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DOCUMENT DE TRAVAIL 1997-011
MODELING PREFERENCES USING STOCHASTIC AND PROBABILISTIC DOMINANCES
Jean-Marc Martel, Kazimierz Zaras
Centre de Recherche sur l’Aide à l’Évaluation et à la Décision dans les Organisations (CRAEDO)
Version originale : Original manuscript : Version original :
ISBN - 2-89524-010-8 ISBN ISBN -
Série électronique mise à jour : One-line publication updated : Seria electrónica, puesta al dia
05-1997
MODELING PREFERENCES USING STOCHASTIC AND PROBABILISTIC DOMINANCES
Jean-Marc MARTEL(1) and Kazimierz ZARAS (2) (1) Faculté des Sciences de l’Administration, Université Laval Cité Universitaire, Québec, Canada G1K 7P4 (2) Université du Québec en Abitibi-Témiscamingue 445, boul. de l’Université, Rouyn-Noranda, (Québec) Canada J9X 5E4
Abstract : Over the last five years our work has focused on applying the concept of Stochastic Dominance (SD) to multiattribute problems. However, we have found that a large number of alternatives still remain incomparable if SD is the only source of preference information. For this reason, we suggest combining the SD concept with the concept of Probabilistic Dominance (PD). From this, we can build precriteria for modeling local preferences. Next, these precriteria are aggregated in order to model global preferences, and finally, depending on which type of multicriterion problem is being analyzed, we use either a kernel concept or a ranking procedure to narrow down global preferences.
Key words : multicriterion analysis, stochastic dominance, probabilistic dominance, modeling preferences.
1. INTRODUCTION In most situations, the construction of the utility function is too complex and unrealistic because it is difficult to obtain complete information about an individual’s preferences, particularly when we are faced with several conflicting and non-commensurable attributes where the trade-off between them is not easy to establish. This means that an improvement in any one attribute cannot offset the value of any of the other attributes. In practice, this kind of situation often arises. Consider, for example, environmental and security impacts. The conflict between attributes conduct to
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oppose partial preferences. It is ideal to look for nondominated relationships, but in general, because of conflicts between attributes, these relationships are few in number. Consequently, we think it reasonable to weaken the unanimity condition of dominance. This argument led us to propose (see Zaras and Martel, 1994) weakening the unanimity condition of classic Multiattribute Stochastic Dominance for a majority attribute condition (MSD n-s ). To treat the attributes separately we need to verify the hypotheses of mutual preference independence, and the preferences over lotteries on attributes X1, ..., Xn , depend only on their marginal probability distributions. Even if these hypotheses are verified, we still have problems modeling preferences in relation to each attribute. Our former proposed modeling of preferences is based on Stochastic Dominance. SD is defined between two cumulative distribution functions Fi and Fj which represent distribution functions for two alternatives ai and aj. Theorems of SD have a long history in the literature on decision-making involving risk. See, for example: Quirk and Saposnik (1962), Hadar and Russell (1969), Hanoch and Levy (1969), Whitmore (1970), Goovaerts (1984), Zaras (1989), Shalit and Yitzhaki (1994). However, this concept of SD is limited. Even if SD is confirmed, all SD situations are not necessarily equivalent from the decision maker’s point of view (see Martel, Azondékon and Zaras, 1991). For a single attribute we still have several uncertain outcomes which might not be detected by Stochastic Dominance. To cover this kind of situation, we can use the concept of PD (Probabilistic Dominance) put forth by Wrather and Yu (1982) and described by Lee, Stam and Yu (1984). We propose formulating a rule that integrates both dominance concepts of preference modeling for each attribute. This rule is defined as a precriterion. Next, we propose an aggregation procedure for modeling global preferences according to non-compensatory logic.
This paper is structured as follows. The problem is formulated in section two. Section three presents the results obtained from combining the Stochastic and Probabilistic Dominances at the level of each attribute. In section four, the local results are aggregated to build global preference relationships, which are then analyzed according to the problematic of the multicriterion problem under risk. Finally, in section five, we apply this approach to a concrete example : A large company in Quebec choosing between ten computer development projects (Zaras and Martel, 1994).
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2. FORMULATION OF THE PROBLEM
The problem may be represented by a (A, X, E) model where the sets of Alternatives, Attributes and Evaluations are defined as follows : A = a finite set of alternatives ai, i=1,2, ..., n; X= a finite set of attributes Xk, k=1,2, ..., m; E = a set of evaluations xik; where Xik is a random variable with the probability function fk(xik) = fik and Fik represents the cumulative distribution function. We assume that the attributes Xk are mutually preferentially independent and that preferences over lotteries on X1, ..., Xn depend only upon the marginal probability distributions (see Dyer and Sarin, 1979). The first hypothesis is necessary in order to consider the attributes separately, the second one is the condition for additive independence and it simplifies the analysis of multiattribute problems involving risk (see Fishburn, 1970). This is also important for modeling preferences in multiattribute non-compensatory logic. In general, these problems, which we will examine, can be classified within the set of the problems whose objective is to choose a best alternative. If A represents a set of alternatives (for example, options for highway routes or nuclear power plant sites), the problem may be stated in the following terms : to choose the best one of the alternatives in the set A, or to identify a subset of best alternatives. By experience we know how difficult it is for analysts to convince the DM that the best solution in keeping with the model is the one that should be adopted. For this reason, we will also examine how to rank chosen alternatives in a decreasing order of preference (ranking procedure). The attributes are defined so that a larger value represents a more preferred alternative. Using Stochastic Dominances to model single-attribute preferences implies implicitly DARA utility functions for each attribute. This means that the utility function u(xk) for each attribute should be continuous, concave, and three times differentiable, such that u’(xk) > 0, u’’(xk) ≤ 0
u’’’(xk) ≥ 0,
and have decreasing absolute risk aversion, which can be expressed by the inequality u’(xk)u’’’(xk) > u’’(xk). SD means one of the three Stochastic Dominances : FSD (First Stochastic Dominance), SSD (Second Stochastic Dominance), and TSD (Third Stochastic Dominance).
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Definition 1 Fik FSDk Fjk if and only if Fik ≠ Fjk, and Fik (xk) ≤ Fjk (xk) for all xk ∈ [ck, dk] where [ck, dk] is the interval of definition of two random variables Xik and Xjk.
Definition 2 Fik SSDk Fjk if and only if Fik ≠ Fjk, and ∫Fik (xk) ≤ ∫ Fjk (xk) for all xk ∈[ck, dk].
Definition 3 Fik TSDk Fjk if and only if Fik ≠ Fjk, and ∫∫Fik (xk) ≤ ∫∫Fjk (xk) for all xk ∈ [ck, dk] and µ( Fik ) ≥ µ( Fjk ).
It is clear that if one of the three SD is verified, the first nonzero values of Fjk - Fik as xk increases from the left must be positive. Therefore if µ( Fik )≥µ( Fjk ) and Fjk - Fik starts negative, neither Fik SDk Fjk nor Fjk SDk Fik is verified. For example, as shown by Lee, Stam, and Yu (1984) (see Figure 1), let Xik and Xjk be such that Pr(Xik = 1) = ¼ , Pr(Xik = 5) = ½, Pr(Xik = 10) = ¼ and Pr(Xjk = 2) = ½, Pr(Xjk = 3) = ½. We can calculate that µ( Fik ) ≥ µ ( Fjk ) and Fjk - Fik starts negative, then Xik and Xjk do not stochastically dominate each other, but it seems that Xik is prefered to Xjk.
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Fk(x) 1.0 Fjk (x)
0.9 0.8
Fik (x)
0.7 0.6 0.5 0.4 0.3 0.2 0.1 1
2
3
4
5
6
7
8
9
10
Xk
Fig.1: Lee, Stam and Yu (1984) example
In fact, we can see in this example that Pr(Xik > Xjk) = Pr(Xik = 5) + Pr(Xik = 10)= 0.75 and the customary way of thinking leads us to conclude that Xik dominates Xjk. The Probability Dominance concept is based on this rationale. Given two random outcomes Xik and Xjk we can define Probabilistic Dominance as follows :
Definition 4 Fik PDk Fjk if and only if :Pr(Xik > Xjk) ≥ β where β∈[ 0.5 ; 1.0]. Let Xik and Xjk be two random outcomes defined as in Wrather and Yu (1982), which are presented in Figure 2, Pr(Xik = 0) = 0.4 , Pr(Xik = 100) = 0.6, and Pr(Xjk = 99) = 1.0. Then we can see that Xik dominates Xjk with probability β = 0.6. Thus, the intuitive meaning of Fik PDk Fjk is that Xik is likely to be better than Xjk ; where « likely » indicates a more than 50 % chance of occurring. In certain situations this is not, in fact, the case. In this example, Xjk seems to be preferred to Xik, and we can verify that Fjk SSDk Fik.
5
F(x) 1.0
0.8
0.6 Fik (x) 0.4
0.2
Fjk (x)
10
90
99 100
Xk
Figure 2 : Wrather and Yu (1982) example
It can be seen that there are situations involving risk where preferences are predicted differently by the two dominance concepts. In such a case, we will give priority to Stochastic Dominance, which is more restrictive. In general, these two concepts of dominance complement each other, and together they offer more possibilities for modeling partial preferences for each attribute. This observation became the basis of the proposal which will be presented in the next section.
3. MODELING PREFERENCES USING STOCHASTIC AND PROBABILISTIC DOMINANCES
Until now, we have modeled preferences for single attributes based on the concept of SD (see Zaras and Martel, 1994; Martel and Zaras, 1995). These local preferences were modeled by means of two binary relations P and R, having the following properties : P (large preference) : irreflexive and asymmetric R (incomparability) : irreflexive and symmetric. When comparing two probability distributions, the DM can be confronted with various situations. His perception will be different for Fik FSDk Fjk where P(Xik > Xjk) = 0.9, and P(Xik > Xjk) = 0.1.
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To express this difference we would like to suggest a precriterion, which will discriminate between Pk (strict preference) and Qk (weak preference), as follows : aik Pk ajk if ∃ xkα∈ Xik such that Pr(Xjk < xkα) > β / (1 - α) and ¬Fjk SDk Fik where β∈[ 0.5 ; 1.0 ] , α∈[ 0 ; 1.0 ) and xkα= sup{xk / Pr(Xik < xk) ≤ α}, aik Qk ajk if ∀ xNα∈ Xik such that Pr(Xjk < xkα) ≤β/ (1 - α) and Fik SDk Fjk
aik Rk ajk for the others. Rk represents relationships of incomparability or indifference; β- is a threshold of presumed habitual preference from the PD. The suggested precriterion cannot distinguish between relationships of indifference and incomparability. In the case of indifference, where Fik = Fjk, neither Fik SDk Fjk nor Fik PDk Fjk according to the precriterion considered. We can say the same about any uncertain option with equal chances of outcomes, compared with the certain equivalent, because Pr(Xjkβ / (1 - α) is more restrictive than Pr (Xik > Xjk ) ≥ β. This precriterion is unable to explain preferences in the case where Pr(Xik > Xjk) is too small and the SD is not satisfied. This is why the famous « Allais Paradox » example is classified by this precriterion as a case of incomparability. Considering the Allais problem, we have four options : a1 = payoff $1 million for certain
payoff $5 million, probability 0.1 a3 = payoff $0, probability 0.9
or
or
payoff $5 million, probability 0.1 a2 = payoff $1 million, probability 0.89 payoff $0, probability 0.01
payoff $1 million, probability 0.11 a4 = payoff $0, probability 0.89.
Do you prefer a1 to a2 , or vice versa ? Alternatively, which of a3 or a4 do you prefer ? We can verify for both problems of choice that neither Pr(Xjk < xkα) > β / (1 - α) condition with β ≥ 0.5, and for at least one α∈[ 0 ; 1.0 ) is fulfilled, nor is the SD condition.
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It can be easily demonstrated that if Xik and Xjk are probabilistically independent, the PD can be established by finding xkα∈ Xik such that Pr(Xjk < xkα) > β / (1 - α) for at least one α∈[ 0 ; 1.0 ). This result arises from the following theorem :
Theorem If ∃ xkα∈ Xik such that Pr(Xjk < xkα) > β / (1 - α) then Pr (Xik > Xjk ) ≥ β, i.e. Fik PDk Fjk . This theorem can be proven as follows (Wrather and Yu, 1982) :
Proof If Pr (Xjk < xkα) > β / (1- α) then (1- α) Pr (Xjk < xkα) > β for α∈[ 0 ; 1.0 ). Knowing that Pr (Xik ≥ xkα) ≥ (1-α) we can say that Pr (Xik ≥ xkα).Pr(Xjk < xkα) ≥ (1- α) Pr (Xjk < xkα) >β. Consequently since Pr (Xik > Xjk ) ≥ (1- α) Pr (Xjk < xkα) >β. This implies that Pr (Xik > Xjk ) ≥ β. This precriterion combines the SD and the PD concepts. We have the strict preference when Pr (Xik < xkα) > β / (1- α) and Fik SDk Fjk, and also when the SDk is not verified (nor Fjk SDk Fik), but Pr (Xik < xkα) > β / (1- α), which extends the SD concept by PD. If these two concepts are in opposition, this rule gives priority to the more restrictive condition of SD, and classifies the relationship under consideration as the weak preference.
4. THE GLOBAL PREFERENCE APPROACH Finally, we would like to build a global preference relationship between each alternative pair in a multiattribute problem involving risk. The suggested procedure for aggregating local preferences is as follows: ai
aj if ¬ ajk Pk aik for all k and if wP+ + wQ+ ≥ wQ- ,
ai ∼ aj for the others; where the two binary relations
and ∼ are defined respectively as large preference and no
preference, and wP+ is the sum of weights for all k where aik Pk ajk; wQ+ is the sum of weights for all k where aik Qk ajk; wQ- is the sum of weights for all k where ajk Qk aik .
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To find the best alternative, or the subset which contains the best alternative, we determine the kernel of the preference graph as in the ELECTRE I method (see Roy and Bouyssou, 1993). The alternatives belonging to the kernel have the following properties : 1. Each element not in the kernel is preferred by at least one element in the kernel. 2. Each element in the kernel is not preferred by any other element in the kernel. Note the cycles that can occur in the preference graph. We can reduce the graph by taking out one of the elements in the cycle, and build a new graph without cycles. If we consider a ranking problematic the alternatives in A should be regrouped into equivalence classes that are as small as possible, and these classes must be put in decreasing order. Let us start by building the preference graph for the aggregation procedure mentioned above. We can arrange this graph in decreasing order, beginning from the alternatives which are not preferred by any other alternative. This is an optimistic ranking. We can also order this graph beginning from the alternatives which do not prefer any other alternative. This is a pessimistic ranking. Both of these pre-orders can vary, because of alternatives which are not preferred, and do not prefer, any other alternative. Thus a comparison between these two pre-orders is important in identifying this kind of alternative. This is why building the intersection of these two pre-orders is often recommended. Finally, we obtain a partial pre-order structure for the set of alternatives A, which corresponds to weak preference order.
5. APPLICATION
We can illustrate the application of the mulicriterion approach with the example of a large company in Quebec choosing between ten computer development projects (Zaras and Martel, 1994). In this example, the objective was to model the DM’s preferences among ten projects, which were evaluated by seven experts on a scale of ten for each of the four following attributes : X1 : human resources required; X2 : discounted profit; X3 : chances of success; X4 : technological trends.
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These evaluations are presented in Table 1 in the form of probability distributions. It was assumed that the first and last attributes have the same importance, 0.09. The most important is the second attribute, 0.55, and the importance of the attribute X3 was assumed to be 0.27. To apply the suggested approach, we must first establish the types of pairwise stochastic dominance relationships for each pair of projects (alternatives) for each attribute. Table 2 shows these relationships between all project pairs where one of three stochastic dominances is confirmed. As we can see, there are some project pairs where such a relationship is not confirmed. We will apply the probabilistic dominance condition to try to clarify some of these relationships. Firstly, for each pair of projects we can verify that if there exists at least one α∈[ 0 ; 1.0) such that Pr(Xjk < xkα) > β / (1 - α) for β greater or equal to 0.5, then we have a strict preference « Pk » relationship. If this condition is not met, then we have a weak preference « Qk » when the SDk is satisfied. In the other cases where the condition Pr(Xjk < xkα) > β / (1 - α) is not satisfied, we have incomparable or indifference « Rk » relationships. The results of this kind of analysis are presented in Table 3, where we can see that application of precriteria gives us more possibilities in modeling of preferences than only the SD condition. If the value of threshold β increases, less of relationships between project pairs will be clarified.
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Table 2 : Observed Dominances X1 1 2 3 4 5 6 7 8 9 10
1 X FSD FSD FSD FSD FSD ? FSD SSD
2 X FSD FSD FSD FSD ? FSD -
3 X FSD SISD
FSD FSD FSD
4 X -
X2
5 6 7 8 9 10 1 ? FSD X ? FSD FSD FSD SSD FSD FSD FSD SSD FSD FSD FSD FSD FSD X SSD FSD ? FSD FSD X SSD FSD FSD FSD TSD X FSD ? X ? SSD FSD X FSD SSD FSD ? FSD X FSD
X3 1 2 3 4 5 6 7 8 9 10
1 2 X FSD X FSD FSD FSD FSD SSD FSD SSD -
3 X -
4 5 6 7 8 9 FSD FSD SSD SSD FSD FSD FSD SSD SSD SSD SSD FSD FSD X FSD FSD FDS SSD X FSD FSD FSD FSD SSD SSD X FSD FSD FSD X FSD X FSD FSD X ? FSD FSD
2 X -
3 4 5 6 7 8 - SISD FSD FSD FSD FSD FSD FSD FSD X FSD FSD FSD FSD FSD X FSD FSD FSD SSD X FSD FSD FSD X FSD FSD X FSD X SSD FSD ? ? FSD FSD
9 10 FSD FSD FSD FSD FSD SSD FSD ? ? X X
X4 10 TSD TSD FSD FSD FSD FSD ? X
1 2 3 4 5 6 7 8 9 X SSD ? FSD X FSD FSD FSD FSD FSD SSD FSD X FSD FSD FSD FSD FSD FSD FSD FSD X FSD FSD FSD FSD FSD FSD X FSD FSD FSD SSD FSD X SSD FSD FSD FSD X FSD SSD X ? FSD X SSD ? ? FSD SSD
10 FSD FSD FSD FSD ? ? X
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Table 3 : Local Preferences X1 1 2 3 4 5 6 7 8 9 10
1 Q P P P P P P Q
2 X P P Q P R P -
X2
3 -
4 -
5 -
6 -
7 -
X P R P Q Q
X -
P X Q Q
P X -
P P Q P X P -
8 Q P P P P P P X P P
9 P R P X -
10 P P P P P X
1 X P P P P Q P P
2 X -
3 P X -
4 P Q X Q -
5 P P X P
X3 1 2 3 4 5 6 7 8 9 10
1 X P P P P P P -
2 X Q Q -
3 X -
4 Q P X Q Q -
5 P X P -
6 P Q Q Q X R
7 P P P P P P X P P P
8 P P P P P P P X P P
9 P P P P X -
10 P P P R X
6 P P P P X -
7 P P P P P X -
8 Q P P P P P P X P P
9 P P P P P P X P
10 P P P P R P X
X4 6 Q P Q X -
7 Q P P P P P X P
8 P P P P P P P X P
9 P P P P P X Q
10 Q P P P P P X
1 X P P P P P P R P
2 X P -
3 P X -
4 Q X -
5 P P P X -
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After identifying the relationships between pairs of projects for each attribute we can begin the aggregation procedure. In the example under consideration we have a lot of strict preference relationships Pk identified between projects. According to the aggregation procedure, the concordance of strict relationships (ai Pk aj) is required for all attributes where they are identified. If there exists one strict relationship (aj Pk ai) which is opposed to the global preference for at least one attribute k, then this attribute has veto power. Identifying a lot of strict preference relationships Pk could imply that the graph representing global preferences is poor. However, in this case, the graph is at least connected, and it can be explored to model global preferences (see Table 4, and Figure 3).
Table 4 : Overall preferences 1 1
2
3
5
6
∼
3
∼
4
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
5
∼
6
7
10
9
10
∼
∼
∼
∼ ∼ ∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
8 9
8
∼ ∼
2
7
4
∼ ∼
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This graph could be richer by increasing the value of threshold β, but this diminishes the advantage of explaning more preferences between project pairs. Then we have a kind of trade-off problem to determine appropriate value of β. From the global preferences (Table 4) we can find the kernel, which consists of two alternatives (projects) a2 and a4 . If we want to rank the projects, we can obtain a partial pre-ordering which is composed of seven levels. On each level we have projects which can be considered equivalent.
1
2
3
4
5
6
7
1
8
9 2
3
5
6 10
4
7
Figure 3 : Resulting partial pre-order. We can see (Figure 3) that the project a4 can be located on levels 1,2, or 3. The same can be said about project a7 , which can be located in the second part of the graph on levels 5, 6, or 7. The sensitivity analysis of relative importance of the attributes may be needed to explain the positions of alternatives a4 and a7 .
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6. CONCLUSION
In this paper we have presented an approach based on the concept of precriteria, which combines the SD and the PD conditions in order to model preferences for each attribute. This type of criterion allowed us to determine strict preference situations where Pr (Xik > Xjk) ≥ β and Fik SDk Fjk, and also to reduce the number of cases of incomparability by introducing a PD condition when SD is not verified. If the SD and PD conditions are in opposition, priority is given to the more restrictive condition, SD, and the relationship is classified as a weak preference. Finally, we suggested an aggregation procedure to construct global preference relationships between each pair of alternatives for a multicriterion problem under risk, and the use of either the kernel concept or the ranking procedure depending on the type of problematic.
7. REFERENCES
Dyer J.S., and Sarin R.K. (1979), « Measurable Multiattribute Value Function », Operation Research, vol. 27 (810-822). Fishburn P.C.(1970), Utility Theory for Decision Making , New York, Wiley. Goovaerts M.J. (1984), Insurance Premium , Elsevier Science, Publishers B. Hadar J.,and Russell W.R. (1969), « Rules for Ordering Uncertain Prospects », American Economic Review, vol. 59 (25-340). Hanoch G., and Levy H.(1969) « The Efficiency Analysis of Choice Involving Risk », Review of Economic Studies, vol. 36 (335-346). Lee
Y.R.,
Stam
A.,
and
Yu
P.L.
(1984)
« Dominance
Concepts
in
Random
Outcomes »,Proceedings of International Seminar on the Mathematics of Multi-Objective Optimization, CISM, Udine, Italy. Martel J.M., and Zaras K.(1995) « Stochastic Dominance in Multicriterion Analysis Under Risk », Theory and Decision, vol. 39 (31-49). Martel JM, Azondékon S, and Zaras K (1991) « Preference Relations in Multicriterion Analysis under Risk » JORBEL, vol. 31, no 3-4 (55-83).
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Quirk J.P., and Saposnik R. (1962) « Admissibility and Measurable Utility Functions », Review of Economic Study, vol. 29 (140-146). Roy B, and Bouyssou D.(1993)-Aide multicritère à la décision : Méthodes et cas, Economica, Paris. Shalit H., and Yitzhaki S. (1994) « Marginal Conditional Stochastic Dominance », Management .Science, vol.40, no. 5 (670 - 684). Whitmore G. A.(1970), « Third-Degree Stochastic Dominance », American Economic Review, vol. 60, no. 27 (457-459). Wrather C., and You PL(1982) « Probability Dominance in Random Outcomes », Journal of Optimization Theory and Application , vol. 36, no. 3 (315-334). Zaras K. (1989) «Dominances stochastiques pour deux classes de fonction d’utilité: concaves et convexes », RAIRO (Recherche Opérationnelle /Operations Research), vol. 23, no. 1 (57-65). Zaras K., and Martel J.M.(1994) « Multiattribute Analysis Based on Stochastic Dominance » in Models and Experiments in Risk and Rationality, by Munier B. and Machina (eds) Kluwer Academic Publishers.
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