330 
 
 No. 1655 COPY 2 
 
 STX 
 
 BEtJK 
 
 FACULTY WORKING 
 PAPER NO. 90-1655 
 
 The Core of an Economy with 
 Differential Information 
 
 jVJ 5 «W 
 
 Nicholas C. Yannelis 
 
 . 
 
 College of Commerce and Business Administration 
 Bureau of Economic and Business Research 
 University of Illinois Urbana-Champaign 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 90-1655 
 
 College of Commerce and Business Administration 
 
 University of Illinois at Urbana-Champaign 
 
 May 1990 
 
 The Core of an Economy with Differential Information 
 
 Nicholas C. Yannelis* 
 
 Department of Economics 
 
 University of Illinois at Urbana-Champaign 
 
 Champaign. IL 61820 
 
 * On different occasions I have benefited from discussions, comments and suggestions by Roko 
 Aliprantis, Kim Border, Don Brown, Mark Feldman. Leo Hurwicz, Charlie Kahn, John Ledyard, 
 Andreu Mas-Colell, Flavio Menezes, Tom Palfrey, Ed Prescott, Aldo Rustichini, David Schmeidler 
 and Sanjay Srivastava. Mark Feldman and Aldo Rustichini both independently brought to my 
 attention the related work of Wilson (1978). My thanks are extended to all the above individuals. 
 Of course, I am responsible for any remaining shortcomings. 
 
ABSTRACT 
 
 We introduce a new core concept for an exchange economy with differential information which is 
 contained in the coarse core concept of Wilson (1978). We prove the existence of (i) a core 
 allocation for an exchange economy with differential information and (ii) an a-core strategy for a 
 game in normal form with differential information. 
 
Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/coreofeconomywit1655yann 
 
1. INTRODUCTION 
 
 An exchange economy with differential information, consists of a finite set of agents 
 each of whom is characterized by a random utility function, an initial endowment, a private 
 information set and a prior. 
 
 The purpose of this paper is to study the following questions: How does one define the 
 notion of the core in an exchange economy with differential information? What is the 
 appropriate core concept? Under what conditions on agent's characteristics is the core 
 nonempty? 
 
 It should be noted that with finitely many states of nature, the existence of core alloca- 
 tions for an economy with differential information, follows easily from the well known result 
 of Scarf (1967), as first shown in a seminal paper by Wilson (1978). However, with a contin- 
 uum of states even if there is symmetric information (i.e., the information set of each agent is 
 the same) the domain of the expected utility becomes infinite dimensional (even if there is 
 only one good in the economy), and consequently Scarfs theorem is not directly applicable. It 
 turns out that in the presence of a continuum of states, functional analytic methods as well as 
 several measure theoretic results seem to be required. 
 
 The paper is organized as follows: Section 2 contains notation and definitions. The 
 model and the main results are presented in Section 3. Sections 4 and 5 contain the proof of 
 our main theorems. Finally Section 6 contains some concluding remarks. 
 
 2. NOTATION AND DEFINITIONS 
 
 2.1. Notation 
 
 R' denotes the /-fold Cartesian product of the set of real numbers R 
 R + denotes the positive cone of R'. 
 
R^_+ denotes the strictly positive elements of R'. 
 
 2 A denotes the set of all nonempty subsets of the set A . 
 
 denotes the empty set. 
 
 / denotes the set theoretic subtraction. 
 
 If X is a linear topological space, its dual is the space X* of all continuous linear func- 
 tional on X, and if/? e X*, and x e X the value of p at x is denoted by p • x. 
 
 2.2. Definitions 
 
 If X and Y are sets, the graph of the set-valued function (or correspondence), 
 4> : X — ► 2 Y is denoted by G^ = {(.v, y) G X xY: ye <f>(x)}. Let (T, T, /z) be a complete, fin- 
 ite measure space, and X be a separable Banach space. The set-valued function <f> : T — ► 2 X 
 is said to have a measurable graph if G^ 6 T (x) (3{X\ where 0(X) denotes the Borcl a-algebra 
 on X and ® denotes product a-algebra. The set-valued function <f> : r — ► 2 X is said to be 
 lower measurable or just measurable if for every open subset V of A', the set 
 {t e T : <f>(t)n V + 0} is an element of T. A well-known result of Debreu [(1966), p. 359] 
 says that if <f> : T —* 2 X has a measurable graph, then <^> is lower measurable. Furthermore, if 
 <f>( • ) is closed valued and lower measurable then <£ : T — ► 2 X has a measurable graph. A 
 theorem of Aumann (1967) which will be of fundamental importance in this paper tells us, 
 that if (7\ T,/i) is a complete, finite measure space, J is a separable metric space and 
 <f> : T — ► 2 X is a nonempty valued correspondence having a measurable graph, then <f>( • ) 
 admits a measurable selection, i.e., there exists a measurable function / : T — * X such that 
 f(t)€<t>(t) ii-a.e. 
 
 Let (7\ T,/i) be a finite measure space and A' be a Banach space. Following Dicstcl- 
 Uhl (1977) the function / : T — * X is called simple if there exist je lt .x 2 , . . . , ,y„ in X and 
 
 «!, at 3 a, in T such that / = £ x,x a ,, where Xa,(^ ) - 1 if / € a,- and x a ,(^ ) = if / g a,-. 
 
 1=1 
 
 A function / : r — ► ,V is said to be ^-measurable if there exists a sequence of simple 
 
functions /„ : T -* X such that lim \\f n {t) - /(Oil = for almost all t e T. A ^-measurable 
 
 n— KX3 
 
 function / : T — ► X is said to be Bochner integrable if there exists a sequence of simple func- 
 tions [f n :n- 1, 2, ... ) such that 
 
 lim / ||/ n (/)-/(OlkMO = 0. 
 
 n—oo l 
 
 In this case we define for each E e T the integral to be / / (t)d^{t) = lim J f n (t)d^(t). 
 
 It can be shown [see Diestcl-Uhl (1977), Theorem 2, p. 45] that, if / : T -* X is a im- 
 measurable function then / is Bochner integrable if and only if J \\f (t)\\d n(t) < oc. It is 
 important to note that the Dominated Convergence Tlieorem holds for Bochner integrable func- 
 tions, in particular, if f n : T — ► X, {n - 1, 2, . . . ) is a sequence of Bochner integrable func- 
 tions such that lim J n (t) = f (t) n-a.e. , and \\f n (t )|| < g(t) \i-a.e. , ( where g : T — * R is an 
 
 fl — * oo 
 
 integrable function), then / is Bochner integrable and lim J \\f n (t)-/(t)\\dfj.(t) = 0. 
 
 n — ► oo * 
 
 For 1 < p < oo, we denote by £ p (£t, A') the space of equivalence classes of .V-valued 
 Bochner integrable functions x : T — » X normed by 
 
 i_ 
 
 ll-V|| p =(/ r ||A-(/)||^M/)) P . 
 
 It is a standard result that normed by the functional || • || p above, L p (n, X) becomes a 
 Banach space [see Dicstel-Uhl (1977), p. 50]. Recall that a correspondence 4> : T — ► 2 X is 
 said to be integrably bounded if there exists a map h e L^fi, R ) such that 
 sup {||.v || : x €<Kt))<h(t)r-a.e. 
 
 A Banach space 4 V has the Radon -Nikodym Property with respect to the measure space 
 (7\ T, /i) if for each /i-continuous measure G : T — ► X of bounded variation there exists 
 g e Li(/i, X) such that G(E) = / g(t)dn(t) for all E e T. A Banach space X has the Radon- 
 Nikodym property (RNP) if X has the RNP with respect to every finite measure space. Recall 
 now [sec Diestel-Uhl (1977, Theorem 1, p. 98)] that if (7*,T,/x) is a finite measure space 
 \<p < oo, and X is a Banach space, then X' has the RNP if and only if 
 
(L P ( M , X)Y = L>, Af*) where -*-+-*-= 1. 
 
 p q 
 
 We will close this section by collecting some basic results on Banach lattices [for an 
 
 excelent treatment see Aliprantis-Burkinshaw (1985)]. Recall that a Banach lattice is a Banach 
 
 space L equipped with an order relation > (i.e. , > is a reflexive, antisymmetric and transitive 
 
 relation) satisfying: 
 
 (i) x > y implies x + z > y + z for every z in L , 
 
 (ii) x > y implies Ajc > \y for all A > 0, 
 
 (iii) for all x,y in L there exists a supremum (least upper bound) x My and an 
 infimum (greatest lower bound) x A y, 
 
 (iv) \x | > \y\ implies ||x|| > ||y|| for all x, y in L. 
 
 As usual x + = x M 0, x~ = (-x ) V and \x \ = x V (-x ) = x + + ,x~ ; we call x + , x~ the 
 positive and negative parts of x, respectively and \x \ the absolute value of x. The symbol |j • || 
 denotes the norm on L. If x, y are elements of the Banach lattice L, then we define the order 
 interval [x, y] as follows: 
 
 [x,y] = {z e L : x < z < y }. 
 
 Note that [x, y] is norm closed and convex (hence weakly closed). A Banach lattice L is said 
 to have an order continuous norm if, x a ^0 in L implies ||xj|0. A very useful result which is 
 going to play an important role in the sequel is that if L is a Banach lattice then the fact that 
 L has order continuous norm is equivalent to the weak compactness of the order interval 
 [x , z] = (y e L : x < y < z) for every x, z in L, [sec for instance Aliprantis-Brown- 
 Burkinshaw (1989), Theorem 2.3.8, p. 104 or Lindenstrauss-Tzafriri (1979, p. 28)]. 
 
 We finally note that Cartwright (1974) has shown that if A' is a Banach lattice with order 
 continuous norm (or equivalenty .V has weakly compact order intervals) then Z-^/i, X), has 
 weakly compact order intervals, as well. Cartwright's theorem will play a crucial role in the 
 proof of our main results. 
 
3. MODEL AND RESULTS 
 
 3.1 The Core of an Exchange Economy with Differential Information 
 
 Let Y be a separable Banach lattice with order continuous norm, whose dual Y* has the 
 RNP. 1 Let (fL F, /z) be a complete cr-finite measure space. 
 
 An exchange economy with differential information T = {(A',, u,-, e,-, F it q { ) : /'=1,2,..., n) 
 is a set of quintuples (X { , u { , e it F,, q { ) where, 
 
 y 
 
 (1) X { : Q — ► 2 + is the random consumption set of agent /, 
 
 (2) iii : fi x X x ■ — ► R is the random utility function of agent /', 
 
 (3) F, is a (measurable) partition 2 of (fi,F) denoting the' private information of agent i, 
 
 (4) e, | : — ♦ F + is the random initial endowment of agent /, e,( " ) is F, -measurable, 
 Bochner integrable and e,(a>) G A'.Cu;) for all /, p-a.e., 
 
 (5) <?, : fi — ► R++ is the prior of agent /', (i.e., q { is a Radon-Nikodym derivative hav- 
 ing the property that / Qi(t)d n(t) =1). 
 
 ten 
 
 Denote by L x . the set of all Bochner integrable and F \-measurable selections from the con- 
 sumption set Xi of agent / , i.e., 
 
 Lx = i x i e L x (y., Y+) '■ x i : fl — ► K + is F, -measurable 
 
 and Xi(ta)sXi(u) n-a.e.}. 
 
 For each /, (/' = 1,2, . . .,«), denote by F,(a;) the event in F, containing the realized state of 
 nature w€fl and suppose that f q i (t)dn(t) > for all /. Given F,(u;) in F, define the 
 
 conditional expected utility of agent i , F, : Q x L x — ► R by 
 
 A basic example of a space which satisfies all these conditions is the Euclidean space It . Remark 6.1 
 Section 6 presents some more examples. 
 
 One could have assumed that F, is a sub (7— algebra of F. 
 
(3.1) VM x { ) = J[ «,-(/, *,(/))<?,(/ | £»)^(0 
 
 where 
 
 (3.2) 9i (/ | £,-(«)) = 
 
 fc(0 
 
 if * £ £ t -(w) 
 
 if / € Ei(w) 
 
 iU)^'^(') 
 
 We are now ready to define the central notions of the paper. 
 
 Definition 3.1.1: We say that x = (x u x 2 , . . . , x n ) e n L x . is a core allocation for T, if 
 
 t=i • 
 
 (0 £*•■ = E*.> and 
 
 1=1 1=1 
 
 (ii) It is not true that there exist S c (1, 2, . . . , n) and (v,), e s e IT L x . such that 
 
 Z)- V i = Yj e i anc * K( w > y*) > Pi( w » *<) f° r a ^ ' e 5" for /i-almost all u> e f2 (where 
 
 tes ies 
 
 Vi is given by (3.1)). 
 
 A couple of comments are in order: Note that x e U L x . implies that each jc,-( • ) is F, 
 
 t=i • 
 
 measurable and therefore the vector x(u) = (x^w), x 2 (u), . . . , x n (<x>)) e U X^oj) is VF, 
 
 t' = l 
 
 i=i 
 
 measurable, (where \l F { denotes the join, the smallest partition containing F u F 2 , . . . , F n ). 
 
 i-l 
 
 Condition (i) above implies thar the markets are cleared in each state of nature, i.e., 
 
 £) .v,(a;) = ]T}e,(ci;) n - a.e.. Condition (ii) shows that no coalition of agents (while each agent 
 i=i i=i 
 
 in the coalition uses his/her own private information) can redistribute their initial endow- 
 ments among themselves for any state of nature and make the conditional expected utility of 
 each agent in the coalition better off. 
 
Note that condition (ii) of definition 3.1.1. implies the following condition: 
 (ii)' It is not true that there exist S c {1,2, ..., n) and y : Q -* II X it y t ( • ) is 
 
 i€S 
 
 A F, -measurable (where A F, denotes the meet, i.e., the maximal partition con- 
 ies i€S 
 
 tained in all of them) such that J] j>»(w) =- J] e i( (J °) p-a.e. and 
 
 %es ies 
 
 Vi(u, }\) > Pi(w» ■*») for a11 » € 5" for /i-almost all u € n. 
 The above blocking notion is the one adopted by Wilson (1978) to define his coarse core con- 
 cept. 3 Note that since each yA • ) is A F, -measurable, the information is verifiable by each 
 
 ies 
 
 member of the coalition. For instance, if we imagine that agents negotiate the terms of a con- 
 tract, then Wilson's definition tells us that a coarse core allocation has the property that no 
 coalition of agents can exchange their own information (in fact, information is verifiable by 
 each member of the coalition) and make each agent in the coalition better off. In other words 
 contracts are realizable because information is verifiable. However, according to our condi- 
 tion (ii) of definition 3.1.1., information is not necessarily verifiable by all the members of the 
 coalition (it is only privately verifiable). The latter makes the core smaller, i.e., any core allo- 
 cation satisfying the definition 3.1.1. is a coarse core allocation (recall that if 
 
 v,( • ) is A F, -measurable, it is also F, -measurable, of course the reverse is not true). Hence, 
 ies 
 
 the theorems that we will prove on the existence of core allocations will imply the existence of 
 coarse core allocations as well. 
 
 Note that if we were to narrow the set of core allocations by replacing F,-measurability 
 of v,( • ) in (ii) of definition 3.1.1, with the V F,-measurability of y : fl — ► II X it then it is easy 
 
 to construct examples which satisfy all the assumptions of Theorem 3.1 below, but the core is 
 empty [see Wilson (1978) for examples to that effect]. We are not aware of any natural set of 
 assumptions on utility functions and initial endowments which will guarantee the existence of 
 such a core. Finally, it is worth pointing out that a core notion which allows for complete 
 
 See also Kobayashi (1980) who has also used the coarse core. 
 
8 
 
 exchange of information among agents in each coalition may not be an appropriate concept, 
 since in most applications, agents do not have an incentive to reveal their own private infor- 
 mation (think of situations of moral hazard or adverse selection). 
 
 n 
 
 Definition 3.1.2: We say that x € Tl L x . is (interim) Pareto optimal if. A 
 
 (0 £.x, = £e,, and 
 «'=i «=i 
 
 (ii) It is not true that there exists y € U L^. such that £;>, = Yi e i an ^ 
 
 1=1 i=l i=l 
 
 V%{u, j\) > Vi(u, x t ) for all i for /i-almost all w e ft (where V { is given by (3.1)). 
 
 Definition 3.1.3: We say that x 6 II L x is individually rational if: 
 
 «=i 
 
 (i) Ex.— E^.and 
 i=i t=i 
 
 (ii) Vi(u, x,) > F,(u>, e t ) for all i and for /x-almost all u e CI, (where V { is given by 
 (3.1)). 
 
 Finally, if the private information set of each agent, is the same (symmetric information, i.e., 
 
 n 
 
 Fi = F for all /) we call any x e II L x . satisfying (i) and (ii) of Definition 3.1.1, symmetric 
 
 »=i * * « 
 
 core allocation for V. 
 
 We are now ready to state our first main result: 
 
 Theorem 3.1: Let T = {(.V,, u it e t , F,, q t ) : / = 1, 2, . . . , n) be an exchange economy 
 with differential information satisfying the following assumptions, for each 
 
 A similar notion is defined by Palfrey and Srivastava (1987). 
 
/, (/ = 1,2, ...,«), 
 
 (a.3.1) X x ■: ft — » 2 Y+ is an integrably bounded, convex, closed, nonempty valued and 
 Fj -measurable correspondence, 
 
 (a.3.2) for each u> e Q, « f -(w, • ) is weakly continuous and integrably bounded, and 
 
 (a.3.3) for each u e Q, «,(o;, • ) is quasi-concave. 
 
 Then a core allocation exists in r. 
 
 The following Corollaries follow directly from Theorem 3.1. 
 
 Corollary 3.1: Let T = {(A',, «,-, e { , F t -, q t ) : i = 1, 2, . . . , n) be an exchange economy 
 with differential information satisfying all the assumptions of Theorem 3.1. Then an individu- 
 ally rational and Pareto optimal allocation exists in T. 
 
 Corollary 3.2: Let V = {(A',, «,-, e f , F it q t ) : i = 1, 2, . . . , n) be an exchange economy 
 with symmetric information, (i.e., F, = F for all i) satisfying all the assumptions of Theorem 
 
 3.1. Then a symmetric core allocation exists in T. 
 
 3.2. The a-Core of a Game in Normal Form with Differential Information 
 
 A game in normal form with differential information B = ((A",-, n,, F,-, q { ) : i = 1,2,...,/?} 
 is a set of quadruples (A',, « f -, F,-, <?,) where 
 
 (1) X, ; : n — •• 2 y is the strategy set-valued function of player /, 
 
 n 
 
 (2) Ui : n x II X t — » R is the random payoff function of player /, 
 
 (3) Fj is a (measurable) partition of (fi, F) denoting the private information of player /', 
 and 
 
10 
 
 (4) q, ; : Q — > R++ is the prior of player z, (i.e., q { is a Radon-Nikodym derivative hav- 
 ing the property that f t&i <?,(/)<z>(0 = 1). 
 
 For each /, (/' = 1, 2, . . . , n) denote by £",(0;) the event in F, containing the true state of 
 nature u eCl and suppose that J t e E . (u) Qi(t)dn(t) > 0. Given £ t -( w ) m Ft define the conditional 
 
 n 
 
 expected payoff of player /, V i ■ : (lx IT L x . —> R by 
 
 1=1 * 
 
 (3.3) VM x) = } teE . H Ui(t,x(f))q t {t \E t (u))dft(t\ 
 
 where #,■(/ |£ t (w)) is defined as in (3.2). 
 
 Before we define the notion of an a-core strategy for the game B we need to introduce some 
 notation. Denote by / the set of players {1, 2, . . . , n). If S c / then (v 5 , ,x 7 / 5 ) denotes the 
 
 n. 
 
 vector z in n L x . where z,- = r, if / e S and z t - = a:, if / £ S. 
 
 Definition 3.2.1: We say that x 6 II L x . = ZL X is an a-core strategy for B if: 
 
 1=1 
 
 (i) It is not true that there exist Sol and {yi) ies e II L x . such that for any 
 
 i€5 
 
 z ; / 5 en L x . , K,-(w, (v 5 , z 7 / 5 )) > K(w, ,x) for all i e 5 for /i-almost all weQ 
 (where F, is given by (3.3)). 
 
 Note that as previously x e II L x . implies that each jc,-( • ) is F, -measurable and consc- 
 
 «=i • 
 
 qucntly the vector x(u) = (.v 1 (a;), . . . , x n (u)) is V F, -measurable. Condition (i) in Definition 
 
 3.2.1 indicates that no coalition of players is able to change its strategy (while each player in 
 the coalition uses his/her own private information) and make the expected utility of each 
 member in the coalition better off, no matter what the complementary coalition chooses to do 
 
11 
 
 (each member in the complementary coalition is also allowed to take advantage of his/her own 
 private information). Following the previous definition of a coarse core allocation for an 
 economy with differential information, we can define an a-coarse strategy for the game B, 
 and show that the set of a-coarse core strategies contains the set of a-core strategies for the 
 game B. 
 
 Since there is no exchange of information among players in each coalition one may sug- 
 gest that it is possible to analyze games in normal form with differential information (or 
 economies with differential information) in a noncooperative setting adopting the notion of a 
 Bayesian Nash equilibrium or correlated equilibrium. However, the latter concepts do not 
 yield Pareto optimal outcomes, contrary to the core or a-core. It seems to us that selecting 
 outcomes out of the Pareto frontier is an attractive property for an allocation mechanism to 
 have. The latter makes the core concept appealing in an economy with differential informa- 
 tion. 
 
 We can now state our second main result. 
 
 Theorem 3.2: Let B = {(X t , u ( , F,, q { ) : i = 1,2,. . . , n) be a game in normal form with 
 differential information satisfying the following assumptions for each player 
 /, (/ = I, 2, ...,«), 
 
 y 
 
 (a.3.2.1) X { : ft — ► 2 + is an integrably bounded, nonempty convex weakly compact 
 valued and F, -measurable correspondence, 5 
 
 (a. 3.2.2) for each u e ft, u t (uj, ■ ) is weakly continuous and integrably bounded, and 
 
 (a.3.2.3) for each u e ft, w,-(w, • ) is quasi concave. 
 
 The assumption that X , ( • ) takes values in the positive cone of Y , is not needed for the proof of this 
 theorem. 
 
12 
 
 Then an a-core strategy exists in B . 
 
 4. PROOF OF THEOREM 3.1 
 
 We first state the well-known core existence result of Scarf (1967) [see also Border 
 (1984) or Yannelis (1985) for recent generalizations] which is going to play a crucial role in 
 the proof of Theorem 3.1. We will first need some notation. 
 
 Let E = {(X { , «,-, e { ) : / = 1, 2, . . . ,n) be an exchange economy, where 
 
 (1) X t cR' is the consumption set of agent /, 
 
 (2) u { : X, ; — » R is the utility function of agent /, and 
 
 (3) e { € X { is the initial endowment of agent i 
 
 Define the set-valued function P, : X t ■ — ► 2 • by P,-(jc,-) = {>>,• 6 X, : /<,(>',) > h, (.*,•)}• Scarfs 
 result asserts that if .V, is a nonempty, closed, convex and bounded from below subset of R', 
 //, is quasi concave and continuous, (i.e., if P, is convex valued and has an open graph in 
 
 n 
 
 X { x X { ), then core allocations exist in E, i.e., there exists x € Tl X { such that: 
 
 n n 
 
 (0 Y,*i - E^.-. and 
 
 1=1 1=1 
 
 (ii) it is not true that there exist 5cfl,2,...,n) and (yA-es € II X { such that 
 
 ies 
 
 T,y'i = E e i and >'i e p *( x i) for a11 » e ' s '- 
 
 We begin the proof of Theorem 3.1 by constructing a new economy 
 G = {(L Yt , P it ei): i = 1,2,...,//} where, 
 
 (i) L x is the consumption set of agent /, 
 
 (ii) P t : L x . — ► 2 * is the preference correspondence of agent / defined by 
 
13 
 
 Pi(x { ) = {y, € L x . : Vffa y.) > Fi(w, x.) for /x-almost all u> e Q), and 
 
 (iii) e, e L x . for all i , is the m/7/7z/ endowment of agent i . 
 
 Note that the existence of a core allocation for G implies the existence of a core allocation for 
 the original economy T = {(Xi, //,• , e iy F,-, q { ) : i = 1, 2, . . . , «}. Hence, all we need to show 
 is that a core allocation exists in the economy G. To this end we first show that for each 
 
 L x . 
 
 i , L x . is closed, bounded convex, nonempty and that P, : L x . — ► 2 * is convex valued having 
 a weakly open graph (i.e., the set G P . = {(x,y) € L x . x L x . : y e P,(x)} is weakly open in 
 ^x, x L x .). 
 
 Note that the fact that L x . is convex, closed and bounded follows directly from assump- 
 tion (a.3.1). To prove that L x . is nonempty, recall that X, ; : f2 — ► 2 + is F, -measurable, 
 nonempty, closed valued and therefore G x . € F, ® 0(Y + ). By the Aumann (1967) measurable 
 selection theorem, we can obtain an F, -measurable function /, : ft — > 7 + such that 
 fi(u)e Xiiu) ii-a.e. Since .V, is integrably bounded, we can conclude that /,• e L t (p,, Y + ). 
 Hence, /, 6 F^ an<3 " this proves that L x is nonempty. 
 
 In order to show that for each /, F, has a weakly open graph, we will first need the fol- 
 lowing claim: 6 
 
 Claim 4.1 For each /, (/' = 1,2,...) and for each u> e fi, Vi(u>, • ) is weakly continuous. 
 
 Proof: Fix /, (/' = I, 2, . . . ,«) and u 6 n and let F,M be an event in F t . Consider the 
 sequence {x t m : m = 1, 2, . . . } in Ly.'cLjOx, K), which converges weakly to x, -e L x ., i.e., 
 p ■ .v, m converges to p ■ x t for any p e LJiji, Y*) - (L x (/x, K))* (recall that F* has the RNP). 
 Note that x/" converges weakly to .v, is equivalent to the fact that p • x- n x A = pxa ' xf* con- 
 verges to p ■ x { \a = PXa ' x i f° r anv P £ £oo(m> Y*), A e F and each condition above implies 
 that y ' ■ xj*x A = y ' Xa ' *F converges toy' ■ x { xa = >' 'xa ' x i f° r anv y' € Y' , A e F . If we 
 
 A similar result is proved in Yannelis-Rustichini (1988). 
 
14 
 
 show that x^xe.^) converges pointwise in the weak topology of X t to x t -x&( w )i then since for 
 each w € H, iii(u), • ) is weakly continuous and integrably bounded the weak continuity of 
 F,(u>, • ) will follow from the Lebesgue dominated convergence theorem. Now if 
 F, = [Ei 1 , £, 2 , . . .} is the partition of agent i , then the fact that x™ and x,- are elements of L x . 
 
 oo oo 
 
 implies that x™ = £} x^x«* » *i ■ £ *i,*Xe* . for *.\* . *<,* in *i and consequently we can 
 
 k=l k=l 
 
 conclude that *rXj%( w ) = E*.\*X|tf ni%( w ) converges weakly to x,x E . M = E**.*X^h^-(«)- This 
 
 A:=l Jk=l 
 
 completes the proof of the claim. 
 
 In view of Claim 4.1 we can now conclude that for each /', P { has a weakly open graph. 
 Moreover, since for each u e Q, w,(w, • ) is quasi-concave so is K,(u>, • ) and therefore, P { is 
 convex valued. We will now construct a suitable family of truncated subeconomies in a finite 
 dimensional commodity space, each of which satisfies the assumptions of Scarfs theorem. 
 Applying Scarfs theorem, we will obtain a net of core allocations for each subeconomy. By 
 taking limits we will show that the existence of a core allocation for each subeconomy implies 
 the existence of a core allocation for the original economy G. 
 
 Let A be the set of all finite dimensional subspaces of Z^Oz, Y + ) containing the initial 
 endowments. For each a e A define the consumption set of agent /, L x a by L x a = L x . n a 
 
 and the preference correspondence of agent /, P, a : L x a — ► 2 'by P { a (x i ) = Pi(x t ) n L x a . 
 We now have an economy C ° = {(L x ° , P, a , e.) : i = 1, 2, . . . , //} in finite dimensional com- 
 modity space, where, 
 
 (4.1) L x . is the consumption set of agent /', 
 
 a 
 a a "X- 
 
 (4.2) P t : L x — ► 2 ' is the preference correspondence of agent /', and 
 
 (4.3) f, € L x ° is the initial endowment of agent /. 
 
15 
 
 It can be easily checked that each economy G " satisfies all the ssumptions of Scarfs theorem 
 
 n 
 
 and therefore there exists x a = (xf , . . . , x n a ) G II L x = L x such that : 
 
 1=1 
 
 (4.4) y>,«=5>,-,and 
 i=i i=i 
 
 (4.5) it is not true that there exist S C {1, 2, . . . ,n) and (>',),- e5 G II L x> a such that 
 
 »€S 
 
 E >\- = E^i and v « e p i W) for a11 ' € 5 - 
 
 i€S i€S 
 
 From (4.4) it follows that for each a G A 
 
 0< E*f -Ee, -e. 
 
 i=i «=i 
 
 Hence for each a G A the vectors x, a lie in the order interval 
 
 n 
 
 [0, e] = {z G II L x . : < z < e }. Since by assumption order intervals in Y are wcaklv com- 
 
 i=i • 
 
 n 
 
 pact, by Cartwright's theorem the order interval [0, e] in II L x . is weakly compact. Direct 
 
 i=i » 
 
 the set A bv inclusion so that {(.xf , x£, ■ ■ ■ , -x n a ) : a G A} forms a net in II L x .. Since all 
 
 i=i * 
 
 the vectors x" lie in the order interval [0, e] which is weakly compact, the net 
 {(.x i , . . . , ,x n a ) : a G A } has a subnet which converges weakly to some vector 
 x u x 2 * ... , -x n in [0, e\ We will show that the vector jc ls . . . , x n is a core allocation for the 
 economy G Denote the convergent subnet by {(x^ m) , . . . , ,x n a(m) ) '• m G A/} where M is a set 
 
 n n 
 
 directed by ">". Since for all m G M, X).x, a(m) = £e, and *i" (m) converges weakly to 
 
 i=i i=i 
 
 n n 
 
 .x, G Lv., we conclude that £.x, = £<?,- We will now complete the proof by showing that: 
 
 i=l i=l 
 
 (4.6) It is not true that there exist Sc{1,2,...,b) and 0\), e5 G II L x . such that 
 E.v, = £?,- and y% e *>.(*,) for all i G 5. 
 
 i '65 i£S 
 
16 
 
 Suppose that (4.6) is false, then there exist S c (1,2, ... , n) and (yA-es € IT L x . such that 
 
 ies ' 
 
 E- v « = S e t an d v,- € Pi(x) for all / € S. Since ,x, a(m) converges weakly to x, and P, has a 
 
 ies ies 
 
 weakly open graph, there exists m e M such that v t - e P, ix-*^) for all w > m and for all 
 i e S. Choose m 1 > m so that , if m > m x , y% 6 L^ 01 m for all i e 5". Then 
 
 v, e Pi a(m) U, a(m) ), for all m >m 1 and for all i e 5. But this contradicts (4.5). Hence (4.6) 
 holds and this completes the proof of the theorem. 
 
 5. PROOF OF THEOREM 3.2 
 
 We begin by stating the a-core existence result of Scarf (1971) which is going to be used 
 in the proof of Theorem 3.2. 
 
 Let N = {(X t , Uf): i = 1, 2, . . . , n) be a game in normal form where, 
 
 (1) A', is a compact, convex and nonempty subset of R', denoting the strategy set of 
 player /', and 
 
 n n 
 
 (2) «,- : II X, ■ — » R is a quasi-concave function on IT X { denoting the payoff of 
 
 i=i i=i 
 
 player /. 
 
 n 
 
 The strategy vector x e IT X { is said to be an a-core strategy for N if: 
 
 t=i 
 
 It is not true that there exist S c {1,2,...,;/} and (v.), es G II X { such that for any 
 
 ies 
 
 z^ s e n X t , h,(v 5 , z 1 ' 8 ) > iii(x) for all i e S. 
 
 As in the proof of Theorem 3.1 we will construct a new game B 
 = {{L Xi , I',) : i = 1,2,...,//} where, 
 
 (a) L x . is the strategy set of player i , and 
 
17 
 
 (b) I 7 ',- : H x n L x . — ► R is the payoff function of player / , ( defined as in (3.3)) 
 
 t=i 
 
 It is easy to see that the existence of an a-core strategy for B implies the existence of an 
 a-core strategy for the original game B = {(X,-, u f -, F,-, q { ) : i = 1, 2, . . . , «}. Our goal is to 
 construct a suitable family of truncated subgames in a finite dimensional strategy space, each 
 of which satisfies all the conditions of the Scarf (1971) theorem. Therefore we will obtain a 
 net of a-core strategies for each subgame. As in the proof of Theorem 3.1, operating a limit- 
 ing argument we can show that the existence of an a-core strategy for each subgame implies 
 the existence of an a-core strategy for the orginal game B. Before we start the outlined con- 
 struction of the family of truncated subgames, we need to make some observations. 
 Note that for each u> e CI, V { (u, ■ ) is weakly continuous (recall Claim 4.1) and by virtue of 
 
 assumption (a.3.2.3) quasi-concave on n L x .. Moreover, note that each L x . is convex and 
 
 t=i ' ■ 
 
 nonempty. However, since Scarfs theorem requires the compactness of each strategy set we 
 will need to prove the following claim which is known as Diestel's theorem. 
 
 Claim 5.1: The set L x is weakly compact in L^n, Y). 
 
 Proof: The proof is based on the celebrated theorem of James (1964) and it is patterned 
 after that of Khan (1982). Note that the dual of L x (ji, Y) is LJjm, Y**) (where w* denotes the 
 w* -topology), i.e., (L x (/i, Y))' = LJjjl, Y*»\ [see, for instance, Tulcea-Tulcea (1969)]. Let x 
 be an arbitrary element af L^/i, Y^*). If we show that x attains its supremum on L x . the 
 result will follow from James' theorem, [James (1964)]. Let, 
 
 Sup V • x = Sup / (if>i(w) ■ x (w))d /i(u>). 
 
 Note that by theorem 2.2 in Hiai-Umegaki (1977), 
 
 Sup / (V\(uO • x(u))dti(u) = f Sup (4>i ■ x(w))dn(u>). 
 
 For each /, define the set-valued function g i ■: Q -» 2 Y by 
 
18 
 
 g { (u) - {y 6 Xtiu) : v • x = Sup fa ■ x). It follows from the weak compactness of X { that 
 
 <£,-ex,-(w) 
 
 for all wen, £,•(&;) is nonempty. For each /, define /,• : Q x Y — > [-00, 00] by 
 
 /•■(<*>, y) = v • .x - Sup 4> • x. It is easy to see that for each fixed u e Q, /.(o;, • ) is continu- 
 
 <*>,ex,(w) 
 
 ous and for each fixed y e Y, /,(-,j) is F, -measurable and hence /(•,•) is jointly 
 F t - -measurable, i.e., for every closed subset V of 
 
 [-00, 00], ff l {V) = {(oj,z)e fix Y : z € A;(w)} belongs to F, <g> B(Y). Since ^ is 
 F, -measurable the set G x . = {(u,x) : .X € A', (a;)} is an element of F t ® B(Y). Moreover, 
 note that G g . = /r*(0) n G x . and since /, _1 (0) and G x . belong to F, <g> B(Y ) so does G g .. It fol- 
 lows from the Aumann measurable selection theorem that there exists an F,- -measurable func- 
 tion z, ■: fi — » 7 such that z,-(w) G ^-(w) n-a.e. Thus, z { e L x . and 
 
 Sup <£,• ■ x = f {zAu) • x(u))d^(uj) = z, • x. Since jc 6 L^/z, yj«) was arbitrarily chosen, 
 
 we conclude that every element of (Li(/i, y))* attains its supremum on L x ., and this completes 
 the proof of the fact that L x . is weakly compact. 
 
 We are now ready to construct a suitable family of truncated subgames. To this end let 
 A be a family of all finite subsets of L x .. For each A e A let L x . denote the closed convex 
 
 hull of A. Then each L x . is a compact, convex, nonempty subset of a finite dimensional 
 
 Euclidean space and u L x . = L x .. Moreover, the set {L x . : A e A) is directed upwards by 
 
 A£A * ' * 
 
 inclusion. For each A e A we have a game B = {(L x . , V { ) : / = 1, 2, . . . , n) where, 
 
 (5.1) L x is the strategy set of player i, and 
 
 a n A 
 
 (5.2) F, : n x II L x — ► R is the payoff function of player /. 
 
 t=i * 
 
 Each B satisfies the assumptions of Scarfs a-core existence theorem and therefore there 
 
 exists x e II L x satisfying the following property: 
 
 t = l 
 
19 
 
 For //-almost all wGfi, it is not true that there exist S c {1, 2, . . . , n) and 
 (v,) l€5 e n L X X such that for each z 1 ' 8 e n L X X , V { x («. (v s , - //s )) > V t \w t x A ) for all 
 i ES. 
 
 Since the set A is directed by inclusion we have constructed a net 
 
 AAA n 
 
 U*i » *2 , . . . , x n ) : A 6 A} of a-core strategies in IT Ly.. Since by Claim 5.1 each L x . is 
 
 AAA 
 
 weaklv compact so is n L x .. Hence the net {(x i , x 2 , . . , x n ) : A € A} has a subset which 
 
 i=i ■ 
 
 converges weaklv to (x u x 2 , ■ ■ ■ , x n ) in II L x .. We must show that x u jc 2 , . . . , x n is an 
 
 i=i * 
 
 a-core strategy for B. Adopting a similar argument with that used in the proof of Theorem 
 3.1, one can now complete the proof of Theorem 3.2. 
 
 6. CONCLUDING REMARKS 
 
 Remark 6.1: In Theorems 3.1 and 3.2, Y is assumed to be a separable Banach lattice 
 with order continuous norm whose dual Y* has the RNP. Basic examples of spaces which 
 satisfy the above properties are: 
 
 (i) the Euclidean space R', 
 
 (ii) the space l p (1 <p < oo) of real sequences [a K : n = 1, 2, . . . } for which the 
 
 1 
 norm K|| p =(£ Kl p ) p is finite, 
 
 n = l 
 
 (iii) the space L P (Q, F,/z) (1 <p < oo) of measurable functions / on the measure 
 
 i_ 
 space (fi,F,/z) for which the norm |[/|L = (/ \f (u)\ p dn(u)) p is finite. 
 
 It is important to give examples of spaces that Theorems 3.1 and 3.2 do not cover: 
 
20 
 
 (iv) L x [0, 1] or Lx(/i), if /z is not purely atomic, c , /«> • £«> [0> 1] and 
 
 (v) the space C(X) of continuous real-valued functions on the infinite compact Haus- 
 dorff space X (with the supremum norm). 
 
 Recall that the spaces in (iv) and (v) do not have the RNP moreover, order intervals are not 
 weakly compact in L^ [0, 1] and C(X). 
 
 Remark 6.2: The separability assumption on Y was used in order to make the Aumann 
 measurable selection theorem applicable. The latter result was used in several steps in the 
 proofs of Theorems 3.1 and 3.2. The relaxation of the separability of Y is possible. In this 
 case however, the consumption set L x . will be the set of all Gel' f and integrable selections from 
 the set-valued function X, : : Q — ► 2 1 ", and one will need to appeal to results on the existence of 
 weak* measurable selections. 
 
 Remark 6.3: Theorem 3.1 and its Corollaries can be easily extended to coalition pro- 
 duction economies provided that the production technology is assumed to be balanced. The 
 proof remains essentially unchanged. 
 
 Remark 6.4: Kahn and Mookerjee (1989), have introduced a core-like concept in order 
 to analyse games in normal form with differential information. Their concept in a two-person 
 game, coincides with the coalitional Nash equilibrium. No existence results are given in their 
 paper. However, it is known [see, for instance, Scarf (1971)] that even if preferences are 
 strictly convex and continuous the jet of coalitional Nash equilibrium strategics may be 
 empty. 
 
 Remark 6.5: We conjecture that the core of a large finite private information economy 
 will converge to the standard Debreu-Scarf (1963) core notion, with the approximation getting 
 finer the larger the private information economy, (this will follow from the law of large 
 numbers provided there is some kind of independence among agents). Hence, we can con- 
 clude that core allocations in large private information economy will become Walrasian. We 
 also conjecture that without the independence assumption among agents, core allocations in a 
 
21 
 
 large private information economy will characterize some kind of rational expectations equili- 
 brium. 7 
 
 Kobayashi (1980, p. 1647) has made a related conjecture for the syndicate problem. Moreover, Srivastava 
 (1984) has shown that a Wilson-type core allocation in a differential information economy becomes a full information 
 core allocation as the number of agents in the economy tends to infinity. 
 
22 
 
 REFERENCES 
 
 Aliprantis, C. D., D. Brown and O. Burkinshaw (1989), Existence and Optimality of Competi- 
 tive Equilibria, Springer-Verlag, Berling Heidelberg, New York. 
 
 Aliprantis, C. D. and O. Burkinshaw (1985), Positive Operators, Academic Press, New York, 
 London. 
 
 Aumann, R. J. (1967), "Measurable Utility and the Measurable Choice Theorem," in La Deci- 
 sion, C.N.R.S. Aix-en-Provence. 
 
 Border, K. (1984), "A Core Existence Theorem for Games Without Ordered Preferences," 
 Econometrica, 52, 1537-1542. 
 
 Cartwright, D. (1974), "The Order Completeness of Some Spaces of Vector-Valued Func- 
 tions," Bulletin of the Australian Mathematical Society, 1 1, 57-61. 
 
 Debreu, G. (1967), "Integration of Correspondences," Proceedings of the Fifth Berkeley Sym- 
 posium on Mathematical Statistics and Probability, Vol. II, Part I, 351-372. 
 
 Debreu, G. and H. Scarf (1963), "A Limit Theorem on the Core of an Economy," International 
 Economic Review, 4, 235-246. 
 
 Diestel, J. and J. J. Uhl (1977), Vector Measures, Mathematical Surveys, No. 15, American 
 Mathematical Society, Providence Rhode Island. 
 
 Hiai, F. and H. Umegaki (1977), "Integrals, Conditional Expectation and Martingales of Mul- 
 tivalued Functions," Journal of Multivariate Analysis, 7, 149-182. 
 
 James, R. (1964), "Weakly Compact Sets," Transactions of the American Mathematical Society, 
 113, 129-140. 
 
 Kahn, C. and D. Mookerjee (1989), "Decentralized Exchange and Efficiency under Moral 
 Hazard." Working paper, University of Illinois and Stanford University. 
 
 Khan, M. Ali (1982), "On the Integration of Set-Valued Mappings in a Nonreflexive Banach 
 Space," Working Paper, Johns Hopkins University. 
 
 Kobayashi, T. (1980), "Equilibrium Contracts for Syndicates with Differential Information," 
 Econometrica, 48, 1635-1665. 
 
 Lindenstrauss, J. and L. Tzafriri (1979), Classical Banach Spaces II, Springer-Verlarg, Berling, 
 Heidelberg, New York. 
 
 Palfrey, T. R. and S. Srivastava (1986), "On Bayesian Implemcntable Allocations," Review of 
 Economic Studies, 178, 193-208. 
 
 Scarf, H. (1967), "The Core of an N-Pcrson Game," Econometrica, 35, 50-69. 
 
 Scarf, H. (1971), "On the Existence of a Cooperative Solution for a General Class of n-Pcrson 
 Games," Journal of Economic Tlieory, 3, 169-171. 
 
23 
 
 Srivastava, S. (1984), "A Limit Theorem on the Core with Differential Information," Econom- 
 ics Letters, 134, 111-116. 
 
 Tulcea, I. A. and C. I. Tulcea (1968), Topics in the Tlieory of Liftings, Springer- Verlag, New 
 York. 
 
 Wilson, R. (1978), "Information, Efficiency, and the Core of an Economy," Econometrica, 46, 
 807-816. 
 
 Yannelis, N. C. (1985), "The Core of an Economy Without Ordered Preferences." Working 
 paper No. 214, University of Minnesota. 
 
 Yannelis, N.C. and A. Rustichini (1988), "Equilibrium Points of Non-cooperative Random 
 and Bayesian Games." Working paper, University of Illinois and Northwestern Univer- 
 sity.