#### Abstract

We prove that the Burkill-Cesari integral is a value on a subspace of and then discuss its continuity with respect to both the and the Lipschitz norm. We provide an example of value on a subspace of strictly containing as well as an existence result of a Lipschitz continuous value, different from Aumann and Shapley’s one, on a subspace of .

#### 1. Introduction

Since the seminal Aumann and Shapley's book [1], it is widely recognized that the theory of value of nonatomic games is strictly linked with different concepts of derivatives. A few papers, up to the recent literature, have investigated these relations (see, e.g., [2–4]). In [1] Aumann and Shapley proved the existence and uniqueness of a value on the space , namely, the space spanned by the powers of nonatomic measures (which, under suitable hypotheses, contains, for instance, games of interest in mathematical economics such as transferable utility economies with finite types). Moreover, in [1, Theorem H], the authors provided an explicit formula for the value of games in in terms of a derivative of their “ideal” set function .

To the best of our knowledge, the more general contribution so far on the link between derivatives of set functions and value theory is Mertens [3]; his results led to the proof of the existence of a value on spaces larger than . A more recent contribution on the same subject is due to Montrucchio and Semeraro [4]. The problem of the existence of a value on the whole space of absolutely continuous games (which contains ) is instead still unsolved and challenging. Therefore, proofs of the existence of a value on other subspaces of , beyond , can represent a step forward, and investigations of this kind appear to be in order.

In Epstein and Marinacci [2] the question of the relation between their refinement derivative and the value was posed and a possible direction sketched; in Montrucchio and Semeraro [4], the authors applied their more general (i.e., without the nonatomicity restriction) notion of refinement derivative to the study of the value on certain spaces of games by extending the potential approach of Hart and Mas-Colell [5] to infinite games.

In a previous paper [6] we had pointed out that, in a nonatomic context, the refinement derivative is connected with the classical Burkill-Cesari (BC) integral of set functions and, for BC integrable functions, the BC integral coincides with the refinement derivative at the empty set. Though less general, the BC integral is analytically more treatable.

Motivated by all these facts, in [6] we have started the study of the BC integral in the framework of transferable utility (TU) games.

In this paper we extend our investigation to develop the connection with the theory of value or of semivalue, also in the light of the problem exposed above. In Section 2 we introduce the general class of BC integrable games and prove that, under natural assumptions, “regular” measure games belong to this class. Moreover, the class of BC integrable games contains a dense subspace of the largely used space , where continuous values and semivalues are largely described in the literature (see, for instance, [7]). In addition, on the subspace of BC integrable games in , the BC integral turns out to be a semivalue. Then, as natural, one considers the subspace of feasible BC integrable games, that is, the space where the BC integral becomes indeed a value. We provide examples of feasible BC integrable games that do not belong to . Actually by means of these examples, we provide a large class of subspaces of where the BC integral is a value, and we also exhibit an example of a subspace of strictly containing on which a value can be defined as a sort of direct sum of the usual Aumann-Shapley value and this new set function.

Unfortunately, the BC integral proves to be not continuous with respect to the norm on the BC integrable games in . As continuity appears to be a crucial property for many questions concerning the value on subspaces of , in Section 3 we specialize to the subspace of the so-called Lipschitz games, where a suitable finer norm (the -norm) is defined and used as an alternative (see [8, 9]). We completely characterize the scalar measure games (where the measure is nonnegative) that belong to and we show that the BC integral on an appropriate subspace is a Milnor (therefore -continuous) semivalue. Then again we turn our attention to the subspace of feasible BC integrable games in and to its closure in the -norm, . In the final part of the paper we consider and somehow characterize the space (namely, the -closure of vector measure games generated by polynomials), and we show that the BC integral is not the unique value on it, in that it does not coincide with the Aumann-Shapley value.

#### 2. A Semivalue on a Space of Burkill-Cesari Integrable Games

From now on we will denote by a standard Borel space (i.e., is a Borel set of a Polish space and the family of its Borel subsets). represents a set of players and the -algebra of admissible coalitions.

A set function such that is called a* transferable utility* (TU) game.

For the sake of brevity we refer the reader to [1, 14] for the terminology concerning TU games: in particular will denote the space of all bounded variation games, endowed with the variation norm . The subspace of nonatomic countably additive measures will be denoted by and the cone of the nonnegative elements of by .

Throughout the paper we will write to mean that absolute continuity by chains holds.

*Definition 1 (see [1]). *A chain is a nondecreasing family of sets:
A* link* of a chain is a set of two consecutive elements . A* subchain* of a chain is any set of links.

A chain will be identified with the subchain consisting of all the links. Given a game and a subchain of a chain , the variation of over is defined as where the sum ranges over all indexes such that is a link in the subchain.

*Definition 2 (see [1]). *If and are two games defined on , is said to be absolutely continuous with respect to if for every there exists a such that, for every chain and every subchain of ,

The space introduced by Aumann and Shapley [1] is the space of all games for which there exists such that is absolutely continuous with respect to .

We also refer the reader to [2] and to our previous paper [6] for details about the Epstein-Marinacci refinement derivative.

A* partition* of a set is a finite family of pairwise disjoint elements of , whose union is . By we will denote the set of all the partitions of . A partition is a* refinement* of another partition if each element of is union of elements of .

As in [11], given a monotone nonatomic game one defines the mesh of a partition as and the Burkill-Cesari (BC) integral of a game with respect to as We denote by BC the space of games such that there exists so that is BC integrable with respect to the mesh .

The BC integral does not depend upon the integration mesh (see Proposition 5.2 in [6]); in other words, for every such that is -BC integrable, the BC integral is the same. Moreover, the BC integral of a game is a finitely additive measure and, as observed in [6], it coincides with the Epstein-Marinacci outer derivative at the empty set (see [2]). Hence, from now on we will use the notation .

As we will see, the space BC contains many games which are of interest in the literature: we begin by recalling a sufficient condition for vector measure games to be in BC, which is an immediate consequence of [6, Theorem 6.1].

Proposition 3. *Let be a nonatomic vector measure, and let be a function with . If is differentiable at 0, then the game , and .*

Anyway, the class of BC integrable games is not limited to smooth measure games. Indeed note that the converse implication of the previous proposition does not hold: consider as in [6, Example 3.2] to be any discontinuous solution to the functional equation Then is additive, and therefore for each and each one has and hence is BC integrable with respect to although is not differentiable at 0.

As for the relation between the spaces and , it is well known (see Theorem C of [1]) that the game (with the Lebesgue measure on ) belongs to . Anyway . To see it, one has to show that is not BC integrable with respect to any mesh determined by some . Indeed is not refinement differentiable at Ø, and then it cannot be BC integrable with respect to ; to get convinced that does not admit outer refinement derivative at Ø, observe that for every partition and every we can provide a refinement such that and [15, Lemma 3.5]. Clearly we can choose quite larger than say . Also we can choose determined by the uniform continuity of on . Thus which shows that the refinement limit does not exist.

In fact , and therefore , whence , while .

So On the contrary, any strongly nonatomic finitely additive measure that is not countably additive provides an example of game belonging to BC (for ), but not to (see the different notions of absolute continuity in [6]).

Therefore, we can now consider the space .

The following result states that the same measure can be used for the absolute continuity and the BC integrability of a game in .

Proposition 4. *The space can be equivalently defined as the space of games such that there exists such that and is BC integrable with respect to .*

*Proof. *The fact that each game for which there exists such that and is BC integrable with respect to lies in is straightforward.

Conversely, let ; then there are such that (since ) and is -BC integrable. Then consider ; evidently and, in view of Proposition 5.2 in [6], is also -BC integrable.

From [1] we recall the following.

*Definition 5. *Let denote the space of automorphisms of , then each induces a linear mapping of onto itself, defined by
for . A subspace that is invariant under for every is called* symmetric*.

Proposition 6. *The space is symmetric.*

*Proof. *We need to prove that for every and every game the game defined in (10) is in ; namely, it is with respect to some nonatomic measure, and it is BC integrable too.

Let be a measure in with respect to which we have and is -BC integrable (thanks to Proposition 4 we can always assume that the default measure is the same).

Fix ; note that preserves set operations; therefore easily is in .

It is also immediate to check that , because transforms chains and subchains into chains and subchains as well.

It remains to prove that is BC integrable with respect to the mesh . Indeed we will prove that
for every .

To this aim, for any and any fixed, one has to find such that for every partition with there holds
Since is BC integrable, to each there corresponds such that for each partition with there follows
Clearly we can rewrite (12) as
We choose ; thus if has , the corresponding partition has since, for each , clearly and hence

According to [7] we recall the following definition.

*Definition 7. *A linear mapping on a symmetric subspace of is called a* semivalue* provided that it satisfies the following properties:V.1(*symmetry*): for each ;V.2 (*positivity*): is positive, that is for every monotone game the measure is nonnegative;V.3(*projection axiom*): is the identity operator on ; when satisfies also the property:V.4(*efficiency*): for each there holds ;

is called a *value* on (compare with [1]).

The following result immediately derives from (11) and the definition of .

Corollary 8. *The mapping is a semivalue on .*

Consider now the space and define the space . Obviously is a value on .

The next example shows that this space contains several games which do not belong to , and through these one finds several new subspaces of on which defines a value.

*Example 9. *Let be a signed measure on a measurable space with range say and, for instance, , and let be defined as
Let and consider the space . Then so that but and is a value on : *Fact I.* ; indeed, since exists, by Proposition 3, and whence immediately .*Fact II. *; indeed, according to Kohlberg [13] a measure game , where is a signed measure, is in if and only if the function is continuously differentiable in . *Fact III. *; for the proof, see the Appendix.

Finally is symmetric, and by linearity and relationship (11), is efficient on ; therefore it is a value on .

It is clear that one can use functions of different forms to provide classes of measure games with signed that are in and in , and hence similar subspaces of on which is a value. These subspaces will not be contained in because the generating game . However, we can go a little further; in fact the next example shows that, by means of a similar construction, there are subspaces in strictly containing on which a value can be defined.

*Example 10. *On equipped with the usual Borel -algebra, consider the signed measure .

Denote , , and and take the function defined by
Define the scalar measure game and take then , where denotes the usual Lebesgue measure.

As in the previous example, and hence are in , (for ) and (again, for the proof, see the Appendix).

Furthermore (and hence ) is in , since is continuous [16].

Thus .

Again take and denote by the smallest linear subspace containing and . Define as follows: for each , set , where is the usual Aumann and Shapley value.

is a value on (see the Appendix for the proof).

Finally, the following example shows that is not continuous on equipped with the variation norm.

*Example 11. *Consider the sequence of scalar measure games , where and . Then as . However, does not converge to .

#### 3. The Operator on Subspaces of Lipschitz Games

In [9] the author considers the class of* Lipschitz games*, that is, games in for which there exists a measure such that both and are monotone games. The reason why these games are called Lipschitz is the fact that the condition can be equivalently labelled in the following form: for every link in there holds
The connection to the Lipschitz condition is made even stronger by the following.

Proposition 12. *For a scalar measure game the following are equivalent:*(1);(2)* is Lipschitz on the interval * (*with ** as Lipschitz constant, for each ** for which* (18)* above is satisfied*)*;*(3)(18)* holds for **, with ** Lipschitz constant for **.*

*Proof. *To prove that (1) implies (2), assume that is a Lipschitz game, and let be a measure for which (18) holds. For simplicity we assume that is a probability measure. Let with say .

Then, by Lyapunov theorem, there exist sets such that
Hence and ; then from (18)
The fact that a Lipschitz function generates a Lipschitz game (where one can precisely choose in (18)) is immediate, so (2) implies (3).

Also (3) implies (1) trivially.

It is immediate to note that . However, the smaller space can be equipped with an alternative norm defined in the following way; for every such that (18) holds, write . Then we set Then is a Banach space when equipped with the above norm.

Again from [9] we quote the following definition.

*Definition 13. *Let and define the following two subsets of :
Then the following two measures exist: , , and immediately (although the symbol should be distinguished from , as the first one refers to setwise ordering while the second one to the order induced by the cone of monotonic games, in the case of measures they actually assume the same meaning).

Let be a linear subspace, and let be a linear operator; we will say that is a* Milnor operator* (MO) provided that for every we have

Consider now the vector subspace of Lipschitz games that are BC integrable.

is strictly included in BC, for there are easy examples of games in .

For instance, consider the function defined as and the scalar measure game , where represents the usual Lebesgue measure. Then with thanks to Proposition 3, but since is not Lipschitz on .

Also the inclusion is a strict one, for there are Lipschitz games that are not in BC. To see this we need the following result, which is a partial converse of Proposition 3.

Proposition 14. *Let the scalar measure game be in ; then the following are equivalent:*(1)* admits right-hand side derivative at* 0*;*(2)* is *- BC* integrable and hence **.*

*Proof. *The implication (1) (2) follows from Proposition 3.

We turn then to the implication (2) (1).

As , there exists such that is -BC integrable. Since we already know that is Lipschitz; hence the ratios are bounded. Assume by contradiction that does not exist. Then it can only happen that
Choose then two decreasing sequences with and
Fix with and . Then there exists such that for each
By means of the continuity of at 0, choose next such that whenever ; also can be chosen so that and such that , where is the parameter of BC integrability.

Choose now the following : by means of Lyapunov theorem, divide into finitely many sets, say , each with , until and then choose ; thus easily .

Then for one has .

We have then, similar to the computation in Proposition 3:
As for the first sum we have the following estimate:
In conclusion
Clearly we can repeat this construction with and find another partition with as above; again
It is then clear that, since , the game is not -BC integrable.

Therefore, for instance, taking defined as the game , since for one has , but, as does not exist, according to the previous result, .

We will need in the sequel the following lemma.

Lemma 15. *The space is symmetric and the following equality holds for every :
*

*Proof. *Fix and choose such that and .

Let . If , then and therefore, by monotonicity, , or else , which is the same as to say that is monotone, and hence .

In a completely analogous way, as , one reaches . In conclusion .

Moreover, whence
To prove the converse inequality, first of all, for , consider the game defined in the following fashion: for every set and set
so that . It is a routine computation, based on the properties of , to show that is a countably additive measure as well.

Again fix and choose such that and .

Take defined above.

Now (or else monotone) implies in turn that is monotone too, and similarly .

Hence
which concludes the proof of relationship (33).

In we have the following result.

Proposition 16. *The BC integral is a Milnor semivalue on .*

*Proof. *Let be any game in , ; then is a monotone game, and hence for each , which in turn implies immediately that , namely, setwise in . As is a measure, this equivalently says that is monotone, that is, . In complete analogy if , then .

Hence for we have necessarily which proves that is a MO.

Finally, we deduce from (11) the symmetry of the operator , and the proof is thus complete.

According to Theorem 1.8 in [9], can be extended to the whole space in such a way that the extension, which we will label as , remains a linear MO. Recall moreover that MO on subspaces of are continuous with respect to the norm [9, Lemma 1.6].

Let denote the -closure of . Then on we have the following.

Theorem 17. * is a Milnor semivalue on .*

*Proof. *If , there exists a sequence in , say that converges to .

Because of (33), for each , we have that . But then too.

Similarly , and then, again by (33), .

In conclusion .

Since powers of probabilities belong to , there immediately follows the following.

Corollary 18. * is a -continuous semivalue on .*

We point out that, as , , and are symmetric subspaces of , there follows from [9, Theorem 3.1] that and are diagonal.

Moreover from [9, Theorem 2.1], there exists a Borel measure on such that the following representation of on holds: where is the ideal extension of the game defined in [1, Theorem G].

Define now the space . Note that . Also define

Proposition 19. * is the -closure of .*

*Proof. *Denote by the -closure of . Indeed it is immediate to prove that : take and a sequence converging in the norm to . Then . Moreover, the convergence of to implies that converges to . Furthermore, converges in variation (and hence setwise) to . Therefore,
so .

For the converse inclusion, take . Hence is the -norm limit of a sequence of elements in . Construct now the sequence
for some . We claim that and converges to .

Indeed, by the feasibility of and by the fact that and , it follows that converges to . Moreover, as , it immediately follows that .

Obviously is a value of , but, unfortunately, on the subspace we lose uniqueness, in that does not agree with the Aumann-Shapley (AS) value.

*Example 20. *Let , where and . Let and be two linearly independent measures in , and , where .

We claim that , but , where denotes the AS value.

The function and we know that and . Hence , while . It is also immediate to check that .

Indeed one can provide infinitely many examples of subspaces of where a value different from the AS one is defined. By means of [9, Theorem 2.1], each probability measure on generates a Milnor semivalue on which clearly becomes a value on the subspace . In our case for the Dirac measure based at , we have precisely . However, we point out that our main interest in this paper is not uniqueness of the value but the fact that the Burkill-Cesari integral and hence the Epstein-Marinacci derivative constitute a value.

It may anyway be of interest to characterize games in . Remember that is the -closure of the linear span of powers of nonatomic probability measures. With the same technique used in Proposition 19 one proves that is in fact the -closure of . It is now easy to characterize these games: in fact if , with , then . So if and only if .

#### Appendix

We will first prove that the measure game in Example 9 is in AC.

Let and let (the total variation of ).

Fix a Hahn decomposition of .

We will prove that .

Fix and choose . Let be a chain and a subchain with . We claim that .

In fact, let be a link in . Then, as , necessarily if do not belong to the same , then they are at most in two contiguous 's.

Indeed, suppose, for instance, that , namely, . Hence and analogously . Thus that is; namely, whence, a fortiori, which contradicts the initial assumption .

Then an easy computation shows that in both cases (either , belong to the same or they lie in two contiguous ’s) and this proves our claim, since .

In a completely analogous way one shows that the game in Example 10 is in .

To prove that in the same example is a value, the only property that needs to be checked is positivity (the others are obvious).

Assume that , is monotone.

We will also use the alternative form , where .

Now by (11) since , while . Hence we need to prove that .

Note first that since , the ideal set function exists, and it actually coincides with [1, Proposition 22.16 p. 152], and that . Hence the ideal function exists too.

Recall now the definition of , , from [16].

In particular, since (for ), we find and similarly for the left-hand side limit. In other words for each .

We note now that, since is a measure game, for every , [1, Proposition 22.16, p. 152], and the proof of Theorem 3 in [16]).

Now, since is monotone, both are nonnegative for each . Hence where in the last summand we do not need to distinguish between and as and hence exists.

But then which is precisely what we wanted to prove.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.