In recent years, there has been increasing awareness of the importance of formal measures of voting power and of the relevance of such measures to real life political issues. Nevertheless, existing measures have been criticized, especially because of their dependence on the unrealistic assumption that different coalitions have equal probabilities. In this paper we show that the classical problem of measuring voting power can be naturally embedded in information theory. This perspective on voting power allows us to extend measures of voting power to cases in which there are dependencies among voters. In doing so, we distinguish between two different notions of a given voter's power-'control' and 'informativeness'-corresponding, respectively, to the average uncertainty regarding the outcome of a vote that remains when all others have voted and the average uncertainty that is eliminated when only the given voter has voted. This distinction settles a number of well-known paradoxes and enables the study of voting power on the basis of actual political behavior at all levels.