Abstract
The paper proposes a general notion of interaction between attributes, which can be applied to many fields in decision making and data analysis. It generalizes the notion of interaction defined for criteria modelled by capacities, by considering functions defined on lattices. For a given problem, the lattice contains for each attribute the partially ordered set of remarkable points or levels. The interaction is based on the notion of derivative of a function defined on a lattice, and appears as a generalization of the Shapley value or other probabilistic values.
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Banzhaf, J.F.: Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev. 19, 317–343 (1965)
Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence, RI (1967)
Choquet, G.: Theory of capacities. Annales de l’Institut Fourier 5, 131–295 (1953)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Orders. Cambridge University Press, Cambridge (1990)
Dilworth, R.P.: Lattices with unique irreducible representations. Ann. Math. 41, 771–777 (1940)
Felsenthal, D., Machover, M.: Ternary voting games. Int. J. Game Theory 26, 335–351 (1997)
Grabisch, M.: The representation of importance and interaction of features by fuzzy measures. Pattern Recogn. Lett. 17, 567–575 (1996)
Grabisch, M.: k-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst. 92, 167–189 (1997)
Grabisch, M., Labreuche, Ch.: Bi-capacities. In: Joint Int. Conf. on Soft Computing and Intelligent Systems and 3rd Int. Symp. on Advanced Intelligent Systems, Tsukuba, Japan (2002, October)
Grabisch, M., Labreuche, Ch.: Bi-capacities for decision making on bipolar scales. In: EUROFUSE Workshop on Informations Systems, pp. 185–190. Varenna, Italy (2002, September)
Grabisch, M., Labreuche, Ch.: Bi-capacities. Part I: definition, Möbius transform and interaction. Fuzzy Sets Syst. 151, 211–236 (2005)
Grabisch, M., Roubens, M.: An axiomatic approach to the concept of interaction among players in cooperative games. Int. J. Game Theory 28, 547–565 (1999)
Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998)
Hammer, P.L., Holzman, R.: On approximations of pseudo-Boolean functions. ZOR-Methods and Models of Operations Research 36, 3–21 (1992)
Labreuche, Ch., Grabisch, M.: Bi-cooperative games and their importance and interaction indices. In: 14th Mini-EURO Conference on Human Centered Processes (HCP’2003), pp. 287–291. Luxembourg (May 2003)
Marichal, J.L.: An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. IEEE Tr. on Fuzzy Systems 8(6), 800–807 (2000)
Monjardet, B.: The consequences of Dilworth’s work on lattices with unique irreducible decompositions. In: Bogart, K., Freese, R., Kung, J. (eds.) The Dilworth theorems, selected papers of Robert P. Dilworth, pp. 192–200. Birkhäuser, Basel (1990)
Rota, G.C.: On the foundations of combinatorial theory I. Theory of Möbius functions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 2, 340–368 (1964)
Shapley, L.S.: A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games, vol. II, number 28 in Annals of Mathematics Studies, pp. 307–317. Princeton University Press, Princeton, NJ (1953)
Sugeno, M.: Theory of fuzzy integrals and its applications. Ph.D. thesis, Tokyo Institute of Technology (1974)
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Grabisch, M., Labreuche, C. Derivative of functions over lattices as a basis for the notion of interaction between attributes. Ann Math Artif Intell 49, 151–170 (2007). https://doi.org/10.1007/s10472-007-9052-7
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DOI: https://doi.org/10.1007/s10472-007-9052-7