This 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].

Synopsis

The concept of a value of a coalitional game, in the spirit of R.J. Aumann and L.S. Shapley, is extended to the case of games with fuzzy coalitions, providing heuristically meaningful insights into the game theoretical context, which have some impact on balance and equilibria analysis in a co-operative environment. Using the suggestive and philosophical power of the concept of fuzzy sets introduced by L.A. Zadeh, this text aims to develop the mathematical machinery of triangular norm-based measures, i.e. valuations preserving binary operations induced by triangular norms on [0, 1 ]. These results show the existence of Aumann-Shapley values for large classes of games with fuzzy coalitions satisfying certain differentiability conditions, and allow the extension of the domain of such values to games with crisp coalitions, and the application to real life situations such as rate problems for services in bulk.

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