Relationships between Non-Bossiness and Nash Implementability

We explore the relationships between non-bossiness and Nash implementability. We provide a new domain-richness condition, weak monotonic closedness, and prove that on weakly monotonically closed domains, non-bossiness together with individual monotonicity is equivalent to monotonicity, a necessary condition for Nash implementation. The result shows an impossibility of Nash implementation in all economies except pure public goods economies, in the sense that it indicates that in all economies except pure public goods economies, it is impossible to implement bossy social choice functions in Nash equilibria, which embody the characteristics inherent in those economies.