This paper identifies several social choice correspondences which are or are not fully implementable in economic environments when agents are incompletely informed about the environment. We show that in contrast to results in the case of complete information, neither efficient allocations nor core allocations define implementable social choice correspondences. We also identify conditions under which the Rational Expectations Equilibrium correspondence is implementable. We extend the concepts of fair allocations and Lindahl allocations to economies with incomplete information, and show that envy-free allocations and Lindahl allocations are implementable under some conditions while fair allocations are not.