Reference : Axiomatizations of quasi-Lovász extensions of pseudo-Boolean functions
 Document type : Scientific journals : Article Discipline(s) : Physical, chemical, mathematical & earth Sciences : Mathematics To cite this reference: http://hdl.handle.net/10993/2790
 Title : Axiomatizations of quasi-Lovász extensions of pseudo-Boolean functions Language : English Author, co-author : Couceiro, Miguel [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] Marichal, Jean-Luc [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] Publication date : Dec-2011 Journal title : Aequationes Mathematicae Publisher : Springer Volume : 82 Issue/season : 3 Pages : 213-231 Peer reviewed : Yes (verified by ORBilu) Audience : International ISSN : 0001-9054 e-ISSN : 1420-8903 City : Basel Country : Switzerland Keywords : [en] Aggregation function ; Discrete Choquet integral ; Lovász extension ; Functional equation ; Comonotonic modularity ; Invariance under horizontal differences ; Axiomatization Abstract : [en] We introduce the concept of quasi-Lov\'asz extension as being a mapping $f\colon I^n\to\R$ defined on a nonempty real interval $I$ containing the origin and which can be factorized as $f(x_1,\ldots,x_n)=L(\varphi(x_1),\ldots,\varphi(x_n))$, where $L$ is the Lov\'asz extension of a pseudo-Boolean function $\psi\colon\{0,1\}^n\to\R$ (i.e., the function $L\colon\R^n\to\R$ whose restriction to each simplex of the standard triangulation of $[0,1]^n$ is the unique affine function which agrees with $\psi$ at the vertices of this simplex) and $\varphi\colon I\to\R$ is a nondecreasing function vanishing at the origin. These functions appear naturally within the scope of decision making under uncertainty since they subsume overall preference functionals associated with discrete Choquet integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-Lov\'asz extensions, we propose generalizations of properties used to characterize the Lov\'asz extensions, including a comonotonic version of modularity and a natural relaxation of homogeneity. A variant of the latter property enables us to axiomatize also the class of symmetric quasi-Lov\'asz extensions, which are compositions of symmetric Lov\'asz extensions with $1$-place nondecreasing odd functions. Permalink : http://hdl.handle.net/10993/2790 DOI : 10.1007/s00010-011-0091-0 Other URL : http://arxiv.org/abs/1011.6302

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