  We present a variational characterization for the R\'{e}nyi divergence of
order infinity. Our characterization is related to guessing: the objective
functional is a ratio of maximal expected values of a gain function applied to
the probability of correctly guessing an unknown random variable. An important
aspect of our variational characterization is that it remains agnostic to the
particular gain function considered, as long as it satisfies some regularity
conditions. Also, we define two variants of a tunable measure of information
leakage, the maximal $\alpha$-leakage, and obtain closed-form expressions for
these information measures by leveraging our variational characterization.
