We study the value of data privacy in a game-theoretic model of trading private data, where a data collector pur- chases private data from strategic data subjects (individu-als) through an incentive mechanism. The private data of each individual represents her knowledge about an underly- ing state, which is the information that the data collector desires to learn. Different from most of the existing work on privacy-aware surveys, our model does not assume the data collector to be trustworthy. Then, an individual takes full control of its own data privacy and reports only a privacy- preserving version of her data. In this paper, the value of ϵ units of privacy is measured by the minimum payment of all nonnegative payment mech- Anisms, under which an individual's best response at a Nash equilibrium is to report the data with a privacy level of ϵ. The higher ϵ is, the less private the reported data is. We derive lower and upper bounds on the value of privacy which are asymptotically tight as the number of data subjects be- comes large. Speciffically, the lower bound assures that it is impossible to use less amount of payment to buy ϵ units of privacy, and the upper bound is given by an achievable pay- ment mechanism that we designed. Based on these funda- mental limits, we further derive lower and upper bounds on the minimum total payment for the data collector to achieve a given learning accuracy target, and show that the total payment of the designed mechanism is at most one individ- ual's payment away from the minimum.