P e r g a m o n Appl. Math. Left. Vol. 11, No. 5, pp. 51-54, 1998

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R e m a r k s on a Soc ia l E q u i l i b r i u m E x i s t e n c e T h e o r e m of G. D e b r e u

S. PARK Department of Mathematics

Seoul National University Seoul 151-742, Korea

(Receired September 1997; accepted October 1997)

Abstract--An acyclic version of the social equilibrium existence theorem of Debreu [I] is obtained. This is applied to deduce acyclic versions of theorems on saddle points, minimax theorems, and the Nash equilibrium. ~) 1998 Elsevier Science Ltd. All rights reserved.

Keywords--Multimap (map), Closed map, Upper semicontinuous (u.s.c.), Lower semicontinuous (l.s.c.), Berge's theorem, Polyhedron, Acyclic, Equilibrium point, Saddle point, Minimax theorem, Nash equilibrium.

1. I N T R O D U C T I O N AND P R E L I M I N A R I E S

In this paper, we give an acyclic version of the social equilibrium existence theorem of Debreu [1]. Our proof is much simpler than the original one. Moreover, our main result is applied to acyclic versions of a saddle point theorem, a minimax theorem, and the Nash equilibrium theorem.

For topological spaces X and Y, a multimap or map F : X --o y is a function from X into the power set 2 Y of Y with nonemptyvalues F(x) C Y for x E X. A map F : X --o y is

said to be closed if its graph Gr (F) = {(x, y) : x E X, y E F(x)} is closed in X x Y; Upper Semicontinuous (u.s.c.) if, for each closed set B C Y, F - ( B ) = {x E X : F(x) N B ~ 0} is closed; Lower Semicontinuous (l.s.c.) if, for each open set B c Y, F - (B) is open; and continuous if it is u.s.c, and l.s.c. If F is u.s.c, with closed values, then F is closed. The converse is t rue whenever Y is compact.

The concepts of upper or lower semicontinuity of extended real-valued functions axe standard. The following is well known [2].

BERGE'S THEOREM. Let X and Y be topological spaces, f : X x Y --. R an extended real-valued

function, F : X -o y a multimap, and

](x)-- for x E X. ueF(x)

(a) H f is u.s.c, and F is u.s.c, with compact values, then ] is u.s.c. (b) I f f is 1.s.c. and F is l.s.c, then ] is 1.s.c. (c) I f f is continuous and F is continuous with compact values, then ] is continuous and G

is u.s.c.

Supported in part by the Nondirected Research Fund, Korea Research Foundation, 1997.

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51

52 S. PARK

A polyhedron is a set in R " homeomorphic to a union of a finite number of compact convex sets in R " . The product of two polyhedra is a polyhedron [1].

A nonempty topological space is said to be acyclic whenever its reduced homology groups over a field of coefficients vanish. The product of two acyclic spaces is acyclic by the Kfinneth theorem.

The following is due to Eilenberg and Montgomery [3] or, more generally, to Begle [4].

LEMMA. Let Z be an acyclic polyhedron and T : Z --o Z an acyc]/c map (that is, u.s.c, with acyclic values). Then T has a fixed point ~ 6 Z; that is, ~ 6 T(~).

2. M A I N RESULTS

Let {Xi}ie~ be a family of sets, and let i 6 I be fixed. Let

X = H X ~ and X ' = H Xj.

If x i 6 X i and j E I \ { i } , let x~ denote the jth coordinate of x i. If # 6 X i and x~ E Xi, let [x ~, xi] 6 X be defined as follows. Its i th coordinate is xi and, for j # i, its j th coordinate is x~.. Therefore, any x 6 X can be expressed as x = [x i, xi] for any i 6 I , where x ~ denotes the projection of x onto X ~.

For A c X, x i e X i, and x, e Xi, let

x A} A(x~)={Y~EX~:[xi,y,]EA} and A(x~)={y~EXi:[y, ~]E .

From now on, assume that I = {1, 2,... ,n}.

The following is a collectively fixed-point theorem equivalent to the lemma.

THEOREM 1. Let {X~}~l be a family of acyclic polyhedra, and T~ : X --o Xi an acyclic map for each i 6 I. Then there exists an ~c E X such that ~ E T~(~) for each i E I.

PROOF. Note that X itself is an aeyclic polyhedron. Define T : X -o X by T(x) = rliexT~(x ) for each z E X. Then T is a acyclic map. In fact, each Ti is u.s.c, for each i E I and hence T is also u.s.c.; see [5, Lemma 3]. Note that each T(x) is acyclic. Therefore, by the lemma, T has a fixed point ~ E X; that is, ~ E T(~), and hence, ~i E Ti(~) for each i E I. This completes our proof.

From Theorem 1, we have the following version of the social equilibrium existence theorem of Debreu [1].

THEOREM 2. Let {X i} ie l be a family of acyclic polyhedra, Ai : X i --0 Xi closed maps, and f i ,g i : Gr(Ai) --* R u.s.c, functions for each i 6 1 such that

(1) gi(x) <_ f i ( * ) , / o r a / /x 6 Gr(A~); (2) %oi(#) = maxveA , ( , , ) g i (# , y ) is a l.s.c, function o f # 6 X i ; and (3) for each i E I and x i 6 X i, the set

M (e) = A, ( e ) : S, > ( e ) }

is acyclic.

Then there exists an equilibrium point ~ E Gr(A~), for all i E I; that is,

5~EAi (5~ ) and f ~ ( 5 ) = max gi(Si , a~) for a l l i E I . aiEA(&i)

PROOF. For each i E I, define a map Ti : X --o Xi by

T~(z) : ( y E A, (x ' ) : f~ (x ' , y ) > ~o, (x ' ) }

Social Equilibrium Existence Theorem 53

for z ~ X. Then, T~ix ) ~ O by (I) since A~(z ~) is compact and g~(x i, .) is u.s.c, on A~(x~). We show that Gr(T~) is closed in X x X~. In fact, let (xa, Ya) e Gr(T~) and (xa, Ya) ~ ( x, Y). Then,

fi (x', y) >_ rima f~ (x~a'y'') >- r-~a qoi (x~)

___ limbo, >_ (e),

and since Gr(Ai) is closed in X ~ × Xi, y~ E A~(x~) implies y E Ai(xi). Hence, (x, y) E Gr(T~). Moreover, each T~(x) = M(x i) is acyclic by i3). Now we apply Theorem I. Then there exists an ~ E X such that ~ E T~(~), for all i E I; that is, ~ E Ai(~ i) and f~(~i ~i) ~_ ~0~(~i). This completes our proof.

REMARK 1. If Xi and M(x~) in (3) are contractible, if f~ = gi is continuous, and if ~0~ is continuous for each i E I, then Theorem 2 reduces to Debreu [1, Theorem]. Note that our proof is much simpler than his.

REMARK 2. Since A~ and g~ are u.s.c., by Berge's theorem, ~0~ are automatically u.s.c. Hence, condition (2) implies continuity of ~0i.

REMARK 3. If Ai and gi are continuous, condition (2) holds immediately by Berge's theorem, and hence, each ~o~ is continuous. This fact is noted by Debreu [1, Remark].

REMARK 4. As was also noted by Debreu, instead of acyclic polyhedra, one might take for example absolute retracts or others.

3. A P P L I C A T I O N S

From Theorem 2, we obtain acyclic versions of a saddle-point theorem and a minimax theorem.

COROLLARY i. Let X, Y be two acyc/ic polyhedra, and f : X x Y -~ R a continuous function. Suppose that for each xo E X and yo E Y, the sets

{ x E X : f (x , yo) = ~ f(¢,yo)}

and

y E Y: f(zo, y)=minf(xo, W)l ~EY

are acyc/ic. Then, f has a saddle point (xo,Yo) E X x Y; that is,

~f(xo, w)=fixo,Yo)=~a~f(~,Yo) •

PROOF. Note that a saddle point is a particular case of an equilibrium point for two agents ( n = 2) in Theorem 2 for a = ( a l , a 2 ) = (z,y), X1 = X, X2 - ' ~ Y, Al(a 1) = X , A 2 ( a 2) = Y ,

f l ( a ) = gl(a) = f i x, y), f2(a) = g2(a) = - f ( x , y). Note also that condition (2) holds by Berge's theorem.

COROLLARY 2. Under the hypothesis of Coro/]ary 1, we have the minimax equa~ty

maxx~x ~ f(x,y) = minztey mxea~ f(x,y).

PROOF. By Corollary I, we have a saddle point (x0, Y0) E X × Y such that

ma~f(x, yo)=fixo,Yo)=~n~f(xo,y) •

54 S. PARK

Since x ~-* minvey f(x, y) and y ~-* maxxex f(x, y) are continuous by Berge's theorem and X and Y are compact, they attain maximum on X and minimum on Y, respectively. Therefore,

min ~a~ f(x, gt) '~ treY _ ~a~ f(x, Yo) = f(xo, glo)

-~ ~:f(xo,N) _< maX=ex ~me mf(x'v)"

On the other hand, we clearly have

min max f ( x , ll) >_ m a x min f ( z , ~/). y x x y

Therefore, we have the conclusion.

From Theorem 2, we have the following generalization of the Nash equilibrium theorem.

COROLLARY 3. Let {X~}~ex be a family of acyc//c po]yhedra and for each i, f~ : X -, R is a continuous funct ion such that

(0) for each x i 6 X i and each a E R , the set

e x , . . f , _> ,,)

is e m p t y or acyc//c.

T h e n there exists a poin t ~ E X such that

f~(~) -=- max f~ (~' , Z/i) for a / / i E I .

PROOF. We apply Theorem 2 wi th f~ = gi and A~ : X ~ --o X~ defined by A~(x ~) = X i for x i E X i.

Then , condi t ion (2) of Theorem 2 follows from Berge 's theorem, and the set in condit ion (3) is n o n e m p t y and acyclic by (0). Therefore, we have the conclusion.

Finally, note t h a t our results in Section 3 generalize corresponding ones in [1,6-11].

R E F E R E N C E S

1. G. Debreu, A social equilibrium existence theorem, Proc. Nat. Acad. $ci. USA 38, 886-893, (1952); Math- emati~l Economics: Twenty Papers of Gerald Debreu, Chapter 2, Cambridge Univ. Press, Cambridge, (1983).

2. C. Berge, Espaces Topologique, Dunod, Paris, (1959). 3. S. Eilenberg and D. Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math 68,

214-222, (1946). 4. E.G. Bogle, A fixed point theorem, Ann. Math. 51, 544-550, (1950). 5. K. Fern, Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci.

USA 38, 121-126, (1952). 6. J. von Neurnann, Zur Theorie der Gesellschafl~piele, Math. Ann. 100, 295-320, (1928). 7. J. yon Neumann, Uber ein 5konomisches Gleichungesystem und eine Verallgemeinerung des Brouwerschen

Fixpunktsatzes, En~ebn/ase eines Mathematischen Kolloquiums 8, 73-83, (1937); Rev. Economic Studies X I I I (33), 1-9, (1945-46).

8. S. Kakutani, A generalization of Brouwer's fixed point theorem, Duke Math. J. 8, 457-459, (1941). 9. J. Nash, Equilibrium points in n-person games, Proc. Nat. Acad. Sci. USA 36, 48-49, (1950).

10. J. NMh, Non-cooperative games, Ann. Math. 54, 286-295, (1951). 11. J. yon Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton Univ. Press,

(1947).