Sufficient conditions for Nash implementation

Given an objective for a group of three or more agents that satisfies monotonicity and no veto power, Maskin (1977) proposes a two-step procedure for constructing a game that implements the objective in Nash equilibrium. The first step specifies the strategy set of the game and three properties of the game rule that are together sufficient to insure Nash implementation of the objective. The second step is the explicit construction of a game that has these properties. An example is presented here that shows that the constructed game of the second step need not have one of the three properties of the first step, and it does in fact not Nash implement the objective in the example. The problem is attributable to restricted preferences. A solution proposed here is to appropriately expand the domain of definition of the objective. This insures that the constructed game has the properties of the game in Maskin's first step, and it therefore Nash implements the original objective.