We introduce a new notion of tractability for multivariate problems, namely -weak tractability for positive and . This allows us to study the information complexity of a -variate problem with respect to different powers of and the bits of accuracy . We consider the worst case error for the absolute and normalized error criteria. We provide necessary and sufficient conditions for -weak tractability for general linear problems and linear tensor product problems defined over Hilbert spaces. In particular, we show that non-trivial linear tensor product problems cannot be -weakly tractable when and . On the other hand, they are -weakly tractable for and if the univariate eigenvalues of the linear tensor product problem enjoy a polynomial decay. Finally, we study -weak tractability for the remaining combinations of the values of and .