Values of Perfectly Competitive Economieshart/papers/val-hgt.pdf · 4 S.Hart isnotsointheNTU-case....

Values of Perfectly Competitive Economies∗

Sergiu Hart†

February 11, 2001

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 The Value Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Proof of Value Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.1 Value Allocations Are Competitive Allocations . . . . . . . . . . . 9

4.2 Competitive Allocations Are Value Allocations . . . . . . . . . . . 11

5 Generalizations and Extensions . . . . . . . . . . . . . . . . . . . . . . 11

5.1 The Non-Differentiable Case . . . . . . . . . . . . . . . . . . . . . 11

5.2 The Geometry of Non-Atomic Market Games . . . . . . . . . . . 12

5.3 Other Non-Atomic TU-Values . . . . . . . . . . . . . . . . . . . . 14

5.4 Other NTU-Values . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.5 Limits of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.6 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.7 Ordinal vs. Cardinal . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.8 Imperfect Competition . . . . . . . . . . . . . . . . . . . . . . . . 17

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

∗Chapter 57 in Handbook of Game Theory, with Economic Applications, Volume III, Editedby R. J. Aumann and S. Hart, Elsevier / North-Holland, 2001. The author thanks Robert J.Aumann and Andreu Mas-Colell for their comments.

†Center for Rationality and Interactive Decision Theory; Department of Economics; andDepartment of Mathematics; The Hebrew University of Jerusalem, 91904 Jerusalem, Israel.E-mail : [email protected] URL: http://www.ma.huji.ac.il/~hart

2 S. Hart

1. Introduction

This chapter is devoted to the study of economic models with many agents, each

of whom is relatively insignificant. These are referred to as perfectly competitive

models. The basic economic concept for such models is the competitive (or Wal-

rasian) equilibrium, which prescribes prices that make the total demand equal to

the total supply, i.e., under which the markets clear. The fact that each agent is

negligible implies that he cannot singly affect the prices, and so he takes them as

given when finding his optimal consumption, or “demand.”

Game-theoretic concepts, particularly “cooperative game-theoretic” solutions

(like core and value), are supported by very different considerations from the

above. First, such solutions are based on the “coalitional game” which, in assign-

ing to each coalition the set of possible allocations to its members, abstracts away

from specific mechanisms (like prices). And second, they prescribe outcomes that

satisfy certain desirable criteria or postulates (like efficiency, stability, or fairness).

Nevertheless, it turns out that game-theoretic solutions may yield the com-

petitive outcomes. The prime example is the Core Equivalence Theorem (see

the chapter of Anderson [1992] in this Handbook) which states that, in perfectly

competitive economies, the core coincides with the set of competitive allocations.

Here we will study another central solution concept, the (Shapley) value (see

the chapter of Winter [2001] in this Handbook). The value, originally introduced

by Shapley [1953] on the basis of a number of simple axioms,1 turns out to measure

the “average marginal contributions” of the players. It is perhaps best viewed as

an “expected outcome” or an “a priori utility” of playing the game.

The first work on values of perfectly competitive economies is due to Shapley

[1964] (see also Shapley and Shubik [1969]), who considered differentiable ex-

change economies (or markets) with transferable utilities (TU), and showed that,

as the number of players increases by replication, the Shapley values converge to

the competitive equilibrium.

1Efficiency, symmetry, dummy player and additivity.

Ch. 57: The Value of Perfectly Competitive Economies 3

To extend this result, two developments were needed. One was to develop the

theory of value for games with a continuum of players (the “limit game”) and,

in particular, to obtain the Shapley [1964] result in this framework; see Aumann

and Shapley [1974] (and the chapter of Neyman [2001] in this Handbook). The

other was to define the concept of value for general non-transferable utility (NTU)

games; see Harsanyi [1963] and Shapley [1969] (and the chapter of McLean [2001]

in this Handbook). This culminated in the general Value Equivalence Theorem of

Aumann [1975]: In a differentiable economy with a continuum of traders, the set of

Shapley [1969] value allocations coincides with the set of competitive allocations.

Next, the non-differentiable case was studied by Champsaur [1975] (limit of

replicas) and by Hart [1977b] (a continuum of traders), who showed that one

direction always holds — all value allocations are competitive — but not neces-

sarily the converse. Together with the work of Mas-Colell [1977] (limit of replicas

in the NTU case), the picture was completed, resulting in the Value Principle:

In perfectly competitive economies, every Shapley value allocation is competitive,

and the converse holds in the differentiable case; see Table 1.

Limit of Replicas Continuum of Agents

Differentiable TU Shapley [1964] Aumann and Shapley [1974]

(Equivalence) NTU Mas-Colell [1977] Aumann [1975]

General TU

(Inclusion) NTU

Champsaur [1975] Hart [1977b]

Table 1: The Value Principle

In the TU-case, the concept of value is clear and undisputed (of course, its

definition in the case of a continuum of players entails many difficulties). This

4 S. Hart

is not so in the NTU-case. There are different ways of extending the Shapley

TU-value, and which one is more appropriate is a matter of some controversy

(see the references in Subsection 5.4). The Value Principle stated above was

exclusively established for the Shapley [1969] NTU-value (also known as the “λ-

transfer value”). Recently, Hart and Mas-Colell [1996a] studied the Harsanyi

[1963] NTU-value in perfectly competitive economies, and showed that these value

allocations may be different from the competitive ones, even in the differentiable

case (and that, moreover, this is a robust phenomenon). Since the Harsanyi NTU-

value is no less appropriate than the Shapley NTU-value (and possibly even more

so), this casts doubt on the general validity of the Value Principle, in particular

when the economy exhibits substantial NTU features; see Subsection 5.4.

The chapter is organized as follows: Section 2 presents the basic model of an

exchange economy with a continuum of agents, together with the definitions of

the appropriate concepts. The Value Principle results are stated in Section 3. An

informal (and hopefully instructive) proof of the Value Equivalence Theorem is

provided in Section 4. Section 5 is devoted to additional material, generalizations,

extensions and alternative approaches.

2. The Model

As we saw in the Introduction, there are various ways to model “perfectly com-

petitive” economies. One is to consider sequences of economies with finitely many

agents, where the number of agents increases to infinity; see Subsection 5.5. An-

other is to study directly the “limit” economy with a continuum of agents. Indeed,

to model precisely the intuitive notion that an individual’s influence on the whole

economy is negligible, one needs infinitely many participants. As Aumann [1964,

p.39] says, “... the most natural model for this purpose contains a continuum

of participants, similar to the continuum of points on a line or the continuum of

particles in a fluid.”

Our basic model is that of a pure exchange economy (or market) with a con-


tinuum of traders, and consists of the following:

E1 The commodities (or goods): ` = 1, 2, ..., L; the consumption set of each agent

is thus2 RL+.

E2 The space of traders : A measure space (T,C, µ), where: T is the set of traders

(or agents); C, a σ-field of subsets of T , is the set of coalitions ; and µ, a

probability measure3 on C, is the population measure (i.e., for every coalition

S ∈ C, the number µ(S) ∈ [0, 1] is the “mass” of S, relative to the whole

space T ).

E3 The initial endowments : For every trader t ∈ T, a commodity vector e(t) =

(e1(t), e2(t), ..., eL(t)) ∈ RL+, where e

`(t) is the quantity of good ` initially

owned by trader t.

E4 The utility functions : For every trader t ∈ T , a function ut : RL+ → R, where

ut(x) is the utility of t from the commodity bundle x ∈ RL+.

We make the following assumptions:

G1 (Measurability) e(t) is measurable in t and ut(x) is jointly measurable in (t, x)

(relative to the Borel σ-field on RL+ and the σ-field C on T ).

G2 (Endowments) Every commodity is present in the market, i.e.,4∫

Te dµÀ 0;

G3 (Utilities) The utility functions ut are all strictly monotonic (i.e., x ¢ y

implies ut(x) > ut(y)), continuous and uniformly bounded (i.e., there exists

M such that |ut(x)| ≤M for all x and t).

2R is the real line, and RL+ is the non-negative orthant of the |L|-dimensional Euclidean spacewhose coordinates are indexed by the elements of L.

3I.e., µ is non-negative and µ(T ) = 1.4For vectors x, y ∈ RL, we take x ≥ y and xÀ y to mean, respectively, x` ≥ y` and x` > y`

for all ` ∈ L. Next, x ¢ y means x ≥ y and x 6= y (i.e., x` ≥ y` for all `, with at least one strictinequality).

6 S. Hart

G4 (Standardness) (T,C, µ) is isomorphic to5 ([0, 1],B, ν), the unit interval [0, 1]

with the Borel σ-field B and the Lebesgue measure ν; in particular, µ is a

non-atomic measure.

All of these assumptions are standard; we will refer to G1–G4 (in addition, of

course, to E1–E4) as the general case. Some assumptions (like uniform bound-

edness of the utility functions) are made for simplicity of presentation only; the

reader should consult the relevant papers for more general setups.

Finally, we introduce an additional set of assumptions known as the differen-

tiable case:

D1 The utility functions ut are all concave and continuously differentiable on6

RL+, and their gradients ∇ut are uniformly bounded and uniformly positive

in bounded subsets of7 RL+.

D2 The initial endowments are uniformly bounded (i.e., there existsM such that

e`(t) ≤M for all ` and t).

An allocation x is a feasible outcome of this exchange economy, i.e., a redis-

tribution of the total initial endowment. That is, x : T → RL+ is an integrable

function, where x(t) ∈ RL+ is agent t’s final commodity bundle, which satisfies:

∫

T

x dµ =

∫

T

e dµ. (2.1)

We now define the two concepts that will be the concern of this chapter:

competitive equilibrium and value. An allocation x is competitive (or Walrasian)

if there exists a price vector p ∈ RL+, p 6= 0 such that, for every agent t ∈ T, the

bundle x(t) is maximal in t’s budget set Bt(p) := {x ∈ RL+ : p · x ≤ p · e(t)} (i.e.,

5There is little loss of generality here; see Aumann and Shapley [1974, Chapter VIII].6Including the boundary of RL+; that is, ut is continuously differentiable in the interior of

RL+, and all its partial derivatives can be continuously extended to the boundary of RL+.7I.e., for every M < ∞ there exist c > 0 and C < ∞ such that c < ∂ut(x)/∂x

` < C for allt, ` and x with ‖x‖ ≤M.


ut(x(t)) ≥ ut(y) for all y ∈ Bt(p)). We refer to e(t) as t’s supply, and to x(t) as

t’s demand ; equality (2.1) then states that total demand equals total supply.

An allocation x is a Shapley [1969] (NTU-)value allocation (see the chapter

of McLean [2001] in this Handbook) if there exists a collection λ = (λt)t∈T of

nonnegative weights (formally, a measurable function λ : T → R+) such that8

∫

S

λtut(x(t)) dµ(t) = ϕvλ(S) for every S ∈ C, (2.2)

where vλ is the TU-game defined by

vλ(S) := max{

∫

S

λtut(y(t)) dµ(t) : y(t) ∈ RL+ for all t ∈ S and

∫

S

y dµ =

∫

S

e dµ}

(2.3)

for all coalitions S ∈ C, and ϕvλ is the “asymptotic value” (see the chapter

of Neyman [2001] in this Handbook) of the game vλ. That is, if utilities were

transferable between the traders at the rates λ, then the value of the resulting

TU-game (2.3) would in fact be achievable without transfers (2.2).

3. The Value Principle

In this section we will state the main results on the relations between value and

competitive equilibria. The following two theorems comprise what is known as

the Value Principle.

Theorem 3.1 (Value Equivalence). In the differentiable case: The set of Shap-

ley value allocations coincides with the set of competitive allocations.

Theorem 3.2 (Value Inclusion). In the general case: The set of Shapley value

allocations is included in the set of competitive allocations (and there are cases

where the inclusion is strict).

8Equivalently, in “density” terms, λtut(x(t)) = (d(ϕvλ)/dµ) (t) for µ-almost every t. Theright-hand side (which is the Radon-Nikodym derivative of ϕvλ with respect to µ) may be viewedas the “value of t;” in our informal arguments, we will abuse notation and write it as ϕvλ(t), so(2.2) becomes λtut(x(t)) = ϕvλ(t) for (almost) every t (just like the case when there are onlyfinitely many agents).

8 S. Hart

In other words: First, value allocations are always competitive. And second,

competitive allocations are value allocations provided the economy is appropri-

ately differentiable (or smooth).

In the framework of the previous section (markets with a continuum of agents),

the “general case” refers to E1–E4 together with G1–G4; the “differentiable case,”

to E1–E4, G1–G4 and D1–D2. Value Equivalence has been proved by Aumann

[1975]. We will provide an informal — and hopefully instructive — proof in the

next section. Value Inclusion has been proved by Hart [1977b]; see Subsections 5.1

and 5.2 below. As stated in the Introduction, parallel results have been obtained

in other frameworks (like limits of sequences, transferable utility, etc.; recall Table

1 and see Section 5).

All of these results use the concept of NTU-value due to Shapley [1969] (defined

in the previous section). However, there are other notions of an NTU-value. A

prominent one is the Harsanyi [1963] NTU-value. We have:

Proposition 3.3. In the differentiable case: The set of Harsanyi value allocations

may be disjoint from the set of competitive allocations.

This is shown by Hart and Mas-Colell [1996a], who provide a robust example

with a unique Harsanyi value which is different from the unique competitive equi-

librium. This clearly raises doubts about either the Harsanyi value or the Value

Principle. In view of the additional literature, Hart and Mas-Colell [1996a] sug-

gest that, since the Harsanyi concept is no less appropriate than the Shapley one

(and perhaps even more so), it follows that the Value Principle may not be valid

in general — particularly when the economy is far removed from the transferable

utility case. See Subsection 5.4.

4. Proof of Value Equivalence

We now provide an informal proof for the Value Equivalence Theorem 3.1, in the

framework of Section 2 (under all the assumptions there, including differentiabil-

ity); in particular, “value” stands for “Shapley NTU-value.”


4.1. Value Allocations Are Competitive Allocations

Let x be a value allocation with corresponding weights λ. We ignore technical

complications and assume that x(t) À 0 and λt > 0 for all t. Also, we will write

“for all t” where in fact it should say “for µ-almost every t,” and we regard a

point t as if it were a “real agent.”9

First, we have∫

T

λtut(x(t)) dµ(t) = vλ(T ) (4.1)

by (2.2) for T together with ϕvλ(T ) = vλ(T ) (since the TU-value is efficient).

Thus the maximum in (2.3) for T is achieved at x. Therefore10

λt∇ut(x(t)) = λt′∇ut′(x(t′)) for all t, t′ ∈ T (4.2)

(this is standard: if not, then a reallocation of goods between t and t′ would

increase the total utility vλ(T )). Let p be the common value of11 λt∇ut(x(t)) for

all t:

∇ut(x(t)) = (1/λt)p for all t ∈ T.

The utility functions ut are concave; therefore ut(y) ≤ ut(x(t)) +∇ut(x(t)) · (y −

x(t)), from which it follows that

if p · y ≤ p · x(t) then ut(y) ≤ ut(x(t)). (4.3)

Next, we claim that the asymptotic value ϕvλ of vλ satisfies

ϕvλ(t) = λtut(x(t)) + p · (e(t)− x(t)). (4.4)

The intuition for this is as follows: The value of t is t’s marginal contribution to

a random coalition Q (this holds in the case of finitely many players by Shapley’s

9As in the case of finitely many agents. More appropriately, one should view dt as the“agent”; we prefer to use t since it may appear less intimidating for some readers.

10We write ∇u(x) for the gradient of u at x, i.e., the vector of partial derivatives(

∂u(x)/∂x`)

`=1,...,L.

11Letting p = (p`)`=1,...,L, one may interpret p` as the Lagrange multiplier (or “shadow price”)

of the constraint∫

Tx` =

∫

Te`.

10 S. Hart

formula, and it extends to the general case since the asymptotic value is the limit

of values of finite approximations). Now a “random coalition” Q, when there is a

continuum of players, is a perfect sample of the grand coalition T . (Assume for

instance that we have a large finite population, consisting of two types of players:

1/3 of them are of type 1, and 2/3 of them of type 2. The coalition of the first

half of the players in a random order will most probably contain, by the Law of

Large Numbers, about 1/3 players of type 1 and 2/3 of type 2. This holds for

“one half” as well as for any other proportion strictly between 0 and 1.) Therefore

the average marginal contribution of t to Q is essentially the same as t’s marginal

contribution to the grand coalition12 T , which we can write suggestively as

ϕvλ(t) = vλ(T )− vλ(T\{t}).

Since vλ(T ) is the result of a maximization problem, its derivative (“in the di-

rection t”) is obtained by keeping the optimal allocation fixed and multiplying the

change in the constraints by the corresponding Lagrange multipliers.13 The opti-

mal allocation is x (by (4.1)); rewriting∫

Tx dµ=

∫

Te dµ as

∫

T\{t}x dµ=

∫

T\{t}e dµ

+(e(t)− x(t)) dµ(t) shows that the change in the constraints is14 e(t)−x(t); and

the Lagrange multipliers are p. Therefore indeed

ϕvλ(t) = λtut(x(t)) + p · (e(t)− x(t)).

In other words, the marginal contribution of t to the total utility vλ(T ) consists

of t’s weighted utility contribution λtut(x(t)), together with his net contribution

of commodities e(t)−x(t), evaluated according to the “shadow prices” p (i.e., the

marginal weighted utilities, which are common to everyone by (4.2)). We have

thus shown (4.4).

12Technically, one uses here the homogeneity of degree 1 of v, hence the homogeneity of degree0 of its derivatives.

13Since the first order condition (that of vanishing derivatives) is satisfied at optimal allo-cations, the change in the optimal allocation is, in the first order, zero. This is the so-called“Envelope Theorem”; see, e.g., Mas-Colell, Whinston and Green [1995, Section M.L].

14In “density” terms.


Comparing (2.2) (see Footnote 8) and (4.4) implies that

p · (e(t)− x(t)) = 0 for all t,

which together with (4.3) shows that x is a competitive allocation relative to the

price vector p.

4.2. Competitive Allocations Are Value Allocations

Let x be a competitive allocation with associated price vector p. From the fact that

x(t) is the demand of t at the prices p (i.e., p ·y ≤ p ·e(t) implies ut(y) ≤ ut(x(t))),

it follows that the gradient ∇ut(x(t)) is proportional to p (again, we assume for

simplicity that x(t)À 0 for all t). That is, there exist λt > 0 such that

λt∇ut(x(t)) = p for all t.

This implies that the maximum in the definition of vλ(T ) (for this collection

λ = (λt)t) is attained at x.

As we saw in the previous subsection (recall (4.4)), the asymptotic value ϕvλ

of vλ satisfies

ϕvλ(t) = λtut(x(t)) + p · (e(t)− x(t)).

But x(t) is t’s demand at p, so p · x(t) = p · e(t) (by monotonicity), and therefore

ϕvλ(t) = λtut(x(t)), thus completing the proof that x is indeed a value allocation

(by (2.2)).

5. Generalizations and Extensions

5.1. The Non-Differentiable Case

In the general case (i.e., without the differentiability assumptions), the Value In-

clusion Theorem says that every Shapley value allocation is competitive, and that

the converse need no longer hold. In fact, in the non-differentiable case the as-

ymptotic value of vλ may not exist (since different finite approximations lead to

12 S. Hart

different limits),15 and so there may be no value allocations at all. Moreover, even

if value allocations do exist, they correspond only to some of the competitive al-

locations; roughly speaking, values “select” certain “centrally located” equilibria.

The proof that every Shapley value is competitive is based on generalizing the

value formula (4.4); see Hart [1977b] and the next subsection.

5.2. The Geometry of Non-Atomic Market Games

A (TU-)market game is a game that arises from an economy as in Section 2 (see

(2.3)).

To illustrate the geometrical structure that underlies the results of this chapter,

suppose for simplicity that there are finitely many types of agents, say n, with a

continuum of mass 1/n of each type. A coalition S is then characterized by its

composition, or profile, s = (s1, s2, ..., sn) ∈ Rn+, where si is the mass of agents of

type i in S. Formula (2.3) then defines a function v = vλ over profiles s; that is,16

v : Rn+ → R. Such a function is clearly super-additive (i.e., v(s+ s′) ≥ v(s)+ v(s′)

for all s, s′ ∈ Rn+) and homogeneous of degree 1 (i.e., v(αs) = αv(s) for every

s ∈ Rn+ and α ≥ 0), and hence concave.

17

Let s = (1/n, 1/n, ..., 1/n) denote (the profile of) the grand coalition. A payoff

vector18 π = (π1, π2, ..., πn), where πi is the payoff of each agent of type i, is in

the core of v if∑n

i=1 πisi = v(s) and∑n

i=1 πisi ≥ v(s) for every s ≤ s. Thus π is

nothing other than a normal vector, or “supergradient” (recall that v is concave)

to v at19 s. In particular, if v is differentiable, then the core has a unique element:

15A necessary condition for the existence of the asymptotic value is that the core possess acenter of symmetry; see Hart [1977a; 1977b]. Note that the convex hull of k points in generalposition does not have a center of symmetry when k ≥ 3 (it is a triangle, a tetrahedron, etc.).

16In fact, only s ≤ (1/n, 1/n, ..., 1/n) matter.17Such a function may be called “conical”: its subgraph {(s, ξ) ∈ Rn+ × R : ξ ≤ v(s)} is a

convex cone.18We only consider type-symmetric payoffs, where agents of the same type receive the same

payoffs.19I.e., the graph of the linear function h(s) := π ·s is a supporting hyperplane to the subgraph

of v (which is a convex cone; see Footnote 17) at the point (s, v(s)) (and thus also on the whole“diagonal” {(αs, v(αs)) : α ≥ 0}).


the gradient ∇v(s) of v at s.

As for the TU-value of v, it is obtained as in Subsection 4.1 (see the considera-

tions leading to (4.4)) as follows. Assume first that v is continuously differentiable.

The profile q of a random coalition Q is (approximately) proportional to that of

the grand coalition, i.e., q ≈ αs for some α ∈ (0, 1) (by the Law of Large Num-

bers). The marginal contribution of an agent of type i to such a coalition Q is

thus given by the partial derivative (∂v/∂si) (q) ≈ (∂v/∂si) (αs), which equals

(∂v/∂si) (s) by homogeneity. Therefore the value of type i is (∂v/∂si) (s), and

the value payoff vector is ∇v(s) — identical to the core.20

If v is not differentiable, then the partial derivatives are replaced by directional

derivatives, which correspond to supergradients, and are again independent of α

by homogeneity. The set of supergradients is convex, therefore averaging (over

random coalitions) implies that the value vector is also a supergradient of v at s

— and so in this case too the value belongs to the core.

Summarizing: In a non-atomic TU-market game, the value belongs to the

core, and is the unique element in the core in the differentiable case. But the core

and the set of competitive allocations coincide — this is the Core Equivalence

Theorem (see the chapter of Anderson [1992] in this Handbook, and note that

differentiability is not required) — which yields the Value Principle in the TU-

case.

Moving now to the NTU-case, note that if π = (π1, π2, ..., πn) is a Shapley

NTU-value with weights21 λ = (λ1, λ2, ..., λn), then the vector (λ1π1, λ2π2, ..., λnπn)

is the TU-value of vλ (see (2.2)) and so, as we have seen above, it is a supergradi-

ent of vλ at s. Therefore∑n

i=1 λiπisi ≥ vλ(s) for every s, implying that π cannot

be improved upon by a coalition of profile s (recall the definition of vλ as a max-

20Another proof of this statement can be obtained using the potential approach of Hart andMas-Colell [1989]: Given v, there exists a unique function P ≡ P (v) : Rn+ → R with P (0) = 0and s · ∇P (s) = v(s) for all s — the potential of v — and moreover ∇P (s) is the value payoffvector. When v is homogeneous of degree 1, the Euler equation implies s · ∇v(s) = v(s), and soP = v and ∇P (s) = ∇v(s) (or: value = core).

21λi is the weight corresponding to agents of type i (recall that, for simplicity, we assumetype-symmetry).

14 S. Hart

imum). In other words, π belongs to the (NTU) core of the economy, and so it is

a competitive allocation by the Core Equivalence Theorem. This establishes the

Value Inclusion result of Theorem 3.2.

For precise presentations of these topics, the reader is referred to Shapley

[1964], Hart [1977a; 1977b] and Hart and Mas-Colell [1996a, Section VII].

5.3. Other Non-Atomic TU-Values

In the non-differentiable case, many of the relevant TU-games may not have an

asymptotic value.22 This happens when different sequences of finite games that

approximate the given non-atomic game have values that converge to different

limits. (The simplest such case is the so-called “3-gloves market”; see Aumann

and Shapley [1974, Section 19] and recall Footnote 15.) Therefore one looks for

other TU-value concepts.

One approach (first used by Aumann and Kurz [1977]) modifies the defini-

tion of the asymptotic value by considering only those finite approximations with

players that are (approximately) equal in size, where “size” is determined by the

underlying population measure µ. The resulting measure-based value (or µ-value)

is shown by Hart [1980] to exist under wide conditions and, moreover, to yield a

competitive allocation (i.e., the Value Inclusion result of Theorem 3.2 continues

to hold for this value). An explicit formula, involving an appropriate normal dis-

tribution, is obtained for the resulting competitive price; this price may then be

viewed as an “expected equilibrium price,” where the expectation is taken over

random samples of the agents.

Another approach, due to Mertens [1988a; 1988b], defines a value concept on

a large space of non-atomic games, which includes all markets (whether differen-

tiable or not); again, the Value Inclusion Theorem 3.2 holds for the Mertens value

as well.

22See however Hart [1977b, Theorem C].


5.4. Other NTU-Values

Extending the concept of the Shapley value from the class of TU-games to the

class of NTU-games is a conceptually challenging problem. One looks for an

NTU-solution concept that, in particular: (i) extends the Shapley [1953] value for

TU-games; (ii) extends the Nash [1950] bargaining solution for pure bargaining

problems; (iii) is covariant with independent rescalings of the utility functions;23

and (iv) satisfies postulates like efficiency and symmetry. However, all this does

not yet determine the solution, and additional constructs are needed.

The two major NTU-value concepts are due to Harsanyi [1963] and to Shapley

[1969]. In fact, Shapley introduced his NTU-value originally as a simplification

of the Harsanyi NTU-value. It then turned out to be of interest in its own right,

and has been used in various models, economic and otherwise. However, further

research has indicated that the Harsanyi concept may well be the more appropri-

ate one, at least in some cases; see Aumann [1985a; 1985b; 1986], Hart [1985a;

1985b], Roth [1980; 1986] and Shafer [1980] for some of the issues related to the

interpretation and appropriateness of these concepts.

The Harsanyi NTU-values of large differentiable economies24 were studied by

Hart and Mas-Colell [1996a]. It is shown, first, that the Value Inclusion result

holds in some cases;25 and, second, that in general the two (non-empty) sets of al-

locations — the Harsanyi values and the competitive equilibria — may be disjoint.

Moreover, this “non-equivalence” is a robust phenomenon, and a result of being

far removed from the transferable utility case (i.e., the lack of substitutability

among the agents’ utilities).

Another NTU-value that has recently been analyzed is the consistent NTU-

23In the TU-case, where there is a common medium of utility transfer, one may multiply allutilities by the same constant without changing the problem. In contrast, in the NTU-case,each player’s utility may be multiplied by a different constant.

24The framework is that of a continuum of agents of finitely many types, as in Subsection5.2. Moreover, as it is shown in Hart and Mas-Colell [1995a; 1995b], the analysis is borne outby limits of finite approximations.

25Specifically, when the Harsanyi value is “tight.”

16 S. Hart

value (see Maschler and Owen [1989; 1992], and also Hart and Mas-Colell [1996b]).

For a first study on the connections between this value and the core, see Leviatan

[1998].

5.5. Limits of Sequences

Rather than analyze the limit economy with a continuum of agents, one may

instead consider sequences of finite economies. The simplest such approach, orig-

inally used in the study of the Core Equivalence Theorem, is that of “replicas”:

Each agent is replaced by r identical agents, and one looks at the limit as r

increases to infinity.

The results here are the same as in Theorems 3.1 and 3.2: Limits of Shapley

value allocations are competitive, and the converse holds in the differentiable case.

See Shapley [1964] (differentiable, TU); Champsaur [1975] (non-differentiable, TU

and NTU); Mas-Colell [1977] and Cheng [1981] (differentiable, NTU); and also

Wooders and Zame [1987a; 1987b] for a “space of characteristics” framework

(non-differentiable, TU and NTU).

5.6. Other Approaches

Among the other approaches to the Value Equivalence result, one should mention

the axiomatic characterization of solutions of large economies due to Dubey and

Neyman [1984; 1997], which captures many of the interesting concepts — compet-

itive equilibrium, core, value — and thus implies their equivalence. Other works

use non-standard analysis (Brown and Loeb [1977]) or fuzzy games (Butnariu

[1987]).

5.7. Ordinal vs. Cardinal

The competitive allocations and the core are clearly “ordinal” concepts: They

depend only on the preference relations of the agents and not on the particular

utility representations. How about the value?


The construction in Section 2 uses given utility functions ut. However, the rel-

ative weights λt are obtained endogenously (as a “fixed-point” where (2.2) holds).

Thus, if we were to apply linear transformations to the functions ut (with dif-

ferent coefficients for different t) the NTU-value allocations would not change.

Moreover, a careful look at the proof of the Value Equivalence result in Section 3

shows that only “local” information is used (e.g., gradients or marginal rates of

substitution), and so applying differentiable monotonic transformations will not

matter either. Therefore, the NTU-value depends only on the preference orders,

and is indeed an “ordinal” concept. In fact, Aumann [1975] (and others) works

directly with collections of utility representations (rather than with one utility

representation and weights λ, as we do here26).

5.8. Imperfect Competition

Perfect competition corresponds to all agents being individually insignificant. Im-

perfect competition results when there are “large” agents. This is modelled by

population measures that are no longer non-atomic; the atoms are precisely the

large agents. An interesting question is whether it is worthwhile for a coalition to

become an atom, i.e., a “monopoly.” For when this is measured by the value, see

Guesnerie [1977] and Gardner [1977] (compare this with the results for the core;

see the chapter of Gabszewicz and Shitovitz [1992] in this Handbook).

For other economic applications, the reader is referred to the chapter of Mertens

[2001] in this Handbook.

References

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Aumann, R. J. [1964], “Markets with a Continuum of Traders,” Econometrica 32,39–50.

26This “λ-approach” is commonly used, and we hope it will thus be easier on the reader.

18 S. Hart

Aumann, R. J. [1975], “Values of Markets with a Continuum of Traders,” Econo-metrica 43, 611–646.

Aumann, R. J. [1985a], “An Axiomatization of the Non-Transferable UtilityValue,” Econometrica 53, 599–612.

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