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A generalization is offered of the power index proposed in the authors' "A New Index of Power for Simple n-Person Games" (International Journal of Game Theory, 1979, 7, 2, 113-123), based on abandonment of the assumption that all minimal winning coalitions form with equal likelihood. The family of indexes is parameterized by allowing minimal winning coalitions to form in accordance with varying probability functions. These indexes are axiomatically characterized & compared with other similarly characterized indexes. A dual family of minimal blocking coalition indexes & their characterization axioms is presented. Modified HA.

Fair districting requires more than compact, contiguous equal-sized districts; namely, sets of districts should also possess certain features. Specifically, they should be neutral (treat all parties alike) and responsive to changes in votes. In order to establish the extent to which these goals can be achieved, we give a precise definition to the concept of neutrality and expand the notion of responsiveness into three characteristics: the range in actual votes over which a districting plan is responsive; the degree of responsiveness in the vicinity of the "normal vote" (i.e., competitiveness); and the constancy of the swing ratio (i.e., the rate at which vote changes yield seat changes) over a range of votes. We show that while all possible values for these features are readily attainable when considered individually, certain combinations of values cannot be achieved. Finally, we identify the nature of the compromises required and the properties that the compromises possess, and show the kinds of trade-offs that result in reasonably fair districting plans.