A Bandit Model for Human-Machine Decision Making with Private Information and Opacity

Applications of machine learning inform human decision makers in a broad range of tasks. The resulting problem is usually formulated in terms of a single decision maker. We argue that it should rather be described as a two-player learning problem where one player is the machine and the other the human. While both players try to optimize the final decision, the setup is often characterized by (1) the presence of private information and (2) opacity, i.e imperfect understanding between the decision makers. In the paper we prove that both properties can complicate decision making considerably. A lower bound quantifies the worst-case hardness of optimally advising a decision maker who is opaque or has access to private information. An upper bound shows that a simple coordination strategy is nearly minimax optimal. More efficient learning is possible under certain assumptions on the problem, for example that both players learn to take actions independently. Such assumptions are implicit in existing literature, for example in medical applications of machine learning, but have not been described or justified theoretically.