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 Γ to implement H (in Nash equilibria), and to satisfy, at the same time, that EΓ, the effectivity function of Γ, is a sub-correspondence of EH (which guarantees that Γ is compatible with E). We also find sufficient conditions for the coincidence of the set of winning coalitions of EΓ and E H, and for EΓ = EH. All our results are sharp as is shown by suitable examples.
- Effectivity function
- Nash equilibrium