Miranda Menéndez, Pedro and Grabisch, Michel
*On vertices of the k-additive monotone core.*
In
IFSA-EUSFLAT 2009 Proceedings.
International Fuzzy Systems Association (IFSA); European Society for Fuzzy Logic and Technology (EUSFLAT), pp. 76-81.
ISBN 978-989-95079-6-8

PDF
Restricted to Repository staff only until 2020. 185kB |

Official URL: http://www.eusflat.org/proceedings/IFSA-EUSFLAT_2009/pdf/tema_0076.pdf

## Abstract

Given a capacity, the set of dominating k-additive capacities is a convex polytope; thus, it is defined by its vertices. In this paper we deal with the problem of deriving a procedure to obtain such vertices in the line of the results of Shapley and Ichiishi for the additive case. We propose an algorithm to determine the vertices of the k-additive monotone core. Then, we characterize the vertices of the n-additive core and finally, we explore the possible translations for the k-additive case

Item Type: | Book Section |
---|---|

Additional Information: | Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, Lisbon, Portugal, July 20-24, 2009. |

Uncontrolled Keywords: | Keywords—Capacities, k-additivity, dominance, core |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 16957 |

References: | O. Bondareva. Some applications of linear programming to the theory of cooperative games. Problemy Kibernet, (10):119–139, 1963. T. Driessen. Cooperative Games. Kluwer Academic, 1988. M. Grabisch. Alternative representations of discrete fuzzy measures for decision making. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5:587–607, 1997. M. Grabisch and P. Miranda. On the vertices of the k-additive core. Discrete Mathematics, (308):5204–5217, 2008. A. P. Dempster. Upper and lower probabilities induced by a multivalued mapping. The Annals of Mathematical Statististics,(38):325–339, 1967. G. Shafer. A Mathematical Theory of Evidence. Princeton University Press, Princeton, New Jersey, (USA), 1976. P. Walley and T. L. Fine. Towards a frequentist theory of upper and lower probability. Ann. of Stat., 10:741–761,1982. M. Wolfenson and T. L. Fine. Bayes-like decision making with upper and lower probabilities. J. Amer. Statis. Assoc.,(77):80–88, 1982. P. Walley. Coherent lower (and upper) probabilities. Technical Report 22, U. of Warwick, Coventry, (UK), 1981. G. Choquet. Theory of capacities. Annales de l’Institut Fourier,(5):131–295, 1953. A. Chateauneuf and J.-Y. Jaffray. Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion. Mathematical Social Sciences,(17):263–283, 1989. G. Owen. Game Theory. Academic Press, 1995. D. Denneberg. Non-additive measures and integral. Kluwer Academic, 1994. M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology, 1974. G. C. Rota. On the foundations of combinatorial theory I. Theory of Möbius functions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, (2):340–368, 1964. M. Grabisch. k-order additive discrete fuzzy measures. In Proceedings of 6th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), pages 1345–1350, Granada (Spain), 1996. L. S. Shapley. Cores of convex games. International Journal of Game Theory, 1:11–26, 1971. T. Ichiishi. Super-modularity: Applications to convex games and to the Greedy algorithm for LP. Journal of Economic Theory, (25):283–286, 1981. P. Miranda and M. Grabisch. k-balanced games and capacities. European Journal of Operational Research, (Accepted). T. Matsui. NP-completeness of non-adjacency relations on some 0-1-polytopes. In Lecture Notes in Operations Research, 1, pages 249–258. 1995. C. Papadimitriou. The adjacency relation on the travelling salesman polytope is NP-complete. Mathematical Programming, 14(1), 1978. D. Radojevic. The logical representation of the discrete Choquet integral. Belgian Journal of Operations Research, Statistics and Computer Science, 38(2–3):67–89, 1998. E. F. Combarro and P.Miranda. On the polytope of non-additive measures. Fuzzy Sets and Systems, 159(16):2145–2162, 2008. P. Miranda, E.F. Combarro, and P. Gil. Extreme points of some families of non-additive measures. European Journal of Operational Research, 33(10):3046–3066, 2006. |

Deposited On: | 31 Oct 2012 09:42 |

Last Modified: | 31 Oct 2012 09:42 |

Repository Staff Only: item control page