Virtual Implementation by Bounded Mechanisms : Complete Information

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 equilibrium outcomes at $\theta$ is $\epsilon $--closed to the socially desirable set $F(\theta)$. Under a domain restriction, we obtain the following result: When there are at least three agents, any $F$ is virtually implementable in Nash equilibrium, as well as in rationalizable strategies, by a bounded mechanism. No ``tail-chasing" construction, common in the constructive proofs of the literature, is used to assure that undesired strategy combinations do not form a Nash equilibrium