On the necessary and sufficient conditions for Nash implementation

In a recent work, Ziad (Ziad, A., 1997. A New Necessary Condition for Nash Implementation. To appear in the Journal Of Mathematical Economics) has defined a new condition for the problem of implementation of a social choice correspondence. He has formulated a family of conditions called Bk-monotonicity that are necessary for Nash implementation, where k is a natural number that indexes a particular condition. The Bk-monotonicity is in general a stronger condition than Maskin monotonicity, but is necessary for implementation. In this note we improve the algorithms proposed by Sjöström (Sjöström, T., 1991. On the necessary and sufficient conditions for Nash implementation. Soc Choice Welfare 8, 333–340) to determine a condition which is necessary and sufficient for implementation. The main result is that in some non implementability cases we do not have to repeat the operation of Sjöström algorithms up to their terminations.