We call a piecewise linear mapping from a planar triangulation to the plane a convex combination mapping if the image of every interior vertex is a convex combination of the images of its neighbouring vertices. Such mappings satisfy a discrete maximum principle and we show that they are oneto-one if they map the boundary of the triangulation homeomorphically to a convex polygon. This result can...

We study in this paper the computation of Choquet optimal solutions in decision contexts involving multiple criteria or multiple agents. Choquet optimal solutions are solutions that optimize a Choquet integral, one of the most powerful tools in multicriteria decision making. We develop a new property that characterizes the Choquet optimal solutions. From this property, a general method to gener...

In this paper we show how the distribution of the discrete Choquet integral can be analytically computed. The advantage of having an analytical expression is that the value of the cumulative distribution function (cdf) can be computed exactly for the Choquet. We also derive an expression for the density of the Ordered Weighted Average (OWA) operator, which is a special case of the Choquet.

In the context of decision under uncertainty, we characterize the 2-additive Choquet integral on the set of fictitious acts called binary alternatives or binary actions. This characterization is based on a fundamental property called MOPI which permits us to relate belief functions and the 2additive Choquet integral. Keywords—Capacity, Möbius transform, Choquet integral, k-monotone function, Be...

The Choquet integral extended to negative functions can be defined in two ways, namley the asymmetric integral (usual Choquet integral) and the symmetric integral (also called the Šipoš integral). No such extension has been defined for the Sugeno integral. In this paper, after recalling the case of Choquet integral, we address the case of Sugeno integral, which we define in a purely ordinal fra...

In [Timonin, 2016] we developed a general axiomatic treatment of a popular multicriteria decision model the Choquet integral. This paper contains extensions of our results to the particular interesting special cases of the Choquet integral, analysis of some aspects of the Choquet integral model learning, and a discussion of the applications of our results in decision theory.

In this paper we give a nesessary and sufficient condition for a Choquet integral model to be decomposable into an equivalent hierarchical Choquet integral model constructed by hierarchical combinations of some ordinary Choquet integral models. The condition is obtained by Inclution-Exclusion Covering (IEC). Moreover we show some properties on the set of IECs.

The usefulness of the Choquet integral for modelling decision under risk and uncertainty is shown. It is shown that some paradoxes of expected utility theory are solved using Choquet integral. It is shown that Choquet expected utility model for decision under uncertainty and rank dependent utility model for decision under risk are respectively same as their simplified version.

In works dealing with capacities (fuzzy measures) and the Choquet integral on finite spaces, it is usually considered that all subsets of the universe are measureable. Hence, all functions are integrable in the sense of Choquet. We consider the situation where some subsets are not measurable (not feasible), so that there are nonintegrable functions. Since this is a severe limitation in applicat...