Sub-Additive Aggregation Functions and Their Applications in Construction of Coherent Upper Previsions

: In this paper, we explore the use of aggregation functions in the construction of coherent upper previsions. Sub-additivity is one of the deﬁning properties of a coherent upper prevision deﬁned on a linear space of random variables and thus we introduce a new sub-additive transformation of aggregation functions, called a revenue transformation, whose output is a sub-additive aggregation function bounded below by the transformed aggregation function, if it exists. Method of constructing coherent upper previsions by means of shift-invariant, positively homogeneous and sub-additive aggregation functions is given and a full characterization of shift-invariant, positively homogeneous and idempotent aggregation functions on [ 0, ∞ [ n is presented. Lastly, some concluding remarks are added.


Introduction
In the subjective probability approach [1], prevision of a random variable is defined by the notion of coherence that assures that the prevision of a random variable X is the amount the subject is willing to bet on X such that neither the bettor nor the banker can win or loose for sure. From a mathematical point of view, linear previsions on a linear space of random variables are coherent if and only if they are linear functionals with values bounded by the infimum and the supremum value of the random variable. From these defining properties, we can prove that a linear coherent prevision is a homogeneous functional. Coherent probabilities are obtained when only indicator functions are considered in the domain. Because of the incomplete and inaccurate information, or to represent some preference orderings (e.g., Example 1 of [2]), in some cases it is appropriate to consider non-linear functionals. Coherent upper and lower previsions can be defined as generalizations of linear previsions. In particular, if a linear prevision is defined with respect to a countably additive probability, coherent upper and lower conditional previsions can be defined by the Choquet integral with respect to the outer and inner measures generated by the countably additive probability, which are the natural extensions of an additive probability defined on a σ-field [3]. A new model of coherent upper previsions defined by the Choquet integral with respect to Hausdorff outer measures is proposed in [4]. Also, coherent upper and lower previsions have a behavioural interpretation: the lower prevision of X can be regarded as the supremal buying price for the random variable X and the upper prevision is an infimal selling price. Nevertheless, we observe that there exist coherent upper previsions defined on the linear space of random variables without linear restrictions; an example is the vacuous upper prevision of a random variable, defined as the supremum of the values assumed by that random variable on Ω. So an open problem is to propose new mathematical tools to define coherent upper previsions that cannot be obtained as extensions of linear previsions.
The main motivation behind this paper is to propose a construction method for coherent upper previsions using aggregation functions [5,6] that are prolific and latter-day part of mathematics that found its place both in the theory and applications.
A functional on a linear space of random variables is a coherent upper prevision if and only if it is bounded by the supremum value of the random variable, sub-additive and positively homogeneous. In this contribution, coherent upper previsions are constructed by aggregation functions, which are interpreted as the gains we obtain with a given resource so that the increase in gains, after increase in resources, can be expressed in terms of aggregation functions. We introduce a new type of transformation of aggregation functions, called a revenue transformation, which represents the best upper bound for possible gains; in the paper we consider only revenue transformations that always assume bounded values. Some properties of revenue transformations are proven; in particular, it is proven that the revenue transformation transforms any aggregation function to some other aggregation function that is sub-additive and is bounded belowed by the transformed function. There already exists a sub-additive transformation of aggregation functions introduced in [7] and is heavily studied by researchers [8][9][10]. This sub-additive transformation always exists and is bounded above by the aggregation function that is being transformed, whereas examples are given in this paper of aggregation functions that do not have a bounded revenue transformation.
The revenue transformation of the Choquet integral, which is an example of aggregation function, is calculated and an open problem is to determine if it is the Choquet integral with respect to a monotone measure. An example of revenue transformation of Choquet integral with respect to a monotone measure µ, which is not sub-modular, is given such that the transformation is the Choquet integral with respect to a sub-modular monotone measure and it can be used to define a coherent upper prevision.
The paper is organized as follows. In Section 2, some basic preliminaries needed later are given. In Section 3, we define a new sub-additive transformation of aggregation functions, called a revenue transformation, and study its properties. Section 4 consists of a construction method of coherent upper previsions based on aggregation functions. This construction method is heavily exemplified in this section. The last section contains some concluding remarks on the topic.

Preliminaries
Let Ω be a non-empty set and let A be a σ-algebra on Ω such that A is finite. With this assumption we may, without loss of generality, assume that Ω = {1, 2, . . . , n} and A = 2 Ω for some n ∈ N. This natural number n is fixed throughout. Note that the motivation of the concepts discussed in this paper given in the literature, see, e.g., [3], including illustrative examples, deal mostly with finite Ω only, therefore we focus on this framework only. Though most of concepts we recall or introduce in this paper remain valid also when a general measurable space (Ω, A) is considered, this is not the case, e.g., with the constructions based on the aggregation functions.
In this setting, every random variable X : Ω → R on (Ω, A) can be represented as n-tuple x ∈ R n with i-th coordinate (denoted by x i ) being equal to X(i) for i = 1, 2, . . . , n. All such vectors (i.e., random variables) form a vector space R n with coordinate-wise addition + and coordinate-wise scalar multiplication (over field R). Note that coordinatewise difference −, coordinate-wise multiplication ·, coordinate-wise division /, coordinatewise supremum ∨, and coordinate-wise infimum ∧ can be also introduced.
Also, a partial order ≤ can be introduced by x ≤ y if and only if x i ≤ y i for all i ∈ {1, 2, . . . , n} and all x, y ∈ R n . By 0 we denote the all-zero vector and by 1 we denote the all-one vector. An indicator vector 1 A of a set A ⊆ Ω is given by its i-th coordinate being equal to 1 A (i), where 1 A is the indicator function of the set A. A positive cone of a vector space R n is the set R n + = {x ∈ R n : x ≥ 0}, i.e., R n Let x ∈ R n be any vector with coordinates x i . There exists a permutation σ : Ω → Ω such that For simplicity, we denote Note that x (i) is just the i-th order statistics of the sample (x 1 , . . . , x n ).
An aggregation function [5,6] is any mapping A : R n + → [0, ∞[ such that A is nondecreasing, i.e., 0 ≤ x ≤ y implies A(x) ≤ A(y); and grounded, i.e., A(0) = 0. In practice, it is convenient to assume that there exists at least one x ∈ R n + such that A(x) = 0. This restriction is not necessary, but A(x) ≡ 0 is not an interesting aggregation function for us and thus such case will be omitted in proofs (with explicit declaration). We say that an aggregation function A is sub-additive, respectively super-additive, if and only if holds for all x, y ∈ R n + . We say that an aggregation function A is positively homogeneous if and only if for all x ∈ R n + and all α ≥ 0. Lastly, we say that an aggregation function for all x, y ∈ [0, 1] n such that x ≤ y and x j = y j = 1 for some j ∈ {1, 2, . . . , n}, with convention 0/0 = 1.

Observe that then
for all x ∈ [0, 1]. See also, e.g., [11,12]. A monotone measure [13] is any set function µ : A → [0, ∞[ that is non-decreasing, i.e., A ⊆ B ⊆ Ω implies µ(A) ≤ µ(B); and grounded, i.e., µ(∅) = 0. Note that a monotone measure can be viewed as an aggregation function defined on 2 Ω with the partial order ≤ being set inclusion. We say that a monotone measure µ is sub-modular if and only if for all A, B ∈ 2 Ω . Sub-modular monotone measures are also sometimes referred to as 2-alternating monotone measures, see, e.g., [14,15]. If A = {x, y, . . . , z} ∈ 2 Ω we will write, for simplicity, µ xy...z instead of µ({x, y, . . . , z}) or µ A instead of µ(A). A conjugate monotone measure for a monotone measure µ is a monotone measure µ : The Choquet integral [16] with respect to a monotone measure µ is the operator Choquet integral is an aggregation function and if µ is sub-modular then Ch µ is a sub-additive aggregation function. In the theory of imprecise probabilities, an extension of Choquet integral, called an asymmetric Choquet integral, that is an operator aCh µ : R n → R given by is used to construct coherent upper previsions, see, e.g., [17].
A coherent upper prevision is a mapping P : R n → R such that conditions hold. Note that condition (CUP2) is positive homogeneity of P and condition (CUP3) represents sub-additivity of P. A coherent lower prevision is any mapping P : R n → R such that P(x) = −P(−x) for all x ∈ R n , where P is some coherent upper prevision.

Revenue Transformation and Its Properties
Let us start with a notion of a revenue transformation. Imagine that the value of an aggregation function A : R n + → [0, ∞[ at y ∈ R n + represents gains with resources represented by y. The increase in gains after increasing resources by x is equal to i.e., the difference between gains with increased resources and original resources. Imagine, that the original resources are not known and we ask ourselves a question: What is the maximal gain (or the best upper bound for possible gains) if we increase our resources by x? The answer is This defines the revenue transformation of A that will be denoted by A. We would like to ensure that A is again an aggregation function and thus we must prevent A to being equal to ∞. Such value is obtainable, see following example. Then one obtains that A(x + y) − A(y) = (x + y) 2 − y 2 = x 2 + 2xy and thus sup y≥0 2xy + y 2 = ∞, for any x > 0. As an example for n-dimensional aggregation function that has a similar behavour is an aggregation function A : R n One of the existence conditions (and we will prove later that it is the only condition) for A to be an aggregation function is that is required to hold for all x ∈ R n + . This defines a sub-class of aggregation functions that will be denoted by A rev . Whenever we refer to any revenue transformation we will always assume that this condition is satisfied and thus we will only work with A ∈ A rev in such a case.
Definition 1. Let A : R n + → [0, ∞[ be an aggregation function such that A ∈ A rev . Then its revenue transformation is a mapping A : R n + → [0, ∞[ given by for all x ∈ R n + .
As we hinted before, any revenue transformation of any aggregation function belonging to A rev yield again an aggregation function. This is stated in the following proposition. Proof. Let us start by proving that A(0) = 0 holds. It is easy to see that Now we will show that A is also non-decreasing. Let x, z ∈ R n + be such that x ≤ z. From the monotonicity of A we have A(x + y) ≤ A(z + y), or, equivalently, A(x + y) − A(y) ≤ A(z + y) − A(y) for any y ∈ R n + and thus Thus, A is an aggregation function.
Now we give some properties of revenue transformations. These are summarized in the following proposition. Proof. (a) To see that A is a sub-additive aggregation function, let x, z ∈ R n + . Then

Proposition 2. Let
and thus A is a sub-additive aggregation function.
To prove (b) it is sufficient to notice that i.e., A ≤ A. Combining this with part (b) of this proposition we obtain that A = A as needed.
To see (d) notice that i.e., the diagonal of A coincides with the diagonal of A if A is shift-invariant. To prove (e), notice that

Now using (d) we have that A(1) = A(1) and thus
i.e., A is indeed a shift-invariant aggregation function if A is. Now, to prove (f) notice that for all x ∈ R n + and all α ≥ 0 if A is positively homogeneous.

Example 2. If we take, for the sake of exemplification, a one-dimensional aggregation function
for any p > 0 then we obtain by item c) of Proposition 2 that A = A if p ∈]0, 1] since A is sub-additive and

Example 3. If we consider a one-dimensional aggregation function
where α ∈ R \ {0} then we obtain by c) of Proposition 2 that A = A if α < 0 since A is sub-additive. On the other hand, if α > 0, the revenue transformation of A does not exist.
for all x ∈ R 2 + , which coincides with Choquet integral Chμ whereμ is given bŷ In general, for n > 2, it is an open problem whether for any monotone measure µ, Ch µ = Ch ν for some monotone measure ν. Clearly, if this is the case, ν(A) = Ch(1 A ) for any A ⊆ Ω. Note that for any µ, ν(A) = Ch µ (1 A ) defines a sub-modular monotone measure.

Example 6.
Let Ω = {1, 2, 3, 4} and let µ be a monotone measure on Ω. Let us continue with Example 2 that we presented in [19]. Note that based on the monotone measure µ introduced there, we can construct a new monotone measure ν : The values of the chosen monotone measure µ and the related monotone measure ν can be found in Table 1. Note that ν is a sub-modular monotone measure and also, if Ch µ is Choquet integral then it would coincide with Ch ν .

Constructions of Coherent Upper Previsions
In this section, we will construct coherent upper previsions using aggregation functions with special properties. Analogous construction process as in [4,19] will be adopted.
for all x ∈ R n defines a coherent upper prevision.
Proof. First of all, note that for all x ∈ R n , x − (inf x)1 ≥ 0, and thus the definition of cup is valid. Since R n is a linear space to prove that cup defines a coherent upper prevision we will prove that cup obeys conditions (CUP1)-(CUP3). To see that (CUP1) holds, notice that for all x ∈ R n . To see (CUP2) it is enough to consider for all α ≥ 0 and all x ∈ R n . Lastly, it remains to show that the condition (CUP3) holds, i.e., to show that cup is sub-additive. Because infimum is a super-additive aggregation function, we know that inf(x + y) ≥ inf x + inf y and thus r = inf(x + y) − inf x − inf y ≥ 0. Note that Now, from − inf(x + y) = − inf x − inf y − r, we obtain that Using the shift-invariance of A and the fact that r ≥ 0 we have that Noticing that inf(x + y) − r = inf x + inf y and by the sub-additivity of A we finally obtain that for all x, y ∈ R n , i.e., cup is sub-additive.
With the fact that A is shift-invariant it is enough to consider only the fact that A(1) = 1.
Assuming this extra condition, the coherent upper prevision can be constructed by which simplifies the construction.

Remark 2.
As another remark, note that a positively homogeneous and idempotent aggregation function A : R n + → [0, ∞[ is fully determined by its restriction to [0, 1] n because for any x ∈ R n + \ {0}, where x/(sup x) ∈ [0, 1] n . Also note that the given restriction A | [0,1] n is a [0, 1]-aggregation function and thus we can characterize all positively homogeneous, shift-invariant and idempotent aggregation functions. To authors' best knowledge, this characterization has not been published anywhere, yet. Coherent lower previsions are obtained by the conjugacy property We can observe that if the aggregation function A, that is considered to define a coherent upper prevision, is linear, then clp(x) = −cup(−x) and a linear prevision is obtained.
Coherent upper probabilities can be obtained by Proposition 3 when only indicator vectors are considered.
The coherent upper probability of an event E ⊆ Ω is defined by and the coherent lower probability is obtained by the conjugacy property, Note that A satisfies all requirements of Proposition 3 and thus A can be used to construct a coherent upper prevision. This construction leads to cup(x) = sup x which is the vacuous upper prevision. Example 8. The Choquet integral Chμ defined in Example 5 by the revenue transformation of Ch µ (which is not a coherent upper prevision because it is not sub-additive), is a coherent upper prevision since it is sub-additive, positively homogeneous and such that Coherent upper probabilities are obtained when only indicator vectors 1 E , for all E ∈ 2 Ω , are considered. These are given by In the following example an aggregation function which cannot be used to define a coherent upper prevision is given.

Example 9.
Let Ω = {1, 2, . . . , n}. The aggregation function A(x) = ∑ n i=1 x 2 i can not be used to construct a coherent upper prevision, because it is not sub-additive; moreover the revenue transformation A does not exists as shown in Example 1.

Example 10.
Let Ω = {1, 2, . . . , n}. The aggregation function A(x) = ∑ n i x i can be used to construct a coherent upper prevision, because it is linear and homogeneous; by Proposition 3 we obtain that cup(x) is the mean and the coherent probability of an event is defined by the counting measure, which is the Hausdorff measure of order 0. In this setting, A = A.

Proof. Observe first that each function
i.e., C is idempotent. Similarly, one can show the positive homogeneity, shift invariantness and sub-additivity of C, which ensures C ∈ K k .
The previous construction method can be used to define coherent upper previsions if C(1) = 0. Also multi-step Choquet integrals [20][21][22], which are multi-step aggregation functions and which are not Choquet integrals, in general, can be used to construct coherent upper previsions.

Conclusions
In this paper a construction method, based on sub-additive positively homogeneous and shift-invariant aggregation functions is proposed to define coherent upper previsions. Moreover a sub-additive transformation of an aggregation function, named revenue transformation is introduced to obtain a sub-additive aggregation function. Under some mild additional constraints, this transformed aggregation function is then used to define a coherent upper prevision. As a distinguished example one can recall the Choquet integral. If the considered monotone measure µ is sub-modular then the related asymmetric Choquet integral Ch µ is a coherent upper prevision. If µ is not sub-modular, clearly, Ch µ is not a coherent upper prevision, but based on our proposal one can define a coherent upper prevision Ch µ . It is an interesting open problem whether the operator Ch µ is comonotone additive and thus it can be represented as a Choquet integral Ch ν , where ν is a sub-modular monotone measure given by ν(E) = Ch µ (1 E ).