Triangular Norm-Based Measures and Games with Fuzzy CoalitionsThis book aims to present, in a unified approach, a series of mathematical results con cerning triangular norm-based measures and a class of cooperative games with Juzzy coalitions. Our approach intends to emphasize that triangular norm-based measures are powerful tools in exploring the coalitional behaviour in 'such games. They not and simplify some technical aspects of the already classical axiomatic the only unify ory of Aumann-Shapley values, but also provide new perspectives and insights into these results. Moreover, this machinery allows us to obtain, in the game theoretical context, new and heuristically meaningful information, which has a significant impact on balancedness and equilibria analysis in a cooperative environment. From a formal point of view, triangular norm-based measures are valuations on subsets of a unit cube [0, 1]X which preserve dual binary operations induced by trian gular norms on the unit interval [0, 1]. Triangular norms (and their dual conorms) are algebraic operations on [0,1] which were suggested by MENGER [1942] and which proved to be useful in the theory of probabilistic metric spaces (see also [WALD 1943]). The idea of a triangular norm-based measure was implicitly used under various names: vector integrals [DVORETZKY, WALD & WOLFOWITZ 1951], prob abilities oj Juzzy events [ZADEH 1968], and measures on ideal sets [AUMANN & SHAPLEY 1974, p. 152]. |
Contents
I | 1 |
TMeasures on Generated Tribes | 7 |
Games with Fuzzy Coalitions | 14 |
The Space | 107 |
Diagonal AumannShapley Value on pFNA | 115 |
Classical Values Revisited | 123 |
Related Topics and Applications | 165 |
Plausibility Measures Possibility Measures | 180 |
Other editions - View all
Triangular Norm-Based Measures and Games with Fuzzy Coalitions D. Butnariu,Erich Peter Klement Limited preview - 2013 |
Triangular Norm-Based Measures and Games with Fuzzy Coalitions D. Butnariu,Erich Peter Klement No preview available - 2014 |
Triangular Norm-Based Measures and Games with Fuzzy Coalitions D. Butnariu,Erich Peter Klement No preview available - 2010 |
Common terms and phrases
A-almost A₁ Aa,ß absolutely continuous AUMANN & SHAPLEY Aumann-Shapley value B₁ Borel measurable bounded variation BUTNARIU chain of fuzzy completing the proof Core(v Corollary corresponding t-conorm countable crisp coalitions crisp subsets Darboux property defined denote diagonal value Dini derivatives dv(t Dv(tX dx(t exists extension finite T-measures finitely T-additive Fréchet differential fundamental t-norms fuzzy coalitions fuzzy sets fuzzy subsets game theory games in pFNA games with crisp games with fuzzy Gâteaux differentiable GDIFFE Hence implies integral Lemma lim sup linear m₁ market game MDIFFE measurable function MERTENS MV algebra n-vector T measure nonatomic nonnegative o-algebra operator ordinal sum proof of Theorem prove representation respect satisfies Section sequence An)nen Shapley value spaces of games subspace of FBV Sv(t T-additive function T-disjoint T-measures T-tribe Theorem 6.2 Too-measure triangular norm-based measures triangular norms tribe v(tX value on pFNA