Some Condition for Scalar and Vector Measure Games to Be Lipschitz

Measure games, that is, transferable utility (TU) games of the form ] = f ∘μ, where μ is a nonatomic measure on σ-algebra Σ of a space Ω and f is a function defined on the range of μ, with f(0) = 0, arise in several contexts including game theory, mathematical economics, and even finance (where, under suitable hypotheses on f and μ, they are termed as distorted probabilities). One of the reasons of their popularity lies in the fact that they generate fundamental spaces of games and that many games of interest, such as, for instance, market games of finite type, fall into this category. Classically in the literature, one distinguishes between scalar measure games where μ is a nonnegative scalar measure and vector measure games where μ is an n-dimensional measure with nonnegative components. The extension to signed measure is also customary. The most classical space related to measure games is pNA, that is, the closed linear subspace of BV generated by all powers (with respect to pointwise multiplication) of nonatomic probability measures. Several results exist, concerning scalar measure games in pNA [1, 2], while the only characterization of vector measure games in this space is the one of Tauman [2]. Besides theBV-norm, one encounters in the literature the


Introduction
Measure games, that is, transferable utility (TU) games of the form ] =  ∘ , where  is a nonatomic measure on -algebra Σ of a space Ω and  is a function defined on the range of , with (0) = 0, arise in several contexts including game theory, mathematical economics, and even finance (where, under suitable hypotheses on  and , they are termed as distorted probabilities).One of the reasons of their popularity lies in the fact that they generate fundamental spaces of games and that many games of interest, such as, for instance, market games of finite type, fall into this category.Classically in the literature, one distinguishes between scalar measure games where  is a nonnegative scalar measure and vector measure games where  is an -dimensional measure with nonnegative components.The extension to signed measure is also customary.
The most classical space related to measure games is , that is, the closed linear subspace of  generated by all powers (with respect to pointwise multiplication) of nonatomic probability measures.Several results exist, concerning scalar measure games in  [1,2], while the only characterization of vector measure games in this space is the one of Tauman [2].
Besides the -norm, one encounters in the literature the ‖ ⋅ ‖ ∞ -norm that defines the subspace  ∞ ⊂  of the socalled Lipschitz games.Then clearly one may define also the space  ∞ as the ‖ ⋅ ‖ ∞ -closure of the space generated by all powers of nonatomic probability measures.In this space, the only characterization of vector measure games we are aware of up to now is due to Milchtaich [3], and it requires the function  to be continuously differentiable.
All these results put in evidence how measure games are difficult to characterize once the differentiability assumption is dropped, though economically significant measure games that do not fall into this category exist in the literature.For example, Milchtaich's characterization shows that if  is piecewise linear, then measure games of the form  ∘  do not belong to  ∞ ; on the other side the linear span of games with  piecewise linear and  vector of mutually singular nonatomic probability measures plays a role for example in value theory [4].
In this paper, we face the problem of characterizing measure games both in  ∞ and in  ∞ .Our starting point is the characterization for scalar measure games in  ∞ given in [5]; there we proved that  ∘  ∈  ∞ if and only if  is Lipschitz.Here we introduce a generalization of the Lipschitz condition, namely, lipschitzianity in link directions, which proves to characterize vector measure 2 International Journal of Mathematics and Mathematical Sciences games  ∘  in  ∞ , when the range of  has finitely many exposed points and which thus covers interesting cases in literature.As a consequence, we extend the characterization in [5] to measure games of the form  ∘  where  is a signed measure.
Another interesting subspace is the class  of Burkill-Cesari () integrable games, introduced in [6].The investigation on this space has been further developed in [5]; there it has been proved that the Burkill-Cesari integral is a ‖ ⋅ ‖ ∞continuous (semi)value (but in general not -continuous) and that it differs from the Aumann and Shapley value.Actually, the  integral and the space  turned out to be fruitful to provide a proper subspace of  strictly larger than  on which a value can be defined; remember that existence and uniqueness of a value on  are well known, while the question is still open on .In force of these results, a better understanding of the structure of the space  ∩  ∞ , starting from its simplest elements, that is, measure games, seems to be an interesting task.
The outline of the paper is as follows.In Section 3 we characterize vector measure games in  ∞ ; although the statement of the result is formally analogous to the one of Tauman, the proof differs from his, and this is essentially due to the difficulties arising in handling the ‖ ⋅ ‖ ∞ -norm on this space.In Section 4 we first characterize a particular class of vector measure games in  ∞ and, as a corollary, also measure games where  is a signed measure are characterized through lipschitzianity.The final part of the section is devoted to the investigation of  integrable Lipschitz measure games; we completely characterize the one-dimensional ones and provide a necessary condition in larger dimensions; also a topological condition is given, to ensure that a Lipschitz measure game is ‖ ⋅ ‖ ∞ -close to a  integrable one.

Preliminaries
In the following we will deal with the following elements, as in [1].
A transferable utility (TU) game ] is a real valued function on Σ such that ](0) = 0.
The set of all nonatomic measures on (Ω, Σ) is denoted by , the cone of nonnegative measures of  by  + , while the set of probability measures in  is indicated by  1 .Given  ∈ , the variation measure is denoted by ||.For a vector measure  = ( 1 , . . .,   ) ∈ ()  , the variation measure is defined by The space of Lipschitz games is denoted by  ∞ for it is a Banach space under the norm ‖]‖ ∞ defined in the following way; for every  ∈  + such that (1) holds, write − ⪯ ] ⪯ .
Analogously to what is usually done in the space , the symbol  ∞ denotes the ‖ ⋅ ‖ ∞ -closure of the space generated by all powers of nonatomic probability measures.
Given a convex subset  of R  , a vector  is called  admissible if  =  −  for some ,  ∈ .A real valued function  defined on  is said to be continuously differentiable on  if for each  admissible  the derivative ( + ℎ)/ℎ exists at each point in the relative interior of  and it can be continuously extended at each point of .The space of continuously differentiable functions on a set  will be denoted by C 1 ().
Given a convex compact subset  of R  , a point  ∈  is said to be exposed if {} is the intersection of  with some supporting hyperplane of .
As in [8], given a monotone nonatomic game , one defines the mesh of a partition  as A game ] is Burkill-Cesari (BC) integrable with respect to   if the following limit exists, for each  ∈ Σ: We denote by  the space of games ] such that there exists  ∈  + so that ] is  integrable with respect to the mesh   .The  integral does not depend upon the integration mesh (see Proposition 5.2 in [6]); in other words, for every  ∈  + such that ] is   - integrable, the  integral remains the same.Moreover, the  integral of a game ] is a finitely additive measure.

Measure Games in 𝑝𝑁𝐴 ∞
In [5] we have obtained a characterization (Proposition 12) of scalar measure games (made through a nonnegative measure) for the whole space  ∞ .Anyway the Lipschitz condition on  is not sufficient to ensure that a measure game belongs to  ∞ , that is, to the ‖ ⋅ ‖ ∞ -closure of the space generated by all powers of nonatomic probability measures.Indeed, in [3] Milchtaich has shown that continuous differentiability is required, and it guarantees that a vector measure game belongs to  ∞ .To our knowledge, this is also the unique characterization of vector measure games in  ∞ so far.It is well known that  ∞ is strictly contained in  and that many games of interest belong to this space [4].
Our first goal is to obtain a characterization for vector measure games in  ∞ , similar to the one given by Tauman for  [2].We point out that the technique of the proof is different, due to the difficulties in dealing with approximations in the ‖ ⋅ ‖ ∞ -norm instead of in the norm.
We begin with a Lemma.
Proof.According to [1] for every  > 0 there exists a polynomial  on () such that the norm ‖ − ‖  < , where one defines and ‖ ⋅ ‖  is the usual uniform norm on continuous functions.Without loss of generality we assume (0) = 0.
From the Mean Value Theorem we have the following: Tauman [2] gave the definition of ‖ ⋅ ‖  -continuity at  for a function  defined on the range of a vector measure  ∈ ( 1 )  , which we recall below.
Note first that without loss of generality we can assume that () has nonempty interior.
Let  ⊂ {1, . . ., } be the set of indexes for which the corresponding link (  ,   ∪   ) does not fulfill the cardinality requirements; then  can be split into two disjoint subsets  1 and  2 , where  1 is the set of indexes for which Ω \ (  ∪   ) is at most countable.
For each  ∈  2 one can choose an uncountable -null set   in Ω \ (  ∪   ) and then replace   with T =   ∪   ; since  ∈  2 , we have   countable, and, therefore, (  ) = 0; thus the replacement does not affect the corresponding summand in (, ,   ).
For each index  ∈ In the sum (  −   , ,   ) there will then appear a sum of the form in (23) which we can estimate with .
Let us now treat the summands of the form and so in the whole |(  −   , ,   )| we can apply the following bound from above: Note that, in proving the "if " part of the above theorem, we have incidentally proven that, for a measure game  ∘  ∈  ∞ , for each  > 0 there exists  > 0 such that ‖( ∘ ) − (  ∘ )‖ ∞ < .
In other words we have proven the following statement.

Measure Games in 𝐴𝐶 ∞
Throughout this section we will deal with maps  : R  + → R with (0) = 0, namely, admissible maps.
For two vectors x, y ∈ R  + we will adopt the notation x ≪ y for the usual componentwise order.We introduce the following definitions, which extend Lipschitz condition when  > 1.
Definition 5.The map  is said to be Lipschitz in increasing directions when  > 0 exists such that for each pair x ≪ y in R  + one has      (x) −  (y)     ≤  ⋅     x − y     .
A map  : R  + → R is then said to be Lipschitz in link directions if there exists  > 0 such that for every -link direction x, y there holds Note that if  is Lipschitz in increasing directions, then it is Lipschitz in -link directions for each -dimensional nonnegative measure .
Let now  ∈ N + and  ∈ ( + )  be fixed, and consider the spaces (i) A = { : R  + → R admissible and Lipschitz in the increasing directions}, (ii) L() = { : R  + → R admissible and Lipschitz in -link directions}.
Note that if  ∈ A, then  ∘  ∈  ∞ for every  ∈ ( + )  , while we can assert the same for  ∈ L() only for precisely .
In fact, for each link  ⊂ , setting We can now state the following.
Each of the conditions of Proposition 7 implies that ] ∈  ∞ .It is, therefore, rather natural to ask whether condition  ∈ L() characterizes vector measure games  ∘  in  ∞ .We already know from Proposition 12 in [5] that this is the case for  = 1.
In the more general case, we have the following result.
Theorem 8. Let  ∈ ( + )  be such that its range () has only finitely many exposed points.Then a vector measure game ] =  ∘  ∈  ∞ if and only if  ∈ L().
Proof.We only need to prove that if ] ∈  ∞ , then  ∈ L().
We know that there are    ≤  Observe that the above result includes two interesting cases: the case of   ⊥   for each pair of components   ,   of  and the case of vector measures  which can be represented as integral measures ( 1 ,  1 ()) where  is a ( − 1)-dimensional simple function.
Also, from Theorem 8 one derives the following corollary that generalizes Proposition 12 in [5] to the case of signed scalar measure games, namely, games of the form ] =  ∘  with  signed measure.
Then  is also Lipschitz with constant .
So far, we have not been able to answer the question whether the condition in Theorem 8 actually characterizes vector measure games in  ∞ .
We now turn our attention to a smaller subspace of  ∞ .In [5,6] we have introduced and studied the space  of Burkill-Cesari () integrable games.In particular in [5] we considered the space  =  ∩  ∞ of Lipschitz games that are indeed  integrable.
Here we will consider the subspace V of vector measure games in .
First of all, observe that  ∞ \  and  \  ∞ are nonempty.Indeed a scalar measure game  ∘  ∈  ∞ if and only if  is Lipschitz on [0, (Ω)], while, according to Proposition 14 in [5],  ∘  ∈  if and only if  admits right hand side derivative at 0. Therefore, for instance, if On the other side, let  be a signed measure with (Ω) = 0, and let ] = ||; then easily ] ∈  ∞ .
In Proof.The proof of the sufficiency goes along the same lines of the proof of Theorem 6.1 in [6].Conversely, since ] ∈  ∞ we know from Corollary 9 that  is Lipschitz; hence the ratios ()/,  ̸ = 0 are bounded.Assume that ] ∈  but   (0) does not exist.We have then the following cases: (i) at least one between   + (0) and   − (0) does not exist; (ii)   − (0) ̸ =   + (0).
The first case can be treated analogously to proof of Proposition 14 in [5], simply working with a set  ⊂ ,  ⊂ , respectively, where (, ) is a Hahn decomposition of Ω.