)}=

which

is closed

closed

finite-dimensional

since

T(y)}

theorem.

= {yES/xEA(y)}

Now

=K’.

where

{yC$‘yG’(x)}. it is the intersection

subspace

spanned

of a closed set F(x) and the by ( x,, x2,...,x,,}. Hence,

K’ is open and condition (ii) of Browder’s theorem is satisfied. Hence, by Browder’s theorem there exists an x,) such that x,, E T( x0) = A( x0) = {y/x g F(y)}, which contradicts the fact that x E F(x) for all x in X. Hence, the family of closed sets F(x) has the finite intersection property and the theorem is proved. Q.E.D. Remark 1. The assumptions of Theorem 2 are quite natural and are the ones usually made in general equilibrium theory. If we assume that X is compact, then condition (b) implies condition (c). Hence, Theorem 2 is a generalization of existing results which require the compactness of X. Remark 2. Theorem 2 can be generalized in a straightforward manner to the case where X is a subset of any Hausdorff topological vector space over the reals. A careful reading of the proof shows that for this infinite-dimensional generalization one needs to assume. in addition to the conditions of Theorem 2, that for each x the finite-dimensional slice (i.e., the intersection with each finite-dimensional subspace) of [Pm ‘(A-)] is a closed set. Remark 3. Further generalizations are possible. For example. condition (c) can be weakened by assuming only that there exists an x,, such that is relatively compact, if a certain condition called the P’(41’ BrezissNirenberggStampacchia condition holds [see Tarafdar and Thompson (1977)]. References Aliprantis. ties,

C. and Social

D. Brown,

science

1982.

working

Equilibria

paper

in markets

427 (California

with

Institute

a Riesz

space

of commodi-

of Technology,

Pasadena,

CA). Bergstrom.

T.. 1975,

transitivity Browder.

F.. 1968. The

spaces, Fan,

Ky.

1961,

D. and

fixed-point

Ann

theory

Annalen,

elements

Arbor.

and

equilibria

in the absence

of

MI).

of multi-valued

mappings

in topological

vector

283-301.

generalization

A. Mas-Colell.

ordered

Sonnenschein,

of

Tychonoffs

fixed-point

An equilibrium

Journal

Demand

of competitive

Sonnenschein,

1975.

preferences,

H.. 1971,

the theory

Tarafdar,

A

of maximal

of Michigan.

theorem,

Mathematische

305-310.

without

York)

existence

Mathematische

Annalen, Gale,

The

(University

theory

equilibrium,

eds., Preferences,

utility

existence

of Mathematical without

transitive

in: J. Chipman. and demand

theorem

Economics preference. L. Hurwicz.

for a general

model

2. no. 1, 9-15. with

applications

M. Richter

to

and

(Harcourt-Brace-Jovanovich.

H.

New,

215-223. E.

Australian

and

H.

Thompson,

Mathematical

Society,

1978, On 220-226.

Ky

Fan’s

minimax

principle,

Journal

of