Bayesian and Dominant Strategy Implementation Revisited

We consider a standard social choice environment with linear utility and one-dimensional types. We show by counterexample that, when there are at least three physical alternatives, Bayes-Nash Incentive Compatibility (BIC) and Dominant Strategy Incentive Compatibility (DIC) need no longer be equivalent. The example with three alternatives is minimal since we do obtain a general equivalence result for settings with only two social alternatives. Our negative result does not mathematically contradict the Manelli and Vincent (2010) equivalence obtained in a one-object auction setting, but it shows that BIC-DIC equivalence is only valid in restrictive environments. Our insights are based on mathematical results about the existence of monotone measures with given monotone marginals.