Every common variant of alpha-mutual information can be rewritten as the multiplicative increase in the maximal generalized mean of an adversary's gain, for suitably chosen scoring rules.
Generalized cutoff rates and R´ enyi’s inf ormation measures,
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Several Representations of $\alpha$-Mutual Information and Interpretations as Privacy Leakage Measures
Every common variant of alpha-mutual information can be rewritten as the multiplicative increase in the maximal generalized mean of an adversary's gain, for suitably chosen scoring rules.