The Choquet integral as a linear interpolator

Michel Grabisch B

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Multi-Criteria Decision Making Game Theory and Voting Systems Advanced Algebra and Logic

Abstract

We show that the Choquet integral is the unique linear interpolator between vertices of the $[0, 1]^n$ hypercube, using the least possible number of vertices. Related results by Lovasz and Singer are discussed, as well as other interpolations. We show that the Choquet integral for bi-capacities can be also casted into this framework. Lastly, we discuss the case of Sugeno integral.

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2004Subjective probability and expected utility without additivity 1996The application of fuzzy integrals in multicriteria decision making 1996Generalized Expected Utility Theory. The Rank-Dependent Model 1989Additive Representations of Preferences 1988Multilinear extensions of games 2000An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria 2003The Choquet integral for the aggregation of interval scales in multicriteria decision making 2016Fuzzy Measures and Integrals in MCDA

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2009A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid 2007Bi-capacities -- Part I: definition, Möbius transform and interaction

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