Journal of Economic Theory 100, 235273 (2001)
doi:10.1006jeth.2001.2810, available online at http:www.idealibrary.com on
The Core of Large Differentiable TU Games 1
Larry G. Epstein
Department of Economics, University of Rochester, Rochester, New York 14627
lepntroi.cc.rochester.edu
and
Massimo Marinacci
Dipartimento di Statistica e Matematica Applicata, Universita di Torino, Italy
massimoecon.unito.it
Received April 19, 2000
For suitable non-atomic TU games &, the core can be determined by computing
appropriate derivatives of &, yielding one of two stark conclusions: either core(&) is
empty or it consists of a single measure that can be expressed explicitly in terms of
derivatives of &. In this sense, core theory for a class of games may be reduced to
2001
calculus. Journal of Economic Literature Classification Number: C71.
Academic Press
Key Words: core; transferable utility games; large games; non-atomic games;
market games; exchange economies; calculus; derivatives.
1. INTRODUCTION
1.1. Outline
We show that for large (non-atomic) TU games satisfying suitable conditions, the core can be determined by computing appropriate derivatives of
the characteristic function &. Further, such computations yield one of two
stark conclusions: either the core of & is empty or it consists of a single
measure. In the latter case, the core measure is expressed explicitly in terms
of derivatives of &. In this sense, core theory for a class of non-atomic
games may be reduced to calculus.
There is simple intuition for a connection between core(&), assumed for
the moment to be nonempty, and the ``derivative'' of &. Denote by 7 the
1
Epstein acknowledges the financial support of NSF (Grant SES-9972442) and Marinacci
that of MURST. We are extremely grateful to an associate editor.
235
0022-053101 35.00
Copyright 2001 by Academic Press
All rights of reproduction in any form reserved.
236
EPSTEIN AND MARINACCI
_-algebra of feasible coalitions; thus &: 7 Ä R 1. Let m be a measure in the
core, so that
m(E)&(E)
for all coalitions E,
and suppose that there exists a coalition A such that
m(A)=&(A).
(1.1)
min(m(E)&&(E))=0=m(A)&&(A).
(1.2)
Then
E#7
In particular, A is a minimizer for m( } )&&( } ) over 7. Analogy with
calculus suggests that, if A is suitably ``interior'', then the ``derivative'' of the
noted function should vanish at A. We propose a definition of derivative
(or more accurately, differential) for characteristic functions &. It satisfies
the following natural property that might be expected of any notion of differentiation: There is a parallel between standard calculus (for real-valued
functions defined on a Euclidean space) and the calculus proposed here
(for real-valued functions defined on a _-algebra of coalitions), in which
additive set functions (measures) play the role of linear functions on a
Euclidean space. Given such an analogy, then, because m is additive, it
equals its differential. Consequently, if we denote the differential of & at A
by $&( } ; A), then the first-order condition corresponding to (1.2) takes the
form
m( } )=$&( }; A).
(1.3)
Finally, suppose that A satisfies (1.1) not only for a particular m in the
core, but for all measures in the core. Conclude that the core of & is
[$&( } ; A)] and we have a singleton core with an explicit representation in
terms of &.
There are two obvious questions regarding this informal argument. First,
how restrictive is the assumption that there exists an ``interior'' A that
satisfies (1.1) for all m in the core? A sufficient condition (see Lemma 4.3)
is that there exists A satisfying
&(A)+&(A c )=&(0).
Such coalitions, called linear coalitions below, exist in many games of interest.
Examples include homogeneous measure games, and more particularly,
market games.
DIFFERENTIABLE TU GAMES
237
The second and more difficult question is whether the first-order condition (1.3) can be justified. After all, the fact that the usual first-order condition takes the form of an equality relies heavily on the linear structure of
Euclidean space. For example, using the obvious notation, if {f (x*) } y>0,
then one can move from x* in the reverse direction &y and reduce thereby
the value of f. By this reasoning, the stated inequality is ruled out if f has
a minimum at x*. However, in the present context where the domain is a
_-algebra, there is no counterpart of a ``reverse direction'' and first-order
conditions take the form of inequalities. Much of the work we do below is
to identify added assumptions on games that lead to (1.3). Primarily, we
define a class of so-called coherent games, for which the informal outline
above can be made rigorous.
As a result, the core of any differentiable coherent game can be determined simply by computing the derivative of its characteristic function.
Moreover, the latter task is arguably as routine as differentiating functions
defined on a Euclidean space. That is because our derivative notion satisfies
counterparts of the rules familiar from calculus (the product and chain
rules, for example). This is the sense in which core theory for differentiable
coherent games is reduced to ``just calculus''.
The introduction of a new calculus into co-operative game theory is our
main contribution: we hope that it will prove to be a fruitful tool in the
study of non-atomic TU games. We demonstrate its usefulness here by
means of the core theorem for abstract (coherent) games that was outlined
above and also by means of applications of this general theorem to the
more concrete class of measure games. In addition, we provide a new
derivation of known results on market games and exchange economies
(uniqueness of the core allocation in suitable exchange economies). This
new derivation seems to us to be of value because of the radically different
and simplifying perspective that it offers relative to the derivations in [2]
and also [3].
1.2. Related Literature
We are not the first to adopt calculus techniques in order to study nonatomic TU games. Aumann and Shapley [2] develop a notion of derivative
for non-additive set functions and apply it to study both the Shapley value
and the core of non-atomic games. The primary intuition underlying their
use of differentiability is that the Shapley value of each infinitesimal player
is her marginal contribution to the worth of a representative coalition
averaged over all such coalitions (see [2, (20.1)]). Because ``marginal contribution'' is most naturally expressed in terms of derivatives, a calculus
approach is intuitive. Note that this intuition differs substantially from that
underlying our analysis; in particular, it focuses on the Shapley value
rather than the core.
238
EPSTEIN AND MARINACCI
One consequence of this difference in motivation is that Aumann and
Shapley take a class of games as the primary object of study, while we
focus on individual games. The first approach is natural for study of the
Shapley value which is defined by its properties over a class; but not so for
study of the core, where it is desirable to have an approach that is
applicable to any given single game. Our notion of derivative is such a
game-by-game tool.
There are other (related) differences. Roughly, the AumannShapley
definition begins with the subspace pNA of games (the closed subspace of
BV spanned by all powers of positive nonatomic measures). The authors
show (Theorem G) that each game or non-additive set function & in pNA
admits a suitable extension to an ``integral'' &*. Because an integral can be
viewed as a functional on the linear space of integrands (real-valued random variables), a Gateaux-like derivative notion can be defined in the
usual way and this is adequate for characterization of the Shapley value
(Theorem H). However, in order to check whether a given game & is differentiable and then to compute its derivative, one must determine whether
& lies in pNA and if so, determine the corresponding integral &*. Neither of
these steps is routine in general. The above analysis admits generalizationsfor example, domains larger than pNA can be accommodated (Section 22 and [7], for example) and the Ga^teaux derivative can be related to
a Frechet-type notion (Section 24)but the above noted difficulties persist.
Even more important than tractability is that at the level of abstract
games, the AumannShapley approach leads to the formulation of assumptions about games through restrictions on their extensions. However, it is
often difficult to understand the meaning of such restrictions in terms of the
assumptions they embody about the underlying game. 2 In contrast, we
define derivative explicitly in terms of the given &. Then we formulate
coherence and other assumptions about games in terms of these derivatives.
This approach permits interpretation much as restrictions on derivatives in
ordinary calculus are often readily interpreted.
Other related literature includes work by Rosenmuller [11], who defines
differentiability for large games. Our derivative notion is inspired by his,
particularly by his use of partitions, but the two notions differ. Though
Rosenmuller is also concerned with the core of TU games (albeit only convex games), neither our results nor the underlying intuition are apparent in
[11].
The notion of derivative that we use here originates in [4] in a decision
theory context, where instead of being a game, the set function & is a
2
Examples of such assumptions include homogeneity (&*(: 1 E )=:&*(1 E )) [2, p. 167] and
concavity of &*( } ) [3].
DIFFERENTIABLE TU GAMES
239
``non-additive probability'' as in [13]. A related notion is developed and
applied to decision theory in [6].
2. PRELIMINARIES
The set of players is 0 and the _-algebra 7 denotes the set of admissible
coalitions. Subsets of 0 are understood to be in 7 even where not stated
explicitly and they are referred to both as sets and as coalitions.
A set function &: 7 Ä R is a game if &(,)=0. A game & is
positive if &(A)0 for all A,
monotone if &(A)&(B) whenever B/A,
superadditive if &(A _ B)&(A)+&(B) for all pairwise disjoint sets A
and B,
continuous at A if lim n Ä &(A n )=&(A) whenever A n A A,
continuous if it is continuous at every A,
additive (or a charge) if &(A _ B)=&(A)+&(B) for all pairwise disjoint
sets A and B,
countably additive if &( i=1 A i )= i=1 &(A i ) for all countable collections of pairwise disjoint sets [A i ] i=1 .
The set of all additive (countably additive) games that are bounded with
respect to the variation norm is denoted FA(CA). An additive game m is
convex-ranged if for all : # (0, 1) and all A # 7 with |m| (A)>0, there exists
B/A such that |m| (B)=: |m| (A). Denote by NA the set of all additive
convex-ranged games. If m=(m 1 , ..., m N ), where each m i is an additive convex-ranged game, then by a version of the Lyapunov Theorem [9,
Theorem 11.4.9], the range R(m)=[m(A): A # 7] is a convex subset of
R N. Throughout the paper when we refer to ``the Lyapunov Theorem'', the
intention is to this version.
The core of & is
core(&)=[m # FA : m(0)=&(0) and m(A)&(A) for all A # 7].
We fix also some terminology and notation for functions defined on a
Euclidean space. Let U be an open subset of R N. A function g: U Ä R is
differentiable at x # U if there is a linear map Dg(x): R N Ä R such that
lim
hÄ0
| g(x+h)& g(x)&Dg(x)(h)|
=0,
|h|
240
EPSTEIN AND MARINACCI
where | } | : R N Ä R is any norm on R N. Call Dg(x) the derivative of g at x.
If Dg(x) exists for all x # U, say that g is differentiable on U. Since Dg(x)( } )
is linear, Dg(x)(h)={g(x) } h for all x # U, where {g(x) # R N is the gradient
of g at x. If A is an arbitrary subset of R N, not necessarily open, say that
the function g: A Ä R is differentiable on A if it can be extended to a differentiable function on some open set U containing A.
3. DIFFERENTIABLE GAMES
*
For any A # 7, let [A j, * ] nj=1
be a finite partition of A. Denote by
j, *
[A ] * the net of all finite partitions of A, where *$>* implies that the
partition corresponding to *$ refines that corresponding to *.
Definition 3.1. A game &: 7 Ä R is differentiable at E # 7 if there
exists a bounded and convex-ranged measure $&( } ; E) on 7 such that
n*
Ä
0, (3.1)
: |&(E _ F j, * &G j, * )&&(E)&$&(F j, *; E)+$&(G j, *; E)| w
*
j=1
for all F/E c and G/E. 3
For all the results to follow, we could adopt a weaker ``one-sided'' definition
of derivative. Define an outer derivative at E, denoted $ +&( }; E), by
n*
: |&(E _ F j, * )&&(E)&$ +&(F j, *; E)| w
Ä
0
*
(3.2)
j=1
and an inner derivative at E, denoted $ &&( } ; E), by
n*
Ä
0,
: |&(E&G j, * )&&(E)+$ &&(G j, *; E)| w
*
(3.3)
j=1
where the convergence is required to hold for each F and G as above.
Given the existence of $ +&( }; E) and $ &&( } ; E), one could define $&( } ; E)
as the sum $ +&( } ; E)+$ &&( } ; E), in which case (3.1) is satisfied when
A&B denotes A & B c =[| # 0 : | # A, | Â B].
Because a difference quotient is not apparent in the defining condition, it may be comforting to make the following observation: For a function .: R 1 Ä R 1 that is differentiable at
some x in the usual sense, elementary algebraic manipulation of the definition of the derivative
&1
.$(x) yields the following expression paralleling (3.1): N
)&.(x)&N &1.$(x)| Ä 0
i=1 |.(x+N
as N Ä .
3
DIFFERENTIABLE TU GAMES
241
E is perturbed by either F or G, but not necessarily when perturbed by
both simultaneously. 4
Two other remarks on the formal definition are: (i) It can be written
alternatively in the form
n*
Ä
0
: |&(E2A j, * )&&(E)&$&(A j, *; E)| w
*
j=1
for all A in 7, where E2A j, * is the symmetric difference. (ii) The requirement that $&( } ; E) be convex-ranged is not needed for the definition per se,
but it is needed for several properties of the associated calculus
(Appendix A) and thus it is built in for convenience.
Turn to interpretation. Roughly, $&( } ; E) approximates the marginal
value of coalitions relative to the base coalition E, in such a way that the
approximation is additive (across disjoint incremental coalitions) and
becomes exact in the limit for ``small'' coalitions. 5 To elaborate, for F disjoint from E, approximate the incremental value of F as follows: Partition
F into arbitrarily small subcoalitions F j, * and compute the marginal value
&(E _ F j, * )&&(E) of each F j, * relative to the base E. Then the sum of these
marginal values equals $&(F ; E), which therefore represents the total
marginal value of F relative to the base coalition E. A similar interpretation
applies when G/E, in which case $&(G; E) is the sum of the marginal
values &(E)&&(E&G j , * ) for an arbitrarily fine partition [G j, * ] of G.
Finally, any coalition A can be written uniquely in the form A=E _ F&G,
where F/E c and G/E, so that
$&(A; E)=$&(F ; E)+$&(G; E),
which provides the sense in which $&(A; E) represents the total marginal
value of A relative to the base coalition E. 6
A crucial feature of our notion of differentiability for games is that it
satisfies many of the properties familiar from calculus, including a form of
the Chain Rule as well as ``sum'' and ``product'' rules (see Appendix A). 7
Bounded and convex-ranged measures play the role of linear functions in
calculus; in particular, if m is such a measure, then it is differentiable at any
4
In the context of measure games g(P), this would permit weakening our assumptions on
g to require only that it have one-sided derivatives.
5
``Smallness'' is measured via fineness of partitions, which relies on the richness of 7;
indeed, if 7 is finite, then only additive games are differentiable.
6
The counterparts for finite games are 7 i # F (&(E _ [i])&&(E)) and 7 i # G (&(E)&&(E&[i]))
for $&(F; E) and $&(G; E) respectively.
7
The appendix also contains some information about the connection with the Aumann
Shapley style derivative.
242
EPSTEIN AND MARINACCI
E and $m( } ; E)=m( } ). Modulo this translation, formulae familiar from
calculus are valid also for games. For example, if &( } ) equals the product
p( } ) q( } ) of two convex-ranged measures, then
$&( } ; E)= p(E) q( } )+q(E) p( } ),
for all E. More general formulae arise in the context of measure games. 8
3.1. Measure Games
The game &: 7 Ä R is a measure game if there exist a vector charge (that
is, a finitely additive vector measure) P=(P 1 , ..., P N ): 7 Ä R N
+ , with
P(0){0 and with each P i : 7 Ä R + bounded and convex-ranged, and a
function g: R(P) Ä R such that
&( } )= g(P( } ))
on 7.
When N=1, &= g(P( } )) is called a scalar measure game.
Lemma 3.2. Let &= g(P): 7 Ä R be a measure game. If g is differentiable at P(E) # R N
+ , then & is differentiable at E and
N
$&( } ; E)={g(P(E)) } P( } )= : g i (P(E)) P i ( } ).
i=1
Though the proof is routine, we provide it here because it may help
clarify for the reader the definition of differentiability and convince her of
our claim that the calculus of games is analogous to the calculus of functions on Euclidean space.
Proof. For x # R N, let |x| =max 1iN |x i |. Given =>0, there exists
$>0 such that
| g(P(E)+h)& g(P(E))&{g(P(E)) } h|
=
|h|
(3.4)
for all |h| $.
Let F/E c and G/E. Since each P i is convex-ranged on the _-algebra
7, by the Lyapunov Theorem there exist finite partitions [F j, *0 ] and
[G j, *0 ] of F and G such that |P(F j, *0 )| + |P(G j, *0 )| $. Hence, for all
*>* 0 ,
|P(F j, * )&P(G j, * )| |P(F j, * )| + |P(G j, * )|
|P(F j, *0 )| + |P(G j, *0 )| $.
8
See Appendix A.3 for an example of a differentiable game that is not a measure game.
243
DIFFERENTIABLE TU GAMES
For convenience, set : j, * =P(F j, * )&P(G j, * ). Then, by (3.4),
| g(P(E)+: j, * )& g(P(E))&{g(P(E)) : j, * |
=
|: j, * |
for all *>* 0 . On the other hand,
n*
n*
j=1
j=1
n*
n*
j=1
j=1
: |: j, * | = : |P(F j, * )&P(G j, * )| : |P(F j, * )| + : |P(G j, * )|
N
n*
N
j=1
i=1
: P i (F j, * )+ :
:
i=1
n*
N
j=1
i=1
: P i (G j, * )2 : P i (0),
and so, for all *>* 0 ,
n*
: | g(P(E)+: j, * )& g(P(E))&{g(P(E)) : j, * |
j=1
n*
=:
j=1
| g(P(E)+: j, * )& g(P(E))&{g(P(E)) : j, * | j, *
|: |
|: j, * |
N
2= DtL: P i (0),
as desired. K
i=1
4. LINEAR SETS AND CORE BOUNDS
Our focus in this paper is to relate core(&) to the derivative of &. We
begin with an important preliminary relation and some immediate implications. The lemma describes inequalities that represent first-order conditions
for (1.2).
Lemma 4.1.
Proof.
Let A # 7 and m # core(&) be such that m(A)=&(A). Then
$&(F ; A)m(F )
for all F/A c
$&(G; A)m(G)
for all G/A.
and
For any F/A c,
n*
$&(F ; A) = : $&(F j, *; A)
j=1
n*
= : [$&(F j, *; A)&&(A _ F j, * )+&(A)]
j=1
244
EPSTEIN AND MARINACCI
n*
+ : [&(A _ F j, * )&&(A)]
j=1
n*
: |$&(F j, *; A)&&(A _ F j, * )+&(A)|
j=1
n*
+ : [&(A _ F j, * )&&(A)]
j=1
n*
: |$&(F j, *; A)&&(A _ F j, * )+&(A)|
j=1
n*
+ : [m(A _ F j, * )&m(A)]
j=1
n*
= : |$&(F j, *; A)&&(A _ F j, * )+&(A)| +m(F).
j=1
*
Thus lim * Ä 0 nj=1
|$&(F j, *; A)&&(A _ F j, * )+&(A)| =0 implies $&(F; A)
m(F).
For G/A,
n*
$&(G; A)= : $&(G j, *; A)
j=1
n*
= : [$&(G j, * )+&(A&G j, * )&&(A)]
j=1
n*
+ : [&(A)&&(A&G j, * )]
j=1
n*
& : |$&(G j, *; A)+&(A&G j, * )&&(A)|
j=1
n*
+ : [&(A)&&(A&G j, * )]
j=1
n*
& : |$&(G j, *; A)+&(A&G j, * )&&(A)|
j=1
n*
+ : [m(A)&m(A&G j, * )]
j=1
n*
= & : |$&(G j, *; A)+&(A&G j, * )&&(A)| +m(G).
j=1
*
Thus lim* Ä 0 nj=1
|$&(G j, *; A)+&(A&G j, * )&&(A)| =0 implies $&(G; A)
m(G). K
245
DIFFERENTIABLE TU GAMES
We make heavy use of this lemma. Also important is the notion of a
linear coalition (or set).
Definition 4.2. The coalition A # 7 is linear with respect to the game
& if &(A)+&(A c )=&(0).
Given the binary partition of 0 into the linear coalitions A and A c, there
is no gain (or loss) in forming the grand coalition. 9 Denote by A the
collection of all linear coalitions; A contains both < and 0. The importance of linear coalitions for our purposes is that they deliver (1.1) which
is an important part of our approach to determining the core. This fact and
more are established in the next lemma.
Lemma 4.3. If [A i ] i # I is a countable partition of 0 satisfying i # I &(A i )=
&(0), then m(A i )=&(A i ) for all i # I and m # core(&). If, in addition, & is superadditive and either (i) & is continuous, or (ii) the partition is finite, then A i # A
for each i in I.
Proof.
Evidently,
0=&(0)&&(0)= : m(A i )& : &(A i )= : [m(A i )&&(A i )],
i#I
i#I
i#I
so that m(A i )=&(A i ) for all i because m(A i )&(A i ) for all i. Next, suppose
that & is superadditive and continuous (the finite partition case is trivial).
For convenience, consider A 1 . We have
\
n
&(0)&(A 1 )+&(A c1 )=&(A 1 )+ lim & . A i
nÄ
i=2
+
n
&(A 1 )+ lim
nÄ
: &(A i )=&(0),
i=2
and so &(A 1 )+&(A c1 )=&(0). K
For linear sets, Lemma 4.1 takes the following stronger form that is a
direct consequence of the two preceding lemmas.
Lemma 4.4.
Let A # A. For any suitably differentiable & and m # core(&),
$&(F; A)m(F )$&(F ; A c )
for all
F/A c
$&(G; A c )m(G)$&(G; A)
for all
G/A.
and
9
A (nonbinary) partition of S with the corresponding property (see the next lemma) is
called an efficient coalition structure by Aumann and Dreze [1].
246
EPSTEIN AND MARINACCI
For any linear set A, these inequalities define bounds on elements of
core(&), that is, for any E,
$&(E & A c; A)+$&(E & A; A c )m(E)
$&(E & A c; A c )+$&(E & A; A).
In particular, because the empty set is linear, one obtains the bound
$&( } ; <)m( } )$&( }; 0).
(4.1)
An implication is that every measure in core(&) is absolutely continuous
with respect to $&( } ; 0). Further implications constitute necessary conditions for nonemptiness of the core; for example, nonemptiness requires that
(for all linear A and for all E)
$&(E & A c; A)+$&(E & A; A c )$&(E & A c; A c )+$&(E & A; A).
5. COHERENT GAMES
Definition 5.1.
A game &: 7 Ä R is coherent at A # 7 if
$&(A; A)>&(A) O $&(A; A c )&(A)
and
$&(A c; A c )>&(A c ) O $&(A c; A)&(A c ).
(5.1)
(5.2)
Defer interpretation for a moment and observe that: (i) & is coherent at A
if and only if it is coherent at A c; and (ii) & is coherent at 0 (and at <)
if and only if
$&(0, 0)&(0)
$&(0; <)&(0).
or
(5.3)
As an example, consider the measure game g(P), where g is differentiable
at 0 and at P(0). Then g(P) is coherent at 0 if
N
N
: g i (P(0)) P i (0)g(P(0))
i=1
or
: g i (0) P i (0)g(P(0)).
i=1
In particular, this is true if g is nonnegative-valued and homogeneous of
degree k # [0, 1] (by Euler's Theorem) or if g is concave, in which case
N
$&(0; 0)= : g i (P(0)) P i (0)g(P(0))=&(0).
i=1
DIFFERENTIABLE TU GAMES
247
To understand the meaning of coherence, recall the intuition sketched in
the introduction surrounding the optimization problem (1.2) and the need
to convert the (first-order) inequalities in Lemma 4.1 into an equality such
as (1.3). Roughly, the obstacle is the apparent lack of a notion of ``reversal
in direction''. The sense in which a reversal is possible is that the subtraction of any G/A from A is tantamount to the addition of G to A c. This
trivial observation can be exploited if $&( } ; A) and $&( } ; A c ) are suitably
relatedand that is the role of coherence. Thus, for example, (5.1) requires
that if $&(A; A) is large in the sense of being larger than &(A), then (ignoring the distinction between strict and weak inequalities) so is $&(A; A c ). 10
In terms of the interpretation offered earlier for derivatives, if the value of
A is less than the total marginal value of A relative to the base coalition
A, corresponding to the effect of removing the players in A, then the same
must be true when the total marginal value is computed relative to A c as
the base coalition, corresponding to the effect of adding the players in A to
A c. (The second condition (5.2) requires the same for A c.) The consistent
relative evaluation of the total marginal contribution of A suggests the
name consistency; we employ ``coherence'' because consistency has other
meanings in co-operative game theory.
At a formal level, one might wonder about the variations of (5.1) and
(5.2) obtained by reversing the directions of all inequalities; after all, these
also express a form of consistency in the evaluation of the total marginal
contribution of A. In fact, these conditions are implied (even without
coherence) if & has a nonempty core, because then, by Lemma 4.4,
$&(A; A c )&(A)
and
$&(A c; A)&(A c ).
The intuition in the introduction relies also on a coalition A having the
property (1.1), which is true for linear sets, by Lemma 4.3. Call the game
& coherent if it is coherent at some linear set A (which presumes differentiability at A and A c ). For coherent games, we do not obtain (1.3); it is not
surprising that the calculus intuition is imperfect. However, we can prove
that the core is either empty or that it is a singleton having an explicit
representation in terms of the derivatives of &.
Theorem 5.2. Let the game &: 7 Ä R be differentiable at some linear
coalition A and suppose that core(&){<. Then the following statements are
equivalent:
10
The counterpart for finite games is that 7 i # A (&(A)&&(A&[i]))>&(A) implies that
7 i # A (&(A c _ [i])&&(A c ))&(A).
248
EPSTEIN AND MARINACCI
(i)
& is coherent at A.
(ii) core(&) equals the singleton [m], where m has the form: There
exist scalar coefficients a, b, c and d such that
a+b=1=c+d,
ab=cd=0
and
(5.4)
m(E)=a $&(E & A; A)+b $&(E & A; A c )+c $&(E & A c; A)
+d $&(E & A c; A c )
(5.5)
for all E in 7.
(iii)
m # core(&) for some measure m having the form (5.5).
In particular, coherence at some linear coalition implies that the core is
either empty or equals a singleton. Conversely, coherence at A is also
necessary for the representation (5.5).
In the latter representation, two coefficients equal 1 and two equal 0,
which leads to 4 possibilities in total, corresponding to the fact that there
are 4 distinct ways in which coherence at A can be satisfied. These and
their corresponding representations are: 11
1. &(A)$&(A; A) and &(A c )$&(A c; A) core(&)=[$&( } ; A)].
(5.6)
2. &(A)$&(A; A) and &(A c )$&(A c; A c ) core(&)=[$&( } & A; A)+$&( } & A c; A c )].
(5.7)
3. &(A)$&(A; A c ) and &(A c )$&(A c; A) core(&)=[$&( } & A; A c )+$&( } & A c; A)].
(5.8)
4. &(A)$&(A; A c ) and &(A c )$&(A c; A c ) core(&)=[$&( } ; A c )].
(5.9)
Note that the first representation corresponds to (1.3), but that the surrounding intuition described in the introduction is consistent also with the
three other representations.
11
These elaborations on the theorem are established in proving the theorem. Naturally, the
stated hypotheses, including nonemptiness of the core, are assumed.
249
DIFFERENTIABLE TU GAMES
Proof. (i) O (ii): We use repeatedly the following elementary
Fact: Given E # 7, if p and q are two measures satisfying
p(B)q(B)
for all
B/E
and
p(E)=q(E).
then p(B)=q(B) for all B/E.
Let m # core(&). Then &(A)=m(A) and &(A c )=m(A c ) by Lemma 4.3.
Therefore, by Lemma 4.4, we have both
$&(F ; A)m(F )$&(F ; A c )
for all F/A c
$&(G; A c )m(G)$&(G; A)
for all G/A,
and
(5.10)
and, taking F=A c and G=A,
$&(A c; A)&(A c )$&(A c; A c ),
(5.11)
$&(A; A c )&(A)$&(A; A).
Now consider in turn each of the possibilities enumerated above.
For 1, deduce from (5.11) that
$&(A; A)=m(A)
and
$&(A c; A)=m(A c ).
(5.12)
From (5.10), $&( } , A)m( } ) within A and $&( } ; A)m( } ) within A c. Then
(5.12) and the Fact imply that $&( } ; A)=m( } ) within A and $&( }; A)=m( } )
within A c, that is, m( } )=$&( } ; A) within 0 (that is, on 7).
For 2, deduce from (5.11) that
$&(A; A)=m(A)
and
$&(A c; A c )=m(A c ).
(5.13)
Argue as above, using (5.10) and the Fact, that $&( } ; A)=m( } ) within A
and $&( } ; A c )=m( } ) within A c. Possibilities 3 and 4 are similar.
(iii) O (i): Once again, argue case by case. If $&( } ; A) # core(&), then
Lemma 4.3 implies $&(A; A)=&(A) and $&(A c; A)=&(A c ), which in turn
implies coherence at A. Similarly for the other cases.
The implication (ii) O (iii) is obvious. K
Several applications of the theorem are provided below in the context of
more concrete (e.g., measure or market) games. Conclude this section with
an application at the level of abstract games.
Corollary 5.3. Let the game &: 7 Ä R be differentiable at A and A c,
where A # A and 0<&(A)<&(0). Assume that core(&){< and that & is
250
EPSTEIN AND MARINACCI
either (i) monotone or (ii) superadditive and differentiable at <. Suppose
finally that & satisfies
$&( } ; A c )=} $&( } ; A)
for some }{0.
(5.14)
Then
core(&)=[$&( }; A)]=[$&( } ; A c )].
(5.15)
The condition (5.14) is a simple formal condition that relates the
derivatives at A and A c, thus connecting with our central intuition. As a
simple illustration, consider the scalar measure game &( } )= g(P( } )), where
g is monotone and both g$(P(A)) and g$(P(0)&P(A)) exist, and are nonzero. Then condition (5.14) is satisfied because
$&( } ; A)=g$(P(A)) P( } )=
=
g$(P(A))
g$(P(A c )) P( } )
g$(P(A c ))
g$(P(A))
$&( } ; A c ).
g$(P(A c ))
Suppose that A is a linear set for &. It follows from the Corollary that if
0<g(P(A))<P(0), then core(&) is empty unless g(x)x for all x in
[0, P(0)], in which case the core is just [P].
Proof of Corollary 5.3. (i) From 0<&(A)<&(0) and Lemmas 4.3 and
4.4, deduce that each of $&(A; A), $&(A c; A), $&(A; A c ) and $&(A c; A c ) is
positive.
Next prove that }=1: By Lemma 4.4,
$&(A c; A)m(A c )=&(A c )$&(A c; A c )=} $&(A c; A),
} $&(A; A)=$&(A; A c )m(A)=&(A)$&(A; A).
(5.16)
(5.17)
Hence, (5.16) implies }1 and (5.17) implies }1, so that }=1 and
$&( } ; A c )=$&( } ; A).
(5.18)
Finally. (5.16) and (5.17) imply that $&(A c; A)=&(A c ) and &(A)=$&(A; A).
Hence & is coherent at A and representation (5.6) applies.
(ii)
By Lemma A.5,
&$(E)# |$&| (E; ,)+&(E) &$&(E; ,)+&(E)0
DIFFERENTIABLE TU GAMES
251
for all E # 7. Thus the game &$ is nonnegative. Furthermore,
&$(A)+&$(A c )=&(A)+&(A c )+ |$&| (A; ,)+ |$&| (A c; ,)
=&(0)+ |$&| (0; ,)=&$(0),
&$(A)=&(A)+ |$&| (A; ,)<&(0)+ |$&| (0; ,),
&$(A c )=&(A c )+ |$&| (A c; ,)<&(0)+ |$&| (0; ,).
Hence, A is linear for &$ and 0<&$(A)<&$(0).
Let m( } ) # core(&). Then m( } )+ |$&| ( } , ,) # core(&$), and so, by the
preceding argument for monotone games,
m( } )+ |$&| ( } ; ,)=$&$( } ; A)=$&( }; A)+ |$&| ( } ; ,),
which evidently leads to (5.15).
K
6. APPLICATIONS
Thus far we have been dealing with abstract games. We turn now to
applications to more concrete specializations.
6.1. Market Games and Exchange Economies
Definition 6.1. A market game is a positive superadditive measure
game g(P): 7 Ä R, where g: R(P) Ä R is homogeneous of degree one. 12
Market games play a fundamental role in co-operative game theory (see
[2, Ch. 6], for example). As an immediate consequence of Theorem 5.2, we
can prove the following result for such games:
Corollary 6.2. Let &= g(P): 7 Ä R be a market game such that g is
differentiable at P(0). Then
{
N
=
core(&)= : g i (P(0)) P i ( } ) .
i=1
(6.1)
Proof. By Euler's Theorem, $&(0; 0)= N
i=1 g i (P(0)) P i (0)=g(P(0))
=&(0). Hence, & is coherent at 0. By Theorem 5.2, see (5.6) in particular,
core(&)=[$&( } ; 0)] if $&( } ; 0) # core(&). Therefore, to complete the proof it
suffices to show that $&( } ; 0) # core(&).
12
Notice that we do not require countable additivity of P.
252
EPSTEIN AND MARINACCI
We know that $&(0; 0)=&(0). Let E # 7 with P(E){0. By the
Lyapunov Theorem, for each t # (0, 1) there exists B t such that P(B t )=
tP(E). Hence, by superadditivity,
g(P(0)&P(B t ))=g(P(B ct )) g(P(0))& g(P(B t ))
=g(P(0))& g(tP(E)),
and by homogeneity,
&(E)=g(P(E))=lim
ta0
lim
ta0
g(P(0))+ g(tP(E))& g(P(0))
t
g(P(0))& g(P(0)&tP(E))
t
={g(P(0)) P(E)=$&(E; 0).
Finally, P(E)=0 implies &(E)=0={g(P(0)) P(E)=$&(E; 0). Conclude
that $&( } ; 0) # core(&). K
An important special case of a market game is provided by an exchange
economy. An exchange economy of finite type consists of 13
1. a measure space (0, 7, +) of agents, where + is a non-atomic
probability measure;
2. a partition [0 i ] K
i=1 /7 of 0;
3. a space R N
+ of goods;
4. a nondecreasing and concave utility function u( } , |): R N
+ Ä R+
for each | # 0 such that u( } , |)=u( } , |$) for all |, |$ belonging to the
same element of the partition;
5. an endowment e: 0 Ä R N
+ such that 0 e d+ # R ++ .
Consider the game defined by
&(E)=max
{| u(x(|), |) d+: x(|) # R
E
N
+
|
x(|) d+
and
E
|
E
=
e(|) d+ ,
(6.2)
where the maximum is taken over all +-integrable allocations x. By [2,
Theorem J], the game is well defined. In particular, concavity of the utility
functions justifies the restriction to type-symmetric allocations x, that is, x
such that x(|)=x(|$) if | and |$ belong to the same element of the partition [2, p. 235].
13
See [2, Ch. 6] for further details and terminology.
253
DIFFERENTIABLE TU GAMES
Because of type-symmetry of the relevant allocations, the above game is
a market game. Indeed, &(E)= g(P(E)) for all E # 7, where:
' i (E)=+(E & 0 i )
` j (E)=
|
for
e j (|) d+
1iK,
for
1 jN,
E
P(E)=(`(E), '(E)),
g(z, y)=max
{
f i ( } )=u( } ; |)
K
for
K
| in 0 i ,
=
i
: y if i (x i ): x # R N
+ and : y x i z ,
i=1
i=1
(z, y) # R N+K
.
+
Under suitable conditions, spelled out in [2, pp. 234241], g is differentiable at P(0). Under those conditions, therefore, the core of a market
game & is the singleton as in (6.1). This is essentially a result of [2, Ch. 6]
and it plays a key role in their analysis of exchange economies. Our contribution is to show how it can be derived from an approach based
primarily on the elementary and familiar perspective provided by calculus.
6.2. The Core of Measure Games
As Hart and Neyman observe [5, p. 32] ``in many applications, one
usually encounters games ... that depend on finitely many measures.'' Therefore, we provide some results that apply to a broad class of measure games.
Definition 6.3. The measure game g(P): 7 Ä R is a Dini measure
g(x)
game if lim inf |x| Ä 0 |x| >&, where | } | is any norm on R N. 14
Special cases include
g(x)
|x|
(i)
all positive measure games, since g0 implies that lim inf |x| Ä 0
(ii)
all measure games g(P) such that g: R N Ä R is differentiable at 0.
0;
We make use also of special categories of sets or coalitions, called
diagonal and pivotal respectively, that we now introduce. Let P: 7 Ä R N be
a finitely additive vector measure with each measure P i convex-ranged.
Given any two distinct sets E$, E" # 7, with P(E$)P(E"), the set
( E$, E") #[E # 7 : P(E)=tP(E$)+(1&t) P(E") for some t # (0, 1)],
14
We call them Dini games because lim inf |x| Ä 0
g(x)
|x|
is the lower Dini derivative of g at 0.
254
EPSTEIN AND MARINACCI
is called the segment joining E$ and E"; it is nonempty by the Lyapunov
Theorem. 15 Call E diagonal if E # (<, 0), that is, if
P(E)=tP(0)
for some 0<t<1,
which expresses a sense in which E is a representative subcoalition of 0.
Say that E is pivotal if there exist linear sets A$ and A" such that
A$ # ( <, E) and A" # ( E, 0).
Remarks. There exists a linear and diagonal set if and only if 0 is pivotal. 16 Thus, assuming that there exists a (linear and) pivotal set is weaker
than assuming that there exists a (linear and) diagonal set; see Theorems
6.4 and 6.5 below. It is easy to see that 0 is pivotal if and only if < is pivotal. For scalar measure games, 0 is pivotal iff there exists a linear set A
such that 0<P(A)<P(0).
Our first result exploits the fact that the coherence required by
Theorem 5.2 is implied by the existence of a linear and diagonal set.
Theorem 6.4. Let &= g(P): 7 Ä R be a Dini measure game with P
countably additive and g bounded below. Let A be a linear and diagonal set
and suppose that & is differentiable at A and at A c. Then
core(&)=<
or
core(&)=[$&( } ; A)].
The Theorem applies in particular to games g(P) where g: R(P) Ä R is
homogeneous of degree one and g(0, ..., 0)=0, because such games admit
many sets that are linear and diagonal. In fact, by the Lyapunov Theorem,
for each : # (0, 1) there exists B : # 7 such that P i (B : )=:P i (0) for
1iN. Therefore,
&(B : )=g(P 1(B : ), ..., P N (B : ))= g(:P 1(0), ..., :P N (0))
=:g(P 1(0), ..., P N (0))=:&(0);
&(B c: )=g(P 1(B c: ), ..., P N (B c: ))= g((1&:) P 1(0), ..., (1&:) P N (0))
=(1&:) g(P 1(0), ..., P N (0))=(1&:) &(0).
Hence, each B : is linear and diagonal. Conclude from the Theorem that if
the core is nonempty (and given differentiability of g at P(0)), then
core(&)=[7 i g i (P(0)) P i ( } )].
15
Adopt the convention that ( E, E) =[<] for all E # 7.
In fact, E # ( <, 0) and < # ( 0, 0) =( <, <), so that we can take A$=E and
A"=<.
16
DIFFERENTIABLE TU GAMES
255
As noted above, there exists a linear and diagonal set if and only if 0 is
pivotal. We can weaken this requirement and assume only that there exists
some pivotal set, not necessarily 0, if we restrict attention to games g(P)
such that either g is convex or g(P) is totally balanced. 17
Theorem 6.5. Let &= g(P): 7 Ä R be a Dini measure game with P
countably additive. Let A be a linear and pivotal set and suppose that & is differentiable at A and at A c.
(i)
If g is convex and bounded below, then
core(&)=<
(ii)
or
core(&)=[$&( } ; A)].
If & is totally balanced, then
core(&)=[m],
where m(E)=$&(E & A; A)+$&(E & A c; A c ), E # 7.
We describe one more result for Dini measure games. Say that the
measure game g(P) is super-homogeneous of degree k>0 at the set E if
g(:P(E)): kg(P(E)) for all : # (0, 1). For example, if g: R(P) Ä R + is
concave with g(0)=0, then g(P) is homogeneous of degree k1 at every E.
This property implies (given auxiliary assumptions) that g(P) is coherent at 0
and hence permits application of Theorem 5.2. More precisely, we can prove:
Theorem 6.6. Let &= g(P): 7 Ä R be a Dini measure game with g differentiable at P(0).
(a) Suppose that & is super-homogeneous of degree k # [0, 1] at P(0)
and that either g is bounded below or that & is superadditive.
(i) If k<1, then core(&)=<.
(ii) If k=1, then core(&)/[$&( }; 0)].
(b) Suppose that g is bounded below and that & is super-homogeneous
of degree 1 at every E # 7. Then the following statements are equivalent:
(i) core(&){<.
(ii) core(&)=[$&( } ; 0)].
(iii)
&(0)=$&(0; 0) and &(E)+&(E c )&(0) for all E # 7.
It is noteworthy that this theorem requires differentiability of g only at the
single point P(0). This feature makes possible the following generalization:
17
Recall that a game is totally balanced iff all its subgames have nonempty cores.
For a scalar measure game, g(P) is convex if and only if g: [0, P(0)] Ä R is convex. This is
not true for N>1. For example, if g(x 1 , x 2 )=- x 21 +x 22 , then by Theorem 5.2, core( g(P))
1- 2(P 1( } )+P 2( } ))]. Hence, g(P) has at most a singleton core and so it is not convex as,
otherwise, it would be additive. In fact, core( g(P))=<.
256
EPSTEIN AND MARINACCI
By the Rademacher Theorem, locally Lipschitzian functions (a large class
that includes convex functions) are differentiable everywhere outside a
Lebesgue measure zero subset DR N; for convex functions, the set D is first
category in R N (see [8, Theorem 1.18] and [10, Theorem 25.4]). Consequently, even without any differentiability requirement on g, Theorem 6.6
holds if g is locally Lipschitzian and if P(0) lies outside the corresponding
``small'' set D. (A similar remark applies to Corollary 6.2.)
APPENDIX A: DERIVATIVES
A.1. Basic Properties
Proposition A.1. For any E # 7, when it exists, the derivative $&( } ; E)
of a game & is unique.
Proof.
have:
Let $ 1 &( } ; E) and $ 2 &( }; E) be derivatives. For any G/E, we
|$ 1 &(G; E)&$ 2 &(G; E)|
}
n*
= : $ 1 &(G j, *; E)&$ 2 &(G j, *; E)
j=1
}
n*
: |$ 1 &(G j, *; E)&$ 2 &(G j, *; E)|
j=1
n*
= : |$ 1 &(G j, *; E)&&(E&G j, * )+&(E)
j=1
+&(E&G j, * )&&(E)&$ 2 &(G j, *; E)|
n*
: |$ 1 &(G j, *; E)&&(E&G j, * )+&(E)|
j=1
n*
+ : |&(E&G j, * )&&(E)&$ 2 &(G j, *; E)| w
Ä
0
*
j=1
and so $ 1 &(G; E)=$ 2 &(G; E). A similar argument holds for all F/E c. By
additivity, it then follows that $ 1 &( } ; E)=$ 2 &( } ; E). K
Proposition A.2. Suppose & 1 and & 2 are two games differentiable at
E # 7. Then both & 1 +& 2 and & 1 & 2 are differentiable at E, and
(i)
$(& 1 +& 2 )( } ; E)=$& 1( } ; E)+$& 2( } ; E),
(ii)
$(& 1 & 2 )( } ; E)=& 2(E) $& 1( } ; E)+& 1(E) $& 2( } ; E).
DIFFERENTIABLE TU GAMES
257
Moreover, if & 2(E){0, then & 1 & 2 is differentiable at E and
$( &&12 )( } ; E)=[& 2(E) $& 1( }; E)&& 1(E) $& 2( }; E)][& 2(E)] 2
(iii)
Proof. We provide the proof of (ii). By adding and subtracting & 1(E)
& 2(E _ F j, * &G j, * ) we obtain:
n*
: |& 1(E _ F j, * &G j, * ) & 2(E _ F j, * &G j, * )&& 1(E) & 2(E)
j=1
&& 2(E) $& 1(F j, * &G j, *; E)&& 1(E) $& 2(F j, * &G j, *; E)|
n*
|& 1(E)| : |& 2(E _ F j, * &G j, * )&& 2(E)&$& 2(F j, * &G j, *; E)|
j=1
}
n*
+ : & 2(E _ F j, * &G j, * )[& 1(E _ F j, * &G j, * )&& 1(E)]
j=1
&& 2(E) $& 1(F j, * &G j, *; E)
}
n*
= |& 1(E)| : |& 2(E _ F j, * &G j, * )&& 2(E)&$&(F j, * &G j, *; E)|
j=1
n*
+ : |[& 2(E _ F j, * &G j, * )&& 2(E)+& 2(E)]
j=1
_[& 1(E _ F j, * &G j, * )&& 2(E)]&& 2(E) $& 1(F j, * &G j, *; E)|
n*
= |& 1(E)| : |& 2(E _ F j, * &G j, * )&& 2(E)&$& 2(F j, * &G j, *; E)|
j=1
n*
+ : |& 2(E _ F j, * &G j, * )&& 2(E)| |& 1(E _ F j, * &G j, * )&& 1(E)|
j=1
n*
+ |& 2(E)| : |& 1(E _ F j, * &G j, * )&& 1(E)&$& 1(F j, * &G j, *; E)|.
j=1
By definition,
n*
Ä
0,
: |& 2(E _ F j, * &G j, * )&& 2(E)&$& 2(F j, * &G j, *; E)| w
*
j=1
n*
: |& 1(E _ F j, * &G j, * )&& 1(E)&$& 1(F j, * &G j, *; E)| w
Ä
0.
*
j=1
258
EPSTEIN AND MARINACCI
On the other hand,
n*
: |& 2(E _ F j, * &G j, * )&& 2(E)| |& 1(E _ F j, * &G j, * )&& 1(E)|
j=1
n*
: |& 2(E _ F j, * &G j, * )&& 2(E)&$& 2(F j, * &G j, *; E)|
j=1
_|& 1(E _ F j, * &G j, * )&& 1(E)&$& 1(F j, * &G j, *; E)|
n*
+ : |$& 2(F j, * &G j, *; E)|
j=1
_|& 1(E _ F j, * &G j, * )&& 1(E)&$& 1(F j, * &G j, *; E)|
n*
+ : |$& 1(F j, * &G j, *; E)|
j=1
_|& 2(E _ F j, * &G j, * )&& 2(E)&$& 2(F j, * &G j, *; E)|
n*
+ : |$& 2(F j, * &G j, *; E)| |$& 1(F j, * &G j, *; E)|.
j=1
Let =>0. There exists * 1 such that, for all *>* 1 ,
n*
: |& 2(E _ F j, * &G j, * )&& 2(E)| |& 1(E _ F j, * &G j, * )&& 1(E)|
j=1
n*
n*
j=1
j=1
= 2 += : |$& 2 | (F j, * &G j, *; E)+= : |$& 1 | (F j, * &G j, *; E)
n*
+ : |$& 2(F j, * &G j, *; E)| |$& 1(F j, * &G j, *; E)|.
j=1
DIFFERENTIABLE TU GAMES
259
Since the derivatives are convex-ranged, there exists * 2 such that
|$& 2 | (F j, * &G j, *; E)= for all *>* 2 and all 1 jn * . Hence, for all
*>* 1 6 * 2 ,
n*
: |& 2(E _ F j, * &G j, * )&& 2(E)| |& 1(E _ F j, * &G j, * )&& 1(E)|
j=1
= 2 += |$& 2 | (F&G; E)+= |$& 1 | (F&G; E)+= |$& 1 | (F&G; E).
Since the derivatives are bounded measures, it follows that
n*
: |& 2(E _ F j, * &G j, * )&& 2(E)| |& 1(E _ F j, * &G j, * )&& 1(E)| w
Ä
0,
*
j=1
which completes the proof of (ii). K
Proposition A.3. Let & be differentiable at E # 7. If & is monotone, then
$&( } ; E) is a nonnegative measure.
Proof.
Let F/E c. Then, by monotonicity,
n*
&$&(F ; E)= : &$&(F j, *; E)
j=1
n*
: [&$&(F j, *; E)+&(E _ F j, * )&&(E)]
j=1
n*
Ä
0,
: | &$&(F j, *; E)+&(E _ F j, * )&&(E)| w
*
j=1
and so &$&(F ; E)0, i.e., $&(F; E)0. Now, let G/E. Again by
monotonicity,
n*
&$&(G; E)= : &$&(G j, *; E)
j=1
n*
: [&$&(G j, *; E)+&(E)&&(E&G j, * )]
j=1
n*
Ä
0,
: | &$&(G j, *; E)+&(E)&&(E&G j, * )| w
*
j=1
and so &$&(G; E)0, i.e., $&(G; E)0. By the additivity of $&( } ; E), it
follows that $&( } ; E)0. K
A.2. Chain Rule
Proposition A.4. Let &: 7 Ä R be differentiable at E # 7, and g: R Ä R
a function differentiable at &(E). Then g(&): 7 Ä R is differentiable at E and
$g(&( } ; E))= g$(&(E)) $&( } ; E).
260
EPSTEIN AND MARINACCI
Proof.
We have
n*
: | g(&(E _ F j, * &G j, * ))& g(&(E))& g$(&(E)) $&(F j, * &G j, *; E)|
j=1
n*
= : | g(&(E _ F j, * &G j, * ))& g(&(E))
j=1
&g$(&(E))[&(E _ F j, * &G j, * )&&(E)]
+ g$(&(E))[&(E _ F j, * &G j, * )&&(E)]
& g$(&(E)) $&(F j, * &G j, *; E)|
n*
: | g(&(E _ F j, * &G j, * ))& g(&(E))& g$(&(E))
j=1
_[&(E _ F j, * &G j, * )&&(E)]| + | g$(&(E))|
n*
_ : |&(E _ F j, * &G j, * )&&(E)&$&(F j, * &G j, *; E)|
j=1
By definition of $&( } ; E),
n*
: |&(E _ F j, * &G j, * )&&(E)&$&(F j, * &G j, *; E)| w
Ä
0.
*
(A.1)
j=1
Set : j, * =&(E _ F j, * &G j, * )&&(E). Then
| g(&(E _ F j, * &G j, * ))& g(&(E))& g$(&(E))[&(E _ F j, * &G j, * )&&(E)]|
=
}
g(: j, * +&(E))& g(&(E))
& g$(&(E)) |: j, * |.
: j, *
}
In particular,
|: j, * | = |&(E _ F j, * &G j, * )&&(E)|
= |&(E _ F j, * &G j, * )&&(E)&$&(F j, * &G j, *; E)
+$&(F j, * &G j, *; E)|
|&(E _ F j, * &G j, * )&&(E)&$&(F j, * &G j, *; E)|
+ |$&(F j, * &G j, *; E)|.
Let '>0. By (A.1), there exists * 1 such that, for all *>* 1 and all
1 jn * ,
|&(E _ F j, * &G j, * )&&(E)&$&(F j, * &G j, *; E)| '2.
261
DIFFERENTIABLE TU GAMES
Moreover, since $&( } ; E) is convex-ranged, there exists a finite partition of
F _ G, indexed by * 2 , such that, for all 1 jn *2 ,
|$&| (F j, *2; E)'2
and
|$&| (G j, *2; E)'2,
and so
|$&(F j, * &G j, *; E)| |$&(F j, *2; E)| + |$&(G j, *2; E)|
|$&| (F j, *2; E)+ |$&| (G j, *2; E)'.
Hence, by setting * $* =* 1 6 * 2 , we have that for all *>* $* ,
|: j, * | |&(E _ F j, * &G j, * )&&(E)&$&(F j, * &G j, *; E)|
+ |$&(F j, * &G j, *; E)| '
for all 1 jn * . On the other hand, since the derivative g$(&(E)) exists,
given any =>0 there exists ' = >0 such that
}
g(: j, * +&(E))& g(&(E))
& g$(&(E)) =
: j, *
}
for all |: j, * | ' = . Therefore, taking '=' = , we have
n*
:
j=1
}
g(: j, * +&(E))& g(&(E))
& g$(&(E)) |: j, * |
: j, *
}
n*
= : |: j, * |,
for all
** '= .
j=1
But, by the definition of $&( }; E), given any =>0, there exists * 0 such that
*>* 0 implies
n*
n*
j=1
j=1
: |: j, * | =+ : |$&(F j, *; E)&$&(G j, *; E)|
n*
=+ : [ |$&(F j, *; E)| + |$&(G j, *; E)| ]K,
j=1
for some K< that is independent of *, F and G, as provided by the
boundedness of the measure $&( } ; E). Conclude that, given any =>0,
n*
:
j=1
}
g(: j, * +&(E))& g(&(E))
& g$(&(E)) |: j, * | K=,
: j, *
}
262
EPSTEIN AND MARINACCI
and so
n*
: | g(&(E _ F j, * &G j, * ))& g(&(E))& g$(&(E)) $&(F j, * &G j, *; E)| w
Ä
0,
*
j=1
as desired. K
A.3. Miscellaneous Facts
Lemma A.5.
Then
Let &: 7 Ä R be a superadditive game differentiable at ,.
$&(E; ,)&(E)
for all E # 7.
Remark. This lemma generalizes the fact that for superadditive real-valued
functions g: [0, 1] Ä R, we have
g$(0) x g(x)
for all x # [0, 1], whenever g$(0)=lim x a 0 g(x)x exists. In fact, let &=g(P):
7 Ä R be the scalar measure game defined through g. The Chain Rule
(Proposition A.4), and Lemma A.5 imply that
g$(0) P(E)=$&(E; ,)&(E)= g(P(E)).
Proof.
For any E # 7,
$&(E j, *; ,)&(E j, * )+ |$&(E j, *; ,)&&(E j, * )|,
and so
n*
$&(E; ,)= : $&(E j, *; ,)
j=1
n*
n*
j=1
j=1
: &(E j, * )+ : |$&(E j, *; ,)&&(E j, * )|
n*
&(E)+ : |$&(E j, *; ,)&&(E j, * )|.
j=1
DIFFERENTIABLE TU GAMES
263
*
Since & is differentiable at ,, lim * nj=1
|$&(E j, *; ,)&&(E j, * )| =0, and so
$&(E; ,)&(E). K
Next we describe an example of a differentiable game that is not a
measure game: Fix any ordered sequence ( p 1 , p 2 , ..., p n , ...) of countably
additive probability measures and define the game & by
&(A)= :
k=1
\
k
` p i (A)
i=1
+<k !.
Then each summand (> ki=1 p i ( } ))k ! is differentiable by the product rule
and the sum of these derivatives yields $&( } ; E). The game & is not a
measure game if there is no finite linear basis for [ p n( } )].
Suppose that all the measures p n assign the same probability p(A) to
some specific A. Then A is a linear set for the transformed game &*=
log(1+&)(&*(A)= p(A)); and &* is differentiable by the Chain Rule.
A.4. Automorphisms
Derivatives are invariant to a ``renaming'' of agents in the following
sense: Call the function %: 0 Ä 0 an automorphism of 0 if it is 7-measurable and bijective, that is, if it is a permutation over 0. Let 3 be the
group of all 7-measurable automorphisms of 0. Given &, each % # 3
induces a game % & defined by
*
% &(E)=&(%(E))
*
for all E # 7 [2, p. 15]. Evidently, for any F and partition [F j, * ], [%(F j, * )]
is a partition of %(F). Furthermore, [%(F *, j )] * is a subnet of the net of all
finite partitions of %(F ): Let [A j ] be any finite partition of %(F). We want
to show that there is some partition of %(F ) of the form [%(F *, j )] which
refines [A j ]. The collection [% &1(A j )] is a finite partition of F, and since
[F *, j ] * is the set of all finite partitions of F, there is some *$ such that
F j, *$ =% &1(A j ) for all j. For all *>*$, [F *, j ] refines [F *$, j ], and so the
finite partition [%(F *, j )] of %(F ) refines [%(F *$, j )]=[A j ].
Lemma A.6. If % is an automorphism and if & is differentiable at %(E),
then % (&) is differentiable at E and
*
$(% &)( } ; E)=% ($&( } ; %(E)))=$&(%( } ); %(E)).
*
*
264
EPSTEIN AND MARINACCI
Proof.
n*
: | (% &)(E _ F j, * &G j, * )&(% &)(E)&$&(%(F j, * ); %(E))
*
*
j=1
+$&(%(G j, * ); %(E))|
n*
= : |&(%(E _ F j, * &G j, * ))&&(%(E))&$&(%(F j, * ); %(E))
j=1
+$&(%(G j, * ); %(E))|
n*
= : |&(%(E) _ %(F j, * )&%(G j, * ))&&(%(E))&$&(%(F j, * ); %(E))
j=1
+$&(%(G j, * ); %(E))|.
The latter converges to 0 if & is differentiable at %(E) because [%(F *, j )] and
[%(G *, j )] are subnets of the nets of all finite partitions of %(F) and %(G)
respectively. K
A.5. Connection with AumannShapley
Let B 1 denote the set [ f # B(0, 7) : 0 f1]. Aumann and Shapley
show (Theorem G, p. 144) that each & in pNA has an extension to a unique
``integral'' &* : B 1 Ä R 1 satisfying specified properties. In particular, for
any measure game &= g(P) where g is a polynomial function and P=
(P 1 , ..., P N ) is nonatomic, then
&*( f )= g(P*( f )),
where P*( f )=(P *(
1 f ), ..., P*
N ( f )) and P *
i ( f )= 0 f dP i for f # B 1 . It
follows from the sum and product rules for differentiation that for all such
polynomial measure games, G/E and F/E c,
$&(F ; E)=
d
&*(1 E +{1 F )| {=0
d{
$&(G; E)= &
and
d
&*(1 E &{ 1 G )| {=0
d{
and both derivatives exist. More generally, for any 0<t1 and set E t
satisfying P(E t )=tP(E),
$&(F ; E t )=
d
&*(t 1 E +{ 1 F )| {=0 ,
d{
DIFFERENTIABLE TU GAMES
265
provided that the range of P( } ) is full-dimensional. Gateaux derivatives like
those on the right are the calculus tool used by Aumann and Shapley.
We do not know how our derivative $& n( } ; } ) behaves along a suitably
convergent sequence of games [& n ]. Thus it is not clear how the above
connection made for polynomial games might extend to games in pNA.
APPENDIX B
This appendix proves Theorems 6.4 and 6.5, beginning with some common lemmas.
B.1. Common Lemmas
Lemma B.1. Let &= g(P): 7 Ä R be a Dini measure game with core(&)
{,. Suppose that either & is superadditive or that g is bounded below. Then
there exists a positive integer k such that the game &*: 7 Ä R defined by
&*=&+k N
i=1 P i is positive and &*(0)>0.
N
Proof. Let | } | p be the norm defined by |x| p = N
i=1 |x i | for all x # R .
N
Since all norms in R are equivalent, it is easy to see that, for any norm | } |,
g(x)
g(x)
>&.
>& lim inf
|x| Ä 0 |x|
|x| p Ä 0 |x| p
lim inf
Therefore, lim inf |x|p Ä 0 g(x)|x| p >& for any Dini game g(P). Assume
first that g is bounded below, i.e., inf x # R(P) g(x)>&. Given M>0, there
exists =>0 such that
g(x)
> &+
|x| p
if
x # R(P)
and 0< |x| p <=.
(B.1)
Let x # R(P), |x| p = and g(x)<0. Then g(x)|x| p g(x)
= inf x # R(P) g(x)=.
Let k$ be a positive integer such that k$(|inf x # R(P) g(x)|=)+M.
Then g(x)|x| p > &k$ for all x # R(P)"[0] such that either 0< |x| p <=
or g(x)<0, implying that g(x)+k$ |x| p >0 for all such x. But g(x)+
k$|x| p >0 also for all x # R(P)"[0] such that g(x)>0. Conclude that
&(E)+k$ N
i=1 P i (E)0 for all E # 7.
Next consider the case where & is superadditive. Given M, choose = as
above satisfying (B.1). For x # R(P)"[0], let E # 7 be such that P(E)=x.
266
EPSTEIN AND MARINACCI
By the Lyapunov Theorem there exists a partition [E l ] Ll=1 7 of E such
that 0< |P(E l )| p <= for each 1lL. Hence,
N
L
&(E)+k$ : P i (E) :
i=1
l=1
\
N
+
&(E l )+k$ : P i (E l ) >0
i=1
since &(E l )+k$ N
i=1 P i (E l )>0 for each 1lL. Conclude that also in
this case &(E)+k$ N
i=1 P i (E)0 for all E # 7.
Finally, assume that P(0){0, so that |P(0)| p >0. Let k>max[k$, &&(0)
|P(0)| p ] and set &*=&+k N
i=1 P i . If &(0)>0, then &*(0)>0 and we are
done. If &(0)0, then &*(E)&(E)+k$ N
i=1P i (E)0 for all E # 7 and
N
K
&*(0)=&(0)+k N
i=1 P i (0)>&(0)+(&&(0)|P(0)| p ) i=1 P i (0)=0.
Lemma B.2.
Let & and &* be as in Lemma B.1. Then:
1. core(&*)=[m+k N
i=1 P i (E): m # core(&)],
2. A is linear with respect to &* if and only if it is linear with respect
to &.
3. & is coherent at A # A if and only if &* is coherent at A.
Proof. The first two assertions are easily proven. For coherence, it
suffices to see that
$&*(A; A)&*(A) $&(A; A)&(A),
$&*(A c; A)&*(A c ) $&(A c; A)&(A c ),
$&*(A; A c )&*(A) $&(A; A c )&(A),
$&*(A c; A c )&*(A c ) $&(A c; A c )&(A c ). K
The subset of core(&) consisting of countably additive measures is the
_-core and is denoted by
core _(&)=[m # CA: m(0)=&(0), and m(A)&(A) for all A # 7].
Lemma B.3. Given a positive game &: 7 Ä R + , every measure
m # core _(&) is non-atomic if at least one of the following two conditions is
satisfied:
(i)
there is some A # A, with &(A)=&(0), such that & is differentiable
at A;
(ii) there is some A # A, with 0<&(A)<&(0), such that & is differentiable at both A and A c.
Proof. For case (ii), suppose that 0<&(A)<&(0). Then m(A)>0 and
m(A c )>0. By definition, both $&( } ; A) and $&( } ; A c ) are convex-ranged.
DIFFERENTIABLE TU GAMES
267
Since & is positive, each m # core(&) is positive. Hence, by Lemma 4.4,
$&( } ; A) and $&( } ; A c ) are positive measures on A and A c, respectively.
Again by Lemma 4.4, for all GA, $&(G; A)=0 O m(G)=0, i.e., m is
absolutely continuous with respect to the finitely additive measure $&( } ; A)
on A. This implies that m has no atoms in A as we proceed to show.
Let m(E)>0 with 7 % EA. By above, $&(E; A)>0. Since $&( }; A) is convex-ranged, there exists a partition E 1, B 1 of E such that $&(E 1; A)=
$&(B 1; A)= 12 $&(E; A). If 0<m(E 1 )<m(E) or 0<m(B1 )<m(E), we are done.
Suppose, in contrast, that either m(E 1 )=m(E) or m(B 1 )=m(E). Wlog, let
m(E 1 )=m(E). Again, there exists a partition E 2 and B 2 of E 1 such that
$&(E 2; A)=$&(B 2; A)= 12 $&(E 1; A). If 0<m(E2 )<m(E 1 ) or 0<m(B 2 )<
m(E 1 ), we are done. Suppose, in contrast, that either m(E 2 )=m(E1 ) or
m(B 2 )=m(E 1 ). Wlog, let m(E 2 )=m(E 1 ). Proceeding in this way, either we
find a set BE such that 0<m(B)<m(E) or we can construct a chain
[E n ] n1 such that $&(E n; A)= 21n $&(E; A) and m(E n )=m(E) for all n1.
Hence, because n1 E n # 7 and n1 E n E, we have $&( n1 E n; A)=0
and m( n1 E n )=m(E)>0, a contradiction. Conclude that there exists some
set BE such that 0<m(B)<m(E), and that m has no atoms in A.
Next replace A by A c and use the convex range of $&( } ; A c ) to deduce
in a similar fashion that m has no atoms in A c. Now, suppose that E is an
atom for m in 7. By definition, either m(E & A)=0 or m(E & A)=m(E). If
m(E & A)=m(E), then E & A is an atom in A, which was ruled out above.
Hence, m(E & A)=0. A similar argument shows that m(E & A c )=0 and so
m(E)=0, a contradiction. Hence, there are no atoms in 7. Because m is
countably additive and non-atomic, it is also convex-ranged.
Finally, for case (i), suppose that &(A)=&(0). Then m(A c )=0. Then if
E is an atom in 7, it must be the case that m(E & A)=m(E)>0, and so
E & A is an atom on A. This can be ruled out as above. K
Lemma B.4. Let &= g(P): 7 Ä R + be a positive measure game with P
countably additive. Then core(&)=core _(&) given at least one of conditions
(i) and (ii) of Lemma B.3.
Proof. Let m # core(v). Suppose (i) is true. Since & is differentiable
at A, it is continuous at A. Suppose that E n A 0. Then E n & A A A and so
m(E n )=m(E n & A) A m(A)=m(0), that is, m # core _(&).
Next suppose (ii) is true. Since & is differentiable at A and A c, & is continuous at both A and A c. Because both are linear sets, this implies that,
for all m # core(&), m(E$n ) A m(A) and m(E"n ) A m(A c ) if E$n A A and E"n A A c.
Therefore, if E n A A, then
m(E n )=m(E n & A)+m(E n & A c ) A m(A)+m(A c )=m(0),
as desired. K
268
EPSTEIN AND MARINACCI
B.2. Proof of Theorem 6.4
Prove the result first for &0. Since the game is differentiable at A and
A c, by Lemmas B.3 and B.4 every m # core(&) is non-atomic and countably
additive.
Suppose that core(&){<. By Lemma B.3, every measure in core(&) is
convex-ranged. For convenience, assume that P(0)=1 # R N. Set x=P(A)
# [0, 1] N and let m # core(&).
Since A # ( <, 0), P(A) Â [0, P(0)]. Hence, by reversing the roles of A
and A c if necessary, we can assume that :P(A)=P(A c ) for some : # (0, 1).
By the Lyapunov Theorem applied to the N+1 dimensional vector
measure (P, m), there exists 7 % E/A such that P(E)=:P(A) and
m(E)=:m(A). Hence,
g(:x)= g(P(E))m(E)=:m(A)=:g(P(A))=:g(x).
(B.2)
By Lemma 4.1, by $&(A c; A)m(A c )=&(A c ). Since g is differentiable at x,
this implies that {g(x) } (1&x) g(1&x). But :x=(1&x). Hence, using
(B.2),
{g(x) } (:x) g(:x):g(x),
which implies {g(x) } x g(x), that is, $&(A; A)&(A). On the other hand,
by Lemma 4.1, $&(A; A)&(A), and so $nu(A; A)=&(A). Then {g(x) } x=
g(x), so that
{g(x) } (1&x)={g(x) } (:x)=:g(x)g(:x)= g(1&x),
that is, {g(x) } (1&x)g(1&x) and hence $&(A c; A)&(A c ). Conclude
that & is efficient at A.
This completes the proof for &0. Let &* be the positive game provided by
Lemma B.1. By Lemma B.2, core(&){< implies core(&*){<. Moreover, it is
easy to check that A # (<, 0) with respect to &* and that &* is differentiable
at A and A c. Hence &* is coherent at A when core(&){<; and & is coherent
at A by Lemma B.2. In particular, it is now easy to see that core(&)/
[$&( } ; A)]. K
B.3. Proof of Theorem 6.5
Prove statement (i) for &0. By Lemmas B.3 and B.4, every m # core(&)
is countably additive and non-atomic.
There exists : # (0, 1) such that P(A$)=:P(A). Apply the Lyapunov
Theorem to (P, m), where m # core(&). There exists 7 % A : /A be such that
P(A : )=:P(A) and m(A : )=:m(A). Because P(A$)=P(A : ) and consequently,
269
DIFFERENTIABLE TU GAMES
P((A$) c )=P(A c: ), we have
m # core(&). In particular,
A : # A.
Hence,
m(A : )=&(A : )
for
all
g(:x)= g(P(A : ))=m(A : )=:m(A)=:g(x).
We want to show that g( ;x)=;g(x) for all 1;:. Suppose, per contra,
that g(x)<;g(x) for some 1;>:. Since g is convex, g(;x);g(x) and
g(:x)= g
:
:
:
\; ;x+ ; g( ;x)<; ;g(x)=:g(x),
a contradiction. Set P(A)=x. For t small enough, (1&t):, so that
lim
ta0
g(x)& g(x&tx)
tg(x)
=lim
= g(x),
t
ta0
t
and so {g(x) x= g(x). Hence $&(A; A)=&(A).
There exists : # (0, 1) such that P(A")=:P(0)+(1&:) P(A). By the
Lyapunov Theorem applied to (P, m), where m # core(&), there exists 7 %
A : /A such that P(A : )=:P(0)+(1&:) P(A) and m(A : )=:m(0)+
(1&:) m(A). Because P(A")=P(A : ) and P((A") c )=P(A c: ), conclude that
A : # A. Hence, m(A : )=&(A : ) for all m # core(&). In particular,
g(:P(0)+(1&:) x)=g(P(A : ))=m(A : )
=:m(0)+(1&:) m(A)
=:g(P(0))+(1&:) g(x).
We want to show that g( ;P(0)+(1&;) x)=;g(P(0))+(1&;) g(x) for
all 0;:. If not, there exists some 0;<: such that g( ;P(0)+
(1&;) x)<;g(P(0))+(1&;) g(x). Then
g(:P(0)+(1&:) x)
\
1&:
:&;
P(0)
( ;P(0)+(1&;) x)+
1&;
1&;
\ + +
1&:
:&;
g(;P(0)+(1&;) x)+
\1&;+ g(P(0))
1&;
1&:
:&;
<
g(P(0))
(;g(P(0))+(1&;) g(x))+
\
1&;
1&; +
=g
=:g(P(0))+(1&:) g(x),
270
EPSTEIN AND MARINACCI
a contradiction. Therefore,
g(x+t(P(0)&x))& g(x) g((1&t) x+tP(0))& g(x)
=
t
t
=
(1&t) g(x)+tg(P(0))& g(x)
t
=
tg(P(0))&tg(x)
t
=g(P(0))& g(x)= g(P(A c )).
Since g is differentiable at x, this implies that {g(x) P(A c )= g(P(A c )), that
is, $&(A c; A)=&(A c ). Conclude that & is coherent at A.
This completes the proof for the case &0. Let &* be the positive game
provided by Lemma B.1 (such a &* exists because g is bounded below). By
Lemma B.2, core(&){< implies core(&*){<. Hence, by what has been
proven above, &* is coherent at A if core(&){<. Again by Lemma B.2, this
implies that & is coherent at A if core(&){<. Consequently, core(&)=
[$&( } ; A)].
Next prove statement (ii). We first prove the following claim:
Claim. Let &= g(P): 7 Ä R be a totally balanced Dini measure game.
Then g( ;x);g(x) for all ; # [0, 1].
Proof of the Claim. Suppose that &0. Let E # 7 be such that
P(E)=x. It is easy to see that the subgame & |E is superadditive. In addition, its dual & |E is subadditive. In fact, for any pair E 1 , E 2 of disjoint sets
in 7 E , there exists a measure m such that m(E 1 _ E 2 )=& |E (E 1 _ E 2 ) and
m(E)& |E (E) for all 7 % EE 1 _ E 2 . Hence, & |E (E 1 _ E 2 )=m(E 1 _ E2 )=
m(E 1 )+m(E 2 )& |E (E 1 )+& |E (E 2 ). Therefore, for all [x i ] ni=1 /R(P) such
that ni=1 x i x, we have
\
n
n
and
+
g x& : x : g(x&x )+(1&n) g(x).
\
+
g
: x i : g(x i )
i=1
i=1
n
n
i
i=1
i
i=1
Moreover, for all :, ; # [0, 1], g(:x) g(;x) if :;: Let E # 7 and
P(E)=;x. By the Lyapunov Theorem there exists 7 % E$/E such that
P(E$)= ;: P(E)=:x. Since & is positive and superadditive, it is monotone.
Thus g(:x)=&(E$)&(E)= g( ;x).
271
DIFFERENTIABLE TU GAMES
We can now use a simple variation of an induction argument of Wasserman and Kadane [14, p. 1729] that we repeat here for the sake of completeness. Let y= k1 x, for some positive integer k. Then g( y) 1k g(x), as
otherwise, g(x)=g(ky)kg( y)>k k1 g(x)= g(x), a contradiction. Similarly,
g(x& y)(1& k1 ) g(x), since otherwise, we have the following contradiction:
\
k&1
1
x (k&1) g x& x +(2&k) g(x)
k
k
+
\ +
1
1
>(k&1) 1&
g(x)+(2&k) g(x)= g(x)g( y).
\ k+
k
g( y)=g x&
Suppose now that for some positive integer h<k, g( kr x) kr g(x) and
g((1& kr ) x)(1& kr ) g(x) for all r # [1, ..., h]. Consider r=h+1. Then
g( kr x) kr g(x): If not, by setting s= k&h
r , we reach the contradiction
g
h
r
r
\\1&k+ x+ = g \s k x+ sg \k x+>s
h+1
h
g(x)= 1&
g(x).
k
k
\ +
Conclude by induction that g(;x);g(x) for all rational numbers ; #
(0, 1). Next, let ; be any real number in (0, 1). Given =>0 and a rational
number q # (0, 1) such that q&=<;q, then g(;x)g(qx)qg(x)
(;+=) g(x) for all =>0. Hence g( ;x);g(x).
If & is not positive, let &* be the positive game provided by Lemma B.2.
This game is totally balanced if & is totally balanced. Thus
N
N
g(;x)= g*( ;x)&k : ;x i ;g*( ;x)&;k : x i =;g(x),
i=1
i=1
which completes the proof of the Claim.
Assume that &>0. By Lemmas B.3 and B.4, all measures in core(&) are
non-atomic and countably additive. There exists : # (0, 1) such that
P(A$)=:P(A). By the Lyapunov Theorem applied to (P, m), where
m # core(&), there exists 7 % A : /A be such that P(A : )=:P(A) and
m(A : )=:m(A). Because P(A$)=P(A : ) and P((A$) c )=P(A c: ), we have
A : # A. Hence, m(A : )=&(A : ) for all m # core(&). In particular,
g(:x)= g(P(A : ))=m(A : )=:m(A)=:g(x).
We want to show that g(;x)=;g(x) for all ;:. If not, then g( ;x)<
;g(x) for some ;>: and
g(:x)= g
:
:
:
\; ;x+ ; g( ;x)<; ;g(x)=:g(x),
272
EPSTEIN AND MARINACCI
a contradiction. Set P(A)=x. For t small enough, (1&t):, so that
{g(x) x=lim
ta0
g(x)& g(x&tx)
tg(x)
=lim
= g(x).
t
ta0
t
Hence $&(A; A)=&(A).
There exists : # (0, 1) such that P(A")=:P(A)+(1&:) P(0). Then,
P((A") c )=:P(A c ). Proceeding as before we get $&(A c; A c )=&(A c ). Hence,
& is coherent at A and, for all m # core(&),
m(E)=$&(E & A; A)+$&(E & A c; A c ).
This completes the proof for &0. Since & is superadditive, we can use
Lemma B.1, and thus the extension to Dini games is evident. K
B.4. Proof of Theorem 6.6
Part (a): We first prove the result for &0. Under (ii), it is easy to see
that {g(P(0)) P(0) g(P(0)), that is $&(0; 0)&(0). Hence, by (5.3) &
is coherent at 0, so that by Theorem 5.2, core(&){< if and only if
core(&)=[$&( } ; 0)]. As to (i), suppose that k<1. By the Lyapunov
Theorem, for each : # (0, 1) there is A : # 7 such that P(A : )=:P(0). We
have,
lim
ta0
g(P(0))& g(P(0)&tP(A : ))
t
lim
g(P(0))&(1&t:) k g(P(0))
t
<lim
g(P(0))&(1&t:) g(P(0))
=:g(P(0)),
t
ta0
ta0
and so {g(P(0)) P(A : )<:g(P(0)), which implies {g(P(0)) P(0)<g(P(0)).
Hence, & is coherent at 0 but $&( }; 0) Â core(&). By Theorem 5.2, core(&)=<.
This completes the proof for positive measure games. Now, let & be any Dini
game and let &* be the positive game provided by Lemma B.1. It is easy to
check that &* is super-homogeneous of degree k # [0, 1] at P(0). Hence,
core(&*)/[$&*( }; 0)] if k=1 and core(&*)=< if k<1. By Lemma B.2,
core(&)/[$&( } ; 0)] if k=1 and core(&)=< if k<1.
DIFFERENTIABLE TU GAMES
273
Part (b): The only non-trivial part is that (iii) implies (ii). By
Theorem 6.6, it suffices to show that $&( } ; 0) # core(&). Since $&(0; 0)=
&(0), this can be shown by proceeding as in the proof of Corollary 6.2,
which needs only that &(E)+&(E c )&(0) for all E # 7. K
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