This note first uses a simple example to show that H. Matsushima's (1991, J. Econ. Theory54, 198–203) regularity condition, to the contrary of his claim, actually does not imply C. d'Aspremont and L.-A. Gérard-Varet's (1979, J. Public Econ.11, 25–45) compatibility condition. It then proves a stronger version of Matsushima's proposition, namely that efficient public decision rules can be truthfully implemented with budget-balancing mechanisms under the weak regularity condition. Journal of Economic Literature Classification Numbers: C72, D82.