Randomized response methods allow different kinds and degrees of privacy depending upon models and parameters. A theoretical approach to efficiency is suggested based on minimizing variance for given privacy requirements. Privacy requirements are defined in units of jeopardy from different points of view and indexed by measures suggested by information theory. Variances of competing models are compared only when the procedures meet the privacy requirements in terms of not exceeding the maximal jeopardizing information allowable for the particular application. The approach is illustrated through developing minimum variance under given jeopardizing information restrictions for a general dichotomous-population-dichotomous-response model.