  We study the privacy-utility trade-off in the context of metric differential
privacy. Ghosh et al. introduced the idea of universal optimality to
characterise the best mechanism for a certain query that simultaneously
satisfies (a fixed) $\epsilon$-differential privacy constraint whilst at the
same time providing better utility compared to any other
$\epsilon$-differentially private mechanism for the same query. They showed
that the Geometric mechanism is "universally optimal" for the class of counting
queries. On the other hand, Brenner and Nissim showed that outside the space of
counting queries, and for the Bayes risk loss function, no such universally
optimal mechanisms exist. In this paper we use metric differential privacy and
quantitative information flow as the fundamental principle for studying
universal optimality. Metric differential privacy is a generalisation of both
standard (i.e., central) differential privacy and local differential privacy,
and it is increasingly being used in various application domains, for instance
in location privacy and in privacy preserving machine learning. Using this
framework we are able to clarify Nissim and Brenner's negative results, showing
(a) that in fact all privacy types contain optimal mechanisms relative to
certain kinds of non-trivial loss functions, and (b) extending and generalising
their negative results beyond Bayes risk specifically to a wide class of
non-trivial loss functions. We also propose weaker universal benchmarks of
utility called "privacy type capacities". We show that such capacities always
exist and can be computed using a convex optimisation algorithm.
