On Vertices of the k-Additive Monotone Core

Given a capacity, the set of dominating k-additive ca- pacities 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. Keywords— Capacities, k-additivity, dominance, core.

[1]  Michel Grabisch,et al.  Alternative Representations of Discrete Fuzzy Measures for Decision Making , 1997, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[2]  Michel Grabisch,et al.  k-Balanced games and capacities , 2010, Eur. J. Oper. Res..

[3]  G. Choquet.  Theory of capacities , 1954 .

[4]  J. Nash.  NON-COOPERATIVE GAMES , 1951, Classics in Game Theory.

[5]  Michel Grabisch,et al.  On the vertices of the k-additive core , 2008, Discret. Math..

[6]  菅野 道夫,et al.  Theory of fuzzy integrals and its applications , 1975 .

[7]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[8]  Alain Chateauneuf,et al.  Some Characterizations of Lower Probabilities and Other Monotone Capacities through the use of Möbius Inversion , 1989, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[9]  G. Rota.  On the foundations of combinatorial theory I. Theory of Möbius Functions , 1964 .

[10]  T. Fine,et al.  Bayes-like Decision Making With Upper and Lower Probabilities , 1982 .

[11]  Krzysztof R. Apt,et al.  Cooperative Games , 2020, A Course in Game Theory.

[12]  L. Shapley.  Cores of convex games , 1971 .

[13]  Christos H. Papadimitriou,et al.  The adjacency relation on the traveling salesman polytope is NP-Complete , 1978, Math. Program..

[14]  Tomomi Matsui,et al.  NP-Completeness of Non-Adjacency Relations on Some 0-1 Polytopes , 1998 .

[15]  T. Fine,et al.  Towards a Frequentist Theory of Upper and Lower Probability , 1982 .

[16]  Tatsuro Ichiishi,et al.  Super-modularity: Applications to convex games and to the greedy algorithm for LP , 1981 .

[17]  Arthur P. Dempster,et al.  Upper and Lower Probabilities Induced by a Multivalued Mapping , 1967, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[18]  Michel Grabisch,et al.  K-order Additive Discrete Fuzzy Measures and Their Representation , 1997, Fuzzy Sets Syst..

[19]  M. Dufwenberg.  Game theory. , 2011, Wiley interdisciplinary reviews. Cognitive science.

[20]  Elías F. Combarro,et al.  Extreme points of some families of non-additive measures , 2006, Eur. J. Oper. Res..

[21]  Endre Pap,et al.  Non-additive measures and integrals , 2006 .

[22]  Elías F. Combarro,et al.  On the polytope of non-additive measures , 2008, Fuzzy Sets Syst..

[23]  G. Rota.  On the Foundations of Combinatorial Theory , 2009 .