  Model explanations improve the transparency of black-box machine learning
(ML) models and their decisions; however, they can also be exploited to carry
out privacy threats such as membership inference attacks (MIA). Existing works
have only analyzed MIA in a single "what if" interaction scenario between an
adversary and the target ML model; thus, it does not discern the factors
impacting the capabilities of an adversary in launching MIA in repeated
interaction settings. Additionally, these works rely on assumptions about the
adversary's knowledge of the target model's structure and, thus, do not
guarantee the optimality of the predefined threshold required to distinguish
the members from non-members. In this paper, we delve into the domain of
explanation-based threshold attacks, where the adversary endeavors to carry out
MIA attacks by leveraging the variance of explanations through iterative
interactions with the system comprising of the target ML model and its
corresponding explanation method. We model such interactions by employing a
continuous-time stochastic signaling game framework. In our framework, an
adversary plays a stopping game, interacting with the system (having imperfect
information about the type of an adversary, i.e., honest or malicious) to
obtain explanation variance information and computing an optimal threshold to
determine the membership of a datapoint accurately. First, we propose a sound
mathematical formulation to prove that such an optimal threshold exists, which
can be used to launch MIA. Then, we characterize the conditions under which a
unique Markov perfect equilibrium (or steady state) exists in this dynamic
system. By means of a comprehensive set of simulations of the proposed game
model, we assess different factors that can impact the capability of an
adversary to launch MIA in such repeated interaction settings.
