Abstract: In differentially private (DP) machine learning, the privacy guarantees of DP mechanisms are often reported and compared on the basis of a single $(\varepsilon, \delta)$-pair. This practise overlooks that DP guarantees can vary substantially even between mechanisms sharing a given $(\varepsilon, \delta)$, and potentially introduces privacy vulnerabilities which can remain undetected.This motivates the need for robust, rigorous methods for comparing DP guarantees in such cases.Here, we introduce the $\Delta$-divergence between mechanisms which quantifies the worst-case excess privacy vulnerability of choosing one mechanism over another in terms of $(\varepsilon, \delta)$, $f$-DP and in terms of a newly presented Bayesian interpretation.Moreover, as a generalisation of the Blackwell theorem, it is endowed with strong decision-theoretic foundations.Through application examples, we show that our techniques can facilitate informed decision-making and reveal gaps in the current understanding of privacy risks, as current practices in DP-SGD often result in choosing mechanisms with high excess privacy vulnerabilities.