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Linz 2013 — Non-Classical Measures and Integrals LINZ 2013 racts Abst 34 Fuzzy Theory Set th Non-Classical Measures Linz Seminar on Seminar Linz Bildungszentrum St. Magdalena, Linz, Austria Linz, Magdalena, St. Bildungszentrum February 26 – March 2, 2013 2, March – 26 February and Integrals Erich Peter Klement Radko Mesiar Abstracts Endre Pap Editors LINZ 2013 — NON-CLASSICAL MEASURES AND INTEGRALS Dedicated to the memory of Lawrence Neff Stout ABSTRACTS Radko Mesiar, Endre Pap, Erich Peter Klement Editors Printed by: Universit¨atsdirektion, Johannes Kepler Universit¨at, A-4040 Linz 2 Since their inception in 1979, the Linz Seminars on Fuzzy Set Theory have emphasized the development of mathematical aspects of fuzzy sets by bringing together researchers in fuzzy sets and established mathematicians whose work outside the fuzzy setting can provide directions for further research. The philos- ophy of the seminar has always been to keep it deliberately small and intimate so that informal critical discussions remain central. LINZ 2013 will be the 34th seminar carrying on this tradition and is devoted to the theme “Non-Classical Measures and Integrals”. The goal of the seminar is to present and to discuss recent advances in non-classical measure theory and corresponding integrals and their various applications in pure and applied mathematics. A large number of highly interesting contributions were submitted for pos- sible presentation at LINZ 2013. In order to maintain the traditional spirit of the Linz Seminars — no parallel sessions and enough room for discussions — we selected those thirty-three submissions which, in our opinion, fitted best to the focus of this seminar. This volume contains the abstracts of this impressive selection. These regular contributions are complemented by six invited plenary talks, some of which are intended to give new ideas and impulses from outside the traditional Linz Seminar community. Radko Mesiar Endre Pap Erich Peter Klement 3 4 Program Committee Radko Mesiar, Bratislava, Slovakia (Chairman) Endre Pap, Novi Sad, Serbia (Chairman) Ulrich Bodenhofer, Linz, Austria Bernard De Baets, Gent, Belgium Didier Dubois, Toulouse, France J´anos Fodor, Budapest, Hungary Llu´ıs Godo, Barcelona, Spain Siegfried Gottwald, Leipzig, Germany Petr H´ajek, Praha, Czech Republic Ulrich H¨ohle, Wuppertal, Germany Erich Peter Klement, Linz, Austria Wesley Kotz´e, Grahamstown, South Africa Daniele Mundici, Firenze, Italy Stephen E. Rodabaugh, Youngstown, OH, USA Susanne Saminger-Platz, Linz, Austria Aldo Ventre, Napoli, Italy Siegfried Weber, Mainz, Germany Local Organizing Committee Erich Peter Klement, Johannes Kepler University Linz (Chairman) Ulrich Bodenhofer, Johannes Kepler University Linz Andrea Breslmayr, Johannes Kepler University Linz Sabine Lumpi, Fuzzy Logic Laboratorium Linz-Hagenberg Bernhard Moser, Software Competence Center Hagenberg Susanne Saminger-Platz, Johannes Kepler University Linz Isabel Tober-Kastner, Software Competence Center Hagenberg Thomas Vetterlein, Johannes Kepler University Linz 5 6 Contents Antoine Billot, Vassili Vergopoulos On the interpretation of ambiguity .....................................11 Humberto Bustince, Javier Fernandez, Radko Mesiar, Anna Koles´arov´a Overlap indices and fuzzy measures ....................................14 Miguel Couceiro, Jean-Luc Marichal On comonotonically modular functions .................................16 Bernard De Baets, Hans De Meyer Convexity and concavity of copulas and co-copulas ......................21 Bernard De Baets, Jana Kalick´a, Hans De Meyer, Radko Mesiar Modular aggregation functions and copulas .............................26 Jeffrey T. Denniston, Austin Melton, Stephen E. Rodabaugh, Sergejs Solovjovs Lattice-valued preordered sets as lattice-valued topological systems .......28 Didier Dubois, Henri Prade, Agn`es Rico Recent results about qualitative monotonic set-functions on finite sets ......35 Fabrizio Durante The multivariate probability integral transform ..........................37 Anton´ın Dvoˇr´ak, Michal Holˇcapek On a convergence of ⊗-fuzzy integrals over MV-algebras .................39 Anatolij Dvureˇcenskij On states on quantum and algebraic structures ..........................42 Yaarit Even, Ehud Lehrer Decomposition-integral: unifying Choquet and the concave integrals ......44 Fabio Fantozzi, Fabio L. Spizzichino On the family of marginal copulas: an analysis motivated by multi-attribute target-based utility ...............48 7 Tommaso Flaminio, Llu´ıs Godo Towards a betting interpretation for belief functions on MV-algebras ......51 Siegfried Gottwald Some new results concerning local and relativized local finiteness in t-norm-based structures ....................................54 Michel Grabisch The core of games with restricted cooperation ...........................55 Michel Grabisch, Agnieszka Rusinowska Modeling influence by aggregation functions ............................57 Salvatore Greco, Radko Mesiar, Fabio Rindone Axiomatic foundations of the universal integral in terms of aggregation functions and preference relations ..........................60 Salvatore Greco, Radko Mesiar, Fabio Rindone Bipolar semicopulas ..................................................63 Salvatore Greco, Fabio Rindone h-k aggregation functions, measures and integrals .......................66 John Harding, Carol Walker, Elbert Walker Partial orders on the truth value algebra of type-2 fuzzy sets ..............71 Silvano Holzer Chebyshev and Jensen inequalities, weak convergence, and convergence in distribution in a non-additive setting .................75 Hykel Hosni, Enrico Marchioni Some notes on ordinal strategic interaction with possibilistic expectation . .82 S´andor Jenei Classification results on residuated lattices – a survey with new results .....85 Tom´aˇsKroupa Games with fuzzy coalitions: modelling partial coalitional memberships . 86 8 Jun Li Convergence theorems in monotone measure theory ......................88 Jean-Luc Marichal, Pierre Mathonet On the cardinality index of fuzzy measures and the signatures of coherent systems ................................................... 92 Jean-Luc Marichal, Bruno Teheux Pivotal decompositions of aggregation functions .........................96 Radko Mesiar, Jun Li, Endre Pap Generalized concave integral .........................................100 Radko Mesiar, Andrea Stupˇnanov´a Universal decomposition integrals ....................................103 Kazimierz Musiał A decomposition theorem for the fuzzy Henstock integral ................105 Endre Pap, Mirjana Strbojaˇ Generalization of Lp space and related convergences ...................106 Jan Paseka On the extensions of Di Nola’s Theorem ...............................110 Beloslav Rieˇcan On non-additive probability theory ....................................114 Carlo Sempi A remark on Archimax copulas .......................................117 Milan Stehl´ık On a constructive approach to complicated probability measures .........118 Ivana Stajner-Papuga,ˇ Tatjana Grbi´c Interval-valued pseudo-integral based measures ........................119 Thomas Vetterlein Triangular norms and tomonoid extensions ............................123 9 Siegfried Weber Problems of conditioning .............................................127 Ronald R. Yager Multi-source uncertain information fusion using measures ...............129 10 On the interpretation of ambiguity Antoine Billot1 and Vassili Vergopoulos2 1 Laboratoire d’Economie Moderne, Institut Universitaire de France Universit´eParis 2, Paris, France [email protected] 2 Centre d’Economie de la Sorbonne Universit´eParis 1, Paris, France [email protected] In his axiomatic treatment of the Subjective Expected Utility (SEU) theory for be- havior in the face of uncertainty, Savage (1954) takes as primitive notions a state space Ω, a set of outcomes X and preferences %Ω over Savage acts, defined as functions map- ping states onto outcomes and whose set is denoted AΩ. Savage designs a set of axioms on %Ω under which there exists a unique probability measure P over Ω and an (almost) unique utility function u : X → R such that a SEU representation is ensured: ∀F,G ∈ AΩ, F %Ω G ⇐⇒ EPu(F) ≥ EPu(G) The Savage axioms include the Sure Thing Principle (STP), which proves to be equivalent to the fundamental property of dynamic consistency (Hammond, 1988; Ghi- rardato, 2002) and requires the existence of a family (%E )E⊆Ω of preferences over Sav- age acts such that: (i) ∀E ⊆ Ω, ∀ f ,g ∈ AΩ, if f = g over E, then f ∼E g (ii) ∀E ⊆ Ω, ∀ f ,g ∈ AΩ, if f %E g and f %Ec g, then f %Ω g At an intuitive level, dynamic consistency requires consistency between strategies that are optimal ex ante and strategies actually implemented ex post. It is therefore usu- ally seen as a normative requirement for behavior under uncertainty. But it also serves a behavioral justification for the optimality of backward induction solutions and for the Bayesian updating rule for probability measures (Al-Najjar and Weinstein, 2009). The Ellsberg paradox challenges the Savage SEU theory and is usually understood as a violation of STP (Ellsberg, 1961). The idea is that people do not always know the ‘true’ probabilities of relevant events and are reluctant to assign any precise value to them. As a result, they do not behave as if they knew these values. When this happens, they are said to perceive ambiguity and their behavior is inconsistent with STP. For instance, in the Schmeidler (1989) model of ambiguity, an individual is characterized by a
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