Evaluating Multiple Guesses by an Adversary via a Tunable Loss Function

We consider a problem of guessing, wherein an adversary is interested in knowing the value of the realization of a discrete random variable $X$ on observing another correlated random variable Y. The adversary can make multiple (say, k) guesses. The adversary's guessing strategy is assumed to minimize a-loss, a class of tunable loss functions parameterized by a. It has been shown before that this loss function captures well known loss functions including the exponential loss (a = 1/2), the log-loss (a = 1) and the 0–1 loss (a = ∞). We completely characterize the optimal adversarial strategy and the resulting expected α-loss, thereby recovering known results for a = ∞. We define an information leakage measure from the k-guesses setup and derive a condition under which the leakage is unchanged from a single guess.