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By F. J. W. Hahne

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Consider now the case when j fj. T. 39 that there exists a Q E {j(J) such that j E Q. 64 that Q fj. a(J). This proves the required assertion. We shall now consider the two possible distinct cases separately. Case i. : f)k. 83, we note that /;0 is coherent on N -{i}. 87, we conclude that la(Ji0)U{j(Ji0) I ::::; la(J)u{j(f)I-1. Suppose now li0 is a self-dual structure. 87 that Ia (Ji0) I ~ la(J) U {j(J) 1- 2. Case ii. : I)k. : f)j. Here we take h = (JD)i0. : ID)m. 39, we note that there exists aTE a(JD) = {j(J) such that mET.

11. Property M. Kaerkes and Radermacher (1977) called a Sperner collection P on N maximal if there does not exist a subset of N not in P which is c-incomparable to all elements of P. Equivalently P is called maximal if and only if for each S E 2N - P there exists some PEP such that either S C P or S :J P. This provides the motivation for the definition of property M for simple games. We say a simple game A on N possesses property M whenever the collection a(A) of minimal winning coalitions is maximal.

In fact he rediscovered some of Isbell's results and also generalized a few of them. 72 and the related results are due to him. 2 a simple form of a structure function. The proof of the uniqueness of the simple form is due to Owen (1972). The multilinear extension of a structure function introduced by Owen(1972) in the game theoretical context is nothing but the reliability function. 5, contains also, a number of new results. 1 Introduction In Chapter 1 we saw how real life reliability systems can be modelled using semicoherent structures.

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