the core of a continuum economy with widespread externalities and finite coalitions: from finite to...

JOURWAE OF ECONOMIC THEORY 49, 135-168 (1989)

The Core of a Continuum Eco idespread Externalities and Finite

ram Finite to Continuum

MAMORU KANEKO

Department of Economics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

AND

MYRNA HOLTZ W~~DBRS

Department of Economics, University of Toronto, Toronto, Ontario, Canada M5S iAl

Received September 25, 1987; revised September 28, 1988

When permissible coalitions in finite economies are constrained to be small relative to the p!ayer set, the continuum model with finite coahtions and its f-core are the limits of large finite economies and their s-cores. We show convergence both of game-theoretic structures-relatively small coalitions in the iinite economies converge to finite coalitions-and of solutions--a-cores converge to the f-core. Our convergence is carried out in the context of exchange economies with widespread externalities where the requirement that coalitions be small is critical. Journal of

Economic Literature Classification Numbers: 021, 022, 026. 0 1989 Academic PESS, IIIC.

1. INTRODUCTION

1.1. General Motivation

The notion of a perfectly competitive economy with re~o~tra~ti one with a very large number of participants, each of whom actively pgr- sues his own interests and each of whom has negligible influe economic aggregates.’ In recontracting, agents meet face to face

I By perfect competition with recontracting we mean to suggest a story of individuals meeting in relatively small groups or market places and engaging in trade. Each individual can possibly frequent many different market places. These activities (rather ?han the actions of a central auctioneer) are envisioned as leading to competitive prices. Thus perfect competition and recontracting are tied together.

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136 KANEKO AND WOODERS

pursuit of their own interests, engage in trade. This suggests that a model describing these notions needs two special aspects: individuals should be effective in pursuit of their own interests through recontracting and simultaneously individuals should be ineffective in influencing any broad economic totals. In other words, in his own recontracting behavior, at the individual level an individual player is nonnegligible to other players, but at the total economy level an individual player is negligible.

To describe these seemingly contradictory aspects of perfect competition with recontracting, we are led to a new model (Kaneko and Wooders [6,7] and Hammond, Kaneko, and Wooders [4]) with a continuum of players, distinct in several respects from the models initiated by Aumann [a]. The main difference between our model and the standard model is the treatment of permissible coalitions. In both models the total player set is a continuum. In our model, coalitions-recontracting groups-are restricted to be finite sets (ones containing only finite numbers of players) while in the standard model, coalitions are restricted to be subsets of positive measure. With coalitions of positive measure we are unable to simulaneously incorporate the two special aspects of perfect competition with recontracting. To discuss this, we need to consider the subtle interpretation of the concept of the individual player.

With coalitions of positive measure, two interpretations of the concept of the individual player might be possible. If we adopt the straightforward interpretation of the individual player as a point in the continuum, then the individual player has negligible influence on economic aggregates-he has no effect on them at all. However, for the same reason, the player is also negligible in the formation of any recontracting group. On the other hand, if we adopt an alternative interpretation of the individual player as arbitrarily “small,” but bigger than a point, so that individuals can be interpreted as having nonnegligible power in forming recontracting groups, then individuals also become nonnegligible relative to the total. Neither of the above interpretations simultaneously captures the two aspects of perfect competition with recontracting-the individual as nonnegligible relative to recontracting groups while negligible relative to the total.

For our approach, since coalitions-finite sets-are groups of individual players, we can adopt only the straightforward interpretation of the individual player as a point in the continuum. Since recontracting groups are finite, players are nonnegligible within these groups. For individual players to be negligible relative to the total economy we need a continuum of players. Thus, we are compelled to use our approach to describe the seemingly contradictory aspects of perfect competition with recontracting.

Our continuum approach is special in that we use absolute magnitudes (the cardinal numbers) and finite sums to describe finite coalitions and their behaviors while we use proportional magnitudes (the measure) to

FROMFINITETO CONTINUUMECONOMIES 137

describe economic aggregates. Our approach requires these two separate, noncomparable, notions of sizes within the same model. This ~benorne~o~ does not (exactly) appear in a finite world. In a finite proportional and absolute sizes are always comparable, that is, the a e magnitudes described by the cardinal numbers are mutually replaceable with the proportional magnitudes described by the counting measure. Now our questions are: does there exist an interpretation of perfect competition with recontracting in a finite world and then, if it exists, what is it? This attempts to provide answers to these questions.

We argue that there are affirmative answers; we can give a finite analogue to the continuum model. The seemingly contradictory aspects of perfect competition with recontracting can be a?~roximately realize large finite setting. However, whether or not a finite analogue is cl the continuum economy depends on how we measure closeness. ~be~efore, one finite analogue does not suffice as a hnitistic interpretation of cement competition with recontracting unless a specific norm, or standard, for the approximation is given. If we cannot specify a natural standard fcr an approximation then one possible approach is to consider sequences of finite analogues instead of a specific one. Our answer is in demo~stratinE the convergence of a sequence of finite economies to the ~o~tin~urn. provide this answer in the context of exchange economies with widespread externalities.

A continuum economy is approximated arbitrarily closely finite economy. The restriction of coalitions in the continu finite sets of players is approximated by the restriction of coalitions to relatively small sets of players in the finite economies. individual level, the effectiveness of the individual is essentially invariant with the size of the economy, while, relative to the entire economy, the individual becomes negligible. The measurements at the individual level correspond to absolute magnitudes and those at the total economy level to proportional magnitudes.

1.2. The Model and Result

When we look at the totality of individual behavior in recontracting, the two separate notions of measurement, absolute m tional magnitudes, must be consistently connected. continuum this connection is made through a “ partition of the player set. Such a partition has members (coalitions) each containing only finite numbers of players. It also has the property that t proportional magnitudes of aggregates of finite coalitions in the partition are consistent with the proportional magnitudes given by the measure. T finite analogue of a measurement-consistent partition is a partition of the


total player set into permissible small coalitions; the measurement-consistency requirement is not a constraint since it is automatically satisfied, In the continuum model, measurement-consistent partitions make a natural connection between the cardinal numbers and the nonatomic measure.

Feasible allocations for the entire economy must be defined consistently with the postulate that cooperation is allowed only in finite coalitions. To do this, we use the measurement-consistent partitions of the continuum of players into finite coalitions. A feasible allocation is defined so that trades are allowed only within coalitions in some measurement-consistent partition. Analogously, in a large finite economy, a feasible allocation must be attainable by cooperation within permissible small coalitions in some partition of the player set into such coalitions. Now we have a concrete picture of the structures to be connected, from the individual player level to the total economy level.

The solution concept used to capture the outcome of recontracting is the core. More precisely, in this paper we adopt thef-core for the continuum economy and the s-core for the finite economies. An allocation is in the f-core if it is feasible for the continuum economy and if it cannot be improved upon by any finite coalition. An allocation is in the a-core if it is feasible for the finite economy and if it cannot be significantly improved upon by any small coalition. Aggregate outcomes of recontracting are allocations in thef-core for the continuum economy and in the s-cores for the finite economies.

We consider convergence not only of the economic environments and of solutions but also of game structures-permissible coalitions. The continuum economy with finite coalitions and the f-core are approximated by finite economies with small permissible coalitions and by approximate cores.

The convergence is obtained in the context of a private goods exchange economy with widespread externalities. By widespread externalities, we mean that the utilities of players depend on the average actions of other players. In this paper we model these externalities by making the preference of a player depend on his own consumption of commodities and also on the distribution of the entire allocation of commodities. The treatment of coalitions as finite allows a coalition to form and to change the allocation for its members without significantly affecting the distribution of the initially given allocation.

The motivation described in Subsection 1.1 might appear just philosophi- cal in the sense that without externalities the equivalence of the core and the Walrasian allocations in a continuum model is obtained either by Aumann’s [2] model (with either or both interpretations of the individual player) or by our model (Hammond, Kaneko and Wooders [4]). However, the situation is quite different in the presence of widespread

FROM FINITE TO CONTINUUM ECONOMIES 139

externalities. Then the equivalence result holds for the f-core and the Walrasian allocations but fails for the standard model; more precisely, the definition of the core itself (with coalitions of positive me~s~~c) becomes problematic. See the discussion in Hammond, Kaneko, and Wooders [ Subsection 2.31. The substantive differences between our approach and the standard approach become especially apparent in consideration of the convergence from finite economies to the continuum.

The main theorem of this paper is the convergence of finite economies ts the continuum economy and of approximate cores to the f-core. In addi- tion, as a corollary to our convergence result and the equivalence result of Hammond, Kaneko, and Wooders [4], we have convergence of approximate cores of the finite economies to the alrasian allocations of the continuum economy.

The reader will observe that our converging sequences are more restric- tively constructed than those in the extant literature, for example Kansas [S]. The reason for the restrictiveness is that we require convergence of economic and game-theoretic structures, while in Kannai (and others) only the convergence of the economic environments is considered. The convergence of game-theoretical structures necessitates the restrictiveness of the converging sequences (see Remark 4.1).

Finally we point out that some modification of the definition of “can improve upon” becomes necessary in finite economies in the presence of widespread externalities. This modification is unnecessary in the cQ~ti~~~ economy, even with widespread externalities, since the formation of a new coalition affects only a negligible set of players. In finite economies, even though coalitions are small, the formation of a new coalition will a other players both directly and indirectly. a coalition forms, some

in some coalitions in the initial par will find that their coali- ave lost some members; these players are kdiresfl~ affected by the

coalition formation The other players are i&irecrky ajjficlecb, via their utilities, by the change in the distribution of the a~~ocat~o~. We define “can improve upon” under the assumption that those players direc may rarily form small coalitions among thcmse~v~s Wash mption that those players only in their behavior. In the convergence process t percentage of pdayers in a coalition and those directly affected by the formation of that coalition is required to tend to zero; smallness of a&ions is essential for convergence to the f-core when we have widespre

The next two sections of this pap ework. The foust sectiora contains the formulation of convergence a.nd our results. Finafly, all proofs are in the fifth section.


2. COMMODITIES AND PREFERENCES

We begin by describing a commodity space permitting indivisible commodities. Then preferences, defined over commodities and distributions of allocations of commodities, are described.

We denote the commodity space by 52 and assume $2 = Z: x R: where Z, is the set of nonnegative integers, I is a finite index set for indivisible commodities, and D is a finite index set for divisible commodities. Either 1 or D may be empty. We endow 52 with the metric jx - y1 = maxlx, - ycl.

The preference relation of a player depends on his own consumption of commodities and also on the distribution of the allocation of commodities. Distributions of allocations are probability measures on commodity space Q. Without any loss of generality, we actually consider the space of probability measures on D * := &2 u {xm >, the Alexandroff compactification of Q. Note that, since 52 is a separable and locally compact metric space, the space sZ* is also metrizable (see Hildenbrand [S, p. 151).

Let B(O*) be the a-algebra of all Bore1 subsets of Q* and let M be the set of all probability measures on B(Q*). Note that the space M is a compact metrizable space with the topology of weak convergence (cf. Hildenbrand [S, p. 49, (30)]). Let pM be a metric for the space M.

The space P of preferences is defined to be the set of all open, irreflexive, and transitive binary relations on Q x M. That is, P = ( > : >- is an open, irreflexive, and transitive subset of (Q x M)2). The expression ([xl, VI], [x2, v’]) E > means that the commodities represented by x1 and the distribution represented by v1 are preferred to those represented by x2 and v2. As is standard, we will simply write [x1, v’] > [x2, v”].

Since Q is a locally compact separable metric space and M is a compact metric space, the product space 52 x M is also a locally compact separable metric space with the metric

d( [xl, VI], [x2, v’]) = max(Ix’ -x2/, pM(vl, v”)). Therefore the space C of all closed subsets of 52 x M endowed with the topology of closed convergence is a compact metrizable space (cf. Hilden- brand [ 5, p. 19, Theorem 21).

Since > is an open subset of (Q x M)2, the complement >” of > is a closed set. It can be proved in the same way as in Hildenbrand [S, p. 96, Theorem 11 that P’ = { >‘: > E P} is a closed subset of C, which implies that PC is a compact metrizable space.2 Let pP be a metric for PC. Because of the natural bijection $( > ) = >-’ between P and PC, we can give the topology and metric of P’ to P. This means that a sequence of preferences (>“} converges to >” in this topology iff pP(& >“), +( >“)) + 0 as v -+ CO.

‘Since we fix the commodity space B we do not need convexity of l2 as in Hildenbrand C5J.

FROM FINITE TO CONTINUUM ECONOMES 141

3. ECONOMIES AND CORES

In this section we describe, in a parallel manner, a finite economy small coalitions and the s-core, and then a continuum economy with coalitions and the f-core.

3.1. Finite Economies with Small Coalitions and the E-Core Let N be the player set, a finite set; fi, an attribute functioq3 a map

from N to Q x P; and B, a bound on the size of ~e~~~s~~b~e ~~a~it~on~~ positive integer. We denote j?(a) by (&(a), >t), or (o(a), 2,) when t meaning is clear.

A finite economy is a triple (N, /3, B), simply a sp~~i~~atio~ of the pla set N, a description given by /I of the endowment and preference relat of each player, and a bound B on permissible coalition sizes (only co tions containing no more than B members will be permitted to form).

We allow trades only within permissible coalitions containing, say, fewer than B players and define feasible allocations arising from cooperation consistently. precisely, we define feasible allocations for the total player

t by all trades only in coalitions in partitions of the player set into bounded coalitions. We call a partition of N with the property that each

element in the partition contains no more than members, a of N.

For comparison with the analogous concept for a co tinuum eco~~rn~, we now provide another description of a B-partition p of N:

for any positive integer k < B, let N$ = UStp, iS, zk S. Then Ni has a partition (N$t}~= r with the properties that there are

q5zk from N& to N”kII . . . . N&, respectively, &&(a)) EP for all QEN;~.

artition of N, the set of all members of k-member coahtions, Npk, into k subsets, N&, . . . . N&, and the members of each subset N$,

are labelled as the tth members of these coalitions. Every k-member co tion has one and only one member in N& for each t. The bijection associates the tth member of a k-member coalition with t of that coalition. The partition {N&j:= 1 is called the partition ~~~~c~ate~ with p and the mappings (&), the bijection associated with p.

Let p be a B-partition. Denote by X(p) the set of all feasible a~l~~~t~Q~~ relative to p, i.e.,

X(p)= i

xeSZN: C x,< C w(a) for ail S~pj ots UES 1

3 This is sometimes called “an economy”


Then the set offeasible allocations X is the union of the feasible allocations X(p) relative to p over all B-partitions p, i.e., X= U, X(p).

It will be convenient to define allocations which are feasible for a subset of players S. We say (x,),,~ is a S-allocation iff X, EQ for all aE S and c aeS x, ,< CaeS co(a). (In the continuum case, S-allocations are defined only for finite subsets of players.)

For each allocation x E LIP, we define the distribution D[x] induced from x by

D[x,(&laE~;aET}i for all Bore1 subsets T of Q*.

Then the expression [x,, D[x]] >* [y,, Dry]] means that a player a prefers the consumption x, in the allocation x to consumption y, in the allocation y.

Because new behavior of a coalition affects the feasibility of the allocations of some players, to define the s-core we must specify the admissible actions of the remaining players when a new coalition forms. For those players who are only indirectly affected (by the change in the distribution) since the feasibility of their allocations is maintained, we can make the Nash assumption that these players make no changes in their actions. In contrast, since the allocations of players who are directly affected by the formation of a new coalition (find that the coalitions they were in have been broken up) may be infeasible, we cannot make the Nash assumption. Instead, we allow all possible feasible actions (reactions) within B-bounded coalitions of directly affected players.

Let p be a B-partition, and let S be a coalition with IS] <B. We call jj a (p, S)-partition iff

p is a B-partition and SEp;

and

REP and RnS=@ imply REP.

The partition ~7 consists of the coalition S, the coalitions R in p that do not intersect S, and arbitrary B-bounded coalitions of directly affected players. Figure 1 provides a simple illustration of two (p, S) partitions. In each case, the indirectly affected players are members of R and the directly affected players are the remaining members of R u S.

Let x be in X(p) for a B-bounded partition p and let ( yJaEs be an S- allocation for a coalition S. We say that an allocation X is compatible with x, P, and (y&s iff X is in X(p) for some (p, S)-partition ~5; for all Rtzpnp, X,=x, for all aER; and X,= y, for all aES.


The allocation X agrees with x on the coalitions that de not intersect the coalition S, agrees with (y,), E s on S, and is feasible for some partition of the remaining players. (For the example j of Fig. 1, @‘np has the unique element R. Then I agrees with x on R, agrees with (y,), E s on S, an arbitrary on the remaining players.)

Let x be a feasible allocation, and let E be a positive number. We say that a coalition S (ISI < B) can E-improve upon x iff there is an S-allocation (YaLs such that for any B-partition p under which x is feasible and any allocation X compatible with x, p, and (ya)oes,

c%, DC-d] >a u[IXm D[Xl; El for all .!2 E s,

where U(x,, D[x); E) = i [z, v] EL! x M: L?( [z, v], Lx,, D[x]]) < E) is the s-ball around [x,, D[x]] and “[X,, D[X]] >, U(xar D[x]; sr means t fx,, D[X]] >a [z, v] for all [z, v] E U(x,, D[x]; 8). Figure 2 and following discussion illustrate the structure behind ‘“e-improve upon”.

In Fig. 2, as in the definition of “r-improve upon,” the allocation x is given. Then, there may be many partitions, say p, p’, JI”~ under which x is feasible. Each one of these partitions determines a set of possible ( partitions, say p, p’, p” for p, and each of these (p, S)-partitions determines

1% / A

X’

yq - ,, X p .

X

‘\

-3, P

P’ .

P”

FIGURE 2.

642149,‘.LO


a possible set of compatible allocations, say 2, X’, 2” for p. The above definition of “s-improve upon” ensures the improvement in the utility of each player in the coalition S for any combination of possible partitions p, (p, S)-partitions p, and compatible allocations X. Thus, our definition of “s-improve upon” is a.conservative one in that the improvement must be for all admissible alternative cases4 (though we make the Nash assumption that the indirectly affected players keep the same allocations of commodities).

The E-core of the finite market (ZV, /3, B) is the set of all feasible allocations that cannot be s-improved upon by any coalition S with JSI QB.

When there are no widespread externalities and E = 0, our definition of the e-core reduces to the usual notion of the core (except for our feasibility condition). When E >O our definition suggests that a coalition will not attempt to improve upon an allocation unless it can be “significantly” (measured in commodities) better off, no matter what admissible actions the directly affected players undertake.

Observe that the s-core depends on the choice of metric. However, we will consider an approximate core property (defined in Section 4) instead of the e-core itself and this property is a topological, rather than a metric, notion. Therefore the dependence of the s-core on the metric creates no substantive problems in our approach.

3.2. Continuum Economies with Finite Coalitions and the f-Core

Let (A, d, p) be a measure space, where A is a Bore1 subset of a com- plete separable metric space; -c$, the g-algebra of all Bore1 subsets of A; and p, a nonatomic measure with 0 -C p(A) < + co. Each element in A is called a player and A is the player set. The measure p represents the distribution of players. The a-algebra is necessary for measurability arguments but does not play any important game-theoretic role.

An attribute function y is a function from A to 52 x P which is measurable in the sense that y-‘(T) E d for any Bore1 subset T of D x P. We write y(a) = (&‘(a), >;) or simply (o(a), >-,) when the meaning is clear.

Let F be the set of all finite subsets of A. Each element S in F is called a finite coalition or simply a coalition. Let FB be the subset of F consisting of finite coalitions with no more than B members. The family F, is the set of permissible coalitions. When B is infinite (N,), FB is simply F.

4 We think of a feasible allocation as actually a pair, x and p, with XEX(I)), but we focus only on allocations. Therefore, for the idea of the model, it sufices to consider some given partition p where XE X(p). For our result, either definition may be used; their differences become irrelevant in large economies. For economy of mathematical definitions, we choose the current presention.


A continuum economy is a triple (A, y, B), comprising a specification of the player set A; a description, given by y, of the endowment and preference relation of each player; and a bound B on permissible coalition sizes. If B= + co, then the continuum economy is denoted by (A, y, Fj.

Recall that only coalitions with no more than members can form, an feasible allocations must be achieved by trades in coalitions in some partition of the set of players into B-bounded coalitions. These partitions are required to be “consistent” with the two notions of measurement described by the cardinal numbers for finite coalitions and by the measure p for broad economic totals.

A partition q of A in FB is measurement-consistent iff for any positive integer k d B

and

AZ = u S is a measurable subset of A; sfq

(SJ=k

each A; has a partition {AZ,) f= r, where each AZ, is measurable, with the following property: there are measure-preserving isomorphism? I/I&, I+/&, ,.., +zk from A;, to A&, . . . . Afk, respectively, such that ($&(a), . . . . $$,(a)> ~q for each aEAEl.

Let II, denote the set of measurement-consistent partitions. When infinite, we denote Ii, simply by IL An element in B will be called a B-partition.

Almost as in the preceeding subsection, for each positive integer k we define AZ as the set of members of k-member coalitions. For %he part~t~~~ 4 to be measurement-consistent, we must be able to partition A;i into subsets {A&):, 1 so that the tth subset AZ, can be viewed as the set of 4t members of k-member coalitions. The isomorphisms associate the first members of k-member coalitions with the tth members for t = 1; . . . . k. measurement-consistency requirement inherent in the measure-preser property of the isomorphisms ensures that the set of first members has same measure as the set of tth members for each t. ithout loss of generality we assume throughout the paper that $& is the identity map,

Paralleling the definition of feasible allocations for a finite economy, we define the set of feasible allocations F(q) for each q E IT, by

F(q) = f~ L(A, Sz): c f(a) < c o(a) for all 5’ E q 0E.S US.7

5 Recall that a function I,/I from a set G in lc4 to a set C in is a measure-presercirzg isomorphism from G to C iff (i) $ is a measure-theoretic isomorphism, i.e., $ is 1 to 1, onto, and measurable in both directions, and (ii) ~(7’) = ~($(a)) for all Tc A with TE .d.


where L(A, 0) is the set of measurable functions from A to Sz. An allocation in F(q) will be called a q-feasible allocation. Let

F= u I;(q) SPflB

and let F* be the set of feasible allocations where

F* = (f6L(A, Q): for some sequence if”} in F, (f”} converges in measure toy}.

Recall that “(f”) converges in measure to F” means that for any E> 0, there is a F such that for all v 3 V, we have ,u({uE,~: If”(a)-f(a)1 >4)-=.

In a finite economy the set of feasible allocations X= U, X(p) is already closed, but this is not necessarily true for the set of (exactly feasible allocations F for a continuum economy, as illustrated via an example in Ham- mond, Kaneko, and Wooders [4]. The set F* of feasible allocations is an idealization of the exactly feasible set. A member of F* is to be interpreted as approximately feasible for arbitrarily close approximation. It is approximately feasible in the sense that for some partition q” of A into permissible (finite) coalitions, there is an f” E F(q”) such that fis “close” to fy for most players.

Finally we note the rather surprising result that although our game structure is quite difference from-that of Aumann [2], our set of feasible allocations F* of the continuum economy (A, y, F) coincides with the Aumann-feasible allocations, i.e.,

F*=jfeL(A,D):jf+j.

(Kaneko and Wooders [7, Lemma 3.1, p. 130]).6 For each function fin L(A, a), we define the distribution D[f] induced

from f by

for all Bore1 subsets T of 0’. Let f be a function in L(A, 0). We say that a coalition S with ISJ Q B can

improue upon f iff for some S-allocation (x,),, s, we have [x,, D[f]] >a

6 This result is essential for the equivalence of the f-core and the Walrasian allocations but is not used in the main theorem of this paper. It is used indirectly in the proof of the corollary to the main theorem.

FROM FINITETO CONTINUUMECONOEVIIES 147

Loa), w-33 f or all aE S. Now the f-core of the economy (A, y, defined to be the set

C = (f~ F*: for some full subset 2 c A, no coalitisn S in A with (S( d B can improve upon j).7

The definition of “can improve upon” is much simpler for the co~t~~~~ than “can &-improve upon” for large finite economies. This is because a coalition S, since it is finite, does not change the distribution of the allocation when it changes its own part of the allocation. Therefore, in our definition of “can improve upon” we can take the d~st~bu~~o~ r>[f] as unchanged by the change from (f(a)faes to (x~)~~~. As in the blitz economy case there will be some players directly affected by the formation of the coalition S. However, these directly affected players are negligible in the continuum case. These aspects enable us to use the standard (game- theoretic) notion of the core in the context of a ~o~t~n~urn of players and finite coalitions, even with widespread externalities.

4. CONVERGENCE OF STRUCTURE AND SOLUTION FROM FINITE TO CONTINLKJM ECONOMIES

In this section we introduce the concepts required for the statement of our results, state our main theorem and a variation of the theorem for t case of no widespread externalities, and apply our main theorem to t Walrasian equilibrium.

4.1. Approximation Sequences and the Approximate Core Property

To define the concept of convergence from finite economies to a continuum, we first embed finite economies within “equivalent” cant economies, called nonatomic representations. With this e attributes and allocations for the finite economies are represe same spaces as those of the continuum economy. The nonatom tation of a finite economy has a continuum of players and h percentage of players with each attribute as the finite econo vergence is defined in terms of the convergence of nonatomic represe~ta~ tions of finite economies.

e now fix a Iimit continuum economy (A’, y”, F) for the rerna~~d~~ of the paper.

’ We could adopt a definition of the f-core requiring that no finite coalition could improve upon an allocation in the f-core. However, this complicates the statement of our results withoil? adding any substance to the meaning.


Let N be a finite set and let /I be an attribute function on N, i.e., fi: N+ 52 x P. A continuum economy (A, y, B) (where A c A0 and p(A) >O) is called a nonatomic representation of the finite economy (N, 8, B) iff there is an onto mapping 5: A + N such that

~((-‘(a))=% for all UEN; (4.1)

Y(a’) = P(a) for all a’ E t-‘(a) and a EN. (4.2)

We call c the representation mapping. Let {(N”, /3”, By)} b e a sequence of finite economies. We call a sequence

((A”, y”, By)} of continuum economies a nonatomic representation of {W”, P”, By)) if (A”, y”, B”) is a nonatomic representation for each v. This definition extends each of the finite economies (N”, B”, B”) to a continuum economy. Each player in N” is replaced by a continuum of players with the same attribute so that the proportion of players having a given attribute remains unchanged. (This is a similar concept to Kannai’s [S] “continuous representation” but we allow the player set to be a proper subset of A0 instead of A0 itself. If a function on A0 is unbounded, by using a subset of A0 as the domain we can approximate the function uniformly by simple functions where, for each simple function, each step has the same size base).

Again let (N, /I, B) be a finite economy and let (A, y, B) be a nonatomic representation with the representation mapping <. Let CI be a function from N to some metric space Y. We call & A + Y the nonatomic extension of CI iff g(a’)=a(a) when a’~(-l(a). Note that the nonatomic extension is uniquely determined by CI and a nonatomic representation (A, y, B), and that y is the nonatomic extension of /? itself. The nonatomic extension Cr is a simple function and a substitute for cx in the nonatomic representation (A, Y, B) of (N P, 8.

Let {(N”, /I”, By)} b e a sequence of finite economies, let (a”} be a sequence of functions from N” to some metric space Y with metric pr, and let {(A”,?“, B”)) b e a nonatomic representation of {(N”, /3”, B”) >. We say that {cx”} expandingly converges to a function CI’: A0 -+ Y with respect to {(A”, Y”, W) iff

p ( ij fi A’)=M’); k=l v=k

(4.3)

the sequence {Cc”>, where Cc” is the nonatomic extension of CI” to (A”, y”, B”), converges uniformly to CC’ on the domains, i.e., sup, E AV ,~~(&“(a), u”(a)) -+ 0 (v --) co). (4.4)

FROM FINITE TO CONTINUUM ECQNOMIES 149

Each function a’ on N” is replaced by a function 2 cm A”. Tken ti” can compared with the limit function CI’, though the domain A” of 2 is s different from A’. Therefore we require convergence both of t (4.3) and of the functions (4.4).

A sequence of finite economies ((IV”, ,lY, By)) is called an a~prQxi~uf~~~ sequence for (A*, y”, I;) iff there is a nonatomic representation ((A”, YP, “) > with the representation mappings (5” 3 suck that

B”W)( = Y”(A”)) c Y’(A’) for all v; (4.3

B”-im asv--+co; 44.6)

{p”) expandingly converges to y0 wit {(A’, ~j’, By)} and with a higher order tka

” x sup,E‘p Id”(a) - 4/(a)l * 0 (v -+ 0); (4.7)

and

The sequence ((A”, y”, B”) 1 will be called an associated nmafomic represen- fafion of ((iv, /I”, By)).

Condition (4.5) is that the attributes of the players in the finite economies are subsets of the attributes of the players of the ~o~ti~~urn economy; the attributes of the players of tke finite economies are converging to those of the continuum economy “from tke inside”. The next condition (4.6) simply allows coalitions to become arbitrarily lar

wever, since commodities are transferable, commodities may umu~ated by some small subset of players and ~on~o~verge~ce of su

accumulated commodities may result. We rule this cut by assuming, in ( the rate of convergence of the endowments 0” is faster than the rare

ivergence of B”. If sizes of permissible coalitions are growing slowly, condition would be satisfied. In the case of widespread externalities we

will, in fact, require stronger conditions on the rate of divergence next condition, (4.8), ensures that each player kas “many” subs call tkis the thickness assumption for the approximation sequence

‘)>. This assumption does not, however, necessarily mean that r has a positive measure of substitutes in the limit economy. f fl”, By)} be a sequence of finite economies and let (x”>

of feasible allocations; i.e., x” is a feasible a~~oca~io~ for “). The sequence (x”} is said to have the a~,~~ox~~~fe core

properfy rff for any E > 0,

tkere is a positive integer v. suck that for all Y 2 wo, x.’ is in the e-core of (N’, B”, B”).

1.50 KANEKO AND WOODERS

The idea is that the sequence itself has a core or the “limit” of the sequence has a core, even though we do not have a limit and even though every economy in the sequence may have an empty core. As indicated earlier, the approximate core property is independent of the particular choice of metric on Q x M.

4.2. The Convergence Theorem

We now give our theorem, designed to show convergence in structures and convergence in solutions.

The following three conditions are used in the main theorem: Let f be a function in L(AO, Q), let ((N”, p”, B”)} b e an approximation sequence for (A’, y”, F), and let (xv) be a sequence of allocations, where xv is feasible for (N”, p”, B’);

W’)* +. WI

as v-00; (4.10)

the sequence {x”} expandingly converges to f with respect to some associated nonatomic representation {(A”, Y’, B”)); (4.11)

the sequence (x”} has the approximate core property. (4.12)

MAIN THEOREM (Upper-Lower Hemicontinuity). The following three statements on a function f in L(A”, Q) are equivalent:

(1) f is in the f -core of the continuum economy (A’, y”, F); (2) There is an approximation sequence ((r\Ty, p”, By)} and a sequence

(x”} of allocations (where xv is feasible for (N”, /3”, B”)) satisfying conditions (4.10), (4.11), and (4.12);

(3) For an approximation sequence ((N”, /I”, B”)) and sequence {x”) of allocations (where xv is feasible for (N”, p”, B”)) with conditions (4.10) and (4.11), condition (4.12) holds.

The first statement of the theorem is clear. The second statement is that there exists an approximation sequence satisfying (4.10), (4.11), and (4.12). In the third statement, instead of the existence of an approximation sequence, it is claimed that any approximation sequence with (4.10) and (4.11) has the approximate core property. (Since we can construct an approximation sequence {(N”, /-I”, B”) ) satisfying (4.10) and (4.11), the third statement is not vacuous). In the “folk language,” (1) * (2) is a form of “lower hemicontinuity” and (3) + (1) is a form of “upper hemicontinuity.“8 The second statement (2) states the existence of an approximation

*C. Kannai [S, p. 8031. For his framework, Kannai gave a counterexample to lower hemicontinuity. In our case, we avoid his counterexample by assuming p(w) c $‘(A’) for all V, and we consider the limit, from the “inside”, to the total.

FROM FINITE TO CONTINUUM ECONOMIES If1

sequence; this existence is quite subtle because of the special required of the sequence.

The theorem provides an answer to the questions raised in the introduction. The idea of perfect competition with recontracting in our contimmm model can be approximately realized in large finite economies wit permissible coalitions. The two special aspects of perfect competition with recontracting, which make the different treatment of small coalitions and total economies necessary, are approximately c y the restriction of permissible coalitions (4.10). Thus the theorem pr es a linitistic interpretation of the continuum model with finite

One of the special properties required of a ation sequence in statements (2) and (3) of the theorem is con (4.10) ensuring that relative to iV the sizes of coalitions become sm ith widespread externalities, for the effects of coalition formation to become reeligible we the percentage of directly affected players to become negligible, i.e., we need (B”)2/jN”j --, 0. To see this, let p” be a B”-partition and let S be a subset of players, S not in p”, with 15’1 d B’. If S forms a coalition, this breaks some coalitions in py. The number of players influenced directly by the formation of S is at most ISI x B”, i.e.,

this leads us to require condition (4.10). When preferences do not depend on the distributions, i.e., there are no

widespread externalities (this is the standard case), we do not need to impose (4.10) in the theorem. In other words, we can delete the references to (4.10) in statements (2) and (3) of the theorem. In fact, in the case of no widespread externalities, this feature that (4.10) can, but do to, be imposed is the reason behind the equivalence, shown in Kaneko, and Wooders [43, of the f-core and the umann core. This can be considered as the reason that the opposing Schmeidler [IO]-Vind [ll j requirements (Schmeidler requires “small” coalitions of positive measure less than E, whereas Vind can require “large” coalitions) botb lea Aumann-core-equilibrium equivalence. Nevertheless, when we consider the convergence to thef-core it is natural to restrict coalition sizes, at least to ensure F/IN”\ 3 0.

Remmk 4.1. In the third part of the Theorem, an a~~roximat~o~ sequence of finite economies depends upon the particularf-core allocation f under consideration. Since we do not assume convexity the f-core allocation f is not necessarily continuous with respect to attributes and does not necessarily satisfy even the equal-treatment property (the property that

152 KANEKOAND WOODERS

players with the same attributes receive the same allocation). Thus under our assumptions the f-core allocations have more “variety” than the attribute functions and we cannot define an approximation sequence based only on attribute functions.

4.3. An Application to Walrasian Equilibrium

The combination of our main theorem and the equivalence theorem in Hammond, Kaneko, and Wooders [4] gives us convergence of the core to the Walrasian equilibrium allocations of the continuum economy. We note that these allocations might not be Pareto-optimal.

In Hammond, Kaneko, and Wooders [4], under the following conditions the equivalence of the f-core and the Walrasian equilibrium allocations is obtained.

(A.0) There is at least one divisible commodity, i.e., the index set D of divisible commodities is nonempty;

(A.l) >a is strictly monotone on I!: ; (A.2) for all [(x,, x,), v] ESZ x M there is a y, E RD, such that

C(O,> YD), VI >a C@,, XLA VI; (A.3) [o(a), v] >a [(x,, 0,), v] for all .X,E 2: and for all probabil-

ity measures v.

THEOREM (Hammond, Kaneko, and Wooders [4]). Under assumptions (A.0) to (A.3) it holds that f is in the f-core of (A’, y”, F) if and only iff is a Walrasian allocation, i.e., for some price vector p E Ry ,

P .f (0) G P . w(a) a.e. in A’;

ae. in A’, Lx, Nfll Sa U(a), DCfll for all x E Q with p.x<p.m(a);

and

s s f< 0. ‘40 A0

COROLLARY. Assume (A.0) to (A.3) hold, The three statements of the theorem on a function f in L(A’, 12) are equivalent to:

(1’) f is a Walrasian allocation for the continuum economy (A’, y”, F).

There is a vast literature on convergence of cores to equilibrium allocations in the case of no widespread externalities (see Anderson Cl] for a recent survey). In many of these papers, the convergence of the core allocations to Walrasian allocations or Walrasian-like allocations is considered


in the context of large, finite economies-convergence to the continuum is not investigated. Nevertheless, this is not the main difference between our convergence results and existing works. The main difference is that we require not only convergence of cores but also convergence of the economic structures (moreover, with widespread externalities).

Although we considered convergence of E-core allocations to allocations as a corollary to our main theorem, our primary co the interpretation of the continuum model with finite coalitions as the himit

economies. Kannai’s [S) paper is related to this aspect of our e considered the convergence from core to core and also from

economies to economies, and obtained several fundamental results. owever, Kannai’s paper is devoted to the interpretation of s

model of a continuum with coalitions of positive measure, and co e of game theoretic structures is not required. this q~a~i~catio~ our approach and Kannai’s approach are counterparts to each other for the continuum models with finite coalitions and with coalitions of measure.

rge economies with no widespread externalities, an important result on the convergence of

Wa~ras~a~ allocations. This result gave an evaluation of a bound on coaii- tion sizes which still permitted a convergence result. His evaluation is shar- per than ours but his paper is different in that feasibility does not require trade only within small coalitions. An interesting open problem is w

as-Cole31 result becomes when the same coalition size bound is used in fining feasible allocations as in “can improve upon.” This is important in

i~ter~ret~t~o~ of the meaning of the result.

Wernark 4.2. In Hammond, Kaneko, and Wooders [La] we prove nonemptiness of the f-core under the assumptions (/LO), (A.l), (k.2), and (A.31 for the case where the widespread externalities depend on the integrals of allocations (rather than their d~strib~tions~. of no widespread externalities can also be covered. From this result and the above corollary we have the existence of the ap core for any approximation sequence for fin the f-core.

n this section, we prove only the equivalence between statements (I) and (2) of the main theorem. When we have existence of an a~~rox~rna~~o~ sequence satisfying (4.10) and (4.11), and a sequence (x”> of a~lo~at~o~~ with the required properties, then (3) j (2) is straightforward. The existence of such a sequence can be proved in tlae same way as the first


the proof of (1) =E- (2). The logical relation (1) + (3) is almost immediate and thus omitted.

5.1. Preliminaries

Before we proceed to the main body of the proofs, we prepare several auxiliary notions and some lemmas. First, we define a typed continuum economy with the rational number property. The rational number property means that the proportional aspects of the economy can be described by rational numbers. This implies that we can find a finite economy with the same proportional aspects.

A continuum economy (A, y, B) is said to be typed iff y(A) is a finite set. The typed continuum economy (A, y, B) is said to have the rational number property iff

d(aEA:y(a)=a)) AA)

is a positive rational number

for all CI E y(A). (5.1)

Let g: A --f Q be a simple function such that [ := (y, g): A -+ (Q x P) x Q has the rational number property (,u( (a E A: [(a) = J})/p(A) is a positive rational number for any il E [(A)). We denote (D x P) x L2 by A.

Let q be a B-partition of A with the associated partition {A&: t = 1, . . . . k and k = 1, . . . . B) and isomorphisms (I@,: t = 1, . . . . k and k = 1, . . . . B}. We say that q has the rational number property wiih respect to 5 iff for all k = 1, . . . . B and ;1= (I,, . . . . 1,) E Ak,

if (5 0 $fl(a), . . . . [ 0 $z(a)) = /z for some a E A&,

is a positive rational number.

Note that if some B-partition q of A has the rational number property, then the economy (A, y, B) also has the rational number property.

Let (N, /I, B) be a finite economy, and let (A, y, B) be a nonatomic representation of (N, p, B) with the representation mapping t. Let p be a B-bounded partition of N. We call a B-bounded partition q of A a nonatomic representation of p iff

5(S) EP and It(s)I = ISI for all SE q. (5.3)

Note that since 5 is an onto mapping, (5.3) implies c(q) = p. A nonatomic representation q is a partition of A and the structure of q mimics that of


p, The partition q, however, contains a continuum of coalitions with the same profile as S for each S EP since a continuum of players t-‘(a) corresponds to a for each a EN.

A B-partition of A with the rational number property with respect to (y, g) can be represented as a B-partition of N in some finite economy (N, 8, B). That is,

LEMMA 1. Let (A, j3, B) be a B-bounded typed continuum economy. Suppose that q is a B-bounded partition of A with the rational number property with respect to [ = (y, g). Then for any E>Q there are a finite economy (N, j?, B) (N c A) and a B-partition p of N such that

(i) (A, y, B) is a nonatomic representation of (N, /3, B) with a representation mapping 4: where t(a) = a for all a E N;

(ii) q is a nonatomic representation qf p;

(iii) (B)2/jNj -cc; and

(iv) the representation mapping 5 preserves the values of [: i.e., (y(d), g(d)) = (y(a), g(a))for all U’E t-‘(a) and a~ N.

(In the finite market, the proportional aspects of a B-bounded partition q are characterized by a finite number of rational numbers, Since rational numbers are ratios of integers, we can multiply those rationals characteriz- ing the partition by one integer and obtain integers describing the same proportional aspects of q. Then we construct a finite economy described by these integers. For completeness, we give a full proof.) e remark that since 5 is a representation mapping, the restriction of 2’ to N is /3, so the part of (iv) concerning y is redundant.

ProoJ Given the E-partition q of A, let (AZ,: t = 1, . . . . k and k= 1, . ..) B} and (I&: t= 1, . . . . k and k= 1, . . . . B) be the associated partition of A and isomorphisms.

Denote the set (A E /I? ([ 0 t&r(a), . . . . [ 0 $&(a)) = I for some a E A&) by Fk 5s (Akl, . ..) A”‘“>. Note that Fk may be empty.’ y the rational number property of q, it holds that

is a positive rational number for all Aks E Fk

and k = 1, . . . . B. 65.4)

91f P=@, then &=O.


Therefore there are positive integers iks (s = 1, . . . . Zk and k = 1, . . . . B)l” such that

for all s = 1, . . . . Ik and k = 1, . . . . B; and

(5.6)

Since Fk## for some k, the above ratios are well-defined. (This means that the rational numbers describing the proportional aspects of (A, y, B) are replaced by natural numbers so that the proportional aspects are preserved.)

Now we define a finite economy (N, /I, B). Denote Az,(lks)= (a E A& : (CO $&(a), . . . . [ 0 $;,(a)) = nks). Then we choose a subset iV& from Azl(nks) (s= 1, . . . . lk and k= 1, . . . . B) so that (N&I = iks. Let NPkts = $4,,(NP,,,) for t = 1, . . . . k, Ns, = U$= 1 N&, JV~ = U:= 1 Nit, and N = lJ,“= r IV;. Then we have, by (5.6),

which is condition (iii). Let /3 be the restriction of y to N. We have attached to each $&(Afl(Lks))

the finite set iV&, so that the proportions of these sets to A and to iV, respectively, are the same, i.e., ~(~~,(A~l(nk”)>)/llo= lNL,l/lN. Since Nils is a subset of A&(1”) ( = I&(A~~(~~~)), we can find an onto mapping t: A~l(lks) + N& such that

and 5(a) = a for all aEN&;

(5.7)

for all a E IV;,,.

Then we define 5: t,bf,(A&(Lks)) + N& by

t(a) = &L~t~KL’(a) for all QE +&(Af,(ik”)). (5.8)

Since I& is a measure-preserving isomorphism, it follows from (5.7) that for each ae N&, ~L(t-‘(a)) =~(~f,(e~‘(~&‘(a)))=~(5-‘(~BT’(a)))=

lo If 1, = 0, then there is no i,, and zz= 1 k x jkz = 0.

FROM FINITE TO CONTINUUM ECONOMIES I.57

~(A)/\N\. Thus we have shown that (A, y, B) is a ~o~atomi~ ~e~r~se~tatio~ of (N, ,8, B) with the representation mapping 5.

From the definition of the representation mapping 4, it is clear that r preserves the values of (y, g).

Since Nit = $E,(Npkr) for all t = 1, . . . . li and s = 1, . . . . lk, the restrictions of $1, > ...? $ik are bijections from Nil to Npkr, . . . . N&. Denote these restrictions by &CC, . . . . &,. Then define a partition p as follows:

p= ((&(a), . . . . #&(a)):aENP,, and k==l, ..~)

Then it suffices to prove that q is a nonatomic representation of p wit representative mapping & Pick an arbitrary coalition S from q /S\ =k. Then S can be represented as S= ($&(a), .“, $&(a)) for some Ada&. It follows from (5.8) that

Since we require an exact form in (5.3) for the definition of a representation of a B-partition of N in a finite economy (N9 ps atomic representation (A, y, B) of (N, /3, B) does not necessarily have a nonatomic representation of a B-partition. However, the ~o~~owi~g lemma holds.

LEMMA 2. Let (N, /?, B) be a finite economy and let x = (-?c,),~,~ be a p-feasible allocation in (N, p, B), where p is a ~-~a~~j~iQ~ @“IV. Let (2, 7, be a nonatomic representation of (N, /?, B) with the re~rese~t~t~on m~~p~~g <, and let g be the nonatomic extension of x to A. Then there is another nor%- atomic representation (A, y, B) of (N, B, B) such that

(i) A is a full subset of 2 and y is the restriction of*7 to A; (ii) (A, y, B) has a nonatomic representation q ofp; (iii) the restriction g of g to A is q-feasible.

(The reason (a, 7, B) is replaced by (A, y, B) is the following: A measurement-consistent partition is constructed by measure-preserving morphisms. However, the existence of a meas~re~preservi~g isomorp between two Bore1 sets with the same measure can be ensured only up to a null set. See Kaneko and Wooders [7, pi l29]. Therefore we have to omit null sets appropriately.)

Proo$ Since (A, y^, B) is a nonatomic representation of (N, 8, the representation mapping 5, for any S := {a,, a?, . ..) ak) EP, (k is a


positive integer), we have ~(2,~) = . . . = ~(6,,), where A,, = 5 -‘(a,) for t = 1, . ..) k. Therefore from Lemma A.1 of Kaneko and Wooders [7], there are measure-preserving isomorphisms $k”,, . . . . $& (mod 0) from A,, to &,, *-., 2, respectively, that is, there are full subsets ail, AZ,, . . . . ai, of &, such that $,“,, . . . . rj,“, are measure-preserving isomorphisms from Ai,, a;,, . . . . 22, to $~r(A~,), $&(2.$), . . . . &&(A!,), respectively. Let A,, = nF=, Ai,. Of course, ,n(A.,) = @(A,,). Here we can assume without loss of generality that a, E +&(A,,) for all t = 1, . . . . k.‘l Define

4(S)= u4!34~ a..> GM>: ~-L*) and 4 = u 4(S). SEP

Let A=lJTEg T and let y be the restriction of 7 to A. It is clear that A is a full subset of 6, and that (A, y, B) is a nonatomic representation of (N, p, B) with the representation mapping <. Then q is also a nonatomic representation of p, since it holds that for any ($~,(u), .-., $iJa)> E q(S) and Scp,

Finally, we prove that the restriction g of g to A is q-feasible. Since g is also a nonatomic extension of (x,),,~, and since 4 is the representation mapping from A to N, we have, for all TE q,

oIxTs(4= c x,6 c 09(u)= c o’(u). 1 aE‘3-f) aEi(T) UET

The next step is to characterize a feasible allocationfE F* in (A’, y”, F) in terms of a sequence of feasible allocations in typed continuum economies with the rational number property. After this step, using the previous lemmas, we will construct an approximation sequence ((N”, p”, II”)} for (A’, Y’, Fh

A sequence of typed continuum economies {(A”, y’, B”)) is called a IZOIZ- atomic approximation sequence for (A’, y”, F) if

(5.9)

Y’M”) = By” for all v; (5.10)

B” + 00 as v--+00; (5.11)

“Indeed, if a,$$$A,,), then we can replace an arbitrary ~E$$(A,,) by a,. The new function plays the same role as the function $z,.

FROM FINITE TO CONTINUUM EC~~~~~~~ 159

fy”] converges uniformly to y” on the domains and with a higher order than B” on Q, i.e., sup iw’“(a) - oyO(~)\ x B”-+Q as v--+00. at.‘%Y (5.12)

Remark 5.1. Of course these conditions correspond to (4.3 )9 (4.5), (4.4), and (4.7) in the definition of an associated nonatomic representation of a sequence of finite economies. The existence of an approximation sequence ((N’, B’, B”) 1 ’ pl tm ies the existence of a nonatomic approximation sequence {(A”, y”, B”)}.

LEMMA 3. ri function f E L(A’, G?) is feasible, i.e., SE F*, for the COB- tirmum economy (A’, y”, F) if and only if there are a subatomic approxima- tioia sequence {(A”, yV, B”)}, a sequence (q”] of B”-partitiolzs of A”, and a sequem-e {g”> of q”-feasible allocations such that

(i) each g” is a simple function on A” and each qy has the rat~~~a~ number property with respect to (y”, 8”);

(ii) the sequence (g”> converges uniformly to f on the domains.

Proof of Sz&kiency. Let (Ed> be a sequence of positive numbers with lim, E, = 0 and E, < I for all n. Since the sequence (w’“) converges uniformly with a higher order than B” by (5.12), there is a v,, for each 11 such that

loYvn(a) - c070(a)\ x Pj”n < E,

efine B;“Y A’ --f D by

for all a E A’“. (5.13)

for all a E A ‘n, gqa) = i

tc(af if CEI maxk?(a) - E,, if CED;

for all a&A*-Avn, g”“(a) = 0

(recall that I and D denote the index sets for the ~ndiv~s~b~e and divisi commodities respectively); and define a”” by

Since (g”] converges uniformly to J; it follows from (5.14) thad {g”> converges in measure to J: Therefore it sufkes to prove that each 2”” is q”*-feasible. For a coalition (a> E 4’” - q’“, it holds that g”*(a) = 0 Q wJ’“fa). Let S be an arbitrary coalition in 4”“. For an indivisible ~~rnrnodi~~ c E I3 we have

642,49:1-l:


since E, < 1 for all n, so o:‘(a) = ozp(a) for all a E A “,. Let c be a divisible commodity in D. If g:(a) = 0 for all a E S,

c g?(a)=O,< c O;“(u). UES ass

If g:(a) > 0 for some a ES, then

d aTs (o:“yu) - lo;‘“(a) - o~“(a)l) d c o;O(u). UES

Thus we have shown that CaeS g”“(u) < C,, s &“(a) for all SE qVn, which means that 2”” is q”n-feasible. 1

Proof of Necessity. Since f is a feasible outcome in the continuum economy (A’, y”, F), there are sequences {fY} and (4”) such that

4” ~fl, fy is @‘-feasible for all v, and (f”} converges in measure to f: (5.15)

We can assume without loss of generality that there is a sequence (B”) of integers such that for all v,

ISI 6 B” for all S~q”.l’ (5.16)

Here we can assume that B” + co as v -+ co. Let (A;,: t = 1, . . . . k and k = 1, . . . . B”} and {$it: t = 1, . . . . k and k = 1, . . . . B”} be the partition of A0 and isomorphisms associated with the partition 4”.

“If there is not such a sequence {B’}, then we can find a sequence {B’} such that

p ( >

u s >p(AO)-2-v for all v. SE@

ISI SB”

Then~‘={S~~“:~S~~B’~u{{a}:a~Sf or some SE y’ with ISI Y B’} and f”, defined by

j$z) = 1

x1-, if aESfor some SET’ with ISI <B’ otherwise,

satisfy conditions (5.14), (5.15) and (5.16).

FROM FINITE TO CONTINUWvl ECONOMIES 141

Let v be an arbitrary positive integer. In the following, we will define a B”-bounded and typed continuum economy (A”, y”, “) and a feasible outcome g” of the economy (A”, y”, B”). Put E = l/2”.

We can assume by (5.15) that

p SEA’: /J’“(a)-f(a)\ >E 4 I)

< $1,

Since A = (L? x P) x Sz is o-compact, we can choose a compact subset A” of A so that

where 5” = (y”, f”). Put ;a”” = (a E [“-‘(A’): if”(a) -f(a)/ < qJ4). T follows from (5.17) and (5.18) that

Since A’ is a compact metric space, there is a finite partition (A;, . . . . /BY,) of A” such that for s = 1, . . . . I,,

1 the radius of AI is less than ___

B” x 2”’ (5.20)

We are now going to define sequences (A”), (y”], (q’), and {g”). Let L, := { 1, 2, . . . . I,). For any k (l,(k<B”) and

r----k--r i = (iI, . . . . ik) E (LJk := L” x . . . x L,,,

choose a subset C;,(i) of A”;Ir :=A;;, n2” so that

C”,,(i)= (a.PI;,: ([“~$~~(a), . . . . i’~$&(a))~A~,x ‘I. xA;~]; (5.21)

~(C~I(i)) is a positive rational number or c;t(i) = & (5.22)

~(C~,(~))B~({U~A”~~: (i”~$;~(a), . ..) i”~ll/;~(a))~A:; x ... xA;~])

1 & --. 4 k.B”.(l,)k’ (5.23 )

l3 Otherwise, we replace the sequence {f’) by an appropriate subsequence.


Take one player a,&(i) arbitrarily from each set C:,(i) as the representative of C;,(i). Define the vth player set A”, the vth function y”: A” --t Q x P, an allocation g”: A” -+ Q, and a measurement-consistent partition q” of A” by

A”= iyi u ii $:,(c;,(i)); k=l ieLCr=l

Y”(a) = YOofwGl(~))) for all a E @;,(C;ll(i)),

ie Lf , t = 1, . . . . k, and k = 1, . . . . B”;

s”(a) =f”($;,(&(Q)) for all a E tj;,(C&(i)),

iELt, t=l,..., k, and k=l,..., B”;

qy = ki, jgk i {tiL(a), . . . . $;,(a)>: a E G,(i)). Y

(5.24)

(5.25)

(5.26)

(5.27)

The first claim follows from these definitions and (5.22).

Claim 1. The partition q” is measurement-consistent and has the rational number property with respect to (y”, g”).

Claim 2. p(U& fl,“=,A”)=p(AO).

Proof. It follows from (5.19), (5.23), and E = l/2” that p(A”) > AL - l/2” for all v. Since

for all k, we have

Cluim 3. The sequence {y”] converges uniformly to y” with a higher order than B” on 0.

Proof: It follows from (5.25) and (5.20) that {y”> converges uniformly to y”. It also follows from those conditions that

B” x sup [wY’(a) - oYo(a)l < & -+ 0 as V+GO. 1 UEAY

Claim 4. The sequence {g”} converges uniformly to J:

FROMFINITETO CONTINUUMECONOMIES 663

ProoJ It follows from the definition of J”, (5.X), and (5.20) that for ah a f ~;,(cv,,m

This means that {g” > converges uniformly to J’I

Finally we can prove

Cfaim 5. Each g” is q”-feasible.

ProoJ: Consider any coalition S in q”. It follows from (5.27) that S ca be represented as S= ($;,(a), +;Ja), . . . . $;,(cJ)) and a E C;,(i). From (5.25) and (5.24) we have

~‘“ofc,W) = ~Yow~~(Q~,GH) for t = 1, . . . . k;

g”(li/;,(Q)) =f”($;,(Q;,(Q)> for f = 1, ...I k. (5.2%)

Since f” is q’-feasible by (5.15),

t follows from (5.28) and (5.29) that

t=1 r=l t=i

5.2. Proof of (l)==+(2)

In this subsection, we prove the equivalence of (I ) and (2) of the main theorem.

(1) s (2). We show that if f is an allocation in the f-core of the cons” tinuum economy (A*, y*, F), then we can find an approximation sequence of finite economies ((N”, fi’, B”)} with conditions (4.10), (4.11), and (4.12).

From Lemma 3, we have a nonatomic approximation sequence {(A”, y”, By)} and sequences (q”} and {g”) with conditions (i) and (ii) of Lemma 3. Let v be an arbitrary integer. Then condition (if of Lemma 3 implies that we can apply Lemma 1 to each continuum economy


(A”, y”, B”), so we can find a finite economy (N”, /J”, B”) (N” c A”) and a B”-partition p” of W such that

(A”, y”, B”) is a nonatomic representation of (N”, p”, B”) with the representation mapping 5” where (“(a) = a for all aENV; (5.30)

q” is a nonatomic representation of p”; (5.31)

(B”)* 1 (N”Iy (5.32)

and

r” preserves the values of (y”, g”) (i.e., (~“(a’), g”(a’)) = (y”(a), g’(a)) for all a’E t”-*(a) and a EN”). (5.33)

Since ((A”, y”, B”)} is a nonatomic approximation sequence of (A’, y”, F) from Lemma 3, the sequence ((N”, p”, By)} is a (finite economy) approximation sequence for (A’, y”, F). It also follows from (5.32) that (B”)‘/)N”I -+ 0 as v -+ co.

Let x” be the restriction of g” to N” for each v. Since g” is q”-feasible, we have

c g”(a)6 1 oY’(4 for all S E q". UES UES

(5.34)

For any Tep”, we can find an SE q” such that t”(S) = T. Since 4;” preserves values of y”, by (5.13), and X” is the restriction of g” to N” it follows from (5.34) that

C x”(a) = C g”(a) < C coy’(a) = C 09’(a). RET acs aes UGT

This means that xv is p”-feasible. From Lemma 3 (ii), (g”} converges uniformly to f, p( lJ,“= 1 n ,“= k A “) =

p(A’) by (5.9), and each g” is the nonatomic extension of x” by (5.33). Hence (x”} expandingly converges to S and satisfies (4.11).

Finally we show (4.12) that the sequence (xv) has the approximate core property. On the contrary, suppose that there is a positive E such that for any vo, we can find a v 2 v. so that some coalition S” with Is”/ < B” can s-improve upon x-‘, i.e., there is an S”-allocation (Y;}~,~ such that for any partition p” under which xv is feasible and for any allocation 2” compatible with x”, P’, and {YZ)~~~,

cv:, navll s,B’ ucx:, ~cx’l1; 8) for all a E S”.


Since the sequence (g”) of nonatomic extensions of (x”} conver uniformly to f and p( U p= I n ,“= k A “) = p(A’), it follows from Hi~de~b~an 15, p. 51, (39)] that the sequence (D[g”]) = {D[x”]) converges weakly to D[f], Since 2:” is different from x” for at most (B”)’ players (see the discussion after the main theorem), and since lim, (B”)2/lN”\ = 0, the sequence ($‘> of nonatomic extensions of {X”) converges in measure toJ: The from Hildenbrand [IS, p. 51, (39)]: (D[X”]) converges weakly to Thus using the fact that yy converges uniformly to yQ on the domai y”(a) = P”(t’(a)) for all atz A”, we have, for large v an all T” c A” wirh (“(TV) = S” and ) T”J = JS”j,

where (z;)~~ *,, is the Y-allocation in the continuum re~rese~tatiQ~ (A”, y”, B”) such that z;= y$,,, for all a E T”. Since y” converges uniformly to y0 with a higher order on Q than B”, we can fin w,, T’ for a large v in the continuum economy (A’, y ficiently close to (z&cTY for [Zi, D[ff] >f [f(a), DCf]] for all a E TV. Thus the coalition T” can improve upon the allocation f in the economy (A”, y”, F) and the measure of the union of such im roving coalitions is positive, i.e., p({‘-‘(Sy)) = p(A”) x ~sY~/~N”~ >O. Therefore f cannot be in the f-core of the continuum economy (A’, y”, F), a contradiction T (4.12) holds.

(2) 3 (1). We now show that a function f in L(A”, f-core of the continuum economy if we can find an ap~roxirnat~Q~ sequence of finite economies ((W, j?“, B”)} with conditions (4.10), (4.11): and (4.12).

From the supposition and Remark 5.1, there are a nonatomic approximation sequence ((a”, y”, By)) and the sequence (2”) o atomic extensions of xv’s to Pi’%. From Lemma 2, each (A“, y”, full continuum subeconomy (A”, y”, B”) with a nonatomic represen of p” for each v. Also, for each economy (A”, y”, B”), the restriction A” is q’-feasible. Since ~(U~z=I n~==,A”)=~(IJ~=, nFzk A”)=p allocation f is feasible in the continuum economy (A’, y*, F) from Lemma 3.

Finally we have to prove that there is a full subset A of A* such that coalition in A can improve upon f in the continuum economy (AO, yO, F), On the contrary, suppose that there is no full subset A of A” with the property that no coalition in A can improve upon f

Consider the set 2 = U,“=, n;=, A”. Since p(A”) =p(A) by (4.3), 2 is a hi subset of A*. Therefore we can find a finite coalition ,Y %n


A” = U,“= 1 n ,“= k A ” such that S can improve upon J; i.e., for some S-allocation (zaLs in the continuum economy (A’, y”, P),

for all a E S. (5.35)

We can assume without loss of generality that S is included in A” for all V.

From (5.35) and the openness of >f, we have, for some E > 0,

U(CL, xfll; 342 WCf(a), m-11; 3E) for all QE S. (5.36)

Since yy converges uniformly to y” on the domains, there is a v1 such that for all v 3 v,,

U(Cz,, m-11; .w-$ Vu-(a), m-11,2&) for all UES. (5.37)

Again since y” converges uniformly to y” on the domains, there is a v2 such that for all v 3 v2, we can find an S-allocation (Y:),~ s with 1~1 -z,/ < E for all a E S in the economy (A”, y”, B”). Since g” converges uniformly to f and ~(A”)-+~(A”), it follows from Hildenbrand [S, p. 51, (39)] that D[g”] converges weakly to J: Therefore we can find a v3 such that for all v 2 v3,

Cs”(ah mfll E WCf(a), Nfll; 6) for all a E S;

and

c.Y:, mgvll E %I, m-11; 8) for all a e S.

Then, by (5.37), we have, for all v 2 max(v,, v2, v3),

cy:, Ng”ll g u(Cg”(a)> ngvll; &I for all ae S; (5.38)

because W(a), Xg”ll E u(Cf@L Kfll;&) * WC&‘(a), ~Ctf’ll~~) c UCf(a), mfll; 2&I.

From the thickness assumption (4.8) and the fact that SC A” for all v, we can find a vq such that for all v > vq, there is a subset r of N” such that some bijection $“: T” --f S preserves the attributes. Therefore for notational simplicity, we assume that T” = S c N” for all v. (This does not cause any problems.)

Let 3” be an allocation compatible with xv, p”, and (~1)~~~ in (N”, /I”, B”), where p” is an arbitrary partition under which xv is feasible. The nonatomic extension g” of 2” to (A”, y”, B”) converges in measure to

FROM FINITE TO CONTINUUM ~CO~O~~E~ 167

f-because jT’( <B” for all v and lim, (B’)Z/\N”J = . (Recall the dimssim after the main Theorem.) It follows from Wildenbrand [5, p. 51, (W)] that D[g”] converges weakly to D[f]. Since g” is the no~atomi~ extension of 2‘“, it holds that D[.2”] = D[g”]. Hence D[i’] also converges weakly to D[f]. Since D[g”] also converges weakly to DES], there is a vs2 max(v,, t’*, E’~, v4) from (5.38) such that for all Y 2 v~,

iv;, DCPll r: u i

[g'(a), Dcg"ll;; >

for all a E: S.

Since g” is the nonatomic extension of xv to A”, we have g”(a) = X: and [xl’] =D[g”‘]. Therefore, for all v >/v,,

l-s~:, D[P 1-J >,B” U Lx:, D[x”]]; f i

for all a E S.

This means that {(N”, j3”, B”)} does not have the approximate core property, which is a contradiction. i

ACKNOWLEDGMENTS

We are grateful to the Natural Sciences and Engineering Council of Canada and to the Social Sciences and Humanities Research Council of Canada for providing the financial support to make our collaborative research possible. Additional support was provided by National Science Foundation Grant SES-83-20454 at the Institute for Mathematical Studies in the Social Sciences, Stanford University. We have benefited from discussions with R. J. Aumann, 6. Debreu, B. Grodal, P. Hammond, Y. Kannai, S. Scotchmer, and M. Shubik.

REFERENCES

‘1. R. M. ANDERSON, Notions of core convergence, in “Contribution to Mathematical Economics, In Honor of Gerard Debreu”, (W. Hiidenbrand and A. Mas Cohel, Eds.), pp. 25-46. North-Holland, Amsterdam.

2. R. 4. AUMANN, Markets with a continuum of traders, Econometricra 32 (1964), 39-50. 3. Deleted in proof. 4. P. HAMMOND, M. KANEKO, AND M. H. WOOBERS, Continuum Economies with Finire

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Press, Princeton, NJ, 1974. 6. M. KANEKCI AND M. H. WOODERS, “Nonemptiness of the core of a game with a continuum

of players and finite coalitions”, Working Paper No. 295, Institute of Socio-Economic Planning, University of Tsukuba, (1985).


7. M. KANEKO AND M. H. WCKIDERS, The core of a game with a continuum of players and finite coalitions; the model and some results, Math. Sot. Sci. 12 (1986), 105-137.

8. Y. KANNAI, Continuity properties of the core of a market, Econometrica 38 (1970), 791415.

9. A. MAS-COLELL, A refinement of the core equivalence theorem, Econ. Left. 3 (1979), 307-310.

10. D. SCHMEIDLER, A remark on the core of an atomless economy, Econometrica 40, (1972), 579-580.

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the core of a continuum economy with widespread externalities and finite coalitions: from finite to...

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