The Permute-and-Flip Mechanism is Identical to Report-Noisy-Max with Exponential Noise

The permute-and-flip mechanism is a recently proposed differentially private selection algorithm that was shown to outperform the exponential mechanism. In this paper, we show that permute-and-flip is equivalent to the well-known report noisy max algorithm with exponential noise. This equivalence is used to simplify the utility and privacy analysis, and extend it to bounded range and concentrated differential privacy.