Implementation via mechanisms with transfers

We consider the following problem: suppose that the social choice rule is given, but the planner has the opportunity to redistribute among agents some numeraire commodity ("money") by compensatory transfers; can the planner find a transfer such that the social choice rule is Nash implementable by the transfer? In this paper, we establish a necessary and sufficient condition for Nash implementation by transfers. Furthermore, we construct a simplified mechanism, which Nash-implements the desired social choice rule by transfers.