We consider MultiCriteria Decision Analysis models which are defined over discrete attributes, taking a finite number of values. We do not assume that the model is monotonically increasing with respect to the attributes values. Our aim is to define an importance index for such general models, considering that they are equivalent to @math -ary games (multichoice games). We show that classical solutions like the Shapley value are not suitable for such models, essentially because of the efficiency axiom which does not make sense in this context. We propose an importance index which is a kind of average variation of the model along the attributes. We give an axiomatic characterization of it.
Grabisch and Lange @cite_13 did not use unanimity games, but took an axiomatic approach to define a Shapley value in a more general context for games over lattices. They define the Shapley value for multichoice game as follows, @math
@cite_13
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Multichoice games have been introduced by Hsiao and Raghavan as a generalization of classical cooperative games. An important notion in cooperative game theory is the core of the game, as it contains the rational imputations for players. We propose two definitions for the core of a multichoice game, the first one is called the precore and is a direct generalization of the classical definition. We show that the precore coincides with the definition proposed by Faigle, and that it contains unbounded imputations, which makes its application questionable. A second definition is proposed, imposing normalization at each level, causing the core to be a convex closed set. We study its properties, introducing balancedness and marginal worth vectors, and defining the Weber set and the pre-Weber set. We show that the classical properties of inclusion of the (pre)core into the (pre)-Weber set as well as their equality remain valid. A last section makes a comparison with the core defined by van den