## Abstract

I define *Weak Implementation* under incomplete information. A social choice set is weakly implementable if the set of equilibrium outcomes of some mechanism is a non-empty subset of the social choice set. Weak implementation is a more natural objective than either full or partial implementation in many cases. I show that there are social choice sets where every subset can be weakly implemented, yet the set cannot be fully implemented. I give a complete characterization of the weakly implementable social choice sets under a weak restriction on preferences. As a corollary, I show that in independent private values environments the set of interim efficient social choice functions is weakly implementable whenever it is partially implementable.

## Keywords

Implementation theory Mechanism design Game theory Full implementation## JEL Classification

C72 D71 D80 D82## Notes

### Acknowledgements

I would like to thank Eddie Dekel, Jeff Ely, Alessandro Pavan and Asher Wolinsky for their invaluable support and feedback. I would also like to thank Chris Li, Matteo Foschi, and a seminar audience at Northwestern for their helpful comments.

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