  We study privacy-utility trade-offs where users share privacy-correlated
useful information with a service provider to obtain some utility. The service
provider is adversarial in the sense that it can infer the users' private
information based on the shared useful information. To minimize the privacy
leakage while maintaining a desired level of utility, the users carefully
perturb the useful information via a probabilistic privacy mapping before
sharing it. We focus on the setting in which the adversary attempting an
inference attack on the users' privacy has potentially biased information about
the statistical correlation between the private and useful variables. This
information asymmetry between the users and the limited adversary leads to
better privacy guarantees than the case of the omniscient adversary under the
same utility requirement. We first identify assumptions on the adversary's
information so that the inference costs are well-defined and finite. Then, we
characterize the impact of the information asymmetry and show that it increases
the inference costs for the adversary. We further formulate the design of the
privacy mapping against a limited adversary using a difference of convex
functions program and solve it via the concave-convex procedure. When the
adversary's information is not precisely available, we adopt a Bayesian view
and represent the adversary's information by a probability distribution. In
this case, the expected cost for the adversary does not admit a closed-form
expression, and we establish and maximize a lower bound of the expected cost.
We provide a numerical example regarding a census data set to illustrate the
theoretical results.
