This paper proposes a conceptual framework for the analysis of reward sharing schemes in mining pools, such as those associated with Bitcoin. The framework is centered around the reported shares in a pool instead of agents and results in two new fairness criteria, absolute and relative redistribution. These criteria impose that the addition of a share to the pool affects all previous shares in the same way, either in absolute amount or in relative ratio. We characterize two large classes of economically viable reward sharing schemes corresponding to each of these fairness criteria in turn. We further show that the intersection of these classes brings about a generalization of the well-known proportional scheme, which also leads to a new characterization of the proportional scheme as a corollary.