Nash Implementation with Lottery Mechanisms

Consider the problem of exact Nash Implementation of social choice correspondences. De\u85ne 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 prefer-ences over alternatives are strict, we show that Maskin monotonicity (Maskin, 1999) is both necessary and su ¢ cient for a social choice cor-respondence to be Nash implementable. We discuss how to relax the assumption of strict preferences. Next, we examine social choice cor-respondences with private components. Finally, we apply our method to the issue of voluntary implementation (Jackson and Palfrey, 2001)