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28
A Discrete Choquet Integral for Ordered Systems
, 2011
"... A model for a Choquet integral for arbitrary finite set systems is presented. The model includes in particular the classical model on the system of all subsets of a finite set. The general model associates canonical nonnegative and positively homogeneous superadditive functionals with generalized be ..."
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A model for a Choquet integral for arbitrary finite set systems is presented. The model includes in particular the classical model on the system of all subsets of a finite set. The general model associates canonical nonnegative and positively homogeneous superadditive functionals with generalized belief functions relative to an ordered system, which are then extended to arbitrary valuations on the set system. It is shown that the general Choquet integral can be computed by a simple Mongetype algorithm for socalled intersection systems, which include as a special case weakly unionclosed families. Generalizing Lovász ’ classical characterization, we give a characterization of the superadditivity of the Choquet integral relative to a capacity on a unionclosed system in terms of an appropriate model of supermodularity of such capacities.
A representation of preferences by the Choquet integral with respect to a 2additive capacity
, 2011
"... In the context of Multiple criteria decision analysis, we present the necessary and sufficient conditions allowing to represent an ordinal preferential information provided by the decision maker by a Choquet integral w.r.t a 2additive capacity. We provide also a characterization of this type of pr ..."
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In the context of Multiple criteria decision analysis, we present the necessary and sufficient conditions allowing to represent an ordinal preferential information provided by the decision maker by a Choquet integral w.r.t a 2additive capacity. We provide also a characterization of this type of preferential information by a belief function which can be viewed as a capacity. These characterizations are based on three axioms, namely strict cyclefree preferences and some monotonicity conditions called MOPI and 2MOPI.
Incremental elicitation of Choquet capacities for multicriteria decision making
 In 21st European Conference on Artificial Intelligence
, 2014
"... Abstract. The Choquet integral is one of the most sophisticated and expressive preference models used in decision theory for multicriteria decision making. It performs a weighted aggregation of criterion values using a capacity function assigning a weight to any coalition of criteria, thus enabling ..."
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Abstract. The Choquet integral is one of the most sophisticated and expressive preference models used in decision theory for multicriteria decision making. It performs a weighted aggregation of criterion values using a capacity function assigning a weight to any coalition of criteria, thus enabling positive and/or negative interactions among criteria and covering an important range of possible decision behaviors. However, the specification of the capacity involves many parameters which raises challenging questions, both in terms of elicitation burden and guarantee on the quality of the final recommendation. In this paper, we investigate the incremental elicitation of the capacity through a sequence of preference queries selected onebyone using a minimax regret strategy so as to progressively reduce the set of possible capacities until a decision can be made. We propose a new approach designed to efficiently compute minimax regret for the Choquet model. Numerical experiments are provided to demonstrate the practical efficiency of our approach. 1
A characterization of the 2additive Choquet integral through cardinal information
, 2010
"... In the context of Multiple criteria decision analysis, we present the necessary and sufficient conditions to represent a cardinal preferential information by the Choquet integral w.r.t. a 2additive capacity. These conditions are based on some complex cycles called cyclones. ..."
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In the context of Multiple criteria decision analysis, we present the necessary and sufficient conditions to represent a cardinal preferential information by the Choquet integral w.r.t. a 2additive capacity. These conditions are based on some complex cycles called cyclones.
A DecisionTheoretic Framework to Select Effective Observation Locations in Robotic Search and Rescue Scenarios
"... Abstract — In some applications, like mapping and search and rescue, robots are autonomous when they are able to decide where to move next, according to the data collected so far. For this purpose, navigation strategies are used to drive the robots around environments. Most of the navigation strateg ..."
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Abstract — In some applications, like mapping and search and rescue, robots are autonomous when they are able to decide where to move next, according to the data collected so far. For this purpose, navigation strategies are used to drive the robots around environments. Most of the navigation strategies proposed in literature are based on the idea of evaluating a number of candidate locations according to an utility function and selecting the best one. Usually, ad hoc utility functions are used to provide a global evaluation of candidates by combining a number of criteria. In this paper, we propose to use a more theoreticallygrounded approach, based on Multi Criteria Decision Making (MCDM), to define exploration strategies for robots employed in search and rescue applications. We implemented our MCDMbased exploration strategies within an existing robot controller and we experimentally evaluated their performance in environments used in the RoboCup Rescue Virtual Robots Competition. I.
Grabisch: A general discrete Choquet integral
"... We consider a collection F of subsets of a finite set N together with a capacity v: F → R+ and call a function f: N → R measurable if its level sets belong to F. Only in this case, the classical definition of the Choquet integral works in our wider context. In this article, we provide a general fram ..."
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We consider a collection F of subsets of a finite set N together with a capacity v: F → R+ and call a function f: N → R measurable if its level sets belong to F. Only in this case, the classical definition of the Choquet integral works in our wider context. In this article, we provide a general framework for a Choquet integral that works also for nonmeasurable functions f and includes the integral proposed by Lehrer as a special case. We show that the general Choquet integral can be computed by a Mongetype algorithm. Moreover, we derive several properties that relate to the extension of capacities on 2N and to supermodularity, in particular.
DecompositionIntegral: Unifying Choquet and the Concave Integrals
, 2013
"... This paper introduces a novel approach to integrals with respect to capacities. Any random variable is decomposed as a combination of indicators. A prespecified set of collections of events indicates which decompositions are allowed and which are not. Each allowable decomposition has a value determ ..."
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This paper introduces a novel approach to integrals with respect to capacities. Any random variable is decomposed as a combination of indicators. A prespecified set of collections of events indicates which decompositions are allowed and which are not. Each allowable decomposition has a value determined by the capacity. The decompositionintegral of a random variable is defined as the highest of these values. Thus, different sets of collections induce different decompositionintegrals. It turns out that this decomposition approach unifies well known integrals, such as Choquet, the concave and Riemann integral. Decompositionintegrals are investigated with respect to a few essential properties that emerge in economic contexts, such as concavity (uncertaintyaversion), monotonicity with respect to stochastic dominance and translationcovariance. The paper characterizes the sets of collections that induce decompositionintegrals which respect each of these properties.
A link between the 2additive Choquet Integral and Belief functions
 IFSAEUSFLAT 2009
, 2009
"... In the context of decision under uncertainty, we characterize the 2additive 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 2additi ..."
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In the context of decision under uncertainty, we characterize the 2additive 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.