By Milan Mares

The publication offers with the mathematical version of coalitional cooperation in accordance with obscure wisdom of its anticipated results. This truth is defined by means of the version of coalitional online game (with or with no transferable software) during which the predicted pay-offs are represented through fuzzy amounts or by means of fuzzy units of real-valued vectors. This method of the fuzziness within the bargaining is new and completes the prevailing literature in this subject by way of substitute perspectives. The presentation doesn't call for deeper than hassle-free wisdom of linear algebra and a few similar issues. The booklet can be utilized as an creation to the topic providing easy recommendations and results.

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If (1, w) is a fuzzy extension of (1, v), if C is the core of (1, v) and if x E C, then the properties of a fuzzy ordering relation t imply that 'Yo( x) = 1. It means that the fuzzy core C F of (1, w) is a fuzzy extension of its deterministic counterpart C as shown in [33]. On the other hand, for every coalition game (1, v) there exists such an extension (1, w) that 'Yo( x) = 1, for some x E RI. It is sufficient to define w(1) "wide" enough. But, if for every L C 1, 1l£(X) < 1 for x =I- v(L), then {x E RI : 'Yo(x) = I} = C as also shown in [33].

9,15]. Hence, the equality also holds between ~({2}, {I, 3}) and d {3}, {I, 2}) which are equal to~. (0 0 ) = 21' 11 mm vsuper(I,w),vsub(1,w) min(vsuper(I,w),vsub(1,w)) = 133. We can see that the possibility that the fuzzy game (I, w) is subadditive is greater than the possibility that it is superadditive and, consequently, the possibility of superadditivity implies also the possibility of additivity. 6. We have already investigated its subadditivity and know that o vsub(1,w) = 9 14' vsub(1,w) = O.

2 + 2 . ),3 1 . ),1 for Player 1 for Player 2. 7. But V(I) V*(I) P(I) = 1,2, are {x E R2 : X2 :::; 4} n {x E R2 : Xl :::; 4} n{ x E R2 : Xl + X2 :::; 6} n {x E R2 : Xl + 2X2 :::; 9}, (R2 - V(I)) U {x E R2 : Xl + X2 = 6} U {x E R2 : Xl + 2X2 = 9}, {xER2: Xl+X2:::;6, 3:::;Xl:::;4} u{xER2: Xl+2x2=9, 1:::;Xl:::;3}. This game is evidently superadditive but not subadditive and its core C C P(I) is equal to {x E R2 : Xl + X2 = 6, 3:::; Xl :::; 4} U {x E R2 : Xl + 2X2 = 9, 2:::; Xl:::; 3}. o Relations between the superadditivity, subadditivity and additivity can be illustrated by the following two examples.

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