Abstract
Non-additive measures, capacities or generally set functions are widely used in decision models, data processing and game theory. In these applications we can find many structures identified as linear transformations or linear operators. The most remarkable of them are Choquet integral, Möbius transform, interaction transform, Shapley value. The main goal of the presented paper is to study some of them recently called event-based linear transformations. We describe them considering the set of all possible linear operators as a linear space w.r.t. their linear combinations and compute the dimensions of its some subspaces. We also study the consensus requirement, i.e. we analyze the condition when the linear operator maps one family of non-additive measures to other family.
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Notes
- 1.
To see Choquet integral as a linear transformation of measures, one need to fix the integrand.
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Acknowledgment
This work has been supported by the grant 18-01-00877 of RFBR (Russian Foundation for Basic Research).
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Bronevich, A.G., Rozenberg, I.N. (2018). Event-Based Transformations of Set Functions and the Consensus Requirement. In: Torra, V., Narukawa, Y., Aguiló, I., González-Hidalgo, M. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2018. Lecture Notes in Computer Science(), vol 11144. Springer, Cham. https://doi.org/10.1007/978-3-030-00202-2_7
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DOI: https://doi.org/10.1007/978-3-030-00202-2_7
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