Emerging cyber-physical systems (CPS) often require collecting end users' data to support data-informed decision making processes. There has been a long-standing argument as to the tradeoff between privacy and data utility. In this paper, we adopt a multiparametric programming approach to rigorously study conditions under which data utility has to be sacrificed to protect privacy and situations where free-lunch privacy can be achieved, i.e., data can be concealed without hurting the optimality of the decision making underlying the CPS. We formalize the concept of free-lunch privacy, and establish various results on its existence, geometry, as well as efficient computation methods. We propose the free-lunch privacy mechanism, which is a pragmatic mechanism that exploits free-lunch privacy if it exists with the constant guarantee of optimal usage of data. We study the resilience of this mechanism against attacks that attempt to infer the parameter of a user's data generating process. We close the paper by a case study on occupancy-adaptive smart home temperature control to demonstrate the efficacy of the mechanism.