Chapter 56 VALUES OF GAMES WITH INFINITELY MANY PLAYERS

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Chapter 56

VALUES OF GAMES WITH INFINITELY MANY PLAYERS
ABRAHAM NEYMAN Institute of Mathematics and Center for Rationality and Interactive Decision Theory, The Hebrew University of Jerusalem, Jerusalem, Israel

Contents
1. Introduction
1.1. An outline of the chapter

2. 3. 4. 5.

Prelude: Values of large finite games Definitions The value on pNA and bv' NA Limiting values
5.1. The weak asymptotic value 5.2. The asymptotic value

6. 7. 8. 9.

The mixing value Formulas for values Uniqueness of the value of nondifferentiable Desired properties of values

games

9.1. Strong positivity 9.2. The partition value 9.3. The diagonal property 10. Semivalues 11. Partially symmetric values
11.1. Non-symmetric and partially symmetric values 11.2. Measure-based values

12. The value and the core 13. Comments on some classes of games
13.1. Absolutely continuous non-atomic games 13.2. Games in pNA 13.3. Games in bv'NA

References

2123 2123 2125 2127 2129 2133 2134 2135 2140 2140 2150 2151 2151 2151 2151 2152 2154 2155 2157 2159 2161 2161 2161 2161 2162

Handbook of Game Theory, Volume 3, Edited by R.J. Aumann and S. Hart @ 2002 Elsevier Science B. V. All rights reserved

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A. Neyman

Abstract

This chapter studies the theory of value of games with infinitely many players. Games with infinitely many players are models of interactions with many players. Often most of the players are individually insignificant, and are effective in the game only via coalitions. At the same time there may exist big players who retain the power to wield single-handed influence. The interactions are modeled as cooperative games with a continuum of players. In general, the continuum consists of a non-atomic part (the "ocean"), along with (at most countably many) atoms. The continuum provides a convenient framework for mathematical analysis, and approximates the results for large finite games well. Also, it enables a unified view of games with finite, countable, or

oceanic player-sets, or indeed any mixture of these.

.

The value is defined as a map from a space of cooperative games to payoffs that satisfies the classical value axioms: additivity (linearity), efficiency, symmetry and positivity. The chapter introduces many spaces for which there exists a unique value, as well as other spaces on which there is a value. A game with infinitely many players can be considered as a limit of finite games with a large number of players. The chapter studies limiting values which are defined by means of the limits of the Shapley value of finite games that approximate the given game with infinitely many players. Various formulas for the value which express the value as an average of margInal contribution are studied. These value formulas capture the idea that the value of a player is his expected marginal contribution to a perfect sample of size t of the set of all players where the size t is uniformly distributed on [0,1]. In the case of smooth games the value formula is a diagonal formula: an integral of marginal contributions which are expressed as partial derivatives and where the integral is over all perfect samples of the set of players. The domain of the formula is further extended by changing the order of integration and derivation and the introduction of a well-crafted infinitesimal perturbation of the perfect samples of the set of players provides a value formula that is applicable to many additional games with ,essential nondifferentiabilities.

Keywords games, cooperative, coalitional form, transferable utility, value, continuum of players, non-atomic JEL classification: D70, D71, D63, C71

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1. Introduction The Shapley value is one of the basic solution concepts of cooperative game theory. It can be viewed as a sort of average or expected outcome, or as an a priori evaluation of the players' expected payoffs. The value has a very wide range of applications, particularly in economics and political science (see Chapters 32, 33 and 34 in this Handbook). In many of these applications it is necessary to consider games that involve a large number of players. Often most of the players are individually insignificant, and are effective in the game only via coalitions. At the same time there may exist big players who retain the power to wield single-handed influence. A typical example is provided by voting among stockholders of a corporation, with a few major stockholders and an "ocean" of minor stockholders. In economics, one considers an oligopolistic sector of firms embedded in a large population of "perfectly competitive" consumers. In all these cases, it is fruitful to model the game as one with a continuum of players. In general, the continuum consists of a non-atomic part (the "ocean"), along with (at most countably many) atoms. The continuum provides a convenient framework for mathematical analysis, and approximates the results for large finite games well. Also, it enables a unified view of games with finite, countable, or oceanic player-sets, or indeed any mixture of these. 1.1. An outline of the chapter In Section 2 we highlight a sample of a few asymptotic results on the Shapley value of finite games. These results motivate the definition of games with infinitely many players and the corresponding value theory; in particular, Proposition 1 introduces a formula, called the diagonal formula of the value, that expresses the value (or an approximation thereof) by means of an integral along a specific path. The Shapley value for finite games is defined as a linear map from the linear space of games to payoffs that satisfies a list of axioms: symmetry, efficiency and the null player axiom. Section 3 introduces the definitions of the space of games with a continuum of players, and defines the symmetry, positivity and efficiency of a map from games to payoffs (represented by finitely additive games) which enables us to define the value on a space of games as a linear, symmetric, positive and efficient map. The value does not exist on the space of all games with a continuum of players. Therefore the theory studies spaces of games on which a value does exist and moreover looks for spaces for which there is a unique value. Section 4 introduces several spaces of games with a continuum of players including the two spaces of games, pNAand bv' NA, which played an important role in the development of value theory for nonatomic games. Theorem 2 asserts that there exists a unique value on each of the spaces pNA and bv'NA as well as onpNAoo' In addition there exists a (unique) value of norm 1 on the space' NA. Section 4 also includes a detailed outline of the proofs. A game with infinitely many players can be considered as a limit of finite games with a large number of players. Section 5 studies limiting values which are defined by

.,.:.:,

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A. Neyman

means of the limits of the Shapley value of finite games that approximate the given game with infinitely many players. Section 5 also introduces the asymptotic value. The asymptotic value of a game v is defined whenever all the sequences of Shapley values of finite games that "approximate" v have the same limit. The space of all games of bounded variation for which the asymptotic value exist is denoted ASYMP. Theorem 4 asserts that ASYMP is a closed linear space of games that contains bv'M and the map associating an asymptotic value with each game in ASYMP is a value on ASYMP. Section 6 introduces the mixing value which is defined on all games of bounded variation for which an analogous formula of the random order one for finite games is well defined. The space of all games having a mixing value is denoted MIX. Theorem 5 asserts that MIX is a closed linear space which contains pNA, and the map that associates a mixing value with each game v in MIX is a value on MIx. Section 7 introduces value formulas. These value formulas capture the idea that the value of a player is his expected marginal contribution to a perfect sample of size t of the set of all players where the size t is uniformly distributed on [0, 1]. A value formula can be expressed as an integral of directional derivative whenever the game is a smooth function of finitely many non-atomic measures. More generally, by extending the game to be defined over all ideal coalitions - measurable [0, 1]-valued functions

on the (measurable) space of players

-

,

value of any game in pNA. Changing the order of derivation and integration results in a formula that is applicable to a wider class of games: all those games for which the formula yields a finitely additive game. In particular, this modified formula defines a value on bv' NA or even on bv'M. When this formula does not yield a finitelx~ additive game, applying the directional derivative to a carefully crafted small perturbation of perfect samples of the set of players yields a value formula on a much wider space of games (Theorem 7). These include games which are nondifferentiable functions (e.g., a piecewise linear function) of finitely many non-atomic probability measures. Section 8 describes two spaces of games that are spanned by nondifferentiable functions of finitely many mutually singular non-atomic probability measures that have a unique value (Theorem 8). One is the space spanned by all piecewise linear functions of finitely many mutually singular non-atomic probability measures, and the other is the space spanned by all concave Lipschitz functions with increasing returns to scale of finitely many mutually singular non-atomic probability measures. The value is defined as a map from a space of games to payoffs that satisfies a short list of plausible conditions: linearity, symmetry, positivity and efficiency. The Shapley value of finite games satisfies many additional desirable properties. It is continuous (of norm 1 with respect to the bounded variation norm), obeys the null player and the dummy axioms, and it is strongly positive. In addition, any value that is obtained from values of finite approximations obeys an additional property called diagonality. Section 9 introduces such additional desirable value properties as strong positivity and diagonality. Theorem 9 asserts that strong positivity can replace positivity and linearity in the characterization of the value on pNAoo, and thus strong positivity and continuity can replace positivity and linearity in the characterization of a value on pNA. A strik-

this value formula provides a formula for the

I

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ing property of the value on pNA, the asymptotic value, the mixing value, or any of the values obtained by the formulas described in Section 7, is the diagonal property: if two games coincide on all coalitions that are "almost" a perfect sample (within 8 with respect to finitely many non-atomic probability measures) of the set of players, their values coincide. Theorem 11 asserts that any continuous value is diagonal. Section 10 studies semivalues which are generalizations of the value that do not necessarily obey the efficiency property. Theorem 13 provides a characterization of all semivalues on pNA. Theorems 14 and 15 provide results on asymptotic semivalues. Section 11 studies partially symmetric values which are generalizations of the value that do not necessarily obey the symmetry axiom. Theorems 15 and 16 characterize in particular all the partially symmetric values on pNA and pM that are symmetric with respect to those symmetries that preserve a fixed finite partition of the set of players. A corollary of this characterization on pM is the characterization of all values on pM. A partially symmetric value, called a JL'-value, is symmetric with respect to all symmetries that preserve a fixed non-atomic population measure JL and (in addition to linearity, positivity and efficiency), obeys the dummy axiom. Such a partially symmetric value is called a JL-value. Theorem 18 asserts the uniqueness of a JL-value on pNA(JL). Theorem 19 asserts that an important class of nondifferentiable markets have an asymptotic JL-value. Games that arise from the study of non-atomic markets, called market games, obey two specific conditions: concavity and homogeneity of degree 1. The core of such games is nonempty. Section 12 provides results on the relation between the value and the core of market games. Theorem 20 asserts the coincidence of the core and the value in the differentiable case. Theorem 21 relates the existence of an asymptotic value of nondifferentiable market games to the existence of a center of symmetry of its core. Theorems 22 and 23 describe the asymptotic JL-value and the value of market games with a finite-dimensional core as an average of the extreme points of the core. 2. Prelude: Values of large finite games This section introduces several asymptotic results on the Shapley value 1/1 of finite games. These results can serve as an introduction to the theory of values of games with infinitely many players, by motivating the definitions of games with a continuum of players and the corresponding value theory. We start with the introduction of a representation of games with finitely many players. Let l be a fixed positive integer and I: JR.i~"JR. a function with 1(0) = O. Given WI, . . . , Wi E JR.~,define the n-person game v = [f; WI, . . . , W d by v(S)

= I(LWi).
IES

""--

A special subclass of these games is weighted majority games where l = 1, 0 < q < L7=1 Wi and lex) = 1 if x ~ q and 0 otherwise. The asymptotic results relate to se-

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A. Neyman

quences of finite games where the function f is fixed and the vector of weights varies. If the function f is differentiable at x, the directional derivative at x of the function f in the direction y E JRf is fy(x) := lims--+o+(f(x + EY) - f(X))/E. The following result follows from the proofs of results on the asymptotic value of non-atomic games. It can be derived from the proof in Kannai (1966) or from the proof in Dubey (1980). PROPOSITION 1. Assume that f is a continuously differentiable function on [0, It

For every E > 0 there is 8 > 0 such that if WI, . . ., Wn E JR~ with (the normalization) andmax7=1 IIWi II < 8, then L7=, Wi = (1,...,1)

!1frv(i)-1'fwi(t,...,t)dtl::;;EIlWill, where v = [f; WI, ..., Wn]. Thus,for every coalition S ofplayers,
I1frV(S)-11 fw(S)(t,...,t)dtl <E,

where 1frv(S) = LiES 1frv(i) and w(S) = LiES Wi. The limit of a sequence of games of the form [f; w~, .. ., W~k]' where f is a fixed function and w~, . . . , W~k is a sequence of vectors in JR~ with L7~1 w7 = (1, . ..,1) and max;l~, IIWi II ~k--+oo 0 can be modeled by a 'game' of the form f 0 (f-tl, ..., f-tf) where f-t = (f-t I, . . . , f-tf) is a vector of non-atomic probability measures defined on a measurable space (I, C). If f is continuously differentiable, the limiting value formula thus corresponds to:
<pCfof-t)(S)

=

l' fp,(S)(t,...,t)dt.

The next results address the asymptotics of the value of weighted majority games. The n-person weighted majority game v = [q; WI, . . ., wn] with WI, ..., Wn E JR+ and
0 < q < L;l~, Wi, is defined by v(S) = 1 if LiES
Wi;?

q and 0 otherwise.

PROPOSITION 2 [Neyman (1981a)]. For every E > 0 there is K > 0 such that if v = [q; W" ..., wn] is a weighted majority game with WI, ..., Wn E JR+, L7=1 Wi = 1 and n. n K maxi=1 . - K maxi=1 Wi, Wi < q < 1 I"'V(S) -

~

Wi
I

< e n}.

for every coalition S c {l,...,

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In particular, if Vk = [qk; w1, . .., W~k] is a sequence of weighted majority games with
w7 ';? 0, L~J~I w7 ~k---+CQ 1, and qk ~k---+CQ q with 0 < q < 1, then for any sequence of coalitions Sk C {I,..., nd, VrVk(Sk) - LiESk wf ~k---+CQ O. The limit of such a sequence of weighted majority games is modeled as a game of the form v = fq 0 ~ where ~ is a non-atomic probability measure and fq (x) = 1 if x ';? q and 0 otherwise. The non-atomic probability measure ~ describes the value of this limiting game:

qJV(S)

= ~(S).

The next result follows easily from the result of Berbee (1981). PROPOSITION 3. Let WI,...,
CQ

Wn,...

be a sequence of positive numbers with

Lwi=l i=1
and 0 < q < 1. There exists a sequence of nonnegative numbers Vri, i ';? 1, with L~I 1/fi= 1, such that for every 8> 0 there is a positive constant 8> 0 and a positive = [q'; u I, . . . , un] is a weighted majority game with n ';? no, Iq' -ql < 8 andL7=I!Wi - uil < 8, then [Vrv(i) -Vril < 8.
Equivalently, let Vk

integer no such that if v

=

[qk; w1, . . . , W~k] be a sequence of weighted majority games

with L7~1 wf ~k---+c<J q. Then the Shapley value 1 and wJ ~k---+CQWj and qk ~k---+c<J of player i in the game Vk converges as k ~ 00 to a limit Vriand L~I Vri = 1. The limit of such a sequence of weighted majority games can be modeled as a game of the form v = fq 0 ~ where ~ is a purely atomic measure with atoms i = 1,2, . . . and ~(i) = Wi. The sequence Vridescribes the value of this limiting game: qJV(S)

=L
iES

Vri.

3. Definitions The space of players is represented by a measurable space (I, C). The members of set I are called players, those of C - coalitions. A game in coalitional form is a real-valued function v on C such that v(0) = O. For each coalition S in C, the number v(S) is interpreted as the total payoff that the coalition S, if it forms, can obtain for its members; it is called the worth of S. The set of all games forms a linear space over the field of real numbers, which is denoted G. We assume that (I, C) is isomorphic to

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A. Neyman

([0, 1], B), where B stands for the Borel subsets of [0,1]. A game v is finitely additive if v(5 U T) = v(5) + veT) whenever 5 and T are two disjoint coalitions. A distribution of payoffs is represented by a finitely additive game. A value is a mapping from games to distributions of payoffs, i.e., to finitely additive games that satisfy several plausible conditions: linearity, symmetry, positivity and efficiency. There are several additional desirable conditions: e.g., continuity, strong positivity, a null player axiom, a dummy player axiom, and so on. It is remarkable, however, that a short list of plausible conditions that define the value suffices to determine the value in many cases of interest, and that in these cases the unique value satisfies many additional desirable properties. We start with the notations and definitions needed formally to define the value, and set forth four spaces of games on which there is a unique value. A game v is monotonic if v(5) ~ veT) whenever 5 J T; it is of bounded variation if it is the difference of two monotonic games. The set of all games of bounded variation (over a fixed measurable space (I, C)) is a vector space. The variation of a game v E G, v II, is the supremum of the variation of v over all increasing chains 51 C 52 C . . . C 5n in C. Equivalently, IIvll = inf{u(I) + w(I): u, ware monotonic games with v = u - w} (inf0 = 00). The variation defines a topology on G; a basis for the open neighborhood of a game v in G consists of the sets {u E G: IIu - v II < E'}where E' varies over all positive numbers. Addition and multiplication of two games are continuous and so is the multiplication of a game and a real number. A game v has bounded variation if IIvll < 00. The space of all games of bounded variation, BV, is a Banach algebra with respect to the variation norm II II,i.e., it is complete with respect to the distance defined by the norm (and thus it is a Banach Space) and it is closed under multiplication which is continuous. Let g denote the group of automorphisms (i.e., one-to-one measurable mappings (9 from I onto I with (9 -1 measurable) of the underlying space (I, C). Each e in g induces a linear mapping (9* of BV (and of the space of all games) onto itself,
II

defined by ((9*v)(5)

= v((95).

A set of games

Q is called symmetric

if (9*Q

=

Q for

all e in g. The space of all finitely additive and bounded games is denoted FA; the subspace of all measures (i.e., countably additive games) is denoted M and its subspace consisting of all non-atomic measures is denoted NA. The space of all non-atomic elements of FA is denoted AN. Obviously, NA c M c FA c BV, and each of the.spaces NA, AN, M, FA and BV is a symmetric space. Given a set of games Q, we denote by A map q;: Q ~ BV is called positive if q;(Q+) C BV+; symmetric if for every (9 E g and v in Q, G*v E Q implies that q;((9*v) = (9* (q;v); and efficient if for every v in Q,

Q+ all monotonic games in Q, and by Q I the set of all games v in Q+ with v(I)

=

1.

(q;v)(I) = v(I).

DEFINITION 1 [Aumann and Shapley (1974)]. Let Q be a symmetric linear subspace of BV. A value on Q is a linear map q;: Q ~ FA that is symmetric, positive and efficient. The definition of a value is naturally extended also to include spaces of games that are not necessarily included in BV. Thus let Q be a linear and symmetric space of games.

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DEFINITION 2. A value on Q is a linear map from Q into the space of finitely additive games that is symmetric, positive (i.e., cp(Q+) C BV+), efficient, and cp(Q nBV) c FA. There are several spaces of games that have a unique value. One of them is the space of all games with a finite support, where a subset T of I is called a support of a game v if v(S) = v(S n T) for every S in C. The space of all games with finite support is denoted FG. THEOREM 1 [Shapley (l953a)]. There is a unique value on FG. The unique value 1/1on FG is given by the following formula. Let T be a finite support of the game v (in FG). Then, 1/1v(S) = 0 for every S in C with S n T = 0, and for t E T, 1/1v({t})

=

1 L[v(P;Z n! R

u (tn

-

v(p;Z)J,

where n is the number of elements of the support T, the sum runs over all n! orders of T, and p;Z is the set of all elements of T preceding t in the order R. Finite games can also be identified with a function v defined on a finite sub field JT of
C: given a real-valued function v defined on a finite subfield

JT of C with v(0) = 0, the Shapley value of the game v is defined as the additive function 1/1: JT ---+ JRdefined by

its values on the atoms of JT; for every atom a of JT,

1/1v(a)=

-

1

n! L...

~R[v(P;-

Ua)

-

v(p;-)J,

where n is the number of atoms of JT, the sum runs over all n! orders of the atoms of JT, and P:: is the union of all atoms of JT preceding a in the order R.

4. The value on pNA and bv' NA Let pNA (pM, pFA) be the closed (in the bounded variation norm) algebra generated by NA (M, FA, respectively). Equivalently, pNA is the closed linear subspace that is Similarly, pM (PFA) is the closed algebra that is generated by powers of elements in M (FA). Let II 1100 be defined on the space of all games G by
Ilvlloo

generated by powers of measure in NA 1 [Aumann and Shapley (1974), Lemma 7.2].

= inf{f-L(I):

f-LE NA+,

f-L- V E BV+,

f-L+ v E BV+}

+ sup{lv(S)I:

S E C}. The defi-

In the definition of IIvlloo, we use the common convention that inf 0

= 00.

nition of
.> -"

\I 1100

e.g., if f is continuously

is an analog of the C1 norm of continuously differentiable functions;
differentiable on [0, 1] and f-L is a non-atomic probability

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A. Neyman

measure, III 0 1L1i00 = max{lf'(t)I: 0 ~ t ~ I} + max{lf(t)l: 0 ~ t ~ I}. Note that lIuv 1100~ Ilu 1100 IIv 1100for two games u and v. The function v r+ II v 1100 defines a topol-

ogy on G; basis for the open neighborhood of a game v in G consists of the sets {u E G: IIu - v 1100 < 8} where 8 varies over all positive numbers. The space of games

a

pNAoo is defined as the II 1100 closed linear subspace that is generated by powers of measures in NA I. Obviously,pNAoo c pNA (111100 ~ 1111). Both spaces contain many games of interest. If IL is a measure in NA I and f: [0,1] -+ JRis a function with f(O) = 0, then
lolL E pNA if and only if f is absolutely continuous [Aumann and Shapley (1974),
0

Theorem C] and in that case III

IL II

= fd If'

and at 1 with f(O) = O},and IL E NA 1 (IL E Ml, IL E FA1). Obviously, pNAoo CpNA c hv'NA c hv'M c hv'FA.
THEOREM 2 [Aumann and Shapley (1974), Theorems A, B; Monderer (1990), Theorem 2.1]. Each of the spaces pNA, hv'NA, pNAoo has a unique value.

derivative of the function f, and foIL E pNAoo if and only if f is continuously differentiable on [0,1]. Also, if IL = (ILl,..., ILn) is a vector of NA measures with range fILeS): S E C} =: R(IL), and I is a C1 function on R(IL) with f(IL(0)) = 0, then foIL EpNAoo [Aumann and Shapley (1974), Proposition 7.1]. Let hv' NA (hv'M, hv' FA) be the closed linear space generated by games of the form foIL where f E hv' =: {f : [0, 1] -+ JR I f is of bounded variation and continuous at 0

(t) Idt where .f' is the (Radon-Nikodym)

I

ments. Let IL E NA I and let 9 (IL)be the subgroupof 9 of all automorphisms e that are IL-measure-preserving, i.e., fLeeS) = IL(S) for every coalition S in C. Then the set of

The proof of the uniqueness of the value on these spaces relies on two basic argu-

all finitely additive games u that are fixed points of the action of 9(1L) (i.e., e*u = u for every e in 9(1L)) are games of the form aIL, a E JR (here we use the standardness assumption that (1, C) is isomorphic to ([0, 1], Bn. Therefore, if Q is a symmetric space of games and cp:Q -+ FA is symmetric, IL E NA l, and f: [0,1] -+ JRis such that foIL E Q, then e* (f 0 IL) = foIL for every e E 9 (IL). It follows that the finitely
additive game cp(f 0 IL) is a fixed point of 9(1L), which implies that cp(f 0 IL)

= aIL.

If,

moreover, cp is also efficient, then cp(f 0 IL) = f(1)IL. The spaces pNA and hv'NA are closed linear spans of games of the form foIL. The space pNAoo is in the closed linear span of games of the form foIL with foIL in pNAoo' Therefore, each of the spaces pNAoo,pNA and hv'NA has a dense subspace on which there is at most one value. To complete the uniqueness proof, it suffices to show that any value on these spaces is indeed continuous. For that we apply the concept of internality. A linear subspace Q of BV is said to be reproducing if Q = Q+ - Q+. It is said to be internal if for all v in Q,

Ilvll

= inf{ u(1)

+ w(1): u, WE Q+ and v

=u-

w}.

Clearly, every internal space is reproducing. Also any linear positive mapping cp from a closed reproducing space Q into BV is continuous, and any linear positive and efficient

Ch.56:

Values afGames with Infinitely Many Players

2131

mapping <pfrom an internal space Q into BV has norm 1 [Aumann and Shapley (1974), Propositions 4.15 and 4.7]. Therefore, to complete the proof of uniqueness, it is sufficient to show that each of the spaces pNA, bv' NA and pNAoo is internal. The closure of an internal space is internal [Aumann and Shapley (1974), Proposition 4.12] and any linear space of games that is the union of internal spaces is internal. Therefore one demonstrates that each of these spaces is the union of spaces that contain a dense internal subspace. There are several dense internal subspaces of pNA [Aumann and Shapley (1974), Lemma 7.18; Reichert and Tauman (1985); Monderer and Neyman (1988), Section 5]. The proof of the internality of bv'NA is more involved [Aumann and Shapley (1974), Section 8]. Note that if Q and QI are two linear symmetric spaces of games and Q ::) QI, the existence of a value on Q implies the existence of a value on QI. However, the uniqueness of a value on one of them does not imply uniqueness on the other. There are, for instance, subspaces of pNA which have more than one value [Hart and Neyman (1988), Example 4]. There are however results that provide sufficient conditions for subspaces of pNA to have a unique value. For a coalition S we define the restriction of v to S, vs, by vs(T) = v(S n T) for every T E C. A set of games M is restrictable if Vs EM for every v E M and for every SEC. Hart and Monderer [(1997), Theorem 3.6] prove the uniqueness of the value on restrictable subspaces of pNAoo that contain NA. This is the non-atomic version of the analogous result for finite games proved in Hart and Mas-Colell (1989) and Neyman (1989). Existence of a value on pNA and bv'NA is proved in Aumann and Shapley, Chapter I. We outline below a proof that is based on the proof of Mertens and Neyman (2001). Assume that v = :E7=1 fi 0 f-Li with fi E bv' and f-Li E NA I. Define <pv(S) = :E7=1 fi (l)f-Li (S). It follows from the continuity at 0 and 1 of each of the functions fi that
n

Lfi(1)f-Li(S)
i=1

-

?-+0+ 6' 0

lim -

1

i

1-?

n

n

[ i=1

Lfi(t+6'f-Li(S))

- L

fief)

i=1

]

dt.

By Lyapunov's Theorem, applied to the vector measure (f-L I, . . . , f-Ln), for every 0 ~ t ~ (S). Therefore, if 1 - 6' there are coalitions T C R with f-Li(T) = t and f-Li(R) = t + 6'f-Li v is monotonic,
n n

vCR) - veT)

= Lfi(t
i=1

+6'f-Li(S?) - Lfi(t) i=l

~ 0,
gj 0 Vj where thus also <pv is

and therefore <pv(S) ~ O. In particular, if 0 = 2:::7=1fi 0 f-Li - :E~=I
fi, gj E bv' and f-Li, Vj E NA I, :E7=1 fi (1)f-Li = :E~=1 gj (1)Vj, and

,,-"

well defined, i.e., it is independent of the presentation. Obviously, <pvE NA c FA and <pv(I) = v(I) and <pis symmetric and linear. Therefore <pdefines a value on the linear and symmetric subspace of bv'NA of all games of the form :E7=1 fi 0 f-Li with n

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A. Neyman

a positive integer, Ii E bv' and fLi E NA 1. We next show that cp is of norm 1 and thus has a (unique) extension to a continuous value (of norm 1) on bv'NA. For each U E FA,
Ilu II

S E C, Icpv(S)1 + Icpv(SC)1~ Ilvll. For each positive integer m let So C SI C S2 C ... C Sm and So c S,' c . . . C S~ be measurable subsets of Sand SC = I \ S respectively with fLi (Sj) = /11~1 fLi (S) and fLi (S5) = m~l fLi (SC). For every 0 ~ t ~ m~1 let It be a measurable subset of I \ (Sm U S'~l) with fLi (It) = t. Define the increasing sequence of coalitions To C 1'1 C . . . C 1'2/11 by To = It, T2j-l = It U Sj U S5-1 and T2j = It U Sj U Sf'

= SUPSEC lueS)

1+ lu(SC) I. Therefore,

it is sufficient

to prove that for every coalition

j = 1,..., m. Obviously,
2m

Ilvll ~ Llv(Tj)
j=1 m-I

- v(Tj-I)!

~
Set e:=

L V(T2j+l)
11

- v(T2j)

+

L V(T2j)
j=l
n

m

- V(T2j-l)

.

j=O

_1
/11

.+ I . Note that

-;

1

{em-I

Jo

~ [ ~Ii(t
[

+e:j +e:fLi(S?) - ~Ii(t Ii (t + e:fLi (S?) -

+ je:)

]

dt

= -; Jo
and similarly
1 (e

1 {I -e

~
/I

~

1/

Ii (t)

]

dt ~m-+oo

cpv(S),

/11

11

11

,

-;Jo

~
1

[
I

~IiCt+e:j)11 /I

~Ii(t+je:-e:fLi(SC))

]

dt

=-;1
As

[ ~Ii(t)-

~Ii(t-e:fLi(SC))

] dt~m-+ooCPV(SC).
/I

11 v(T2j+l) - v(T2j)

= LIi(t
i=l

+e:j +e:fLi(S?) - LIiCt
i=1

+ je:),

and
v(T2j) - V(T2j-I)

=L
Il

/I

Ii ct + 2je:) - Lfi
i=1

(t + 2je: -

e:fLi (SC)),

i=l

I

Ch. 56:

Values of Games with Infinitely Many Players

2133

we deduce that for each fixed 0 ~ t ~ s,

'~[t
+
and therefore

Ii (t +
Ji(t

?j+

?/li(S))

-

~ [t

+

ej)

-

t

t

Ji(t

+ j?)]
~ Ilvll

Ii (I + je - eJ-ti(S'))]

l<pv(S) I + l<pv(SC) I ~ IIv II. It follows

that the map

<p,

which is defined on

a dense subspace of bv' NA and with values in NA, is linear, symmetric, efficient, and of norm 1, and therefore <p has a unique extension to a continuous value on bv'NA. The bounded variation of the functions Ii is used in the above proof to show that the
integrals

f: I (x) dx, 0 ~ a < b ~ 1, are well

defined.

Therefore,

the proof also demon-

strates the result of Tauman (1979), i.e., that there is a unique value of norm 1 on In' NA: the closed linear space of games that is generated by games of the form I 0 fJ- where 1 fJ- E NA and I E In' = {I: [0,1] ---+IR I 1(0) = 0, I integrable and continuous at 0 and I}. The integrability requirement can also be dropped. Mertens and Neyman (2001) prove the existence of a value (of norm 1) on the spaces' AN :J ' NA: the closure in the variation distance of the linear space spanned by all games I 0 fJ-where fJ-E AN 1 :J NA 1 and I is a real-valued function defined on [0,1] with 1(0) = 0 and continuous at 0 and 1. Moreover, the continuity of the function I at 0 and 1 can be further weakened by requiring the upper and lower averages of I on the intervals [0, s] and [1 - s, 1] to converge to 1(0) = 0 and 1(1), respectively. The existence of a value (of norm 1) on pNA and bv'NA also follows from various constructive approaches to the value, such as the limiting approach discussed immediately below and the value formulas approach (Section 7). Other constructive approaches include the mixing approach (Section 6) that provides a value on pNA. Each constructive approach describes a way of "computing" the value for a specific game. The space of all games that are in the (feasible) domain of each approach will form a subspace and the associated map will be a value. Each of the approaches has its advantages and each sheds additional interpretive light on the value obtained. We start with limiting value approaches.

5. Limiting values Limiting values are defined by means of the limits of the Shapley value of the finite games that approximate the given game. Given a game v on (I, C) and a finite measurable field n, we denote by Vn the restriction of the game v to the finite field 7T.The real-valued function Vn (defined on n)
0"'

"'

2134

A. Neyman

can be viewed as a finite game. The Shapley value for finite games 1/!vrr on T[ is given by
( 1/fVn ) (a)

=

1
n!

[v (P:: U a) L R

-

v (P::) ],

where n is the number of atoms of T[, the sum runs over all orders R of the atoms of T[, and P7! is the union of all atoms preceding a in the order R. 5. J. The weak asymptotic value The family of measurable finite fields (that contain a given coalition T) is ordered by inclusion, and given two measurable finite fields T[ and T[t there is a measurable finite field T[ff that contains both T[ and T[t. This enables us to define limits offunctions of finite measurable fields. The weak <isymptotic value of the game v is defined as limn 1/fvn whenever the limit of 1/fvn (T) exists where T[ ranges over the family of measurable finite fields that include T. Equivalently, DEFINITION 3. Let v be a game. A game cpv is said to be the weak asymptotic value of v if for every SEe and every E:> 0, there is a finite subfield T[ of C with SET[ such that for any finite sub field T[' with T[t ~ T[, l1/fvn/(S) - cpv(S) I < E:. REMARKS. (1) Let v be a game. The game v has a weak asymptotic value if and on!y if for every S in C and E:> 0 there exists a finite subfield T[ with SET[ such that for any finite subfield T[t with T[t ~ T[, l1/fvn/(S) -1/fvn(S)1 < 8. (2) A game v has at most one weak asymptotic value. (3) The weak asymptotic value cpv of a game v is a finitely additive game, and cpvII ~
II

IIvll whenever

v E BV.

THEOREM 3 [Neyman (1988), p. 559]. The set of all games having a weak asymptotic value is a linear symmetric space of games, and the operator mapping each game to its weak asymptotic value is (],v(],lue on that space. If ASYMP* denotes all games with bounded variation having a weak asymptotic value, then ASYMP* is a closed subspace of BV with bvtPA c ASYMP*. REMARKS. The linearity, symmetry, positivity and efficiency of 1/f (the Shapley value for games with finitely many players) and of the map v 1-+ Vn imply the linearity, symmetry, positivity and efficiency of the weak asymptotic value map and also imply the linearity and symmetry of the space of games having a weak asymptotic

value. The closedness of ASYMP* follows from the linearity of 1/f and from the inequalities l11/fvnII ~ IlvnII ~ IIvll. The difficult part of the theorem is the inclusion
bvt FA C ASYMP*, on which we will comment later. The inclusion bvt FA C ASYMP* implies in particular the existence of a value on bv~FA.

Ch.56:

Values afGames with Infinitely Many Players

2135

5.2. The asymptotic

value

The asymptotic value of a game v is defined whenever all the sequences of the Shapley values of finite games that "approximate" v have the same limit. Formally: given T in C, a T -admissible sequence is an increasing sequence (Jq, H2, . ..) of finite subfields of C such that T E H) and Ui Hi generates C. DEFINITION 4. A game cpvis said to be the asymptotic value of v, if limk-;-oo 1jrvnk(T) exists and = cpv(T) for every T E C and every T -admissible sequence (Hi)~)' REMARKS. (1) A game v has an asymptotic value if and only if limk-;-oo 1/Ivnk(T) exists for every T in C and every T -admissible sequence P = (Hk)~), and the limit is independent of the choice of P. (2) For any given v, the asymptotic value, if it exists, is clearly unique.
(3) The asymptotic value cpv of a game v is finitely additive, and IIcpvll ~ IIvll when-

ever v E BV. (4) If v has an asymptotic value cpv then v has a weak asymptotic value, which = cpv. (5) The dual of a game v is defined as the game v* given by v*(S) = v(I) - v(I\S). and therefore v has a (weak) asymptotic value if and Then 1jrv; = 1/Ivn = 1jr(v-+;,v*)n, only if v* has a (weak) asymptotic value if and only if (v-+;.v*) has a (weak) asymptotic value, and the values coincide. The space of all games of bounded variation that have an asymptotic value is denoted ASYMP. THEOREM 4 [Aumann and Shapley (1974), Theorem F; Neyman (198la); and Neyman (1988), Theorem A]. The set of all games having an asymptotic value is a linear symmetric space of games, and the operator mapping each game to its asymptotic value is a value on that space. ASYMP is a closed linear symmetric subspace of BV which contains bvl M. REMARKS. (1) The linearity and symmetry of 1/1(the Shapley value for games with finitely many players), and of the map v -+ Vn, imply the linearity and symmetry of the asymptotic value map and its domain. The efficiency and positivity of the asymptotic value follows from the efficiency and positivity of the maps v -+ Vn and Vn -+ 1jrvn. The closedness of ASYMP follows from both the linearity of 1/1 and the inequalities
111/Ivn II

,'-"

(2) Several authors have contributed to proving the inclusion bvl M C ASYMP. Let FL denote the set of measures with at most finitely many atoms and Ma all purely atomic measures. Each of the spaces, bvlNA , bvlFL and bvl Ma is defined as the closed subspace of B V that is generated by games of the form f 0 I-Lwhere f E bvl and I-Lis a probability measure in NA 1, FL 1, or M; respectively.

~

Ilvlln

~

Ilvll.

2136

A. Neyman

Kannai (1966) and Aumann and Shapley (1974) show that pNA C ASYMP. Artstein (1971) proves essentially that pMa C ASYMP. Fogelman and Quinzii (1980) show that pFL C ASYMP. Neyman (198la, 1979) shows that bv'NA C ASYMP and bv'FL C ASYMP respectively. Berbee (1981), together with Shapley (1962) and the proof of Neyman [(1981a), Lemma 8 and Theorem A], imply that bv'Ma C ASYMP. It was further announced in Neyman (1979) that Berbee's result implies the existence of a partition value on bv'M. Neyman (1988) shows that bv'M C ASYMP. (3) The space of games, bv'M, is generated by scalar measure games, i.e., by games that are real-valued functions of scalar measures. Therefore, the essential part of the inclusion bv'M C ASYMP is that whenever f E bv' and fJ, is a probability measure,

f 0 fJ,has an asymptotic value. The tightness of the assumptions on f and fJ,for f

0 fJ,

to have an asymptotic value follows from the next three remarks. (4) pFA rf ASYMP [Neyman (1988), p. 578]. For example, fJ,3 rf ASYMP whenever fJ, is a positive non-atomic finitely additive measure which is not countably additive. This illustrates the essentiality of the countable additivity of fJ, for f 0 fJ, to have an asymptotic value when f is a polynomial. (5) For 0 < q < 1, we denote by fq the function given by fq (x) = 1 if x ~ q and an asymptotic value if and only if fJ, is positive [Neyman (1988), Theorem 5.1]. There are games of the form fq 0 fJ" where fJ,is a purely atomic signed measure with fJ,(I) = 1, for which !l 0 fJ, does not have an asymptotic value [Neyman (1988), Example5.2]. These two comments illustrate the essentiality of the positivity of the measure fJ,.
2

j4 (x) =

0 otherwise.

Let fJ, be a non-atomic

measure

with total mass 1. Then fq

0 fJ, has

(6) There are games of the form f 0 fJ" where fJ, E NA 1 and f is continuous at 0 and

1 and vanishes outside a countable set of points, which do not have an asymptotic value [Neyman (1981a), p. 210]. Thus, the bounded variation of the function f is essential for f 0 fJ, to have an asymptotic value. (7) The set of games DIAG is defined [Aumann and Shapley (1974), p. 252] as the set of all games v in BV for which there is a positive integer k, measures fJ,1, . . ., fJ,k E NA I, and 8 > 0, such that for any coalition SEe with fJ,i(S) - fJ,j (S) :( 8 for all 1 :( i, j :( k, v(S) = O. DIAG is a symmetric linear subspace of ASYMP [Aumann and Shapley (1974)]. (8) If fJ,I, fJ,2, fJ,3, are three non-atomic probabilities that are linearly independent, then the game v defined by v(S) = min[fJ,] (S), fJ,2(S), fJ,3(S)] is not in ASYMP [Aumann and Shapley (1974), Example 19.2]. (9) Garnes of the form fq 0 fJ, where fJ, E M I and 0 < q < 1 are called weighted
majority games; a coalition S is winning, i.e., v(S)

= 1, if

and only if its total fJ,-weight

is at least q. The measure fJ, is called the weight measure and the number q is called the quota. The result bv'M C ASYMP implies in particular that weighted majority games have an asymptotic value. (10) A two-house weighted majority game v is the product of two weighted majority garnes, i.e., v = (fq 0 fJ,)(fc 0 v), where fJ" v E Ml;O < c < 1, and 0 < q < 1. If fJ" v E

Ch. 56:

Values of Games with Infinitely Many Players

2137

NA 1 with f-L -:;FV and q -:;F 1/2, v has an asymptotic value if and only if q -:;F c. Indeed, q < c, (fq 0 f-L)(fc 0 v) - (fc 0 v) E DIAG and therefore (fq
0 f-L)(fc
0

if

v) E

bv'NA + DIAG c ASYMP,

and if q = c -:;F 1/2, (fq 0 f-L)(fc 0 v) '/:. ASYMP [Neyman and Tauman (1979), proof of Proposition 5.42]. If f-L has infinitely many atoms and the set of atoms of v is disjoint to the set of atoms of fJ" (fq 0 f-L)(fc 0 v) has an asymptotic value [Neyman and Smorodinski
f-L E

(1993)].
1 and

(11) Let A denote the closed algebra that is generated by games of the form f
where

0

f-L

NA

f

E bv' is continuous.

It follows from the proof of Proposition 5.5

in Neyman and Tauman (1979) together with Neyman (1981a) that A c ASYMP and
that for every U E

A and v E bvlNA, uv E ASYMP.

The following should shed light on the proofs regarding asymptotic and limiting values. Let n be a finite subfield of C and let v be a game. We will recall formulas for the Shapley value of finite games by applying these formulas to the game Vn. This will enable us to illustrate proofs regarding the limiting properties of the Shapley value 1jrvn of the finite game Vn. Let A (n), or A for short, denote the set of atoms of n. The Shapley value of the game Vn to an atom a in A is given by
1jrvn(a)

=

1
JAI!

:L[v(PaR R

Ua) - v(pJ<-)J,

where IAI stands for the number of elements of A, the summation ranges over all orders R of the atoms in A, and PI! is the union of all atoms preceding a in the order R.

An alternative formula for 1jrvn could be given by means of a family Xa, a E A(n), of
i.i.d. real-valued random variables that are uniformly distributed on (0, 1). The values of Xa, a E A(n), induce with probability 1 an order R on A; a precedes b if and only if Xa < Xb. As Xa, a E A, are i.i.d., all orders are equally likely. Thus, identifying sets with their indicator functions, and letting I (Xa ( Xb) denote the indicator of the event Xa ( Xb, the value formula is

1jrvn(a) = E[V(

:LbI(Xb
bEA

(Xa?)

-

V(:LbI(Xb
bEA

<

Xa?)

1

Thus, by conditioning on the value of Xa, and letting S(t), 0 ( t ( 1, denote the union of all atoms b of n for which Xb ( t, we find that

-

1jrvn(a)

= 11 E(v(S(t)

Ua) - v(S(t) \a?)dt.

2138

A. Neyman

Assume now that the game v is a continuously differentiable function of finitely many non-atomic probability measures; i.e., v = f 0 fJ-, where fJ-= (fJ-J, . . ., fJ-n) E (NA 1)n and f : R(fJ-) -+ R is continuously differentiable. Then
n i=1

v(S(t) Ua) - VeSel)\a) ="L.t -(Y)fJ-i(a), ac. <;1
where Y is a point in the interval [fJ-(S(t) \ a), fJ-(S(t) U a)]. Thus, in particular, Iv(S(t) U a) - veSel) \ a)1 is bounded. L(/ElT For any scalar measure v, VeSel)) is the sum LaEA(lT) v(a)I(Xa ~ f), i.e., it is a sum of IA(n)1 independent random variables with expectation tv (a) and variance t(1 t)v2(a). Therefore, E(v(S(t))) = tv (I) and

af

Var(S(t)) = tel - t) ..L

v2(a) ~ t(1-

t)v(I) maxi v(a): a E A(n)}.

(/EA(lT)

Therefore, if fJ-= (fJ-l, . . . , fJ-11) is a vector of measures, and (nk)~1 is a sequence of finitefields with max{lfJ-j(a)I: 1 ~ j ~ n, a E A (nk)} -+k-+oo 0, fJ-(S(t)) converges in distribution to t fJ-(I); so if T is a coalition with T E nk for every k, then
n

..L
lIcl " aEA(lTk)

[V(Sk(t) Ua) - V(Sk(t) \a)] -+k-+OO..L -(tfJ-(I))fJ-i(T),
. 1=1 aXi

af

and therefore

limk-+oo

1/fVlTk(T)

exists

and

k-+oo

lim

1/fVlTk(T)

=

10

r' f

JJ-(T)

(t fJ-(I) ) dt.

For any T-admissible sequence (nk)~1 and non-atomic measure v, max{v(a): a E A(nk)} -+k-+oo O. Therefore any game v = f 0 fJ-,where f is a continuously differentiable function defined on the range of fJ-, {fJ-(S): S E C}, has an asymptotic value cpv whenever fJ-= (fJ-l , . . . , fJ-11) is a vector of non-atomic measures. Also, given any E > 0 and a non-atomic element v of FA +, there exists a partition n such that max{v(a): a E A(n)} < E. Therefore v = f 0 fJ-has a weak asymptotic
value cpv whenever fJ- = (fJ-I , . . . , fJ-n) is a vector of non-atomic bounded finitely additive

measures. In both cases the value is given by
cpv(S)

=

10

r'

111(0') (tfJ-(I)) dr.

The following tool which is of independent interest will be used later on to show how to reduce the proof of bvlNA c ASYMP (or bvl M c ASYMP) to the proof that fq °fJ-EASYMPwhere fq(x) =I(x ?;q),O<q < 1.,andfJ-ENAl (orfJ-EM1).

Ch. 56:

Values of Games with Infinitely Many Players

2139

Let v be a monotonic game. For every a > 0, let va denote the indicator of v ~ a, i.e., va (S) = I (v(S) ~ a), i.e., va (S) = 1 if v(S) ~ a and 0 otherwise. Note that for every coalition S, v(S) = 1000va (S) da. Then for every finite field J[, ljr(va):rr da. Therefore if (J[k)~1 is aT -admissible seljrv:rr= 1000ljr(va):rr da = 10V(1) quence such that for every a > 0, limk-..oo 1fr(va):rrk (T) exists, then from the Lebesgue's bounded convergence theorem it follows that limk-..oo 1frV:rrk (T) exists and
(T) = lim 1frV:rrk k-..oo

l

V(I)

a

(k-..oo lim 1frv~k(T))

da.

Thus, if for each a > 0, va = I (v ~ a) has an asymptotic value <pva , then v also has an asymptotic value <pv,which is given by 1000<pva(T) da = <pv(T). The spaces ASYMP and ASYMP* are closed and each of the spaces bv' NA, bv'M and bv' FA is the closed linear space that is generated by scalar measure games of the form f 0 /-l with f E bv' monotonic and /-l E NA 1, /-l E M 1, /-l E FA 1 respectively. Also, each monotonic f in bv' is the sum of two monotonic functions in bv', one right continuous and the other left continuous. If f E bv', with f monotonic and left continuous, then (f 0 /-l)* = go /-l, with g E bv' monotonic and right continuous. Therefore, to show that bv'NA C ASYMP (bv'M c ASYMP or bv'FA C ASYMP*),

it suffices to show that f
(f
/-l)a(S)

continuous function f in bv' and /-l E NA 1 (/-l E M1 or /-l E FA 1). Note that
0

0 /-l E

ASYMP (or E ASYMP*) for any monotonic and right

= I(f(/-l(S))

~ a) = I (/-l(S) ~ inf{x: f(x) ~ an = fq(/-l(S)),

where q = inf{x: f(x) ~ a}. Thus, in view of the above remarks, to show that bv'NA C ASYMP (or bv'M c ASYMP), it suffices to prove that fq °/-l E ASYMP for any 0< q < 1 1 and /-l E NA (or /-l E Ml), where fq (x) = 1 if x ~ q and = o otherwise. The proofs of the relations fq °/-l E ASYMP for 0 < q < 1 and /-l E NA 1 (or /-l E M or /-l E M 1) rely on delicate probabilistic results, which are of independent mathematical

d

interest. These results can be formulated in various equivalent forms. We introduce them that are uniformly distributed on the open unit interval (0, 1). sequence W = (wi)7=1 or W = (Wi)~l of positive numbers, process, SwO, or SO for short, given by S(t) = Li WJ(Xi a process is called a pure jump Poisson bridge). The proof
1 /-l E NA uses the following

as properties of Poisson bridges. Let X 1, X2, . .. be a sequenceof i.i.d. random variables
With any finite or infinite we associate a stochastic
:::;;t) for 0:::;; t :::;;1 (such

of fq

0 /-l E

ASYMP for

result, which is closely related to renewal

theory.

PROPOSITION 4 [Neyman (1981a)]. For every 8 > 0 there exists a constant K > 0, such that if W1, ..., Wn are positive numbers with L7=1 Wi = 1, and K max7=1 Wi < q < 1 - K max7=1 Wi then
n .' -"

~]Wi
i=1

- Prob(S(Xi)

E [q, q + wd)1 < 8.

2140

A. Neyman for f.L E Ma uses the following result, due to

The proof of fq 0 f.L E ASYMP Berbee (1981).

PROPOSITION 5 [Berbee (1981)]. For every sequence (Wi)~l with Wi > 0, and every 0 < q < L~I Wi, the probability that there exists 0 ~ t ~ 1 with Set) = q is 0, i.e.,

Prob(3t, S(t) = q) = O. 6. The mixing value
An order on the underlying space I is a relation R on I that is transitive, irreflexive, and complete. For each order R and each s in I, define the initial segment I (s; R) by 1(8; R) = {t E I: sRt}. An order R is measurable if the a-field generated by all the initial segments I (s; R) equals C. Let ORD be the linear space of all games v in BY such that for each measurable R, there exists a unique measure cp(v; R) satisfying cp(v; R) (I (s; R))

= v(I

(s; R))

forallsEI.

Let f.Lbe a probability measure on the underlying space (I, C). A f.L-mixing sequence is a sequence (el, fEh, . ..) of f.L-measure-preserving automorphisms of (I, C), such that for all S, T in C, limk-+oo f.L(S n ek T) = f.L(S)f.L(T). For each order R on I and auto{} sRt. If v morphism (9 in 9 we denote by eR the order on I defined by eseRet is absolutely continuous with respect to the measure f.L we write v ? f.L. DEFINITION 5. Let v E ORD. A game cpv is said to be the mixing value of v if there is a measure f.Lvin NA I such that for all f.Lin NA 1 with f.Lv ? f.Land all f.L-mixing sequences

((9 I, (92, . . .), for all measurable orders R, and all coalitions S, limk-+00 cp (v; ek R) (S)

exists and

= (cpv)(S).

The set of all games in ORD that have a mixing value is denoted MIX. THEOREM 5 [Aumann and Shapley (1974), Theorem E]. MIX is a closed symmetric linear subspace of BY which contains pNA, and the map cp that associates a mixing value cpv to each v is a value on MIX with IIcpll~ 1.

7. Formulas for values
Let v be a vector measure game, i.e., a game that is a function of finitely many measures.

Such games have a representation of the form v = f

0 f.L,

where

f.L

= (f.L I , . . . , f.Ln)

is a

vector of (probability) measures, and f is a real-valued function defined on the range . R(f.L) of the vector measure f.Lwith 1(0) = O.

Ch.56:

Values of Games with Infinitely Many Players

2141

When f is continuously differentiable, and JL = (JL I, . . . , JLn) is a vector of nonatomic measures with JLi(1) =1= 0, then v = f 0 JL is in pNAx (C pNA c ASYMP) and the value (asymptotic, mixing, or the unique value on pNA) is given by Aumann and Shapley (1974, Theorem B), q;v(S) - q;(f
0

JL)(S) = 11 f~(s) (t JL(1)) dt

(where f~(s) is the derivative of f in the direction of JL(S)); i.e.,
n

q;(fOJL)(S)

=

1

"JLi(S)
i=1

L.

10

-(tJLI(1),...,tJLn(l))dt.

af

axI

The above formula expresses the value as an average of derivatives, that is, of marginal contributions. The derivatives are taken along the interval [0, JL(1)], i.e., the line segment connecting the vector 0 and the vector JL(1). This interval, which is contained in the range of the non-atomic vector measure JL, is called the diagonal of the range of JL, and the above formula is called the diagonal fonnula. The formula depends on the differentiability of f along the diagonal and on the game being a non-atomic vector measure game. Extensions of the diagonal formula (due to Mertens) enable one to apply (variations of) the diagonal formula to games with discontinuities and with essential nondifferentiabilities. In particular, the generalized diagonal formula defines a value of norm 1 on a closed subspace of BV that includes all finite games, bvl FA, the closed algebra generated by bvl NA, all games generated by a finite number of algebraic and lattice operations I from a finite number of measures, and all market games that are functions of finitely many measures. The value is defined as a composition of positive linear symmetric mappings of norm 1. One of these mappings is an extension operator which is an operator from a space of games into the space of ideal games, i.e., functions that are defined on ideal coalitions, which are measurable functions from (1, C) into ([0, 1], B). The second one is a "derivative" operator, which is essentially the diagonal formula obtained by changing the order of integration and differentiation. The third one is an averaged derivative; it replaces the derivatives on this diagonal with an average of derivatives evaluated in a small invariant perturbation of the diagonal. The invariance of the perturbed distribution is with respect to all automorphisms of (1, C). First we illustrate the action and basic properties of the "derivative" and the "averaged derivative" mappings in the context of non-atomic vector measure games. The derivative operator [Mertens (1980)] provides a diagonal formula for the value of vector measure games in bvlNA; let JL= (JLI,..., JLn) E (NA+)n and let f: R(JL) ~ JR
.' ~ 1 Max and min.

2142 with 1(0)

A. Neyman

value of

I

= 0 and I
0 fl,

0 fl, in bvfNA. Then

I

is given by

is continuous at fl,(/) and fl,(0) = 0, and the

<p(1 0 fl,)(S)

= =

8---70+ 8

lim
lim

1
-

1-8

10
1-8

[I (tfl,(/) + 8fl,(S)) [/(tfl,(/) +8fl,(S))

I(tfl,(/))] - I(tfl,(/)

dt -8fl,(S))]dt.

8---70+ 28

-

1

1

8

The limits of integration in these formulas, 1 - 8 and 8, could be replaced
respectively whenever fl,(/) and fl,(0).
Let fl,

I

is extended

to JRn in such a way that

I

remains

by 1 and 0 continuous at

= (fl,I,..., fl,n) E (NA+)n be a vector of measures. Consider the class F(fl,) of all functions I: R(fl,) -+ JR with 1(0) = 0, I continuous at 0 and fl,(/), 10 fl, of bounded variation, and for which the limit

8---7028

Urn

~

18

1-8[J(tfl,(/) +8fl,(S)) - I (tfl,(/) -~fl,(S))]dt

exists for all Sin C. Note that IE F(fl,) whenever I is continuously differentiable with 1(0) = o. An example of a Lipschitz function that is not in F (fl,) where fl, is a vector of 2 linearly independent probability measures is I(XI, X2) = (X] - X2) sin log IxI - x21. Denote this limit by Dv(S) and note that Dv is a function of the measures fl,l, . . . , fl,n; i.e., Dv = 10 fl, for some function I that is defined on the range of fl,. The derivative operator D acts on all games ofthe form 10 fl" fl, E (NA +)n , and I E F(fl,); it is a value of norm 1 on the space of all these games for which D(I 0 fl,) E FA. Let v = 10 fl" where fl, E (NA +)n and IE F(fl,). Then D(I 0 fl,) is of the form j 0 fl" where I- is linear on every plane containing the diagonal [0, fl,(/)], I(fl,(/)) = I(fl,(/)) and III 0 fl, II ~ III 0 fl, II. There is no loss of generality in assuming that the measures fl, I, . . . , fl,n are independent (i.e., that the range of the vector measure R(fl,) is fulldimensional), and then we identify I with its unique extension to a function on JRnthat is linear on every plane containing the interval [0, fl,(/)].

The averaging operator expresses the value of the game

I

0 fl,

(= D(I

0 fl,) for

I in

F(fl,)) as an average, E(L7=1 ~? (Z)fl,i), where Z is an n-dimensional random variable whose distribution, P ~, is strictly stable of index 1 and has a Fourier transform ~ to be described below. When fl,1, . . . , fl,n are mutually singular, Z = (Z], . . . , Zn) is a vector of independent "centered" Cauchy distributions, and then

,Cy) = exp( - EI";(l?)

= exp( E

III";II).

Ch. 56:

Values of Games with infinitely Many Players

2143

Otherwise,

let vENA + be such that f-L 1, . . . , f-Ln are absolutely

to v; e.g., v

=L
=

f-Li . Define
11

NIL : JRI1 -+ JRn by

continuous

with respect

NIL(Y)

1

~(df-LddV)Yi

dv.

TheIl z is an n-dimensional random variable whose distribution has a Foqrier transform
~z(y) = exp( -NIL(Y))' Let Q be the liIlear space of games generated by games of the form v ---:f 0 f-L where -:(f-L1, . . . , f-Ln) E (NA+)n is a vector of linearly independent non-atomic measures f-L and f E F(f-L). The space Q is symmetric aIld the m:ip cp; Q -+ FA given by
c

cp(f

0

f-L) = Ep/L(JIL(S) (z))

-:- Ep/L

(t :t
i=l
I

(z),q (S)

)

is a value of norm 1 on Q and therefore has an extension to a value of norm 1 on the closure Q of Q. The space Q contains bv'NA, the closed algebra generated by bv'NA, all games generated by a finite number of algebraic :ind lattice operatioIls from a finite number of non-atomic measures and all games of the form f 0 f-L where f-LE (NA+)n and f is a continuous COIlcave fUIlction on the range of f-L. To show that Ep/L(]IL(S)(z)) is well defined for any f in F(f-L), OIle shows that] is Lipschitz aIld therefore on the one hand [fez + 8J-L(S)) - f(z)]j8 is bounded and contiIluous, and on the other h,:nd, given S in C, for almost every Z (with respect to the Lebesgue measure on JRn) fILes) (z) exists and is bounded (as a function of Z). The distrib_utioIl PILis absolutely continuous with respect to ~ebesgue measur~ and therefore Ep/l (f1L(S)(z)) is well defined and equals lim6-".0 Ep/L([f (z + 8f-L(S)) - f (z) ]18) (using the Lebesgue bounded convergence theorem). To show that Ep/l (flL(s) (z)) is finitely additive in S, assume that S, T are two disjoint coalitions. For any 8> 0, let P~ be the translation of the measure PIL by -8f-L(S), i.e.,

P;(B) = PIL(B +8f-L(S)).
Since P~ converges in norm to PIL as 8 -+ 0 (i.e., IIP~ - FILII-+6-".00), and [fez + 8f-L(T)) - ](z)]18 is bounded, it follows that

6-".0
.c ,> ~.

lim Ep/l

([] (z + 8f-L(S)+ 8f-L(T)) - ] (z + 8f-L(S)) ]18)
-

= 6-".0 limEp8([](z+8f-L(T)) /L
= limEpJ[J(z 6-".0 +8f-L(T))

](z)]18) ](z)]18) = Ep/L(flL(T)(z)).

2144

A. Neyman

Therefore
Ep" (]fI(SUT)

(Z))

=

lim Ep" ([](z + 8M(S)+ 8M(T)) 0--+0

- ](z + 8M(S))+ l(z + 8M(S)) - l(z) ]/8)
= Ep,~(]fI(S) (z)) + Ep" (]fI(T) (z)).

We now turn to a detailed description of the mappings used in the construction of a value due to Mertens on a large space of games. Let BU, C) be the space of bounded measurable real-valued functions on U, C), and
B (I, C)

sets). A function v on BtU, C) is constant sum if vel) + v(l - f)

t

=

{f E B (I, C): 0 ~ I ~ I} the space of "ideal coalitions"

(measurable

fuzzy

IE BtU, C). It is monotonic if for every f, g E B{U, C) with f ~ g, v(f) ~ v(g). It isfinitely ~dditive if v(f + g) = v(f) + v(g) whenever f, g E B{U, C) with 1+ g ~ 1. It is of bounded variation if it is the difference of two monotonic functions, and the

= v(1)

for every

variation norm IIv IIlBv (or increasing sequences 0 ~
on BtU, C) by (G*f)(s)

II fl

v

~ h ~ ... ~ 1 in BtU, C). The symmetry group 9 acts

II

for short) is the supremumof the variation of v over all

= I(Gs).

DEFINITION 6. An extension operator is a linear and symmetric map 1jffrom a linear symmetric set of games to real-valued functions on BtU, C), with (1jfv)(O) = 0, so that (1jfv)(1) = vU), II1jfvlllBv ~ IIvll, and 1jfv is finitely additive whenever v is finitely additive, and 1jfv is constant sum whenever v is constant sum. It follows that an extension operator is necessarily positive (i.e., if v is monotonic so is 1jfv) and that every extension ope.!ator y! on a linear subspace Q of BY has a unique extension to an extension operator 1jfon Q (the closure of Q). One example of an extension operator is Owen's (1972) multilinear extension for finite games. For another example, consider all games that are functions of finitely many

non-atomic probability measures, i.e., Q = {f 0 (MI..., Mn): n a positive integer, Mi E NA I and f : R(f1, I, . . . , Mn) -+ IP&. with f (0) = O},where R(M 1, . . . , Mn) = {(f1, 1(S), . . . ,
f1,11 (S)): S E C} is the range of the vector measure (MI, . .., Mn). Then Q is a linear and symmetric space of games. An extension operator on Q is defined as follows: for f1, = (MI, . . . , f1,11) E (NA 1)'1 and g E B{ U, C), let M(g) = (il g dMI, . . . , II g dMn) and Then it is easy to verify that 1jf is well defined, lindefine 1jf(f 0 f1,)(g) = f(M(g)). ear, and symmetric, that 1jfv(1) = vU), 1jfv is finitely additive whenever v is, and 1jfv
is constant sum whenever v is. The inequality l11jfvlllBv ~ Ilvll follows from the re-

sult of Dvoretsky, Wald, and Wolfowitz [(1951), p. 66, Theorem 4], which asserts that
for every M E (NAI)11 and every sequence 0 ~ !I ~ h ~ ~ fk ~ 1 in BtU, C), there exists a sequence (0 C Sj C S2 C . .. C Sk C I with f1,(fi) = M(Si). This ex'"

tension operator satisfies additional properties:

1jf(f

0

v)

=f

0

1jfv whenever
(1jfv)

f: IP&. -+ IP&. with f(O)

and 1jf(v v u)

=

v (1jfu). The NA-topology on BU, C) is the minimal topology

(1jfv)(S) = v(S), 1jf(vu) = (1jfv)(1jfu), = 0, 1jf(v /\ u) = (1jfv) /\ (1jfu)

Ch. 56:

Values of Games with Infinitely Many Players

2145

for which the functions I J + ~(f) are continuous for each ~ E NA. For every game v E pNA, lfrv is NA-continuous. Moreover, as the characteristic functions Is E Bt (I, C),
S E C, are NA-dense in B~ (I, C), lfrv is the unique NA-continuous function on B~ (I, C) with lfrv(S) = v(S). Similarly, for every game v inpNA' - the closure in the supremum norm of pNA - there is a unique function v : B (I, C) ~ IR.which is NA -continuous and

whenever I: IR.~ IR.is continuous with f(O) = O. Another example is the following natural extension map defined on bv' M. Assume that ~ E M1 with a set of atoms A = A(~). Let v = I 0 ~ with I E bv' and ~ E NA 1. ~i with ~i E M1 and Ii E bvl we define lfrv(g) = L7=1 Vi(g). It is possible to show that IllfrvIIIBV ~ II v II and thus in particular v is well defined (i.e., it is independent of the representation of v) and can be extended to the closure of all these finite sums, i.e., it defines an extension map on bvl M. The space bv' NA is a subset of both bv'M and Q and the above two extensions that were defined on Q and on bvl M coincide on bvl NA, and thus induce (by restriction) an extension operator lfr on bv' NA and on pNA. The extension on bvl NA can be used to provide formulas for the values on bvl NA and on pNA. Given v in Q, we denote by v the extension (ideal game) lfrv. We also identify S with the indicator of the set S, xs, and the scalar t also stands for the constant function in B{ (I, C) with value t. Then, if v E pNA, for almost all 0 < t < 1, for every coalition S, the derivative av(t, S) =: lim8-+0[V(t) + 8s) - v(t)]j8 exists [Aumann and Shapley (1974), Proposition 24.1] and the value of the game is given by
For g E B~(I, C) define lfrv(g) = E(/(~(gAC + LaEA Xaa?) where (Xa)aEA are independent {a, l}-valued random variables with E(Xa) Ac

tiCS) = v(S). Moreover, the map v J+ V is an extension operator on pNA' and it satisfies: vu = vii,v 1\ u = v 1\u and v v u = v V u for all v, u E pNA' and (I ~ v) - 10 v

~

=

=I g(a).

\

A and If v =

L7=1 Vi, Vi = Ii

0

cpv(S) =

11

av(t, S) dt.

For any game v in bvl NA, the value of v is given by cpv(S)

= =

1 1-8 lim [v(t + 8S) - v(t)] dt 8-+0+ 8 0

1

8-+028

lim-

1

1-181

1

181

[v(t+8S)-v(t-8S)]dt.

More generally, given an extension operator lfr on a linear symmetric space of games Q, we extend v = lfrv, for v in Q, to be defined on all of BU, C) by vel) =

v (max[O, min(1, f)]) (and then we can replace the limits 181and 1 - 181in the above
integrals with 0 and 1 respectively, and define the derivative operator 1/f D by
lfrDV(X)

.<, :..,

=

8-+0

hm

.

10

1v(t+8X)-v(t-8X)
28

dt

~

2]46

A. Neyman

whenever the integral and the limit exists for all X in B(I, C). The integral exists whenever v has bounded variation. The derivative operator is positive, linear and symmetric. Assuming continuity in the NA-topology of v at 1 and 0, VrDalso satisfies
Vrov(a + (3X)

every plan in B(I, C) containing the constants and VrDis efficient, i.e., VrDv(l) = v(1).
The following [due to Mertens (1988a)] will (essentially) extend the previously mentioned extension operators to a large class of games that includes bv' FA and all games of the form I 0 ~ where ~ = (~I, .. ., ~n) E NA. Intuitively we would like to define Vr E vel) for I in Bt (I, C) as the "limit" of the expectation v(S) with respect to a distribution P of a random set that is very similar to the ideal set I. Let :.Fbe the set of all triplets a = (n, v, 8) where n is a finite subfield of C, v is a finite set of non-atomic elements of FA and 8 > O. :.Fis ordered by a = (n, v, 8) ~ a' =

=

av(1)

+ (3[VrDV](X)

for every a, (3 E IR, so that VrDV is linear

on

(n', vi, 8') if n' =:) n, v' =:) v and 8' ~ 8. (:.F,~) is filtering-increasing with respect to this order, i.e., given a = (n, v, 8) and a' = (n', v', 8') in:.F there is a" = (nil, v", 8") with a" ~ a and a" ~ a'. For any a = (n, v, 8) in :.F and I in Bt(I, C), Pa,f is the set of all probabilities P with finite support on C such that ILSEC SpeS) - II IEp(S) - 11< 8 uniformly on in n with TI n T2 = 0, T[ n S is independent of T2 n S, and for all ~ in v~(S n T]) =
I (i.e., for every tEl, I LSEC P(S)I(t E S) - J(t)1 < 8), and such that for any Tl, T2

=

IT' I d~. That Pa,f f. 0 follows from Lyapunov's convexity theorem on the range of a vector of non-atomic elements of FA. For any game v, and any I E Bt(I, C), let v(f)=lim sup Ep(v(S))=lim
/

aEF PEP ex,--

aEF

sup

Lv(S)P(S)

PEP ex,f S C E

and

Q(f) = Jim
aEF

PEPex,f

inf

Ep (v(S)).

The definition of v is extended to all of B(I, C) by v(1) = v (max(O,min(1, I))).

Note that if v is a game with finite support, then for any I in Bt(I, C), v(f) = Q(J) coincides with Owen's multilinear extension, i.e., letting T be the finite support of v, v(f)

=

.seT SES

Il (1 ~,[Il I(s) sET\S

-

I(S))] v(S)

and whenever v is a function of finitely many non-atomic elements of FA, v], . . . , vn, i.e., v = g 0 (VI, .. ., vn) where g is a real-valued function defined on the range of (Vj,...,vn),thenforany IEBt(I,C) V(f)=Q(f)=g(! fdVI,...,! fdvn). .

Ch. 56: Let V8

Values of Games with Infinitely Many Players

2147

= {X E

for which sup{ti(X) - .!!.(X): X E Vd -+ 0 as 8 -+ 0+. Obviously, D => D8 where D8 is the set of all games v for which ti(X) = .!!. (X) for any X E V8. Next we define the derivative operators C{! D; its domain is the set of all games v in D for which the integrals (for sufficiently small 8 > 0) and the limit
[C{!DV](X) = tim. ,-;.0 1 0
.

t

Bt (1, C): sup X - inf X ~ d. Let D be the space of all games v in BV

1 ti(t+TX)-ti(t-TX) 2T

dt

exist for every X in B(1, C). The integrals always exist for games in BV. The limit, on the other hand, may not exist even for games that are Lipschitz functions of a non-atomic signed measure. However, the limit exists for many games of interest, induding the concave functions of finitely many non-atomic measures, and the algebra generated by bvl FA. Assuming, in addition, some continuity assumption on ti at 1 and 0 (e.g., ti(f) -+ ti(1) as inf(f) -+ 1 and ti(!) -+ ti(O) as sup(f) -+ 0) we obtain that <(JDV obeys the following additional properties: C{!DV(X)+ C{!Dv(1- X) = v(1) whenever v is a constant sum; [C{!Dv](a+ bx) = av(1) + b[C{!DV] (x) for every a, b E R THEOREM 6 [Mertens (1988a), Section 1]. Let Q
Q is a closed symmetric is a value on Q. space that contains

DIFF, DIAG and bVI FA and C{! D : Q -+ FA

=

{v E DomC{!D:

C{!DV E FA}.

Then

For every function w : B (1, C) -+ JR.in the range of C{! D, and every two elements X, h in B(I, C), we define Wh(X) to be the (two-sided) directional derivative of w at X in the direction h, i.e.,

Wh(X) = lim [w(X + 8h) - w(X - 8h) ]/(28)
8-;.0

..c :.:-.

whenever the limit exists. Obviously, when w is finitely additive then Wh(X) = w (h). I,...,J.-Ln) For a function w in the range of C{! D and of the form w = foIL where IL = (J.-L is a vector of measures in NA, f is Lipschitz, satisfying f(alL(I) + bx) = af(J.-L(I? + bf (x) and for every y in L (R(J.-L? - the linear span of the range of the vector measure IL - for almost every x in L (R(J.-L? the directional derivative of f at x in the direction y, fy(x) exists, and then obviously Wh(X) = fJk(h) (J.-L(X?. The following theorem will provide an existence of a value on a large space of games as well as a formula for the value as an average of the derivatives (C{! DOC{! E V)h (X) = Wh(X) where X is distributed according to a cylinder probability measure on B(I, C) which is invariant under all automorphisms of (1, C). We recall now the concept of a cylinder measure on B(1, C). The algebra of cylinder sets in B(I, C) is the algebra generated by the sets of the = (J.-LI, . . . , ILn) is any vector of firm IL-I (B) where B is any Borel subset of JR.nand J.-L measures. A cylinder probability is a finitely additive probability P on the algebra of

L

2148

A. Neyman

, . . . , ~n) , p 0 ~ -I is a countably additive probability measure on the Borel subsets of JRT1. Any cylinder probability P on B (I, C) is uniquely characterized by its Fourier transform, a function on the dual Let P be the cylinder probability measure that is defined by F(~) = Ep(exp(i~(X)). on B(I, C) whose Fourier transform Fp is given by Fp(~) = exp( -II~II). This cylinder measure is invariant under all automorphisms of (I, C). Recall that (I, C) is isomorphic to [0, 1] with the Borel sets, and for a vector of measures ~ = (~1, . . ., ~T1)' the Fourier

cylinder sets such that for every vector measure ~

= (~I

and P 0 ~-I is absolutely continuous with respect to the Lebesgue measure, with moreover, continuous Radon-Nikodym derivatives. Let Q M be the closed symmetric space generated by all games v in the domain of cP D
such that either cP DV E FA, or cP D (v) is a function of finitely many non-atomic measures. THEOREM 7 [Mertens (1988a), Section 2]. Let v E QM. Then for every h in B(I, C), (cp [) (v)) h (X) exists for P almost every X and is P integrable in x, and the mapping of each game v in QM to the game cpv given by cpv(S)

transform of P 0 ~-l,

Fpojk-' is given by FPorl (y) = exp(-Njk(Y))'

=

f

(CPD(v))s(X)dP(X)

is a value of norm 1 on QM. REMARKS. The extreme points of the set of invariant cylinder probabilities on B(I, C) have a Fourier transform FI71,a (~) = exp(im~(1) - a II~II) where m E JR, a ~ O. More precisely, there is a one-to-one and onto map between countably additive measures Q on JRx JR+ and invariant cylinder measures on B (I, C) where every measure Q on JRx JR+ is associated with the cylinder measure whose Fourier transform is given by
FQ(fL)

=

1JR:x JR:+ FI71,a(~) dQ(m,

a).

The associated cylinder measure is non degenerate if Q(r = 0) = 0, and in the above value formula and theorem, P can be replaced with any invariant nondegenerate cylinder measure of total mass 1 on B(I, C). Neyman (2001) provides an alternative approach to define a value on the closed space

spanned by all bounded variation games of the form f

0

~ where ~ is a vector of non-

atomic probability measures and f is continuous at ~(0) and ~(I), and QM. This alternative approach stems from the ideas in Mertens (1988a) and expresses the value as a limit of averaged marginal contributions where the average is with respect to a distribution which is strictly stable of index 1. For any JR"-valued non-atomic vector measure ~ we define a map CPt from Q(~) the space of an games of bounded variation that are functions of the vector measure ~ and are continuous at ~(0) and at ~(I) - to BV. The map CPt depends on a small positive constant l5 > 0 and the vector measure ~ ='(~l,..., ~T1)'

Ch. 56:

Values afGames with Infinitely Many Players

2149

For 8 > 0 let h (t) = I (38 ~ t < 1 - 38) where I stands for the indicator function. The essential role of the function h is to make the integrands that appear in the integrals used in the definition of the value well defined.

Let fL = (fL l, . . . , fLn) be a vector of non-atomic probability measures and f : R(fL) -+ JRbe continuous at fL(0) and fL(1) and with f 0 fL of bounded variation. It follows that for every x E 2R(fL) - fL(1), SEe, and t with h (t) = 1, t fL (1) + 8x and t fL(1) + 8x + 8fL(S) are in R(fL) and therefore the functions t r-+ h (t) f (t fL (1) +
+ 8x + 8fL(S)) are of bounded variation on [0, 1] and thus in particular they are integrable functions. Therefore, given a game f 0 fL E Q (fL), the function Ff,f-/-> defined on all triples (8, x, S) with 8 > 0 sufficiently small (e.g., 8 < 1/9), x E JRnwith 8x E 2R(fL) - fL(1), and SEe by 8x) and t r-+ h(t)f(tfL(1)

.

Ff.fL(8,x,S) = r h(t) 10
is well defined.

1

f (t fL(1) + 82x + 83 fL(S)) - f (t fL(1) + 82x)

83

dt,

Let p~ be the restriction of PfL to the set of all points in {x E JRn: 6X E 2R(fL) fL(1)}. The function x r-+ Ff,fL(6, x, S) is continuous function CPtdefined on Q (fL) x C by CP~(fOfL,S)= and bounded and therefore the

l

AF(fL)

Ff,fL(6,x,S)dP;(x),

where AF(fL) stands for the affine space spanned by R(fL), is well defined. The linear space of games Q(fL) is not a symmetric space. Moreover, the map CPt violates all value axioms. It does not map Q (fL) into FA, it is not efficient, and it is not symmetric. In addition, given two non-atomic vector measures, fL and v, the operators CPt and cP~ differ on the intersection Q (fL) n Q (v). However, it turns out that the violation of the value axioms by CPt diminishes as 6 goes to 0, and the difference CPt (f 0 fL) - cP~(g 0 v) goes to 0 as 6 -+ 0 whenever f 0 fL = go v. Therefore, an appropriate limiting argument enables us to generate a value on the union of all the spaces Q (fL). Consider the partially ordered linear space ?, of all bounded functions defined on the open interval (0,1/9) with the partial order h >- g if and only if h(6) ?: g(6) for all sufficiently small values of 6 > O. Let L: ?, -+ JR be a monotonic (i.e., L(h) ?: L(g) whenever h >- g) linear functional with L(I) = 1. Let Q denote the union of all spaces Q(fL) where fL ranges over all vectors of finitely many non-atomic probability measures. Define the map cp: Q -+ JRCby cpv(S) = L(cpt (v, S)) whenever v E Q(fL). It turns out that cp is well defined, cpv is in FA (whenever v E Q) and cp is a value of norm 1 on Q. As cp is a value of norm 1 on Q and the continuous extension of any value of norm 1 defines a value (of norm 1) on the closure, we have:

.,'~-

PROPOSITION 6. cpis a value of norm 1 on Q.

2150

A, Neyman

8. Uniqueness of the value of nondifferentiable games
The symmetry axiom of a value implies [see, e.g., Aumann and Shapley (1974), p. 139] that if f-L = (fl, I, . . . , fl,n) is a vector of mutually singular measures in NA 1, f: [0, 1]n -+ R and v = f a f-Lis a game which is a member of a space Q on which a value cp is defined, then
11

cpv = Lai(f)fl,i. i=1

= =1 symmetric in the n coordinates all the coefficients ai (f) are the same, i.e., ai (f) = f (1, . . . , 1). The results below will specify spaces of games that are generated by * ve~tor measure games v = f a fl, where fl, is a vector of mutually singular non-atomic probability measures and f : [0, 1]/1-+ ffi.and on which a unique value exists. Denote by M~ the linear space of function on [0, l]k spanned by all concave Lipschitz functions f: [0, l]k -+ ffi.with f(O) = 0 and fy(x) ~ fy(tx) whenever x is in
The efficiency of cp implies that
ai (f)

'L7

f (1, . . . , 1), and if in addition

f

is

the interior piecewise

of [0, l]k, Y E ffi.~ and 0 < t ~ 1. M: is the linear space of all continuous linear functions f on [0, l]k with f(O) = O.

DEFINITION 7 [Haimanko (2000d)]. CONs (respectively, LINs) is the linear span of games of the form f a f-Lwhere for some positive integer k, f E M~ (respectively, f E M~) and f-L = (f-L 1, . . . , fl,k) is a vector of mutually singular measures in NA 1. The spaces CONI' and LINs are in the domain of the Mertens value; therefore there is

a value on CONs and on LIN~. If f E M~ or f E M:, the directional derivative fy (x)
exists whenever x is in the interior of [0, l]k and Y E ffi.k, and for each fixed x, the directional derivative of the function Y ~ fy (x) in the direction Z E ffi.k,
' I1m fy+z(x)-fy(x)

~o+

8

,

measures in NA I and cp M is the Mertens value,
CPM (f a f-L)(S) = 11 af(th; x; fl,(S)) dt

exists and is denoted alex; y; Z), If f-L = (f-LI,..., fl,k) is a vector of mutually singular

where lk = (1, . . ., 1) E ffi.k and X = (X 1, . . ., Xk) is a vector of independent random variables, each with the standard Cauchy distribution.

THEOREM 8 [Haimanko (2000d) and (2001b)]. The Mertens value is the unique value on CONI' and on LINs.

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9. Desired properties of values The short list of conditions that define the value, linearity, positivity, symmetry and efficiency, suffices to determine the value on many spaces of games, like pNAoo, pNA and bvlNA. Values were defined in other cases by means of constructive approaches. All these values satisfy many additional desirable properties like continuity, the projection axiom, i.e., <pv= v whenever v E FA is in the domain of <p,the dummy axiom, i.e.,
<pv(SC)

= 0 whenever

S is a carrier of v, and stronger versions of positivity.

9.1. Strong positivity One of the stronger versions of positivity is called strong positivity: Let Q ~ BV. A map <p:Q ~ FA is strongly positive if <pv(S) ;? <pw(S) whenever v, w E Q and veT U S/) veT) ;? weT U S/) - weT) for all SI ~ Sand T in C. It turns out that strong positivity can replace positivity and linearity in the characterization of the value on spaces of "smooth" games. THEOREM 9 [Monderer and Neyman (1988), Theorems 8 and 5]. (a) Any symmetric, efficient, strongly positive map from pNAoo to FA is the AumannShapley value. (b) Any symmetric, efficient, strongly positive and continuous map from pNA to FA is the Aumann-Shapley value. 9.2. The partition value Given a value <pon a space Q, it is of interest to specify whether it could be interpreted as a limiting value. This leads to the definition of a partition value. A value <p on a space Q is a partition value [Neyman and Tauman (1979)] if for every coalition S there exists aT-admissible sequence (Jik)~1 such that <pv(T) =
limk-+oo

It follows that any partition value is continuous (with norm ~ 1) and coincides with the asymptotic value on all games that have an asymptotic value [Neyman and Tauman (1979)]. There are however spaces of games that include ASYMP and on which a partition value exists. Given two sets of games, Q and R, we denote by Q R the * closed linear and symmetric space generated by all games of the form u v where u E Q and v E R. THEOREM 10 [Neyman and Tauman (1979)]. There exists a partition value on the closed linear space spanned by ASYMP, bvlNA bvlNA and A bvlNA bvlNA. * * * 9.3. The diagonal property

1jJV:rrk (T).

..;~'.

-

A remarkable implication of the diagonal formula is that the value of a vector measure game f 0 I-Lin pNA or in b vINA is completely determined by the behavior of f near the

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A. Neyman

diagonal of the range of f.-L. We proceed to define a diagonal value as a value that depends only on the behavior of the game in a diagonal neighborhood. Formally, given a vector
of non-atomic probability :( measures f.-L

=

(f.-Ll, . . . , f.-Ln) and

E > 0, the E - f.-Ldiagonal

neighborhood, V(f.-L,E), is defined as the set of all coalitions S with

for all 1 :( i,.i

n. A map cp from a set Q of games is diagonal if for every E - f.-L

If.-Li(S)

- f.-L j (S) I < E

diagonal neighborhood V(f.-L,E) and every two games v, win Q n BV that coincide on V(f.-L,E) (i.e., v(S) = w(S) for any Sin V(f.-L,E)), cpv = cpw. There are examples of non-diagonal values [Neyman and Tauman (1976)] even on reproducing spaces [Tauman (1977)]. However, THEOREM 11 [Neyman (1977a)]. Any continuous value (in the variation norm) is diagonal. Thus, in particular, the values on (pNA and) bv'NA [Aumann and Shapley (1974), Proposition 43.1], the weak asymptotic value, the asymptotic value [Aumann and Shapley (1974), Proposition 43.11], the mixing value [Aumann and Shapley (1974), Proposition 43.2], any partition value [Neyman and Tauman (1979)] and any value on a closed reproducing space, are all diagonal. Let DIAG denote the linear subspace of BV that is generated by the games that vanish on some diagonal neighborhood. Then DIAG C ASYMP and any continuous (and in particular any asymptotic or partition) value vanishes on DIAG.

10. Semivalues Semivalues are generalizations and/or analogies of the Shapley value that do not (necessarily) obey the efficiency, or Pareto optimality, property. This has stemmed partly from the search for value function that describes the prospects of playing different roles in a game. DEFINITION 8 [Dubey, Neyman and Weber (1981)]. A semivalue on a linear and symmetric space of games Q is a linear, symmetric and positive map 1jf: Q --+ FA that satisfies the projection axiom (i.e., 1jfv = v for every v in Q n FA). The following result characterizes all semivalues on the space FG of all games having a finite support. THEOREM 12 [Dubey, Neyman and Weber (1981), Theorem 1].
(a) For each probability measure

~ on

[0, 1], the map 1jf~ that is given by

1jf~v(S) =

11

av(t, S)d~(t),

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2153

where iJ is Owen's multilinear extension of the game v, defines a semivalue on FG. Moreover, every semivalue on FG is of this form and the mapping ~ -+ Vrl; is I-I. (b) Vrl; is continuous (in the variation norm) on G if and only if ~ is absolutely continuous W.r.t. the Lebesgue measure A. on [0, 1] and d~ IdA. E LcdO, 1]. In that case, IIVrI;IIBv= Ild~/dA.lloo' Let W be the subset of all elements g in Loo[O, 1] such that fa1get) dt

=

1.

THEOREM 13 [Dubey, Neyman and Weber (1981), Theorem 2]. For every g in W, the map Vr g : pNA -+ FA defined by
Vrg v(S)

= 11 aiJ(t, S)g(t) dt

is a semivalue on pNA. Moreover, every semivalue on pNA is of this form and the map g -+ Vr g maps W onto the family of semivalues on pNA and is a linear isometry. Let We be the subset of all elements g in W which are (represented by) continuous
functions on [0, 1]. We identify W (We) with the set of all probability measures

~ on

[0, 1] which are absolutely continuous w.r.t. the Lebesgue measure A. on [0, 1], and their Radon-Nikodym derivative d~ IdA. E W (E We). DEFINITION 9. Let v be a game and let
<pv is said to be the weak asymptotic

~-semivalue

~ be

a probability measure on [0, 1]. A game
of v if, for every SEe and every

8 > 0, there is a finite sub field n of C with S E n such that for any finite subfield n' with n' ::) n, IVrI;Vn/(S) - <pv(S)! < 8. REMARKS. (1) A game v has a weak asymptotic ~-semivalue if and only if, for every S in C and every 8 > 0, there is a finite subfield n of C with S E n such that for any subfield n' with n'::) n, IVri;vn/(S) -Vrl;vn(S)1 < 8. (2) A game v has at most one weak asymptotic ~-semivalue. (3) The weak asymptotic ~-semivalue <p v of a game v is a finitely additive game and lI<pvll:::;IIvlllld~/dA.lloo whenever~ E W. Let ~ be a probability measure on [0, 1]. The set of all games of bounded variation and having a weak asymptotic ~-semivalue is denotedASYMP*(~).
THEOREM 14. Let ~ be a probability measure on [0, 1]. (a) The set of all games having a weak asymptotic ~-semivalue

is a linear symmetric

..."'-'c

space of games and the operator that maps each game to its weak asymptotic

~-semivalue

is a semivalue

on that space.

2154

A. Neyman

(b) ASYMP*(~)

is closed <? ~ E W <? pFA C ASYMP*(~)

<? pNA C ASYMP*(~).

(c) ~ E We =} bvlNA C bvlFA C ASYMP*(O.
The asymptotic ~-semivalue of a game v is defined whenever all the sequences of the

~-semivalues
the asymptotic

of finite games that "approximate"

v have the same limit. qJV is said to be
sequence

DEFINITION 10. Let (JTi)~ I'

~-semivalue

~ be

a probability measure on [0,1]. Agame
of v if, for every T
E

C and every T -admissible

the following limit and equality exists:

Ie-HX!

lim 1/f~v][k(T)

= qJv(T).

REMARKS.

(1) A game v has an asymptotic

and every S-admissible sequence P
is independent of the choice of P.
(2) For any given v, the asymptotic

= (JTi)~l

~-semivalue

if and only if for every S in C the limit, limk-HxJ 1/f~V][i (S), exists and
if it exists, is clearly unique.

~-semivalue,

(3) The asymptotic ~-semivalue qJV of a game v is finitely additive and IIqJvllBV~ IlvIIBvlld~/dAlloo (when ~ E W). (4) If v has an asymptotic ~-semivalue qJV, then v has a weak asymptotic

~-semivalue
is

(= qJv).
The space of all games of bounded variation that have an asymptotic denoted ASYMP(~).

~-semivalue

THEOREM 15. (a) [Dubey (1980)]. pNAoo C ASYMP(~) and if qJV denotes the asymptotic semivalue ofv EpNAoo' then IIqJvlloo~ 2l1vlloo. (b) pNA cpM CASYMP(O whenever~ E W. (c) bvlNA c bvlM cASYMP(~) whenever~ EWe.

~-

11. Partially symmetric values Non-symmetric values (quasivalues) and partially symmetric values are generalizations and/or analogies of the Shapley value that do not necessarily obey the symmetry axiom. A (symmetric) value is covariant with respect to all automorphisms of the space of players. A partially symmetric value is covariant with respect to a specified subgroup of automorphisms. Of particular importance are the trivial group, the subgroups that preserve a finite (or countable) partition of the set of players and the subgroups that preserve a fixed population measure. The group of all automorphisms that preserve a measurable partition II, i.e., all automorphisms e such that es = S for any SEll, is denoted g(II). Similarly, if II is a cr-algebra of coalitions (II C C) we denote by 'g(II) the set of all automorphisms e

.

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2155

such that e 5 = 5 for any 5 E JT. The group of automorphisms measure M on the space of players is denoted Q(M).
11.1. Non-symmetric and partially symmetric values

that preserve a fixed

DEFINITION 11. Let Q be a linear space of games. A non-symmetric value (quasivalue) on Q is a linear, efficient and positive map 1/1: Q -+ FA that satisfies the projection axiom. Given a subgroup of automorphisms 1i, an 1i-symmetric value on Q is a non-symmetric value 1/1on Q such that for every e E 1i and v E Q, 1/1(e*v) = e*(1/1v). Given a a-algebra JT, a JT-symmetric value on Q is a Q(JT)-symmetric value. Given a measure M on the space of players, a M-value is a Q(p,)-symmetric value. A coalition structure is a measurable, finite or countable, partition JT of I such that every atom of JT is an infinite set. The results below characterize all the Q (JT)symmetric values, where JT is a fixed coalition structure, on finite games and on smooth games with a continuum of players. The characterizations employ path values which are defined by means of monotonic increasing paths in B (I, C). A path is a monotonic function y: [0,1] -+ B~(I, C) such that for each s E I, y(O)(s) = 0, y(l)(s) = 1, and the function t f---?> yct)(s) is continuous at 0 and 1. A path y is called continuous if for every fixed s E I the function t f---?> Y ct) (s) is continuous; NA -continuous if for every ME NA the function t -+ M(YCt? is continuous; and II-symmetric (where JT is a partition of I) if for every 0 :( t :( 1, the function s f---?> y ct) (s) is JT -measurable. Given an extension operator 1/1on a linear space of games Q (for v E Q we use the notation

i

v = 1/1 v) and a path y, the y -path extension of v, Vy, is defined [Haimanko (2000c)] by
Vy(f)

= v(y*(f))
=
y(f(s?(s). For a coalition

where y*: B~(I, C) -+ B~(I, C) is defined by y*(f)(s) 5 E C and v E Q, define
cp~(v)(5)

=

1
-

1-8

8

1
8 cP~

[vyCtI +85)

- vy(tI)]dt

and DIFF(y)
CPr(v) (5)

is defined as the set of all v E Q such that for every 5 E C the limit
(v) (5) exists and the game CPr(v) is in FA.

= lim8--+o+

PROPOSITION 7 [Haimanko (2000c)]. DIFF(y) is a closed linear subspace of Q and the map CPris a non-symmetric value on DIFF(y). If JT is a measurable partition (or a a-algebra) and y is JT-symmetric then CPris a JT-symmetric value. In what follows we assume that Q JT is a fixed coalition structure.

= bv' NA

with the natural extension operator and

2156

}. Neyman

THEOREM 16 [Haimanko (2000c)]. (a) y is NA-continuous {} pNA C DIFF(y). (b) Every non-symmetric value on pNA(A), A E NA], is a mixture of path values. (c) Every n -symmetric value (where n is a coalition structure) on pNA (on FG) is a mixture (4 n -symmetric path values. REMARKS [Haimanko (2000c)]. (1) There are non-symmetric values on pNA which are not mixtures of path values. (2) Every linear and efficient map 1ft:pNA -+ FA which is n -symmetric with respect to a coalition structure n obeys the projection axiom. Therefore, the projection axiom of a non-symmetric value is not used in (c) of Theorem 16. The above characterization of all n -symmetric values on pNA(A) as mixtures of path values relies on the non-atomicity of the games in pNA. A more delicate construction, to be described below, leads to the characterization of all n -symmetric values on pM. For every game v in bv'M there is a measure A E M1 such that v E bv'M(A). The

set of atoms of A is denoted by A(A). Given two paths y, y', define for v E bv'M(A), f E B~(I, C)
Vy.yl(f) = Vyll

where y" = y [I - A(A)] + y'[A(A)] and v -+ v is the natural extension on bv'M. It
can be verified that Vy,yl is well defined on bv'M, i.e., the definition is independent of

the measure A.For a coalition S E C and v E bv'M, define
qJ~,y/(V)(S)

1

'-1:

= --;;

1I:

[Vy,yl (t 1+ ES) - Vy,yl(t I)] dt.

We associate with bv'M and the pair of paths y and y' the set DIFF(y, y') of all games v E bv'M such that for every S E C the limit qJy,y/(V)(S) = liml:---+o+ qJ~II(V)(S) exists and the game qJy,y/(V) is in FA. THEOREM 17 [Haimanko (2000c)]. (a) DIFF(y, y') is a closed linear subspace of bv'M and the map qJy,yl: DIFF(y, y') -+ FA is a non-symmetric value. (b) Given a measurable partition (ora (J'-algebra) n, qJy,yl is n -symmetric whenever s f-+ y (t)(s) and s f-+ y' (t)(s) are n measurable for every t. (c) {f n is a coalition structure, any n -symmetric value on pM is a mixture of maps

qJy,y'where the paths y and y' are continuous and n -symmetric.

-

REMARKS [Haimanko (2000c)]. (1) If y is NA-continuous and the discontinuities of the functions t f-+ y' (t) (s), s E I, are mutually disjoint, pM C DIFF(y, y'). In particular if y and y' are continuous, pM C DIFF(y, y') and therefore if in addition y (t) and y'(t) are constant functions for each fixed t, y and y' are Q-symmetric and thus qJy,yl

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2157

defines a value of nonn 1 on pM. These infinitely many values on pM were discovered by Hart (1973) and are called the Hart values. (2) A surprising outcome of this systematic study is the characterization of all values on pM. Any value on pM is a mixture of the Hart values. (3) If n is a coalition structure, any n -symmetric linear and efficient map 1/1: bv' M -? FA obeys the projection axiom. Therefore, the projection axiom of a nonsymmetric value is not used in the characterization (c) of Theorem 17. Weighted values are non-symmetric values of the fonn <{Jy where y is a path described by means of a measurable weight function w: I -? JR++ (which is bounded away from 0): for every y(s)(t) = tw(s). Hart and Monderer (1997) have successfully applied the potential approach of Hart and Mas-Colell (1989) to weighted values of smooth (continuously differentiable) games: to every game v E pNAoo the game (Pwv)(S) = f01 v(t; S) dt is inpNAoo and <{Jy v(S) = (Pw v)ws (I). In addition they define an asymptotic w-weighted value and prove that every game in pNAoo has an asymptotic w-weighted value. Another important subgroup of automorphisms is the one that preserves a fixed (population) non-atomic measure J-L. 11.2. Measure-based values

Measure-based values are generalizations and/or analogues of the value that take into account the coalition worth function and a fixed (population) measure J-L.This has stemmed partly from applications of value theory to economic models in which a given population measure J-L is part of the models. Let J-L be a fixed non-atomic measure on (I, C). A subset of games, Q, is J-L-symmetric if for every <9 E Q(J-L)(the group of J-L-measure-preserving automorphisms), <9*Q = Q. A map <{J from a J-L-symmetric set of games Q into a space of games is J-L-symmetric if
<{J<9*v

=

<9*<{Jv for every <9* in Q(J-L) and every game v in Q.

DEFINITION 12. Let Q be a J-L-symmetric space of games. A J-L-value on Q is a map from Q into finitely additive games that is linear, J-L-symmetric, positive, efficient, and
obeys the dummy axiom (i.e., <{Jv(S) = 0 whenever the complement of S, Sc, is a carrier

of v). REMARKS. The definition of a J-L-value differs from the definition of a value in two aspects. First, there is a weakening of the symmetry axiom. A value is required to be covariant with the actions of all automorphisms, while a J-L-value is required to be covariant only with the actions of the J-L-measure-preserving automorphism. Second, there is an additional axiom in the definition of a J-L-value, the dummy axiom. In view of the other axioms (that <{JV is finitely additive and efficiency), this additional axiom could be viewed as a strengthening of the efficiency axiom; the J-L-value is required to obey <{Jv(S)= v(S) for any carrier S of the game v, while the efficiency axiom requires it

2158

A. Neyman

only for the carrier S = I. In many cases of interest, the dummy axiom follows from the other axioms of the value [Aumann and Shapley (1974), Note 4, p. 18]. Obviously, the value on pNA obeys the dummy axiom and the restriction of any value that obeys the dummy axiom to a fl,-symmetric subspace is a fl,-value. In particular, the restriction of the unique value on pNA to pNA(fl,), the closed (in the bounded variation norm) algebra generated by non-atomic probability measures that are absolutely continuous with respect to fl" is a fl,-value. It turns out to be the unique fl,-value on pNA (fl,). THEOREM 18 [Monderer (1986), Theorem A]. There exists a unique fl,-value on pNA(fl,); it is the restriction to pNA(fl,) of the unique value o11;pNA. REMARKS. In the above characterization, the dummy axiom could be replaced by the projection axiom [Monderer (1986), Theorem D]: there exists a unique linear, fLsymmetric, positive and efficient map ?J :pNA(fL) -+ FA that obeys the projection axiom; it is the unique value onpNA(fl,). The characterizations of the fl,-value onpNA(fl,) is derived from Monderer [(1986), Main Theorem], which characterizes all fL-symmetric, continuous linear maps from pNA(fl,) into FA. Similar to the constructive approaches to the value, one can develop corresponding constructive approaches to fL-values. We start with the asymptotic approach. Given T in C with fl,(T) > 0, a fl,-T -admissible sequence is a T -admissible sequence (IT!, lT2,...) for which limk-+oo(min{fL(A): A is an atom of lTk}/max{fl,(A): A is an
atom of lTk })

=

1.

DEFINITION 13. A game ?JVis said to be the fl,-asymptotic value of v if, for every T in C with fl,(T) > 0 and every fl,-T-admissible sequence (lTi)~l' the following limit and equality exists
k-+oo

lim 1jrVTCk (T)

= ?Jv(T).

REMARKS. (1) A game v has a fl,-asymptotic value if and only if for every T in C with fl,(T) > 0 and every fl,-T-admissible sequence P= (lTi)~l' the limit, limk-+oo 1jrVTCk (T) exists and is independent of the choice of P. (2) For a given game v, the fl,-asymptotic value, if it exists, is clearly unique. (3) The fl,-asymptotic value ?JVof a game v is finitely additive and lI?JvII:::;; IIv IIwhenever v has bounded variation. The space of all games of bounded variation that have a fl,-asymptotic value is denoted by ASYMP[fL]. THEOREM 19 [Hart (1980), Theorem 9.2]. The set of all games having a fl,-asymptotic value is a linear fL-symmetric space of games and the operator mapping each game to its fl,-asymptotic value is a fl,-value on that space.' ASYMP[fl,] is a closed fL-symmetric

Ch. 56:

Values of Games with Infinitely Many Players
0

2159

linear subspace of BV which contains all functions of the form f

(fJ-l, . . . , fJ-1l)where derivatives,

fJ-J, . . ., fJ-1lare absolutely continuous W.r.t. fJ- and with Radon-Nikodym dfJ-i / dfJ-, in L2 (fJ-) and f is concave and homogeneous of degree 1.

12. The value and the core The Banach space of all bounded set functions with the supremum norm is denoted BS, and its closed subspace spanned by pNA is denoted pNAI. The set of all games in pNAI that are homogeneous of degree 1 (i.e., v (a!) = av(!) for all f in B{U, C) and all a in [0, 1]), and superadditive (i.e., v(S U T) ~ v(S) + veT) for all S, T in C), is denoted HI. The subset of all games in HI which also satisfy v ~ 0 is denoted H~. The core of a game v in HI is non-empty and for every S in C, the minimum of fJ-(S) when fl, runs over all elements fJ-in core (v) is attained. The exact cover of a game v in HI is the game min{fJ-(S): fJ-E Core(v)} and its core coincides with the core of v. A game v E HI that equals its exact cover is called exact. The closed (in the bounded variation norm) subspace of BV that is generated by pNA and DIAG is denoted by pNAD. THEOREM 20 [Aumann and Shapley (1974), Theorem I]. The core of a game v in pNAD n HI (which obviously contains pNA n HI) has a unique element which coincides with the (asymptotic) value ofv. THEOREM 21 [Hart (1977a)]. Let v E H~.lfv has an asymptotic value cpv, then cpv is the center of symmetry of the core of v. The game v has an asymptotic value cpv when either the core of v contains a single element v (and then cpv = v), or v is exact and the core of v has a center of symmetry v (and then cpv = v). REMARKS. (1) The second theorem provides a necessary and sufficient condition for the asymptotic value of an exact game v in H~ to exist: the symmetry of its core. There are (non-exact) games in H~ whose cores have a center of symmetry which do not have an asymptotic value [Hart (1977a), Section 8]. (2) A game v in H~ has a unique element in the core if and only if v E pNAD. (3) The conditions of superadditivity and homogeneity of degree 1 are satisfied by all games arising from non-atomic (i.e., perfectly competitive) economic markets (see Chapter 35 in this Handbook). However, the games arising from these markets have a finite-dimensional core while games in H~ may have an infinite-dimensional core. The above result shows that the asymptotic value fails to exist for games in H~ whenever the core of the game does not have a center of symmetry. Nevertheless, value theory can be applied to such games, and when it is, the value turns out to be a "center" of the core. It will be described as an average of the extreme points of the core of the game v E H~.

0° -.

2160

A. Neyman

Let v E H~ have a finite-dimensional core. Then the core of v is a convex closed subset of a vector space generated by finitely many non-atomic probability measures, v I, . . . , Vll' Therefore the core of v is identified with a convex subset K of ffi./1,by a = (al,...,an) E K if and only if Lai/Li E Core(v). For almost every z in ffi./1there exists a unique a(z) = (a] (z),..., an(Z)) in K with L7=1 ai(Z)Zi = minCL~I=laiZ( (al,...,an)EK}. THEOREM 22 [Hart (1980)]. If VI,..., V/1are absolutely continuous with respect to P, and dVifdp, E L2 (p,), then v E ASYMP[p,] and its p,-asymptotic value ({JfL v is given by
({JjLV=

[

}nfo/l

Lai(z)vi

dN(z)

where N is a normal distribution on ffi.n with 0 expectation and a covariance matrix
that is given by

EN (Zi Zj) Equivalently,

=

j( )( )
dVi

dVJ

/

dp,

dp,

dp,.
({IN is given by

N is the distribution

on ffi./1whose Fourier transform

~N(Y)

= exr( -

J (~Yi (~:)

r dl')

THEOREM23 [Mertens (1988b)]. The Mertens value of the game v E H~ with afinitedimensional
({Jv =

core, ({Jv, is given by (Lai(Z)Vi) dG(z)

ill

where G is the distribution on ffi.n whose Fourier transform ({JGis given by ({JG (y)
exp( - Nv (y)), v
for the value.

= (v I, . . . , vn)

and N v is defined as in the section regarding formulas

=

It is not known whether there is a unique continuous value on the space spanned by all games in H~ which have a finite-dimensional core. We will state however a result that proves uniqueness for a natural subspaceo Let M F be the closed space of games spanned by vector measure games fop, E H~ where p, = (P,I, . . ., P,k) is a vector of mutually singular measures in NA] . THEOREM 24 [Haimanko (200] a)]. The Mertens value is the only continuous value onMF.

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2161

13. Comments

on some classes of games

13.1. Absolutely continuous non-atomic games A game v is absolutely continuous with respect to a non-atomic measure f-L if for every 8 > 0 there is 8 > 0 such that for every increasing sequence of coalitions

~k ~k - V(S2i-l)1 < 8. SI C S2 C... C S2k with L..,i=] f-L(S2i) - f-L(S2i-l) < 8, L..,i=llv(S2i) + such that v is abA game v is absolutely continuous if there is a measure f-L E NA

solutely continuous with respect to f-L. A game of the form I 0 f-L where f-L E NA I is absolutely continuous (with respect to f-L)if and only if I: [0, f-L(I)] ~ ~ is absolutely continuous. An absolutely continuous game has bounded variation and the set of all absolutely continuous games, AC, is a closed subspace of BV [Aumann and Shapley (1974)]. If v and u are absolutely continuous with respect to f-LE NA + and vENA + respectively, then vu is absolutely continuous with respect to f-L+ v. Thus AC is a closed subalgebra of BV. It is not known whether there is a value on AC. 13.2. Games in pNA

The space pNA played a central role in the development of values of non-atomic games. Games arising in applications are often either vector measure games or approximated by vector measure games. Researchers have thus looked for conditions on vector (or scalar) measure games to be in pNA. Tauman (1982) provides a characterization of vector measure games in pNA. A vector measure game v = I 0 f-L with f-L E (NA 1)/1 is in pNA if and only if the function that maps a vector measure v with the same range

as f-Lto the game I 0 v, is continuous at f-L,i.e., for every 8 > 0 there is 8 > 0 such that if v E (NA 1)/1has the same range as the vector measure f-Land L7 =] II f-Li - Vi II < 8 0 0 v then II I f-L - I II < 8. Proposition 10.17 of [Aumann and Shapley (1974)] provides
a sufficient condition, expressed as a condition on a continuous
function I: ~/1 ~ ~, for a game of the form valued

I

non-decreasing

real-

0 f-L where

f-L is a vector

of

non-atomic measures to be inpNA. Theorem C of Aumann and Shapley (1974) asserts that the scalar measure game I 0 f-L(f-LE NA]) is in pNA if and only if the real-valued function I: [0, 1] ~ ~ is absolutely continuous. Given a signed scalar measure f-Lwith range [-a, b] (a, b > 0), Kohlberg (1973) provides a sufficient and necessary condition
I

on the real-valued function I: [-a, b] ~ ~ for the scalar measure game 10 f-Lto be in pNA. 13.3. Games in bvfNA Any function I E bvf has a unique representation as a sum of an absolutely continuous function lac that vanishes at 0 and a singular function Is in bvf, i.e., a function whose variation is on a set of measure zero. The subspace of all singular functions in bvf is denoted Sf. The closed linear space generated by all games of the form I 0 f-L where

'" '-,.
.? ~"

2]62

A. Neyman

I IE s' and f-L E NA is denoted s'NA. Aumann and Shapley (1974) show that bv'NA is the algebraic direct sum of the two spaces pNA and s' NA, and that for any two games,

u E pNA and v E s' NA,

II

u

+v =
II

is defined as the variation of Ion Therefore, if (fi )~l is a sequence

is a sequence of non-atomic probability measures, the series .L~l Ii 0 f-Li converges to a game v E bv' NA. If the functions Ii are absolutely continuous the series converges to a game in pNA. However, not every game in pNA has a representation as a sum of scalar measure games. If the functions Ii are singular, the series .L~I Ii 0 f-Li converges to a game in .'I' NA, and every game v E .'I' NA is a countable (possibly finite) sum; v = .L~ I Ii 0 Ili, where Ii E s' with II v II = .L~ I Ii II and f-Li E NA I. A simple game is a game v where v(S) E {O, I}. A weighted majority game is a (simple) game of the form
II

u + II v II. The norm of a function I E bv', II I II, [0, 1]. If f-LE NA 1 and IE bv' then 1110 f-LII= Ilfli. offunctions in bv' with .L~l Il/i II < 00 and (f-Li)~]
II II

v(S) =

{~

if f-L(S) ? q if f-L(S) < q

or
v(S) =

{~

if f-L(S) > q if f-L(S) ~q

I

where f-LE }fl+ and 0 < q < f-L (I). The finitely additive measure f-Lis called the weight
measure and q is called the quota. Every simple monotonic game in bv' NA is a weighted majority game, and the weighted measure is non-atomic.

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