Double Implementation in Dominant Strategy Equilibria and Ex Post Equilibria with Private Values

We consider the implementation problem under incomplete information and private values. We investigate double implementability of social choice functions in dominant strategy equilibria and ex post equilibria. We show that the notion of an ex post equilibrium is weaker than the notion of a dominant strategy equilibrium. Then, this double implementability notion is not trivial even under private values. We define a new strategic axiom that is stronger than “strategy-proofness,” but weaker than “secure strategy-proofness.” We call it “weak secure-strategyproofness.” We show that a social choice function is doubly implementable if and only if it is weakly securely-strategy-proof