Motivated by a distributed task-encoding problem, two closely related families of dependence measures are introduced. They are based on the Rényi divergence of order α and the relative α-entropy, respectively, and both reduce to the mutual information when the parameter α is one. Their properties are studied and it is shown that the first measure shares many properties with mutual information, including the data-processing inequality. The second measure does not satisfy the data-processing inequality, but it appears naturally in the context of distributed task encoding.