Some implications of the restricted distributivity of aggregation operators with absorbing elements for utility theory

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Some implications of the restricted distributivity of aggregation operators with absorbing elements for utility theory Dragan Joˇci´c a , Ivana Štajner-Papuga b,∗ a Higher School of Professional Business Studies, Novi Sad, Serbia b Department of Mathematics and Informatics, University of Novi Sad, Serbia

Received 15 September 2014; received in revised form 8 June 2015; accepted 9 June 2015

Abstract The issue of restricted distributivity (conditional distributivity), i.e., distributivity on the relaxed domain, is crucial for many different areas such as utility theory and integration theory. This paper considers restricted distributivity of a continuous nullnorm with respect to a continuous t-conorm and some applications of the obtained results in utility theory. © 2015 Published by Elsevier B.V. Keywords: Absorbing element; Nullnorms; Restricted distributivity; Utility function; Threshold

1. Introduction Aggregation operators play an important role in many different theoretical and practical fields (fuzzy set theory, theory of optimization, operations research, information theory, engineering design, game theory, voting theory, integration theory, etc.), particularly in various approaches to the problem of decision making (see [2,12,13,18]). The classical approach to decision making under uncertainty assumes that uncertainty is represented by probability distributions. This approach corresponds to the well-known classical utility theory that is based on the notion of mathematical expectation. Its axiomatic foundations, that were given by Von Neumann and Morgenstern [21], rely on the notion of probabilistic mixtures [14]. In order to generalize decision theory to non-probabilistic uncertainty, the approach focused on the generalized mixture sets emerged as a highly acceptable answer. In [7] Dubois et al. have extended the notion of mixtures to decomposable (pseudo-additive) measures (see [11,22,25]) that include some well-known set functions such as probability and possibility measures [26]. In axiomatic foundations of generalized mixtures crucial role play pairs of t-norms and t-conorms that satisfy the distributivity law. This fact has led to new kind of mixtures called possibilistic mixtures, that form the basis of the possibilistic utility theory (see [8]). * Corresponding author.

E-mail addresses: [email protected] (D. Joˇci´c), [email protected] (I. Štajner-Papuga). http://dx.doi.org/10.1016/j.fss.2015.06.003 0165-0114/© 2015 Published by Elsevier B.V.

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Based on the characterization of pairs of continuous t-norms and t-conorms that satisfy the distributivity law on the relaxed domain [18], Dubois, Pap, Prade in [9] have shown that beyond the possibilistic and probabilistic mixtures, the only remaining possibility is hybridization, i.e., a mixture is possibilistic under a certain level a, 0 ≤ a ≤ 1, and probabilistic above it. The hybrid utility function by means of hybrid mixtures was given in [9]. It should be stressed that one of the closely related problems is the problem of independence of events for decomposable measures. Dubois, Pap, Prade in [9,10] have shown that the same condition, i.e., relaxed distributivity, must be satisfied between the t-conorm characterizing the pseudo-additive measures and the t-norm expressing independence. Mixtures and generalized independence (called separability) are closely linked within utility theory (see [6,9]). Namely, this connection allows the construction of an algorithm for calculating the value of measures in the compound lottery, and hence the algorithm for calculating the hybrid mixture. Also, an interesting application of this conditional distributivity on two Borel–Cantelli lemmas and independence of events for decomposable measures was given in [5]. Therefore, it is obvious that the special pairs of aggregations operators that satisfy distributivity law are highly useful in utility theory. The aim of this paper is to extend on the previous research from [9] towards aggregation operators with non-trivial absorbing element (annihilator), that can be applicable in utility theory for modeling behavior of a decision maker. Thus, the focus is now on the distributivity equation F (x, S(y, z)) = S(F (x, y), F (x, z)),

x, y, z ∈ [0, 1],

S(y, z) < 1,

where F is a continuous nullnorm and S is a continuous t-conorm and consequences its solutions to utility theory. This paper is based on [16]. The paper is organized as follows. Section 2 contains preliminary notions concerning nullnorms, restricted distributivity and hybrid utility function. Properties of the utility function UF based on a continuous nullnorm F with the non-trivial absorbing element k are given in the third section. Section 4 is devoted to the analysis of behavior of the decision maker with respect to the utility function UF . Some concluding remarks are given in the fifth section. 2. Preliminaries A short overview is first given in this section regarding some of the basic notions that are essential for this topic (see [4,9,13,18,19]). 2.1. Aggregation operators First, let us recall the basic definition of an aggregation operator on [0, 1]. Definition 1. (See [13].) An aggregation operator is a function A(n) : [0, 1]n → [0, 1] that is non-decreasing in each variable and that fulfills the following boundary conditions A(n) (0, . . . , 0) = 0

and

A(n) (1, . . . , 1) = 1.

Of course, the previous definition of aggregation operators can be extended to an arbitrary real interval [a, b]. The integer n represents the number of input values of the observed aggregation operator. Since the topic of this paper are the binary aggregation operators, they will be denoted simply by A instead of A(2) . Many additional properties such as continuity, associativity, commutativity, idempotency, decomposability, autodistributivity, bisymmetry, existence of neutral and absorbing elements, etc., are often required for aggregation operators, depending on the background in which the aggregation is performed (see [13]). The first type of aggregation operators that will be used in this paper is the aggregation operator with an absorbing element, namely the nullnorm. Nullnorms were introduced in [4] as solutions of the Frank equation for uninorms. Definition 2. (See [4].) The nullnorm F is a binary aggregation operator on [0, 1] that is commutative, associative and for which there is an element k in [0, 1] such that F (x, 0) = x, for x ≤ k

and

F (x, 1) = x, for x ≥ k.

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Fig. 1. A nullnorm with k ∈ (0, 1).

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Fig. 2. Nullnorm from Example 4.

It is clear that k from the previous definition is an absorbing element for F , i.e., F (x, k) = k for all x ∈ [0, 1]. Therefore, it is obvious that the absorbing element (if one exists) has a decisive influence on the aggregation score, and it can be used like an eliminating score or like a veto. The definitions of triangular norms and triangular conorms are special cases of the previous definition. Now, for k = 0 operator F is a t-norm denoted by T and for k = 1 it is a t-conorm denoted by S (see [18]). The representation of nullnorms for k ∈ (0, 1), i.e., for a nullnorm that is not a t-norm, nor a t-conorm is given by the following theorem from [19]. Theorem 3. (See [19].) Let F be a nullnorm such that k ∈ (0, 1). Then ⎧ x y  2 ⎪ ⎨ kS k , k if (x, y) ∈ [0, k] ,  y−k F (x, y) = k + (1 − k)T x−k if (x, y) ∈ [k, 1]2 , 1−k , 1−k ⎪ ⎩ k otherwise,

(1)

where T is a t-norm, and S is a t-conorm. Applying the previous theorem on some well-known t-norms and t-conorms, some interesting examples of nullnorms can be constructed. Example 4. Operator F given by ⎧ ⎪ ⎨ max(x,  y) F (x, y) = max 15 , x + y − 1 ⎪ ⎩1 5

if (x, y) ∈ [0, 15 ]2 , if (x, y) ∈ [ 15 , 1]2 , otherwise,

is a continuous nullnorm obtained by (1) where a t-norm T is the Lukasiewicz t-norm TL (x, y) = max (0, x + y − 1), t-conorm S is the maximum and annihilator k = 15 (see Figs. 1 and 2). Some more general families of operators with an absorbing element were studied in [20]. 2.2. Restricted distributivity Now, let us recall the functional equations that are called left and right distributivity laws ([1], p. 318). Definition 5. Let F , G be two arbitrary binary aggregation operators. F is distributive over G, if the following two laws hold: (DL) F is a left distributive over G, i.e., F (x, G(y, z)) = G(F (x, y), F (x, z)), and

for all x, y, z ∈ [0, 1]

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(DR) F is a right distributive over G, i.e., F (G(y, z), x) = G(F (y, x), F (z, x)),

for all x, y, z ∈ [0, 1].

Of course, for commutative F (DL) and (DR) coincide. Since the problem of non-restricted distributivity of a t-norm over a t-conorm gives us only a trivial solution, that is, the t-conorm in the question has to be SM = max, (see [12,18]). It was necessary to relax the domain of distributivity in the following manner [18]. Definition 6. A t-norm T is restrictedly distributive (RD) over a t-conorm S if for all x, y, z ∈ [0, 1] the following holds T (x, S(y, z)) = S (T (x, y), T (x, z)) whenever S(y, z) < 1. This type of distributivity is also known as the conditional distributivity (see [18]) and, although the domain is only weakly relaxed, the class of pairs of operators that fulfill (RD) is much wider. The following theorem illustrates this for a continuous t-norm and a continuous t-conorm (see [18], pp. 138–140). Theorem 7. (See [18].) A continuous t-norm T and a continuous t-conorm S satisfy (RD), if and only if exactly one of the following cases is fulfilled: (i) S = SM (ii) there is a strict t-norm T ∗ and a nilpotent t-conorm S ∗ such that additive generator s of S ∗ satisfying s(1) = 1 is also a multiplicative generator of T ∗ , and there is an a ∈ [0, 1) such that for some continuous t-norm T ∗∗ the following holds:

 y−a a + (1 − a)S ∗ x−a , if (x, y) ∈ [a, 1]2 , 1−a 1−a S(x, y) = (2) max(x, y) otherwise, and

⎧ ∗∗  x y  2 ⎪ ⎨ aT a, a  if (x, y) ∈ [0, a] , y−a T (x, y) = a + (1 − a)T ∗ x−a if (x, y) ∈ [a, 1]2 , 1−a , 1−a ⎪ ⎩ min(x, y) otherwise.

(3)

Remark 8. (i) Due to isomorphisms between strict t-norms and the product t-norm TP (x, y) = xy and nilpotent t-conorms and the Lukasiewicz t-conorm SL (x, y) = min{x + y, 1}, the previous result is often reduced to the pair

 y−a a + (1 − a)SL x−a , if (x, y) ∈ [a, 1]2 , 1−a 1−a S(x, y) = (4) max(x, y) otherwise, and

⎧ ∗∗  x y  2 ⎪ ⎨ aT a, a  if (x, y) ∈ [0, a] , x−a y−a T (x, y) = a + (1 − a)TP 1−a , 1−a if (x, y) ∈ [a, 1]2 , ⎪ ⎩ min(x, y) otherwise,

(5)

given by Fig. 3 (see [18]). (ii) Further generalizations of Theorem 7 can be found in [15], where T is replaced by a uninorm, and in [23], where T is replaced by a pseudo-multiplication. The following theorem is an extension of Theorem 7 to the case with a continuous nullnorm F with the non-trivial absorbing element k (see [17]). This result is the base for the analysis of a decision makers behavior when a threshold, i.e., an absorbing element, is imposed, which is the main issue discussed in this paper.

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Fig. 3. Restricted distributivity: t-norm and t-conorm.

Fig. 4. Restricted distributivity: nullnorm and t-conorm.

Theorem 9. (See [17].) A continuous nullnorm F with an absorbing element k ∈ (0, 1) and a continuous t-conorm S satisfy (RD) if and only if exactly one of the following cases is fulfilled: (i) S = SM (ii) there is an a ∈ [k, 1) such that S is of the form (4) and F is given by: ⎧   2 kS1 xk , yk ⎪ ⎪  if (x, y) ∈ [0, k] , ⎪ ⎪ y−k ⎪ , if (x, y) ∈ [k, a]2 , k + (a − k)T1 x−k ⎪ ⎪ ⎪ a−k a−k ⎨ x−a y−a if (x, y) ∈ [a, 1]2 , F (x, y) = a + (1 − a)TP 1−a , 1−a ⎪ ⎪ min(x, y) if k ≤ min(x, y) ⎪ ⎪ ⎪ ⎪ ≤ a ≤ max(x, y), ⎪ ⎪ ⎩ k otherwise,

(6)

where S1 is a continuous t-conorm, and T1 is a continuous t-norm. Proof of Theorem 9 is largely based on Theorem 3, that gives the form of nullnorms with respect to t-norms and t-conorms, and on Theorem 7 that is ground braking result for restricted distributivity. The complexity of this structure is illustrated in Fig. 4. Remark 10. The previous problem can also be observed for non-restricted distributivity. However, only the trivial solution is obtained, i.e., the only possibility is S = SM (see [3]). 2.3. Hybrid utility function Let X be a fixed non-empty finite set of outcomes. First, let us recall the definition of the S-measure.

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Definition 11. (See [22].) Let S be a t-conorm and let  be a σ -algebra of the subsets of X. A mapping m :  → [0, 1] is called S-decomposable measure, shortly S-measure, if m(∅) = 0, m(X) = 1 and if for all A, B ∈  with A ∩ B = ∅ we have m(A ∪ B) = S(m(A), m(B)). Remark 12. (i) Each S-measure m : P (X) → [0, 1] is uniquely determined by values m({x}) for x ∈ X. (ii) In the general case when X is an arbitrary non-empty set (also infinite) there is an additional condition for m, namely it has to be continuous from below. (iii) The probability and possibility measures are special cases of S-measures obtained for S = SL and S = SM , respectively. Let (S, T ) be a pair of continuous t-conorm and t-norm satisfying (RD) and s a set of ordered pairs (α, β) defined in the following way s = {(α, β)|α, β ∈ [0, 1], S(α, β) = 1}. In [9] is defined set S,a ⊂ s by S,a = {(α, β)| α, β ∈ (a, 1), α + β = 1 + a or min(α, β) ≤ a, max(α, β) = 1}. The following definition from [9] is an extension of definition of the mixture sets investigated in [7] and [14]. Definition 13. (See [9].) A hybrid mixture set is a quadruple (G, M, T , S) such that (S, T ) is a pair of continuous t-conorms and t-norms that satisfies (RD), G is a set of S-measures defined on X, and M : G2 × S,a → G is a function (hybrid mixture operation) given by M(m, m ; α, β) = S(T (α, m), T (β, m )). The optimistic hybrid utility function by means of hybrid mixtures is given by the following (see [9]): U (u1 , u2 ; μ1 , μ2 ) = S(T (u1 , μ1 ), T (u2 , μ2 )),

(7)

where u1 , u2 are two utilities with values in [0, 1] and μ1 , μ2 are two degrees of plausibility from S,a . Also, the pessimistic hybrid utility function U is introduced in [9] using the utility function U in the following way U (u1 , u2 ; μ1 , μ2 ) = 1 − U (1 − u1 , 1 − u2 ; μ1 , μ2 ). The next definition from [9] extends the notion of independence to S-measures. Definition 14. (See [9].) Two events A and B are said to be T -separable if m(A ∩ B) = T (m(A), m(B)) for a t-norm T . It is stated in [9,10] that the only reasonable S-measures admitting of the independence concept, are based on the restricted distributive pairs (S, T ) of t-conorms and t-norms from Theorem 7 (see Fig. 3). Therefore, we have: • probability measures and T = TP (a = 0); • possibility measures and T is any t-norm (a = 1);

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• suitably normalized hybrid set-functions m such that there is an a ∈ (0, 1) which gives for A and B disjoint  m(A) + m(B) − a if m(A), m(B) > a, m(A ∪ B) = max(m(A), m(B)) otherwise, for separability

⎧ (m(A)−a)(m(B)−a) ⎪ ⎨ a +  1−a m(B) m(A ∩ B) = a · T ∗∗ m(A) , a a ⎪ ⎩ min(m(A), m(B))

if m(A), m(B) > a, if m(A), m(B) ≤ a, otherwise,

with the normalization condition: m({x}) = 1 + (card({x, m({x}) > a}) − 1) · a.

(8)

{x},m({x})>a

It is well-known that any probability distribution on a finite set X can be represented as a sequence of binary lotteries (see [24]). This result was generalized in [9] to S-measure. Turning the S-measure into a sequence of binary trees leads to the necessity of solving the following system of equations T (μ, v1 ) = α1 ,

T (μ, v2 ) = α2 ,

S(v1 , v2 ) = 1

(9)

for given α1 and α2 , where the t-norm T expresses separability for S-measures. The third condition expresses normalization (with no truncating effect for t-conorm S allowed). Assuming that T ∗∗ = TM equations (9) were solved in [9] and exhibited in the analytical forms (μ, v1 , v2 ). 3. Hybrid utility function with a threshold The focus of this section is on the continuous nullnorm F from the previous section that can be replaced by a continuous t-norm T in the hybrid utility function (7). Now, let us observe the following hybrid utility function UF (u1 , u2 ; μ1 , μ2 ) = S(F (u1 , μ1 ), F (u2 , μ2 )), where u1 , u2 are two utilities with values in [0, 1] and μ1 , μ2 are two degrees of plausibility from S,a . The examination of the behavioral characteristics of this new utility function follows ([16]). Case I Let μ1 > a, μ2 > a, i.e., μ1 + μ2 = 1 + a. Now the following subcases have to be considered. 1. Let u1 > a, u2 > a. Then

 (u1 − a)(μ1 − a) (u2 − a)(μ2 − a) UF (u1 , u2 ; μ1 , μ2 ) = S a + ,a + . 1−a 1−a Since a +

(ui −a)(μi −a) 1−a

> a for i = 1, 2 we obtain (u1 − a)(μ1 − a) (u2 − a)(μ2 − a) + 1−a 1−a u1 (μ1 − a) + u2 (1 − μ1 ) = . 1−a

UF (u1 , u2 ; μ1 , μ2 ) = a +

2. Let u2 > a ≥ u1 . Then: (a) if u1 ≤ k then

 (u2 − a)(μ2 − a) (u2 − a)(μ2 − a) UF (u1 , u2 ; μ1 , μ2 ) = max k, a + =a+ , 1−a 1−a

(b) if k < u1 then

 (u2 − a)(μ2 − a) (u2 − a)(μ2 − a) UF (u1 , u2 ; μ1 , μ2 ) = max u1 , a + =a+ . 1−a 1−a

(10)

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3. Let u1 > a ≥ u2 . As in the previous case we have UF (u1 , u2 ; μ1 , μ2 ) = a +

(u1 − a)(μ1 − a) . 1−a

4. Let u1 ≤ a, u2 ≤ a. Then: (a) if u1 ≤ k, u2 ≤ k, then UF (u1 , u2 ; μ1 , μ2 ) = S(k, k) = max(k, k) = k; (b) if u2 ≤ k < u1 , then UF (u1 , u2 ; μ1 , μ2 ) = S(u1 , k) = max(u1 , k) = u1 ; (c) if u1 ≤ k < u2 , then UF (u1 , u2 ; μ1 , μ2 ) = S(k, u2 ) = max(u2 , k) = u2 ; (d) if k < u1 , k < u2 , then UF (u1 , u2 ; μ1 , μ2 ) = S(u1 , u2 ) = max(u1 , u2 ). Case II Let μ1 ≤ a, μ2 = 1 (analysis of the case μ2 ≤ a, μ1 = 1 can be done in an analogous way). Now the following subcases are being considered. 1. Let μ1 ≤ k. Then the following holds. (a) If u1 > a, u2 > a, then UF (u1 , u2 ; μ1 , μ2 ) = max(k, u2 ) = u2 . (b) If u1 ≤ a, u2 ≤ a, then: i. if u1 ≤ k, u2 ≤ k, then UF (u1 , u2 ; μ1 , μ2 ) = max(S1 (u1 , μ1 ), k) = k; ii. if u2 ≤ k < u1 , then UF (u1 , u2 ; μ1 , μ2 ) = max(k, k) = k; iii. if u1 ≤ k < u2 , then UF (u1 , u2 ; μ1 , μ2 ) = max(S1 (u1 , μ1 ), u2 ) = u2 ; iv. if k < u1 , k < u2 , then UF (u1 , u2 ; μ1 , μ2 ) = max(k, u2 ) = u2 . (c) If u1 ≤ a < u2 , then: i. if u1 ≤ k, then UF (u1 , u2 ; μ1 , μ2 ) = max(S1 (u1 , μ1 ), u2 ) = u2 ; ii. if u1 > k, then UF (u1 , u2 ; μ1 , μ2 ) = max(k, u2 ) = u2 . [(d)] (d) If u2 ≤ a < u1 , then: i. if u2 ≤ k, then UF (u1 , u2 ; μ1 , μ2 ) = max(k, k) = k; ii. if u2 > k, then UF (u1 , u2 ; μ1 , μ2 ) = max(k, u2 ) = u2 . 2. Let μ1 > k. Then the following holds. (a) If u1 > a, u2 > a, then UF (u1 , u2 ; μ1 , μ2 ) = max(μ1 , u2 ) = u2 . (b) If u1 ≤ a, u2 ≤ a, then: i. if u1 ≤ k, u2 ≤ k, then UF (u1 , u2 ; μ1 , μ2 ) = max(k, k) = k; ii. if u1 ≤ k < u2 , then UF (u1 , u2 ; μ1 , μ2 ) = max(k, u2 ) = u2 ; iii. if u2 ≤ k < u1 , then UF (u1 , u2 ; μ1 , μ2 ) = max(T1 (u1 , μ1 ), k) = T1 (u1 , μ1 ); iv. if k < u1 , k < u2 , then UF (u1 , u2 ; μ1 , μ2 ) = max(T1 (u1 , μ1 ), u2 ). (c) If u1 ≤ a < u2 , then: i. if u1 ≤ k, then UF (u1 , u2 ; μ1 , μ2 ) = max(k, u2 ) = u2 ; ii. if u1 > k, then UF (u1 , u2 ; μ1 , μ2 ) = max(T1 (u1 , μ1 ), u2 ) = u2 . (d) If u2 ≤ a < u1 , then: i. if u2 ≤ k, then UF (u1 , u2 ; μ1 , μ2 ) = max(μ1 , k) = μ1 ; ii. if k < u2 , then UF (u1 , u2 ; μ1 , μ2 ) = max(μ1 , u2 ). 4. Behavior of a decision maker with respect to the utility function UF The main part of this paper is the following discussion of the previously described behavioral characteristics for the hybrid utility function given by (10). Now, we are considering behavior of a potential decision maker that can correspond to a nullnorm with a non-trivial absorbing element, i.e., of a decision maker that insists on the certain truncation level. Case I (Fig. 5) represents a situation when the decision maker is highly uncertain about the state of the world, i.e., both μ1 and μ2 are high and the two corresponding states x1 and x2 have high plausibility (if the state of the world xi occurs, its utility is ui and plausibility for this situation is μi ). For the first three subcases the situation is the same as for the hybrid utility function U (see [9]). So in the subcase 1 the decision maker’s opinion is probabilistic, whereas in the subcases 2 and 3 the decision maker looks forward to the best outcome. However, the subcase 4 is of special

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Fig. 5. Case I: both states of the world have high plausibility.

Fig. 6. Case II: the first state of the world is unlikely.

interest for further investigation because it shows that there are four possibilities. Subcase 4a shows that when rewords are less than k in both states, the value of utility function UF is k. Subcase 4b (analogously 4c) is when the reword is greater than k in state x1 but less than k in the other state. Then, the decision maker looks forward to reword u1 . Subcase 4d is when rewords are greater than k in both states, then the decision maker is possibilistic and again focuses on the best outcome. Therefore, in the worst case 4a when both rewords are lower than k, the decision maker looks forward to reword k. So, we can conclude that, in the subcase 4 the utility function UF is possibilistic with a minimal value k. This property opens the door for further discussion on utility functions with a previously imposed threshold by the absorbing element k ∈ (0, 1) of a continuous nullnorm F . Case II (Fig. 6) represents a situation when state x1 is unlikely. We distinguish two subcases depending on the degree of plausibility μ1 . From the subcase 1 when the degree of plausibility μ1 ≤ k, it can be seen that the value of the utility function UF is either u2 or k. When the plausible reword is greater than k then decision maker looks forward to this reword. When the plausible reword is lower than k, then the decision maker focuses on the threshold k. Also, for the subcase 1 the value of the resulting utility does not depend on the degree of plausibility μ1 ≤ k. So, it can be assumed that the plausibility of the state x1 is at least k. Therefore, for UF threshold k also appears in the S-measure m. Hence, we can conclude that it makes sense to construct S-measure m (see Example 16) with a previously imposed threshold. Subcase 2 is obtained for the degree of plausibility k < μ1 ≤ a, i.e., when state x1 is more plausible than in the subcase 1. It follows that (subcase 2a, 2c) we have the same situation as in the subcase 1 when u2 > a (plausible reword is good). In the subcase 2d when u2 > k the decision maker hopes that state x1 will prevail if u2 is really unwanted, while when u2 ≤ k state x1 is the preferred one. In the subcase 2b when u1 ≤ k, i.e., when the least plausible reword is lower than k, we have the same situation as in the subcase 1. When k < u1 ≤ a the decision maker looks forward to reword (subcase 2b) that is better than in the subcase 1. Therefore, again the previously imposed threshold appears and utility function UF in the case II is possibilistic with a minimal value k. From the previous analysis, the optimistic attitude of the decision maker using the utility function UF is obvious. Remark 15. Due to the obtained threshold k in the S-measure m, and the structure of the nullnorm F on the square [k, 1]2 , it makes sense to use the nullnorm F in Definition 14 for expressing separability for S-measure m. In following example an S-measure with a threshold k is used for construction of the hybrid utility function UF with the same threshold.

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Fig. 7. S-measure tree from Example 16.

Fig. 8. Corresponding S-measure binary tree from Example 16.

Example 16. Let us suppose that k = 0.15 and a = 0.3. For X = {x1 , x2 , x3 , x4 , x5 } from Theorem 9 we define S-measure m : P (X) → [0, 1] for singletons as: m1 = 0.75,

m2 = 0.45,

m3 = 0.4,

m4 = 0.24,

m5 = 0.15

(see Fig. 7). Due to (8) we must have m1 + m2 + m3 = 1 + 2a = 1.6. Of course, m12345 = m(X) = 1, m(∅) = 0, and the other values are: • for two points sets we have: m12 = 0.9,

m13 = 0.85,

m24 = m25 = 0.45,

m14 = m15 = 0.75, m34 = m35 = 0.4,

m23 = 0.55,

m45 = 0.24;

• for three points sets we have: m123 = 1,

m124 = m125 = 0.9, m234 = m235 = 0.55,

m134 = m135 = 0.85, m245 = 0.45,

m145 = 0.75,

m345 = 0.4;

• for four points sets we have: m1234 = m1235 = 1,

m1245 = 0.9,

m1345 = 0.85,

m2345 = 0.55.

Now, as in [9], the utility of a compound lottery can be computed by decomposing the S-measure into the sequence of binary trees (see Fig. 8) and applying the utility function UF recursively from the bottom to the top of the binary tree expansion, as it can be seen in the following example. Example 17. Let m be the S-measure from Example 16, F a nullnorm restrictively distributive over t-conorm S and UF corresponding utility function given by (10). For the following five utilities u1 = 0.8,

u2 = 0.1,

u3 = 0.4,

u4 = 0.2,

u5 = 0.3

we can calculate the corresponding utility using the binary tree given in Fig. 8 and we present the procedure in Fig. 9.

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Fig. 9. Utility binary tree.

• • • •

For u4 the case II 1. (b) iv. is applicable, i.e., u4 = max(0.2, 0.15) = 0.2. For u3 the case II 2. (c) ii. is applicable, i.e., u3 = max(0.4, T1 (0.24, 0.2)) = 0.4. For u2 the case I 2. (a) is applicable, i.e., u2 = 0.3 + (0.4−0.3)(0.58−0.3) = 0.34. 1−0.3 For UF the case I 1. is applicable, i.e., UF =

0.8(0.75 − 0.3) + 0.34(1 − 0.75) = 0.64. 1 − 0.3

5. Conclusion The aim of this paper was to present the possible impact of solutions of distributivity equations (on the relaxed domain) for a continuous nullnorm and a continuous t-conorm on the utility theory. It has been shown that if one uses a continuous nullnorm F with non-trivial absorbing element k, instead of a continuous t-norm T , in the generalized definition of hybrid utility function, its value cannot be less than k, i.e., a threshold that is imposed by the absorbing element of a continuous nullnorm is present. Also, the same threshold k appears in the S-measure m. This property can be very useful for modeling the behavior of some decision makers, therefore further investigations will go into this direction. Additionally, a further field of interest for study is separability in the framework of hybrid possibilistic-probabilistic measures with a previously imposed threshold. Acknowledgements This paper has been supported by the Ministry of Science and Technological Development of Republic of Serbia, project 174009 and by the Provincial Secretariat for Science and Technological Development of Vojvodina (Republic of Serbia), project “Mathematical models of intelligent systems and their applications” 114-451-2388/2011-02. References [1] [2] [3] [4] [5] [6] [7] [8]

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