We consider the problem of conducting a survey with the goal of obtaining an unbiased estimator of some population statistic when individuals have unknown costs (drawn from a known prior) for participating in the survey. Individuals must be compensated for their participation and are strategic agents, and so the payment scheme must incentivize truthful behavior. We derive optimal truthful mechanisms for this problem for the two goals of minimizing the variance of the estimator given a fixed budget, and minimizing the expected cost of the survey given a fixed variance goal.
In this paper, we consider the problem of estimating a potentially sensitive (individually stigmatizing) statistic on a population. In our model, individuals are concerned about their privacy, and experience some cost as a function of their privacy loss. Nevertheless, they would be willing to participate in the survey if they were compensated for their privacy cost. These cost functions are not publicly known, however, nor do we make Bayesian assumptions about their form or distribution. Individuals are rational and will misreport their costs for privacy if doing so is in their best interest. Ghosh and Roth recently showed in this setting, when costs for privacy loss may be correlated with private types, if individuals value differential privacy, no individually rational direct revelation mechanism can compute any non-trivial estimate of the population statistic. In this paper, we circumvent this impossibility result by proposing a modified notion of how individuals experience cost as a function of their privacy loss, and by giving a mechanism which does not operate by direct revelation. Instead, our mechanism has the ability to randomly approach individuals from a population and offer them a take-it-or-leave-it offer. This is intended to model the abilities of a surveyor who may stand on a street corner and approach passers-by. We initiate the study of markets for private data, through the lens of differential privacy. Although the purchase and sale of private data has already begun on a large scale, a theory of privacy as a commodity is missing. In this paper, we propose to build such a theory. Specifically, we consider a setting in which a data analyst wishes to buy information from a population from which he can estimate some statistic. The analyst wishes to obtain an accurate estimate cheaply, while the owners of the private data experience some cost for their loss of privacy, and must be compensated for this loss. Agents are selfish, and wish to maximize their profit, so our goal is to design truthful mechanisms. Our main result is that such problems can naturally be viewed and optimally solved as variants of multi-unit procurement auctions. Based on this result, we derive auctions which are optimal up to small constant factors for two natural settings: When the data analyst has a fixed accuracy goal, we show that an application of the classic Vickrey auction achieves the analyst's accuracy goal while minimizing his total payment. When the data analyst has a fixed budget, we give a mechanism which maximizes the accuracy of the resulting estimate while guaranteeing that the resulting sum payments do not exceed the analyst's budget. In both cases, our comparison class is the set of envy-free mechanisms, which correspond to the natural class of fixed-price mechanisms in our setting. In both of these results, we ignore the privacy cost due to possible correlations between an individual's private data and his valuation for privacy itself. We then show that generically, no individually rational mechanism can compensate individuals for the privacy loss incurred due to their reported valuations for privacy. This is nevertheless an important issue, and modeling it correctly is one of the many exciting directions for future work.