Lexicographic refinements of Sugeno Integrals - IRIT

Lexicographic refinements of Sugeno Integrals Didier Dubois and Hélène Fargier IRIT, 118 route de Narbonne 31062 Toulouse Cedex, France {dubois, fargier}@irit.fr

Abstract. This paper deals with decision-making under uncertainty when the worth of acts is evaluated by means of Sugeno integral on a finite scale. One limitation of this approach is the coarse ranking of acts it produces. In order to refine this ordering, a mapping from the common qualitative utility and uncertainty scale to the reals is proposed, whereby Sugeno integral is changed into a Choquet integral. This work relies on a previous similar attempt at refining possibilistic preference functionals of the max-min into a so-called big-stepped expected utility, encoding a very refined qualitative double lexicographic ordering of acts.

1

Introduction

In the framework of decision under uncertainty, it has been pointed out that information about preference and uncertainty in decision problems cannot always be quantified in a simple way, but only qualitative evaluations can sometimes be attained. As a consequence, the topic of qualitative decision theory is a natural one to consider [1]. A trade-off between purely symbolic and purely numerical approaches to ranking decisions consists in using a single qualitative scale for assessing beliefs and utilities. In [2] two possibilistic qualitative criteria, an optimistic and a pessimistic one, whose definitions only require a common linearly ordered scale for utility and uncertainty have been proposed. Later, a wider family of decision criteria called monotonic utilities and that encompasses both possibilistic decision criteria, have been investigated [3]. It is based, mathematically speaking, on a Sugeno integral [11]. Unfortunately, monotonic utilities in general and possibilistic utilities in particular, may suffer from a lack of decisiveness power: the principle of Pareto efficiency is not respected: namely, when two actions have the same potential consequence in some given, likely state of the world, they may be equally preferred by qualitative decision criteria, although one may have much better consequences than the other, in the remaining states of the world. Some authors tried to use the idea of lexicographic refinements with Sugeno integral. Murofushi [7] considered refining Sugeno integral with respect to a capacity by a vector of Sugeno integrals with respect to a sequence of capacities. This supposes a much richer information than the one available in decision making under uncertainty. Grabisch [5] proposed to use the fact that Sugeno integral

2

Didier Dubois and Hélène Fargier

is a median to refine it by cancelling the median terms when equal and comparing the values around the median via a leximin. Here we try to generalize the weighted extension of the leximin and the leximax to possibilistic integrals, proposed by Fargier and Sabbadin [4]. These authors have indeed shown that in the possibilistic case the order induced on decisions can be refined by expected utility, so that the strict consistency with the Pareto principle can be recovered. It can thus be asked if the same question can be solved for Sugeno integrals since prioritized minimum and maximum are special cases of fuzzy integrals. The paper is structured as follows. In Section 2, we first give some background on qualitative preference functionals, and point out the “drowning effect” that generates a lack of decisiveness power. It is shown that the only situations where Sugeno integral satisfies the principle of efficiency are degenerate. Section 3 presents lexicographic refinements of Sugeno integrals. In particular, it shows that for any Sugeno integral with respect to a capacity, there exists a Choquet integral with respect to the same ordering of events that refines it. Another kind of refinement based on Moebius transforms is then considered. Proofs are omitted for the sake of brevity.

2 2.1

Limitations of Qualitative Preference Functionals Definitions

Let S be a set of potential states of the world and X be a set of possible consequences. F = X S denotes the set of potential acts, associating to each possible state s a consequence f (s) ∈ X. In this paper, S and X are supposed to be finite. Three particular sub classes are worthwhile noticing. – Constant acts: each such act is identified with some x ∈ X, i.e., ∀s ∈ S, x(s) = x – We will also often refer to the notion of compound act . For any set of states A, f Ag is the act defined by: f Ag(s) = f (s) for all s ∈ A, and f Ag(s) = g(s) for all s ∈ A. – For any pair of consequences x and y in X, xAy will denote for short the act defined by: xAy(s) = x for all s ∈ A, and xAy(s) = y for all s ∈ A. Such a compound act will be called a binary act. Consider again the set F of acts. In our framework, we assume that it is possible to evaluate uncertainty and preferences by means of a finite totally ordered scale (L, ≤) whose top and bottom elements are denoted > and ⊥ respectively. We will also write αi the elements of L, with (α0 = > > α1 > . . . > αl = ⊥). The mapping from the set of consequences to L is a utility function µ : X → L. It is supposed that the top and bottom elements of L are in µ(X) = {µ(x), x ∈ X}. If not, just add an ideal consequence denoted > and a totally bad consequence denoted ⊥ to X, that will be identified with the bounds of L. Uncertainty is assumed to be captured by means of a set function γ : 2S → L which is a monotonic measure (or a capacity), i.e. is such that: γ(∅) = ⊥,

Lecture Notes in Computer Science

3

γ(S) = >, A ⊆ B ⇒ γ(A) ≤ γ(B). This kind of set-function is very general and represents the minimal requirement for the representation of partial belief. When numerical, this family includes probability measures and most other well-known representations of partial belief (including belief and plausibility functions, necessity and possibility measures...). In the following, we assume without loss of generality that for any A ⊆ S, there exists x such that γ(A) = µ(x) (if there is no such x in the original consequence set, just add an element to X having this utility value). This is a classical assumption of existence of a certainty equivalent for each lottery. The utility of an act f can then be defined as a Sugeno integral [11], a qualitative counterpart of weighted sum, where the sum is replaced by a sup (a max in the finite case) and the product by an inf (a min in the finite case): Sγ,µ (f ) = max min(αi , γ(Fi )) i

(1)

where Fi = {s, µ(f (s)) ≥ αi )}. Sugeno integral computes the median of its arguments. For a binary act xAy where x y, Sγ,µ (xAy) is the median value in the set {µ(x), µ(y), γ(A)}. This Sugeno integral thus defines a weak order on F (i.e. complete and transitive relation) representing the preferences of the Decision Maker over acts: f sug γ,µ g ⇔ Sγ,µ (f ) ≥ Sγ,µ (g).

(2)

When there is no ambiguity about γ and µ, we simply use the notation sug . 2.2

Limitations of qualitative preference functionals

As said in the introduction, monotonic utilities suffer from a lack of decisiveness power and can even fail to satisfy the principle of efficiency of Pareto. This general principle says that, if f is as least as good as g on each state, and better than g on some non null state s, then f should be strictly preferred to g. We put a restriction on null events, which by definition do not play any role in the decision. Formally, let be a preference relation on F: Null Events An event A is said to be null1 with respect to a preference relation on acts iff ∀f, g, h ∈ F, f Ah gAh Remark that if A and B are null, so is A ∪ B and reciprocally. So, it can be said that a state s is null iff {s} is null. 1

If is defined by a Sugeno integral, A null implies γ(A) = ⊥ but γ(A) = ⊥ does not imply that A is null. For instance let S = {s1 , s2 , s3 } and let γ be the necessity measure built on the possibility distribution π(si ) = >, ∀si . Consider the acts h = f = >{s1 }> and g = ⊥{s1 }>. Obviously, γ({s1 }) = ⊥. But s1 is not a null state; indeed, Sγ,µ (f {s1 }h) = Sγ,µ (f ) = > > Sγ,µ (g{s1 }h) = Sγ,µ (g) = ⊥. In the context of Sugeno integral, A null rather means that γ(A ∪ B) = γ(B) ∀B.

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Weak Pareto Dominance Act f weakly Pareto-dominates g (denoted f P areto g) iff ∀s not null , f (s) ≥ g(s). Pareto Strict Preference: f dominates g according to Pareto (f P areto g) iff f P areto g and not g P areto f The principle of efficiency of Pareto is then classically defined as an agreement with Pareto’s strict preference. Pareto Efficiency Principle: A preference relation on F satisfies the principle of Pareto efficiency iff f P areto g =⇒ f g Let us go back to the weak order induced by Sugeno integral. It is obviously in agreement with weak dominance, but not with strict dominance. Consider for instance an event A which is not certain but quite likely (γ(A) > ⊥) and consequence x such µ(x) = γ(A) > ⊥. Then consider acts f, g that both ensure x on A but differ on the opposite event f : f (s) = x if s ∈ A, f (s) = ⊥ if s ∈ / A, g : g(s) = x whatever s ∈ S. Then Sγ,µ (g) = µ(x), but also Sγ,µ (f ) = max(min(µ(x), γ(A)), min(µ(⊥), γ(S))) = µ(x). Hence f ∼ g. This means that the fact that µ(g(s)) > µ(f (s)) on A is not taken into account. Technically, the weak utility of ⊥ is ”drowned” by the coefficient min(µ(x), γ(A)) in the computation of Sγ,µ (f ). More generally, the standard expression of Sugeno integral (1) uses two operators that are monotonic but not strictly (namely, max and min), hence two nested drowning effects. The drowning effect is also often understood as an incapacity to obey the well-known Sure-Thing Principle (STP) [9]. STP: ∀f, g, h, h0 , f Ah gAh ⇔ f Ah0 gAh0 This principle indeed insures that identical consequences do not influence the relative preference between two acts, hence the impossibility of a drowning effect. The Pareto efficiency and Sure Thing principles are close to each other. Indeed, when the preference is complete and transitive (it is a weak order), as it is the case here, the STP is a sufficient condition for Pareto-efficiency. It has been shown by Marichal [6] that the STP is generally not compatible with Sugeno integrals. We can moreover prove that the Sugeno integral is almost incompatible not only with the STP, but also with the less demanding principle of Pareto efficiency. This is the first result of this paper. Theorem 1 Under the assumption of existence of certainty equivalents for binary acts, then: sug γ,µ is Pareto-efficient if and only if there exist a unique state s∗ such that ∀A, γ(A) = > if s∗ ∈ A, γ(A) = ⊥ if s∗ ∈ / A. This means that Sugeno integral cannot be efficient unless applied when there is no uncertainty at all. These impossibility results are not necessarily damning.


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It is now established [4] that two of the Sugeno integrals, namely those that are defined upon a possibility distribution can be refined by an expected utility. This means that, when γ is a possibility measure, there always exist a probability distribution p and a (risk-prone) utility function u∗ such that f sug γ,µ g implies f EU + g, EU + being the expected utility based on p and u∗ . This expected utility confirms all strict preferences decided by the qualitative rule, but can break ties left by the latter. As any expected-utility-based preference relation, it satisfies the Sure Thing Principle and thus the principle of Pareto efficiency. An alternative (risk-averse) utility function u∗ and an expected utility EU − can similarly be built if γ is a necessity measure. It can thus be asked if the same question can be solved for discrete Sugeno integrals since prioritized minimum and maximum are special cases of fuzzy integrals. At a first glance, the answer is negative. One basic reason why prioritized maximin and minimax aggregations can be refined by a weighted average with fixed weights is that these operations do not violate the STP in a drastic way. Indeed the ordering relations induced by possibility and necessity measures satisfy a weaker independence condition: Axiom WSTP:∀f, g, h, h0 , f Ah gAh ⇒ f Ah0 gAh0 . So modifying two acts by altering their common consequences never results in a strong preference reversal. On the contrary, such a preference reversal is clearly possible for Sugeno integrals because for a fuzzy measure γ and three sets A, B, C, where C is disjoint from both A and B, one may have γ(A) > γ(B) and γ(B ∪ C) > γ(A ∪ C). This feature makes it impossible to refine rankings of acts induced by Sugeno integrals by means of another functional which satisfies the Sure Thing Principle. In particular, a Sugeno integral with respect to a given fuzzy measure cannot be refined by some expected utility with respect to a single probability distribution. Several lines can nevertheless be explored in order to partially recover efficiency, as shown in the next section.

3

Toward lexicographic refinements of the Sugeno integral

The general idea is to define refinements of sug γ,µ , i.e. relations such that: g =⇒ f g. For the reason laid bare just before, if sug f sug γ,µ violates γ,µ the WSTP, none of its refinements can satisfy it. We can nevertheless try to satisfy the so-called Comonotonic Sure Thing Principle. Recall that two acts f, g are comonotonic iff there exists a single permutation σ on the states of S that rearrange the values of both µ(f ) and µ(g) in non-decreasing order, i.e. such that: µ(f (sσ(1) )) ≤ µ(f (sσ(2) )) ≤ · · · ≤ µ(f (sσ(n) )) µ(g(sσ(1) )) ≤ µ(g(sσ(2) )) ≤ · · · ≤ µ(g(sσ(n) )) Hence the Comonotonic Sure Thing Principle stipulates: CSTP ∀f, g, h, h0 comonotonic , ∀A ⊆ S : f Ah gAh ⇐⇒ f Ah0 gAh0

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Sugeno integral generally does not satisfy CSTP, but it obeys its weak form forbidding preference reversals. WCSTP ∀f, g, h, h0 comonotonic , ∀A ⊆ S : f Ah gAh ⇐⇒ f Ah0 gAh0 It should thus be possible to get refinements that satisfy CSTP. The first idea consists in going back to the basics of Fargier and Sabbadin’s work, and the so-called big-stepped transformation, that can be applied to any max-min form. The second idea is to exploit the similarity between expressions of Sugeno integral and of discrete Choquet integral [10][8]. In particular, while Choquet integrals are additive for comonotonic acts, Sugeno integrals are both maxitive and minitive for such acts. The natural idea is thus to look for a Choquet integral as a refinement of Sugeno integrals. Finally, the qualitative Moebius transform of a fuzzy measure can be turned into a probabilistic mass function via a transformation that directly yields a Choquet integral. 3.1

Lexicographic refinements of maximin aggregations → − − Let → a = (a1 , ..., an ) and b = (b1 , ..., bn ) be some vector of evaluations using a common and finite ordered scale L = (α0 = > > α1 > . . . > αl = ⊥). A usual way of escaping the drowning effect is to refine the ordering on vectors induced by the max (resp. min) aggregation using the leximax (resp. → − → leximin) ordering. Let − a , b ∈ Ln . Then ∀j, a(j) = b(j) → − → a lmax − (3) a ⇔ or ∃i, ∀j < i, a(j) = b(j) and a(i) > b(i)   ∀j, a(j) = b(j) → − → − ∃i, ∀j > i, a(j) = b(j) and a(i) > b(i) (4) a lmin b ⇔ or  → → where, for any − w ∈ Ln , w(k) is the k-th greatest element of − w (i.e. w(1) ≥ . . . ≥ w(n) ). In practice, applying a leximin (resp. leximax) comparison on vectors comes down to rearranging their components in increasing (resp. decreasing) order, then comparing the ordered vector lexicographically, hence the name leximin (resp. leximax). Following [4] the same type of approach can be used to refine the ranking induced by a max-min aggregation of entries in matrices. Let us consider any weak order (i.e. complete and transitive relation) on vectors of Lm . The definition of leximin and leximax procedures can be applied to matrices n × m, since the rows of the matrices can be rearranged in increasing and decreasing order according to . So, denoting (resp. ≡) the strict (resp. symmetric) part of , we can compare any two matrices [a] and [b] according to Leximin() and Leximax(): ( [a] lmax() [b] ⇔ or

→ − → ∀j, − a (j) ≡ b (j) → − → − → → ∃i, ∀j < i, − a (j) ≡ b (j) and − a (i) b (i)

(5)

( [a] lmin() [b] ⇔ or


7

→ − → ∀j, − a (j) ≡ b (j) → − → − → → ∃i, ∀j > i, − a (j) ≡ b (j) and − a (i) b (i)

(6)

→ where for any matrix [w] ∈ (Lm )n , − w (k) is the k-th greatest line of [w] accord→ − → − ing to (i.e. w (1) . . . w (n) ). Now, just let be the leximin or the leximax ranking of vectors (this is possible, since these relations are complete and transitive). Then, nested lexicographic ordering procedures Leximax(lmin ), Leximin(lmax ) can be recursively defined, in order to compare L-valued matrices. Consider for instance the relation lmax(lmin ) obtained by the procedure Leximax(lmin ). It applies to matrices of dimensions n × m with coefficients in (L, ≥). In practice, the comparison comes down to rearranging [a] and [b] such that terms in each row are reordered increasingly w.r.t. ≥ and rows are arranged lexicographically top-down in decreasing order. Let [a? ] and [b? ] be rearranged matrices [a] and [b]. Let a∗i. (resp. b∗i. ) be row i of a∗ (resp. b∗ ). Then: ∀i, a?i· =lmin b?i· [a] lmax(lmin ) [b] ⇔ or ∃k ≤ p s.t. ∀i < k, a?i· =lmin b?i· and a?k· >lmin b?k· Relation lmax(lmin ) is a weak order. [a] 'lmax(lmin ) [b] if and only if [a∗] = [b∗ ], i.e. both matrices have the same coefficients up to the above described rearrangement. As expected , lmax(lmin ) refines the ranking obtained by the max-min aggregation: max min aij > max min bij implies [a] lmax(lmin ) [b]. i

j

i

j

and especially, if [a] Pareto-dominates [b] in the strict sense (∀i, j, aij ≥ bij and ∃i∗ , j ∗ such that ai∗ j ∗ > bi∗ j ∗ ), then [a] lmax(lmin ) [b]. We can now show that this ordering can be encoded by a (double) big-stepped transformation, generalizing the result of [4] established for m = 2 Theorem 2 There exist a transformation χ : L → [0, +∞) such that: X Y X Y [a] lmax(lmin ) [b] ⇐⇒ χ(ai,j ) > χ(bi,j ) i=1,n i=1,m

i=1,n i=1,m

As a matter of fact, the following transformation can be used to capture the leximax(leximin) ordering: χ∗ (αl ) = 0;

χ∗ (αi ) =

v N M i +1

, i = 0, k − 1

(7)

where N ≥ max(2, n), M ≥ max(2, m) and v is any positive normalization factor. For instance for M = 2 the series: Nv , Nv2 ; Nv4 , Nv8 ,... can be chosen. The choice of N, M, v does not really matter provided that M, N satisfy the constraints N ≥ max(2, n) and M ≥ max(2, m).

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3.2


Capacity-preserving refinements of Sugeno Integrals

In the standard expression of Sugeno integral (equation (1)), the two operators max and min are monotonic but not strictly, hence two nested drowning effects. The simplest idea to refine Sugeno integral is to use the Leximax(lmin ) procedure. This leads to the following decision rule: f lsug g ⇐⇒ [f ]γ,u lmax(lmin ) [g]γ,u

(8)

where ∀f ∈ X S , [f ]γ,u is a (l + 1) × 2 matrix on L = (α0 > α1 > . . . > αl ) with coefficients fi1 = αi and fi2 = γ(Fαi ), i = 0, l. The properties of lmax(lmin ) are thus inherited: Theorem 3 lsug is a complete and transitive relation that refines the ranking of acts induced by Sγ,µ . Moreover, f ∼lsug g iff ∀α, γ(Fα ) = γ(Gα ) The last point indicates that acts equivalent with respect to stochastic dominance will not be discriminated by lsug . Now, being a Leximax(lmin ) procedure, lsug can be encoded by a sum of products. We can for instance use the “big-stepping” function χ provided by equation 7. Let us set N = l + 1, M = 2 and choose the normalization factor so that χ(γ(S)) = 1. We can now immediately derive a new evaluation function E lsug , that provides a refinement of the ranking induced by Sγ,µ : E lsug (f ) =

X

χ∗ (α) · χ∗ (γ(Fα ))

(9)

α∈L

Theorem 4 f lsug g ⇐⇒ E lsug (f ) ≥ E lsug (g) It should be noticed that E lsug (1L A0L ) = χ∗ (γ(A)) and more generally that: ∀x P y : E lsug (xAy) ≥ E lsug (xBy)

⇐⇒ xAy lsug xBy ⇐⇒ γ(A) ≥ γ(B)

(10)

i.e. when utility degrees are Booleans, the comparison of events in the sense of E lsug is perfectly equivalent to the one in terms of γ — that is why we say that this refinement preserves the capacity. More generally, the procedure is perfectly unbiased in the sense that the original information, i.e. the ordinal evaluation of the likelihood of the events on L and the one of the utility degrees of the consequence on the same scale is preserved. As expected, lsug is ordinally equivalent to a Choquet integral, namely the one based on the utility u0 = χ∗ ◦ u and the capacity ν = χ∗ ◦ γ. Theorem 5 f lsug g ⇐⇒ Chχ∗ ◦γ,χ∗ ◦µ (f ) ≥ Chχ∗ ◦γ,χ∗ ◦µ (g) where Chχ∗ ◦γ,χ∗ ◦µ (f ) =

X αi ∈L

χ∗ (αi ) · (χ∗ (γ(Fαi )) − χ∗ (γ(Fαi−1 )))


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The intuition behind this result is that the ranking of acts is not modified when replacing χ∗ (γ(αi )) by χ∗ (γ(αi )) − χ∗ (γ(αi+1 )) in the definition of E lsug (equation 9) since when γ(αi+1 ) is negligible with respect to γ(αi ). We thus get the Choquet integral. Corollary 1 lsug satisfies the comonotonic Sure Thing Principle. It can be noticed that when the capacity is a possibility measure Π (resp. a necessity measure N ), lsug does not recover the ranking of acts provided by expected utility EU + (resp. EU − ). The procedures share the same principles, but they are different. The ordering lsug is lmax(lmin ) applied to a (l + 1) × 2 matrix, whose lines are pairs (α, Π(Fα )). On the other hand the definition of EU + is based on to an expression tailored to possibility measure (SΠ,u (f ) = maxs min(u(f (s)), π(s))). EU + thus applies lmax(lmin ) to an |S| × 2 matrix, whose lines are pairs (π(s), µ(f (s))). The contrast between the two approaches appears clearly when comparing binary acts >A⊥. lsug considers that the information about the likelihood of events contained capacity should be respected, and indeed >A⊥ lsug >B⊥ iff + Π(A) ≥ Π(B). On the contrary, EU does not preserve this order over events but refines it: the possibility ordering of event becomes a big-stepped probability ordering. 3.3

A state-based refinement

As suggested in the previous section, different formulations of the Sugeno integral may lead to different refinements. For instance, the idempotence of min and max enable the following expression: Sγ,µ (f ) = max min(µ(f (s)), γ(Fµ(f (s)) )) s∈S

We can then use the χ∗ transformation (again, with N ≥ |S|). X E lstates (f ) = χ∗ (µ(f (s))) . χ∗ (γ(Fµ(f (s)) ))

(11)

s

The following equivalent formulation is more practical. X E lstates (f ) = |{s, µ(f (s)) = αi }| . χ∗ (αi ) . χ∗ (γ(Fi ))

(12)

i

Let lstates be the preference ordering induced by E lstates . It is a refinement of sug and does generally not satisfy the STP. But it satisfies the comonotonic STP. Interestingly, it holds that if sug satisfies WSTP, then lstates satisfies STP. E lstates induces an order on binary acts that can be different from the one encoded by γ. Indeed, E lstates (>A⊥) = |A|.χ(γ(A)). So, A ∼lstates B ⇐⇒ (γ(A) = γ(B) and |A| ∼ |B|), and A lstates B ⇐⇒ (γ(A) > γ(B) or (γ(A) =

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γ(B) and |A| > |B|). We get the refinement of the original ranking of events according to γ by the ranking in terms of cardinality. Moreover, it turns out that lsug and lstates are not comparable: lstates is not a refinement of lsug , nor is lsug a refinement of lstates , as shown by the following counter example. Example 1 Let

f g π s1 0.8 0.8 0.8 s2 0.6 0.8 0 s3 0.7 0.6 1 Then Sγ,µ (f ) = Sγ,µ (g) = 0.8. The vectors of pairs (αi , γ(Fi )) are: ((0.8, 0.8), (0.7, 1), (0.6, 1)) for f and ((0.8, 0.8), (0.7, 0.8), (0.6, 1)) for g. Hence E lsug (f ) > E lsug (g). Now use pairs (µ(f (s)), γ(Aµ(f (s)) ). We get: ((0.8, 0.8), (0.7, 1), (0.6, 1)) for f and ((0.8, 0.8), (0.8, 0.8)(0.6, 1)) for g. Hence E lstates (g) > E states (f ). E lsug and E lstates make opposite rankings, hence one cannot refine the other. On this example, the choice of E lsug is closer to the intuition than the one of E , because g is better than f only on an impossible state while f is as least as good as g on each non-impossible state. Clearly, the problem with E states is that impossible states may influence the decision. states

3.4

Refinement using Moebius transforms

Another approach to the same problem may start from the expression of Sugeno integral involving all subsets of S: Sγ,µ (f ) = max min(γ # (A), uA (f )) A⊆S

(13)

where uA (f ) = mins∈A u(f (s)) and γ # (A) = γ(A) if γ(A) > maxB(A γ(B), and 0L otherwise. γ # is the qualitative Moebius transform of γ 2 . The above expression of the Sugeno integral has the standard maxmin form w.r.t. a possibility distribution (on the power set of S). Consider the increasing transformation χ∗ that changes a max-min aggregation into a sum-of-products encoding of its Leximax(lmin ) refinement X EU lex# (f ) = χ∗ (uA (f )).χ∗ (γ # (A)) (14) A∈2S

Notice that here the referential is neither S nor L, but 2S ; so, in the definition of χ∗ , we set N = 2Card(S) . We normalize the transformation in such a way that P ∗ # S A∈2S χ (γ (A)) = 1. So, the function m∗ : 2 7→ [0, 1]: m∗ (A) = χ∗ (γ # (A)) 2

It is a qualitative counterpart of the following expression of the Choquet integral: P Chν (f ) = A⊆S mν (A) × uA (f ) where mν is the Moebius transform of a numerical capacity ν.


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is a positive mass assignment. Note that m∗ is a big-stepped mass function in the sense that: X m∗ (A) > 0 =⇒ m∗ (A) > m∗ (B). (15) B⊆S, s. t.m∗ (B) 0L then m∗ (A) > maxB(A m∗ (B). Now, it is easy to show that χ∗ (uA (f )) = χ∗ (mins∈A u(f (s))) = mins∈A χ∗ (u(f (s))). Then: X EU lex# (f ) = m∗ (A) . min χ∗ (u(f (s))) (16) A⊆S

s∈A

P is a Choquet integral w.r.t. a belief function. Letting Bel∗ (A) = B⊆A m∗ (B) be the induced belief function, the obtained Choquet integral reads: X EU lex# (f ) = ChBel∗ (f ) = χ(αi ).(Bel∗ (F αi+1 ) − Bel∗ (F αi )) (17) i=0,...,m−1

This shows that any Sugeno integral can be refined by a Choquet integral w.r.t a belief function. In summary: Theorem 6 For any Sugeno integral Sγ,µ , there exist a Choquet integral ChBel∗ ,u∗ with respect to a belief function Bel∗ and a utility function u∗ such that: Sγ,µ (f ) > Sγ,µ (g) =⇒ ChBel∗ ,u∗ (f ) > ChBel∗ ,u∗ (g) Contrary to the Choquet integral presented in the Section 3.2, the capacity γ is generally not preserved under the present transformation. The resulting Choquet integral is always pessimistic, and sometimes much more, sometimes not more refined than the original criterion. Two extreme particular cases are interesting to consider: – If γ is a possibility measure Π, then γ # (A) is positive on singletons of positive possibility only. In other words, γ # coincides with the possibility distribution of Π and the Moebius expression of the Sugeno integral coincides with the expression of the optimistic possibilistic criterion. So m∗ is a regular big-stepped probability and the Choquet integral collapses to the regular expected utility EU + . – On the contrary if γ is a necessity measure N , ChBel∗ ,u∗ does not collapse at all with the pessimistic expected utility EU − . Indeed, if γ is a necessity measure N , γ # (A) is positive on alpha-cuts of the possibility distribution only. So the mass assignment m∗ is positive on the nested family of sets Ai , and the belief function Bel∗ is a necessity measure ordinally equivalent to the original one. In this case, the resulting Choquet integral preserves the necessity measure. Only the “max-min” framing of the Sugeno integral has been turned into a “sum-product” framing: the transformation has preserved the nature of the original capacity and the refinement identified in Section 3.2 is retrieved.

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4


Conclusion

This paper tries to bridge the gap between qualitative and quantitative criteria for decision-making under uncertainty with a view to increase their discrimination power. It is shown that qualitative criteria can be refined by symmetric lexicographic procedures. These procedures can be simulated by quantitative criteria using big-stepped scale transformations that preserve the orderof -magnitude and negligibility phenomena pervading the qualitative criteria. When uncertainty is encoded by means of qualitative possibility and necessity measures, a big-stepped expected utility criterion is known to provide a maximal refinement. Here we study the case when uncertainty is encoded by a general qualitative capacity and the criterion is a Sugeno integral. Our results indicate that a Choquet integral is the natural choice for defining refined rankings. Numerous questions remain open and are not considered here by lack of space, for instance the detailed study of the refinements of a capacity and the question of the (non) unicity of the maximal refinement. Can the capacity-preserving refinement be improved by refining the capacity in a second step? How to relate the various refinements obtained by the various expressions of Sugeno integral? Lastly, finding complete act-driven axiomatics of the new decision rules proposed here is also in order, by putting together Savage axioms and Sugeno integral axioms in some way.

References 1. J. Doyle and R. H. Thomason. Background to qualitative decision theory. AI Magazine, 20(2):55–68, 1999. 2. D. Dubois and H. Prade. Possibility theory as a basis for qualitative decision theory. In Proceedings of IJCAI’95, pages 1925–1930, 1995. 3. D. Dubois, H. Prade, and R. Sabbadin. Qualitative decision theory with Sugeno integrals. In Proceedings of UAI’98, pages 121–128, 1998. 4. H. Fargier and R. Sabbadin. Qualitative decision under uncertainty: Back to expected utility. Artificial Intelligence, 164:245–280, 2005. 5. M. Grabisch. Some lexicographic approaches to the Sugeno integral. In Proceedings International Conference on Information Processing and Management of Uncertainty (IPMU), 2006. 6. J.-L. Marichal. An axiomatic approach of the discrete Sugeno integral as a tool to aggregate interacting criteria in a qualitative framework. IEEE transactions on fuzzy systems, 9(1):164–172, 2001. 7. T. Murofushi. Lexicographic use of Sugeno integrals and monotonicity conditions. IEEE Transactions on Fuzzy Systems, 9(6):783–794, 2001. 8. R. Sarin and P. P. Wakker. A simple axiomatization of nonadditive expected utility. Econometrica, 60(6):1255–1272, 1992. 9. L.J. Savage. The Foundations of Statistics. Wiley, New York, 1954. 10. D. Schmeidler. Integral representation without additivity. Prooceedings of the American Mathematical Society, 97(2):255–261, 1986. 11. M. Sugeno. Theory of fuzzy integral and its applications. PhD thesis, Tokyo Institute of Technology, Tokyo, 1974.