The Burkill-Cesari Integral on Spaces of Absolutely Continuous Games

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 .


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][3][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, 2 International Journal of Mathematics and Mathematical Sciences 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.

A Semivalue on a Space of Burkill-Cesari Integrable Games
From now on we will denote by (Ω, Σ) a standard Borel space (i.e., Ω is a Borel set of a Polish space and Σ the family of its Borel subsets).Ω represents a set of players and Σ the algebra of admissible coalitions.A set function ] : Σ → R such that ](Ø) = 0 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 { −1 ,   }.
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 Λ of a chain C, the variation of ] over Λ is defined as where the sum ranges over all indexes  such that { −1 ,   } is a link in the subchain.
Definition 2 (see [1]).If ] and  are two games defined on Σ, ] is said to be absolutely continuous with respect to  if for every  > 0 there exists a  > 0 such that, for every chain  and every subchain Λ 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  ∈ Σ is a finite family of pairwise disjoint elements of Σ, whose union is .By Π() we will denote the set of all the partitions of .A partition  ∈ Π() is a refinement of another partition  ∈ Π() 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  + Ø (], ⋅) (see [2]).Hence, from now on we will use the notation  + Ø (], ⋅).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  : Σ → R  be a nonatomic vector measure, and let  : R  → R be a function with (0) = 0.If  is differentiable at 0, then the game ] =  ∘  ∈ BC, 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]  : R → R to be any discontinuous solution to the functional equation Then ] =  ∘  is additive, and therefore for each  ∈ Σ and each  ∈ Π() one has and hence ] is BC integrable with respect to   although  is not differentiable at 0. As for the relation between the spaces  and BC, it is well known (see Theorem C of [1]) that the game ] = √  (with  the Lebesgue measure on [0, 1]) belongs to .Anyway ] ∉ BC.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   ∈ Π(Ω) and every  > 0 we can provide a refinement   = { 1 , . . .,   ,   } such that ( 1 ) = ⋅ ⋅ ⋅ = (  ) and (  ) <  [15,Lemma 3.5].Clearly we can choose  quite larger than say ♯  .Also we can choose  = () determined by the uniform continuity of which shows that the refinement limit does not exist.
Therefore, we can now consider the space  = BC ∩ .
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.
From [1] we recall the following.Definition 5. Let G denote the space of automorphisms of (Ω, Σ), then each  ∈ G induces a linear mapping  * of  onto itself, defined by for  ∈ Σ.A subspace that is invariant under  * for every  ∈ G is called symmetric.
Proposition 6.The space  is symmetric.
Proof.We need to prove that for every  ∈ G 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 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  ∈ Σ.
The following result immediately derives from (11) and the definition of  + Ø .

Obviously 𝜕 +
Ø 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  + Ø defines a value.
It is clear that one can use functions of different forms to provide classes of measure games  ∘  with signed  that are in \ and in  0 , and hence similar subspaces of  on which  + Ø 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.
Define the scalar measure game ] =  ∘  and take then  = ] + , where  denotes the usual Lebesgue measure.As in the previous example, ] and hence  are in  0 ,  ∉  (for ] ∉ ) and  ∈  (again, for the proof, see the Appendix).
Ψ is a value on  (see the Appendix for the proof).Finally, the following example shows that  + Ø is not continuous on  equipped with the variation norm.

The Operator 𝜕 + Ø 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 Σ there holds
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: ] * = g.l.b. ] , ] * = l.u.b. ] , 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  = BC ∩  ∞ of Lipschitz games that are BC integrable.
is strictly included in BC, for there are easy examples of games in BC \  ∞ .
For instance, consider the function  : [0, 1] → R defined as and the scalar measure game ] =  ∘ , where  represents the usual Lebesgue measure.Then ] ∈ BC with 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 ] =  ∘ ,  ∈  + ,  ̸ = 0 be in  ∞ ; then the following are equivalent: (1)  admits right-hand side derivative at 0; (2) ] is   -BC integrable and hence ] ∈ .
We have then, similar to the computation in Proposition 3: As for the first sum we have the following estimate: Clearly we can repeat this construction with   ñ and find another partition  * ∈ Π() with   ( * ) < (/3) as above; again It is then clear that, since ℓ 1 ̸ = ℓ 2 , the game ] is not   -BC integrable.
We will need in the sequel the following lemma.
To prove the converse inequality, first of all, for  ∈ , consider the game  −1 *  defined in the following fashion: for every  ∈ Σ set  =  −1 () ∈ Σ and set so that  * [ −1 * ] = .It is a routine computation, based on the properties of , to show that  −1 *  is a countably additive measure as well.
In  we have the following result.
Since powers of probabilities belong to , there immediately follows the following.

Obviously ∂+
Ø is a value of  ∞ , but, unfortunately, on the subspace  ∞ ∩  ∞ we lose uniqueness, in that ∂+ Ø does not agree with the Aumann-Shapley (AS) value.
The function  ∈  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 [0, 1] generates a Milnor semivalue   on  ∞ which clearly becomes a value on the subspace   = {] ∈  ∞ :   (])(Ω) = ](Ω)}.In our case for the Dirac measure  0 based at 0, we have precisely   0 =  ∞ ∩  ∞ .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.