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 $α$-loss, a class of tunable loss functions parameterized by $α$. It has been shown before that this loss function captures well known loss functions including the exponential loss ($α=1/2$), the log-loss ($α=1$) and the $0$-$1$ loss ($α=\infty$). We completely characterize the optimal adversarial strategy and the resulting expected $α$-loss, thereby recovering known results for $α=\infty$. 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.