Full-truthful implementation in Nash equilibria

We consider full-truthfulNash implementation, which requires that truth telling by each agent should be a Nash equilibrium of a direct revelation mechanism, and that the set of Nash equilibriumoutcomes of the mechanismshould coincide with the f-optimal outcome. We showthat restricted monotonicity together with an auxiliary condition called boundedness is both necessary and sufficient for full-truthful Nash implementation. We also prove that full-truthful Nash implementation is equivalent to secure implementation (Saijo et al. (2005)). This gives us an alternative characterization of securely implementable social choice functions.