The paper identifies a necessary and sufficient condition for a deterministic local optimum to be locally improved upon by a stochastic deviation. When this condition is satisfied, a method to construct the stochastic allocations that increase the objective is provided. This technique is applied to a number of adverse selection and moral hazard problems.
View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00969344 Published, Journal of Economic Theory, 2014, 151, 493-507