MOPIC Properties in the Representation of Preferences by a 2-Additive Choquet Integral
In the context of Multiple Criteria Decision aiding (MCDA), we present necessary conditions to obtain a representation of a cardinal information by a Choquet integral w.r.t a 2-additive capacity. A cardinal information is a preferential information provid
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Abstract. In the context of Multiple Criteria Decision aiding (MCDA), we present necessary conditions to obtain a representation of a cardinal information by a Choquet integral w.r.t a 2-additive capacity. A cardinal information is a preferential information provided by a Decision Maker (DM) containing a strict preference, a quaternary and indifference relations. Our work is focused on the representation of a cardinal information by a particular Choquet integral defined by a 2-additive capacity. Used as an aggregation function, it arises as a generalization of the arithmetic mean, taking into account the interaction between two criteria. Then, it is a good compromise between simple models like arithmetic mean and complex models like general Choquet integral. We consider also the set of fictitious alternatives called binary alternatives or binary actions from which the Choquet integral w.r.t a 2-additive capacity can be entirely specified. The proposed MOPIC (MOnotonicity of Preferential Information for Cardinal) conditions can be viewed as an alternative to balanced cyclones which are complex necessary and sufficient conditions, used in the characterization of a 2-additive Choquet integral through a cardinal information. Keywords: MCDA BETH
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Preference modeling
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Choquet integral
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MAC-
Introduction
MultiCriteria Decision Aid (MCDA) aims at representing the preferences of a Decision-Maker (DM) on a set of alternatives (or actions, options) X evaluated over a finite set of attributes or criteria N = {1, . . . , n} (n > 1) often conflicting. An alternative can be identified as an element x = (x1 , . . . , xn ) of the Cartesian product X = X1 × · · · × Xn , where X1 , . . . , Xn represent the set of points of view or attributes. The Multi-Attribute Utility Theory (MAUT) is one of the decision models usually used to represent the preferences of the DM. In practice, MAUT elaborates a preference relation over X by asking to the DM some pairwise comparisons of alternatives on a finite subset X of X. X is called a learning data set (or reference set) and has a small size in general. Hence we get a preference relation X on X . The question is then: how to construct c Springer International Publishing Switzerland 2015 T. Walsh (Ed.): ADT 2015, LNAI 9346, pp. 222–235, 2015. DOI: 10.1007/978-3-319-23114-3 14
MOPIC Properties in the Representation of Preferences
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a preference relation X on X, so that X is an extension of X ? To this end, people usually suppose that X is representable by an overall utility function: x X y ⇔ F (U (x)) ≥ F (U (y))
(1)
where U (x) = (u1 (x1 ), . . . , un (xn )), ui : Xi → R is called a utility function, and F : Rn → R is an aggregation function. Usually, we consider a family of aggregation functions characterized by a parameter vector θ (e.g., a weight distribution over the criteria). The parameter vector θ can be deduced from the knowledge of X , that is, we determine the possible values of θ for which (1) is fulfilled over X . The aggregation function F we study is
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