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A previous work by Friedman et al. (Theory and Decision, 61:305–318, 2006) introduces the concept of a hierarchy of a simple voting game and characterizes which hierarchies, induced by the desirability relation, are achievable in linear games. In this paper, we consider the problem of determining all hierarchies, conserving the ordinal equivalence between the Shapley–Shubik and the Penrose–Banzhaf–Coleman\ud power indices, achievable in simple games. It is proved that only four hierarchies are\ud non-achievable in simple games.Moreover, it is also proved that all achievable hierarchies\ud are already obtainable in the class of weakly linear games. Our results prove that\ud given an arbitrary complete pre-ordering defined on a finite set with more than five elements, it is possible to construct a simple game such that the pre-ordering induced by the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices coincides with the given pre-ordering.Postprint (published version

Abstract-The first step in a reliability optimization process is to make a reliability assessment for each component in the system. If this assessment is made in a qualitative way, by grouping together components with the same reliability, and establishing a prevalence order among groups, is there a way to decide which components have the greatest Birnbaum measure without computing the exact value of this measure? In this paper, three relations between components are introduced and studied, and it is proved that they are useful for selecting the components that have the biggest effect on the system reliability in the sense of Birnbaum. An algorithm that uses the results in the paper to select these important components is also provided.Index Terms-Birnbaum's measure of reliability importance, important components, pre-ordering relation.

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