Some Condition for Scalar and Vector Measure Games to Be Lipschitz
We provide a characterization for vector measure games in , with vector of nonatomic probability measures, analogous to the one of Tauman for games in , and also provide a necessary and sufficient condition for a particular class of vector measure games to belong to .
Measure games, that is, transferable utility (TU) games of the form , where is a nonatomic measure on -algebra of a space and is a function defined on the range of , with , 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 .
Besides the -norm, one encounters in the literature the -norm that defines the subspace of the so-called 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 , 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 .
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 ; 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 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  to measure games of the form where is a signed measure.
Another interesting subspace is the class of Burkill-Cesari () integrable games, introduced in . The investigation on this space has been further developed in ; 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.
In the following we will deal with the following elements, as in .
denotes a measurable space isomorphic to (where denotes the Borel -algebra on ).
A transferable utility (TU) game is a real valued function on such that .
The set of all nonatomic measures on is denoted by , the cone of nonnegative measures of by , while the set of probability measures in is indicated by . Given , the variation measure is denoted by . For a vector measure , the variation measure is defined by .
A game is said to be Lipschitz if there exists such that for every link in it holds 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 . Then set To simplify notation, given a game , and for fixed we will denote as usual by the class of finite partitions of consisting of elements in ; for every , say, , let . Then denote For a game define, according to , the measures Then It is clear that if is a monotone game, then are nonnegative and in this case .
Note that for every , , one has On the other side, since are measures, there are Hahn decompositions , which allow us to rewrite (5) as Hence, if for a game one has for each and every , then clearly .
Given a vector measure and denoting its range by , a vector measure game is a game of the form , where is a real valued function defined on the range of and . In the case , this type of games is called scalar measure games.
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 , 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 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 .
Given a convex compact subset of , a point is said to be exposed if is the intersection of with some supporting hyperplane of .
As in , given a monotone nonatomic game , one defines the mesh of a partition as
A game is Burkill-Cesari () integrable with respect to if the following limit exists, for each : 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 ); 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.
3. Measure Games in
In  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  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 .
Our first goal is to obtain a characterization for vector measure games in , similar to the one given by Tauman for . 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.
Lemma 1. Let , . Then for every there exists a polynomial in variables such that
Proof. According to  for every 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 .
Fix , , . From the Mean Value Theorem we have the following: Now for each and hence Thus
Tauman  gave the definition of -continuity at for a function defined on the range of a vector measure , which we recall below.
Given , denote with the set of measures having the same range of and, for , define Fixed , a function on (with , is -continuous at , if for every there exists such that for each there follows . Tauman proved that a vector measure game , , belongs to if and only if is -continuous at .
We translate the same definition using the -norm; that is, we say that a function on is -continuous at if for every there exists such that for each there follows . Then we are able to prove that the analogous characterization of Tauman holds for vector measure games in .
Theorem 2. is -continuous at if and only if .
Proof. We first prove the sufficient condition; namely, we assume that is -continuous at .
Note first that without loss of generality we can assume that has nonempty interior.
As in , fix such that the cube and for define Thus and according to [2, Lemma 7] .
We also need the following structure Lemma [2, Lemma 9].
Lemma 3. Let and let be a link with both uncountable. Then, for every and for every there exists with for each with .
Let be momentarily fixed, and let be any link in , with , uncountable. Consider the difference where . We have Now by Lemma 3 applied with , , and for every there is such that and . Hence, continuing the computation (18) we reach By the assumption corresponding to there exists such that for one has .
Let us now fix , and let , , be fixed also, in such a way that , are uncountable.
Then we have Since we know that , and from (6), for each . Hence for each pair that satisfies the cardinality conditions at each link that proves that approximates in the -norm.
To conclude our proof, we will prove that for every , , can be replaced by a choice , , so that each link fulfills the cardinality requirements for each link, and .
To this aim, let be fixed, with . Without loss of generality we can assume that each is not -null.
Let be the set of indexes for which the corresponding link does not fulfill the cardinality requirements; then can be split into two disjoint subsets and , where is the set of indexes for which is at most countable.
For each one can choose an uncountable -null set in and then replace with ; since , we have countable, and, therefore, ; thus the replacement does not affect the corresponding summand in .
For each index choose an uncountable -null set and replace with , with .
Then we can replace with setting and analogously with where and finally set . By construction then .
To prove the converse implication, according to [1, Lemma 7.2 page 41], it is enough to prove that monomials of the form , with , , , are -continuous at each .
We first prove that if , , , , then In fact, for each summand we have where the last inequality follows from Lipschitz condition on and the fact that . Then in we have further , whence for the whole sum we obtain (23).
Let now as usual , , , be fixed. Consider the single summand in 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 . Clearly and so in the whole we can apply the following bound from above: Now each term again by Lipschitz condition and the choice of . Also since and for the same reason. Hence, because of (23), Finally from which we conclude according to (25) and (23) which only depends upon . Hence the proof is complete.
Note that, in proving the “if” part of the above theorem, we have incidentally proven that, for a measure game , for each there exists such that .
On the other side, as , we can apply Lemma 1 and find a polynomial such that .
In other words we have proven the following statement.
Theorem 4. Let . The following are equivalent. (i) is in ;(ii)for each there exists a polynomial vanishing at 0 such that
4. Measure Games in
Throughout this section we will deal with maps with , namely, admissible maps.
For two vectors we will adopt the notation for the usual componentwise order. We introduce the following definitions, which extend Lipschitz condition when .
Definition 5. The map is said to be Lipschitz in increasing directions when exists such that for each pair in one has
Definition 6. Let be a measure; a pair is said to be a -link direction when there exists a link in such that , .
A map is then said to be Lipschitz in -link directions if there exists such that for every -link direction there holds Note that if is Lipschitz in increasing directions, then it is Lipschitz in -link directions for each -dimensional nonnegative measure .
Let now and be fixed, and consider the spaces (i) admissible and Lipschitz in the increasing directions},(ii) admissible and Lipschitz in -link directions}.Note that if , then for every , while we can assert the same for only for precisely .
Now, given , let and define the subsets of . Consider Note that if is a vector measure game in , then . Indeed, let be such that and consider the Jordan decomposition with , . Then there are two disjoint sets which support , respectively. This means that vanishes on ; hence, .
In fact, for each link , setting , since , one has
We can now state the following.
Proposition 7. Let and . The following are equivalent: (i);(ii)there exists such that ;(iii).
Proof. (i) implies (ii), since, if , then immediately .
The implication (ii) (iii) is immediate.
Finally, let . This immediately implies that there exists such that -a.e., and this in turn implies that on . On the other side, as mentioned in Section 2, we have that for every pair of disjoint sets . Then immediately, on each -link direction and every link such that , one finds that says that .
Each of the conditions of Proposition 7 implies that . It is, therefore, rather natural to ask whether condition characterizes vector measure games in . We already know from Proposition 12 in  that this is the case for .
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 .
Proof. We only need to prove that if , then .
Let be the exposed points of , and let be sets in such that , .
Consider the partition generated by , say, , and assume that each has nonnull -measure.
If denotes the algebra generated by , we have that ; in fact by additivity each and from Lyapunov Theorem is convex; hence where the last equality is deduced from Straszewicz Theorem.
Let be a fixed -link direction. Then , , where, for each , , . Moreover, easily .
Set now .
Fix and .
We know that there are such that , . Choose then such that for the -dimensional measure one has , and let with
This is possible for and for .
Then , is a link and hence But , . Thus (38) becomes Now Thus Since , we have that .
In conclusion we have that with .
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 where is a -dimensional simple function.
Corollary 9. Let be a nonatomic signed measure, and let . Then if and only if is Lipschitz on .
Proof. Consider the nonatomic two-dimensional measure and the function defined by . Thus . Since is a rectangle, if , from Theorem 8, namely, a constant exists, such that for each -link direction , 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  we considered the space of Lipschitz games that are indeed integrable.
Here we will consider the subspace 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 , while, according to Proposition 14 in , if and only if admits right hand side derivative at 0.
Therefore, for instance, if and , .
On the other side, let be a signed measure with , and let ; then easily .
Proposition 10. Take and let . Then if and only if is Lipschitz on and exists.
Proof. The proof of the sufficiency goes along the same lines of the proof of Theorem 6.1 in .
Conversely, since we know from Corollary 9 that is Lipschitz; hence the ratios , are bounded. Assume that but does not exist. We have then the following cases: (i)at least one between and does not exist;(ii). The first case can be treated analogously to proof of Proposition 14 in , simply working with a set , , respectively, where is a Hahn decomposition of .
Assume then that .
Let then and let us fix with , so that . Choose . 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 integrability.
Choose now the following ; by means of Lyapunov Theorem, divide into finitely many sets, say , each with , until and then choose ; thus easily for Then for one has .
We have then As for the first sum we have the following estimate: In conclusion Clearly we can repeat this construction with to find a partition with as above; again Hence and On the other side, we can choose a decomposition with and then for each choose symmetrically with so that .
Then we can define to produce a decomposition of the whole with but . Since , the game is not - integrable.
Up to this point we have been able to characterize scalar measure games and signed measure games in by means of the existence of . What can be said for more general vector measure games?
We will see that differentiability at zero is not necessary, because we are considering admissible functions defined on the whole positive orthant of , while with integrability we are taking into account the -admissible directions.
For example, consider on the classical example of nondifferentiable map which is not differentiable on the whole orthant, being not continuous at . However, if and , then reduces to a line segment and since it can be represented as with which is differentiable.
Indeed the following necessary condition derives from the integrability.
Proposition 11. Let be in , and let be in . If , then admits directional derivative along every admissible direction (i.e., such that for some and some ; moreover, the convergence is uniform with respect to .
The proof of the existence of each directional derivative is substantially the same as that of Proposition 10, while the uniformity of the limit is deduced from the assumption of integrability, where the defining limit is uniform with respect to .
Observe that the requirement in the previous statement could be weakened by requiring that for each radial direction that crosses at a point , the ratios are bounded in .
Unfortunately, even with this weakening, the condition expressed in the above Proposition does not characterize vector measure games in . To get convinced, we present the following example.
Example 12. Consider as above which is not differentiable at , since it is not continuous, despite the fact that it admits directional derivative at every direction on the positive orthant, given by
A simple computation in fact provides, for , For the sake of simplicity set . Then we want to show that the ratios can be made arbitrarily small, somehow independently on the values of .
Easily which in turn is smaller than for ; one immediately checks that the map is increasing for .
Consider now defined as with .
Then consider the “reverse” function defined as ; then the subset of given by is a zonoid, that is, the range of a nonatomic measure (see ), and the admissible directions for such have slopes not exceeding ; this is enough to achieve the required uniformity for the vector measure game .
We will now prove that the weakened assumption is satisfied.
Fix an admissible direction , that is, such that span intersects at a point .
Observe then that all the ratios along the direction are bounded.
However the game is not integrable; in fact, let be a set for which ; whichever , we choose, we can always find a subset of such that (thanks to the Hereditarily overlapping boundary property ).
Now one can always decompose into sets in such a way that , .