Distinguishing Distributions When Samples Are Strategically Transformed

Often, a principal must make a decision based on data provided by an agent. Moreover, typically, that agent has an interest in the decision that is not perfectly aligned with that of the principal. Thus, the agent may have an incentive to select from or modify the samples he obtains before sending them to the principal. In other settings, the principal may not even be able to observe samples directly; instead, she must rely on signals that the agent is able to send based on the samples that he obtains, and he will choose these signals strategically. In this paper, we give necessary and sufficient conditions for when the principal can distinguish between agents of ``good'' and ``bad'' types, when the type affects the distribution of samples that the agent has access to. We also study the computational complexity of checking these conditions. Finally, we study how many samples are needed.