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In the last years we have witnessed the appearance of a variety of strategies to design optimal location privacy-preserving mechanisms, in terms of maximizing the adversary's expected error with respect to the users' whereabouts. In this work, we take a closer look at the defenses created by these strategies and show that, even though they are indeed optimal in terms of adversary's correctness, not all of them offer the same protection when looking at other dimensions of privacy. To avoid "bad" choices, we argue that the search for optimal mechanisms must be guided by complementary criteria. We provide two example auxiliary metrics that help in this regard: the conditional entropy, that captures an information-theoretic aspect of the problem; and the worst-case quality loss, that ensures that the output of the mechanism always provides a minimum utility to the users. We describe a new mechanism that maximizes …