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The interactions of a group of non-identical voting units may be studied by applying the concept of the Shapley value from n-person co-operative game theory. In this theory one assumes that voting units, or players, may form coalitions of various kinds in order to achieve success in the game, and one may assign a "value" to each such coalition based on what it can accomplish against arbitrary coalitions of the remaining players. The relative value of an individual player is calculated by considering how much he brings to each coalition he might join, that is, by how much the value of that coalition increases because of his membership, and summing this figure over all coalitions of which he could be a part.

AbstractOne–sided confidence regions for continuous cumulative distribution functions are constructed using empirical cumulative distribution functions and the generalized Kolmogorov–Smimov distance. The band width of such regions becomes narrower in the right or left tail of the distribution. To avoid tedious computation of confidence levels and critical values, an approximation based on the Poisson process is introduced. This approximation provides a conservative confidence region; moreover, the approximation error decreases monotonically to 0 as sample size increases. Critical values necessary for implementation are given. Applications are made to the areas of risk analysis, investment modelling, and analysis of fault–tolerant systems.