Constitutional implementation

Abstract.

We consider the problem of implementing a social choice correspondence H in Nash equilibrium when the constitution of the society is given by an effectivity function E. It is assumed that the effectivity function of $H,E^{H}$, is a sub-correspondence of E. We found necessary and efficient conditions for a game form $\Gamma$ to implement H (in Nash equilibria), and to satisfy, at the same time, that $E^{\Gamma}$, the effectivity function of $\Gamma$, is a sub-correspondence of $E^{H}$ (which guarantees that $\Gamma$ is compatible with E). We also find sufficient conditions for the coincidence of the set of winning coalitions of $E^{\Gamma}$ and $E^{H}$, and for $E^{\Gamma}=E^{H}$. All our results are sharp as is shown by suitable examples.