Approximations of Lovász extensions and their induced interaction index

Discrete Applied Mathematics

Volumn 156, Issue 1, 2008, Pages 11-24

  Marichal, Jean Luc ^a   Mathonet, Pierre ^b  

^a UNIVERSITY OF LUXEMBOURG   (Luxembourg)

^b UNIVERSITY OF LIÈGE   (Belgium)

Author keywords

Discrete Choquet integral; Interaction index; Least squares approximation; Lov sz extension; Pseudo Boolean function

Indexed keywords

BOOLEAN FUNCTIONS; GAME THEORY; INDEXING (OF INFORMATION); PROBLEM SOLVING; TRIANGULATION;

DISCRETE CHOQUET INTEGRALS; INTERACTION INDEX; PSEUDO BOOLEAN FUNCTIONS; UNIQUE AFFINE FUNCTIONS;

APPROXIMATION THEORY;

EID: 35649025384     PISSN: 0166218X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.dam.2007.07.011     Document Type: Article

References (24)

1. Akhiezer N. The Classical Moment Problem and Some Related Questions in Analysis (1965), Oliver & Boyd, Edinburgh, London
2. Algaba E., Bilbao J., Fernández J., and Jiménez A. The Lovász extension of market games. Theory and Decision 56 1-2 (2004) 229-238
3. Banzhaf J.F. Weighted voting doesn't work: a mathematical analysis. Rutgers Law Rev. 19 (1965) 317-343
4. Choquet G. Theory of capacities. Ann. Inst. Fourier (Grenoble) 5 131-295 (1955) 1953-1954
5. Denneberg D. Non-additive measure and integral, Theory and Decision Library, Series B: Mathematical and Statistical Methods vol. 27 (1994), Kluwer Academic Publishers, Dordrecht
6. Dubey P., and Shapley L.S. Mathematical properties of the Banzhaf power index. Math. Oper. Res. 4 (1979) 99-131
7. Fujimoto K., Kojadinovic I., and Marichal J.-L. Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices. Games Econom. Behav. 55 1 (2006) 72-99
8. Fujishige S. Submodular Functions and Optimization, Annals of Discrete Mathematics. second ed. vol. 58, (2005), Elsevier, Amsterdam
9. Grabisch M. k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems 92 2 (1997) 167-189
10. Grabisch M., and Labreuche C. How to improve acts: an alternative representation of the importance of criteria in MCDM. Internat. J. Uncertain. Fuzziness Knowledge-Based Syst. 9 2 (2001) 145-157
11. Grabisch M., Marichal J.-L., and Roubens M. Equivalent representations of set functions. Math. Oper. Res. 25 2 (2000) 157-178
12. M. Grabisch, T. Murofushi, M. Sugeno (Eds.), Theory and applications, in: Fuzzy Measures and Integrals, Studies in Fuzziness and Soft Computing, vol. 40, Physica-Verlag, Heidelberg, 2000.
13. Grabisch M., and Roubens M. An axiomatic approach to the concept of interaction among players in cooperative games. Internat J. Game Theory 28 4 (1999) 547-565
14. Hammer P., and Holzman R. Approximations of pseudo-Boolean functions; applications to game theory. Z. Oper. Res. 36 1 (1992) 3-21
15. Hammer P., and Rudeanu S. Boolean Methods in Operations Research and Related Areas (1968), Springer, Berlin, Heidelberg, New York
16. Hiriart-Urruty J.-B., and Lemaréchal C. Convex Analysis and Minimization Algorithms, I, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences) vol. 305 (1993), Springer, Berlin
17. L. Lovász, Submodular functions and convexity, in: Mathematical Programming, 11th International Symposium, Bonn 1982, 1983, pp. 235-257.
18. Marichal J.-L. On Choquet and Sugeno integrals as aggregation functions. Fuzzy Measures and Integrals, Stud. Fuzziness Soft Comput., vol. 40 (2000), Physica, Heidelberg 247-272
19. Marichal J.-L. Aggregation of interacting criteria by means of the discrete Choquet integral. Aggregation Operators: New Trends and Applications (2002), Physica, Heidelberg 224-244
20. Marichal J.-L., and Roubens M. The chaining interaction index among players in cooperative games. Advances in Decision Analysis (Mons 1997) (1999), Kluwer Academic Publishers, Dordrecht 69-85
21. Rota G.-C. On the foundations of combinatorial theory. I: theory of Möbius functions. Z. Wahrscheinlichkeitstheorie Verw. Geb. 2 (1964) 340-368
22. M. Roubens, Interaction between criteria and definition of weights in MCDA problems, in: Proceedings of the 44th Meeting of the European Working Group "Multiple Criteria Decision Aiding", October 1996.
23. Shapley L. A value for n-person games. Contributions to the Theory of Games II (Annals of Mathematics Studies, vol. 28) (1953), Princeton University Press, Princeton, NJ 307-317
24. Singer I. Extensions of functions of 0-1 variables and applications to combinatorial optimization. Numer. Funct. Anal. Optim. 7 (1984) 23-62