Constrained implementation

Consider a society with two sectors (issues or objects) that faces a design problem. Suppose that the sector-2 dimension of the design problem is fixed and represented by a mechanism , and that the designer operates under this constraint for institutional reasons. A sector-1 mechanism constrained implements a social choice rule φ in Nash equilibrium if for each profile of agents' preferences, the set of (pure) Nash equilibrium outcomes of the mechanism played by agents with those preferences always coincides with the recommendations made by φ for that profile. If this mechanism design exercise could be accomplished, φ would be constrained implementable. We show that constrained monotonicity, a strengthening of (Maskin) monotonicity, is a necessary condition for constrained implementation. When there are more than two agents, and when the designer can use the private information elicited from agents via to make a socially optimal decision for sector 1, constrained monotonicity, combined with an auxiliary condition, is sufficient. This sufficiency result does not rule out any kind of complementarity between the two sectors.