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Constrained implementation. (English) Zbl 1422.91233
Summary: 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 $$\Gamma^2$$, and that the designer operates under this constraint for institutional reasons. A sector-1 mechanism $$\Gamma^1$$ constrained implements a social choice rule $$\varphi$$ in Nash equilibrium if for each profile of agents’ preferences, the set of (pure) Nash equilibrium outcomes of the mechanism $$\Gamma^1\times\Gamma^2$$ played by agents with those preferences always coincides with the recommendations made by $$\varphi$$ for that profile. If this mechanism design exercise could be accomplished, $$\varphi$$ 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 $$\Gamma^2$$ 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.
##### MSC:
 91B14 Social choice 91A40 Other game-theoretic models
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##### References:
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