  We consider active learning under incentive compatibility constraints. The
main application of our results is to economic experiments, in which a learner
seeks to infer the parameters of a subject's preferences: for example their
attitudes towards risk, or their beliefs over uncertain events. By cleverly
adapting the experimental design, one can save on the time spent by subjects in
the laboratory, or maximize the information obtained from each subject in a
given laboratory session; but the resulting adaptive design raises
complications due to incentive compatibility. A subject in the lab may answer
questions strategically, and not truthfully, so as to steer subsequent
questions in a profitable direction.
  We analyze two standard economic problems: inference of preferences over risk
from multiple price lists, and belief elicitation in experiments on choice over
uncertainty. In the first setting, we tune a simple and fast learning algorithm
to retain certain incentive compatibility properties. In the second setting, we
provide an incentive compatible learning algorithm based on scoring rules with
query complexity that differs from obvious methods of achieving fast learning
rates only by subpolynomial factors. Thus, for these areas of application,
incentive compatibility may be achieved without paying a large sample
complexity price.
