Triangular Norm-Based Measures and Games with Fuzzy by Dan Butnariu, Erich Peter Klement (auth.)

By Dan Butnariu, Erich Peter Klement (auth.)

This publication goals to offer, in a unified technique, a chain of mathematical effects con­ cerning triangular norm-based measures and a category of cooperative video games with Juzzy coalitions. Our process intends to stress that triangular norm-based measures are robust instruments in exploring the coalitional behaviour in 'such video games. They no longer and simplify a few technical elements of the already classical axiomatic the­ in simple terms unify ory of Aumann-Shapley values, but additionally offer new views and insights into those effects. in addition, this equipment permits us to acquire, within the video game theoretical context, new and heuristically significant info, which has an important impression on balancedness and equilibria research in a cooperative atmosphere. From a proper perspective, triangular norm-based measures are valuations on subsets of a unit dice [0, 1]X which look after twin binary operations precipitated by means of trian­ gular norms at the unit period [0, 1]. Triangular norms (and their twin conorms) are algebraic operations on [0,1] that have been prompt by way of MENGER [1942] and which proved to be beneficial within the concept of probabilistic metric areas (see additionally [WALD 1943]). the belief of a triangular norm-based degree was once implicitly used below a number of names: vector integrals [DVORETZKY, WALD & WOLFOWITZ 1951], prob­ talents oj Juzzy occasions [ZADEH 1968], and measures on perfect units [AUMANN & SHAPLEY 1974, p. 152].

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2n}, if iE{0,1, ... ,2 n -1}, A(x) = I} if i = 2n. {Ai/2n,(i+l)/2n {x E X I From Claims 1 and 2 it follows that the step functions 54 II. TRIANGULAR NORM-BASED MEASURES are all &Tv-measurable. ; m(A· Gn,i) = m(A) ~ 2ft tn dri" = v A and (tn) '\. A. Taking into account ~o • 2ft L ;n' ri" (Hn,i) ~ L m(A . Hn,i) = m(A). 2), completing the proof of the Theorem. 3 Remark. 9 it is easily seen that the Too -measures on a generated Ta -tribe with T. 9), for each x E X can be written in the form Jt(x, [a,,8[) = ,8 - a.

5. 3(iv)) the conditions (i), (ii) and (iii) are no longer equivalent. 1 Throughout this section X denotes a nonempty set, Tat-norm, and Sits corresponding t-conorm. For a subset 9' ~ [0, l]X we consider functions m : 9' [-00, +00] which assume at most one of the values -00 and +00. Definition. Let 9' be aT-clan. 1) O. m(A) + m(B). 3) [-00, +00] is said to be monotone if m(A) ~ m(B) whenever A ~ B. 2 Remark. 1) is finitely T. -additive without being necessarily a T. -valuation. This shows that T-valuations are particular additive functions in the sense of SCHMIDT [1982, p.

N + I} and I:::; k -1 define h,l = {l, ... ,k -I} \ {I}. 2) and the monotonicity of Sand T together imply (8 iElk,1 Ai) T Ak = 0. 4) 40 II. 5) A; = Ai) + A k • iE1k+l,1 iEh,1 Now, put k = n and I ~ n - 1. 5). 5) again. Continuing untill = k - 1 we again obtain the desired result. In order to show (iv) =} (i), choose k E {1,2, ... ,n} and put h = {1,2, ... ,n} \ {k}. 25) yields (8 Ai) T Ak + (8 Ai) S Ak = 8 Ai + Ak o which immediately implies the T-disjointness. 5 Corollary. 25) holds, and let (An)nEN be a countable family of fuzzy subsets of X.

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