A social choice rule (SCR) $F$ maps preference profiles to lotteries over some finite set of outcomes. $F$ is virtually implementable in (pure and mixed) Nash equilibria provided that for all $\epsilon 0$, there exists a mechanism such that for each preference profile $\theta$, its set of Nash...

A social choice correspondence (SCC) F is implementable in rationalizable strategies provided that there exists a mechanism such that for each state θ, the support of its set of rationalizable outcomes is equal to the socially desirable set F(θ). We find that r -monotonicity is a necessary...