# Values of Non-Atomic Games

R. J. AUMANN
L. S. SHAPLEY
Pages: 348
https://www.jstor.org/stable/j.ctt13x149m

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Preface
(pp. ix-2)
4. Introduction
(pp. 3-10)

Interaction between people—as in economic or political activity—usually involves a subtle mixture of competition and cooperation. Thus bargaining for a purchase is cooperative, in that both sides want to consummate the transaction, but also competitive, in that each side wants terms that are more favorable to itself, and so less favorable to the other side. People cooperate to organize corporations, then compete with other corporations for business and with each other for positions of power within the corporation. Political parties compete for the voter’s favor, but cooperate in forming ruling coalitions and in “logrolling.” Often it is impossible...

5. Chapter I. The Axiomatic Approach
(pp. 11-92)

The symbol$\left\| {} \right\|$for norm is used in many different senses throughout the book; but it is never used in two different senses on the same space, so no confusion can result. In particular, whenx; is in a euclidean space of finite dimension (i.e. it is a finite-dimensional vector), then$\left\| x \right\|$will always mean the maximum norm, i.e.

$\left\| x \right\| = \frac{{\max }}{i}\left| {{x_i}} \right|.$

It is important to distinguish notationally between functions and their values. For example, ifμis a measure, then$\left\| x \right\|$is its total variation, whereas is simply the absolute value of the real number$\mu \left( S \right)$

Occasionally it will be necessary...

6. Chapter II. The Random Order Approach
(pp. 93-125)

Shapley’s original paper (1953a) contained two equivalent approaches to the valuation of games with finitely many players: one based on axioms, and one based on determining the expected marginal payoff to a player when the players are brought into coalition in a random order (see Appendix A). Chapter I was devoted to a generalization of the first of these approaches to games with a continuum of players. The present chapter will be devoted to investigating analogues of the second approach.

It turns out that a direct generalization of the random order approach is impossible; this will be shown in Sections...

7. Chapter III. The Asymptotic Approach
(pp. 126-140)

In this chapter we shall treat an approach, due to Kannai, in which a non-atomic game is regarded as a limit of games with finitely many players. We shall find that the asymptotic value presently to be defined is similar in its properties to the mixing value. This kinship may be attributed to the fact that the value for finite games, though originally defined axiomatically (Shapley, 1953a), is inextricably involved with the notion of random order.

In this section we define the concept of “asymptotic value” and state the basic theorem (Theorem F), which relates it to the value defined...

8. Chapter IV. Values and Derivatives
(pp. 141-166)

Let us recall formula (3.1) for the value of a “vector measure game,” i.e. a set function of the form$\int ^\circ \mu ,$, whereμis ann-dimensional vector measure and$\int {}$is a real function of n real variables. The formula reads

$\varphi \left( {f^\circ \mu } \right)\left( S \right) = {\int_0^1 {{f_u}} _{\left( S \right)}}\left( {t\mu \left( 1 \right)} \right)dt,$

where,${f_{{u_{\left( S \right)}}}}$is the derivative offin the directionμ(S) This formula is of central importance in the study of values. The purpose of this chapter is to reformulate and generalize it, and thereby also to gain a better insight into what the formula says.

Formula (3.1) may be intuitively understood as follows: Suppose the players could...

9. Chapter V. The Value and the Core
(pp. 167-174)

The main object of Chapter V is the proof of

Theorem I.Let υ be a superadditive set function in pNA that is homogeneous of degree 1. Then the core of õ has a unique member, which coincides with the value of v.

Several of the terms used in the statement of Theorem I may not be familiar to the reader. A set functionυissuperadditiveif for disjoint S and T,

$\nu \left( {S \cup T} \right) \ge \nu \left( T \right).$

It ishomogeneous of degree1 if

$v*({}_\alpha x{}_s) = \alpha v(S)$

for all a in [0,1] and all sets S ε e, where v* is the extension defined by...

10. Chapter VI. An Application to Economic Equilibrium
(pp. 175-251)

In this chapter we will apply the theory developed in the previous chapters to certain economic models. These models may be interpreted either as productive economies similar to—but more general than—the one described in the introduction to Chapter V,¹ or as exchange economies with transferable utility.² Our chief result is that under fairly wide conditions, the set function derived from such a model is inpNA,that there is a unique point in its core, and that this unique point coincides with the value. We shall also define the notion of competitive equilibrium for such economies, and show...

11. Chapter VII. The Diagonal Property
(pp. 252-280)

Let$v = {f^^\circ }\mu$whereμis a vector ofNAmeasures andfis continuously differentiable with f(0) = 0. Then according to Theorem B,vis inpNA,and

$(\alpha v)\left( S \right) = \int_0^1 {{f_{\mu \left( s \right)}}} (t\mu (1))dt.$

In Section 3 we have already pointed out a remarkable implication of this formula, namely that φv pv is completely determined by the behavior off near the diagonalofμonly. In particular, if f vanishes in some neighborhood of the diagonal, then φv must vanish identically. Moreover, we know (refer to (3.3)) that φv is determined by its behavior “near the diagonal” even if v = f °...

12. Chapter VIII. Removal of the Standardness Assumption
(pp. 281-294)

In this chapter we shall thoroughly investigate how much of what we have done remains valid when the Standardness Assumption (2.1) is removed or modified. Though the situation is rather complicated, the upshot is that, for most of the results, one can either substitute a weaker form of (2.1), or dispense with it altogether.

The Standardness Assumption is used in five key places in this book: in the proofs of Proposition 6.1 and Lemma 12.5; implicitly in the definitions of the mixing and asymptotic values (Sections 14 and 18); and again implicitly in the quotations from Aumann and Perles (1965)...

13. Appendix A. Finite Games and Their Values
(pp. 295-300)
14. Appendix B. e-Monotonicity
(pp. 301-303)
15. Appendix C. The Mixing Value of Absolutely Continuous Set Functions
(pp. 304-314)
16. References
(pp. 315-320)
17. Index of Special Spaces and Sets
(pp. 321-322)
18. Index
(pp. 323-334)
19. Back Matter
(pp. 335-335)