  Differential privacy has been widely applied to provide privacy guarantees by
adding random noise to the function output. However, it inevitably fails in
many high-stakes voting scenarios, where voting rules are required to be
deterministic. In this work, we present the first framework for answering the
question: "How private are commonly-used voting rules?" Our answers are
two-fold. First, we show that deterministic voting rules provide sufficient
privacy in the sense of distributional differential privacy (DDP). We show that
assuming the adversarial observer has uncertainty about individual votes, even
publishing the histogram of votes achieves good DDP. Second, we introduce the
notion of exact privacy to compare the privacy preserved in various
commonly-studied voting rules, and obtain dichotomy theorems of exact DDP
within a large subset of voting rules called generalized scoring rules.
