Consider the problem of exact nash implementation of social choice correspondences. Define a lottery mechanism as a mechanism in which the planner can randomize on alternatives out of equilibrium while pure alternatives are always chosen in equilibrium. When preferences over alternatives are strict, we show that maskin monotonicity (maskin in rev econ stud 66: 23–38, 1999) is both necessary and sufficient for a social choice correspondence to be nash implementable. We discuss how to relax the assumption of strict preferences. Next, we examine social choice correspondences with private components. Finally, we apply our method to the issue of voluntary implementation (jackon and palfrey in j econ theory 98: 1–25, 2001).