We consider the (random) entropy of systems of random transformations, where the transformations are chosen from the set of generators of a ${\bf Z}^d $ action. Every such random group action is generated via a cocycle. We show that the classical definition gives unsatisfactory entropy results in the higher-dimensional case, i.e.\ when $d \geq 2$. We propose a new definition of the random entropy for random group actions which agrees with the classical definition in the one-dimensional case, and which gives satisfactory results in higher dimensions. We identify the random entropy by a concrete formula which makes it possible to compute the random entropy in certain cases. We also show that for a random group action the associated cocyle is recurrent if and only if the random entropy is zero. This implies a Polya type dichotomy for this situation