  We suggest generalized robustness for quantifying nonlocality and investigate
its properties by comparing it with white-noise and standard robustness
measures. As a result, we show that white-noise robustness does not fulfill
monotonicity under local operation and shared randomness, whereas the other
measures do. To compare the standard and generalized robustness measures, we
introduce the concept of inequivalence, which indicates a reversal in the order
relationship depending on the choice of monotones. From an operational
perspective, the inequivalence of monotones for resourceful objects implies the
absence of free operations that connect them. Applying this concept, we find
that standard and generalized robustness measures are inequivalent between
even- and odd-dimensional cases up to eight dimensions. This is obtained using
randomly performed CGLMP measurement settings in a maximally entangled state.
This study contributes to the resource theory of nonlocality and sheds light on
comparing monotones by using the concept of inequivalence valid for all
resource theories.
