Incentive design for adaptive agents

We consider a setting in which a principal seeks to induce an adaptive agent to select a target action by providing incentives on one or more actions. The agent maintains a belief about the value for each action---which may update based on experience---and selects at each time step the action with the maximal sum of value and associated incentive. The principal observes the agent's selection, but has no information about the agent's current beliefs or belief update process. For inducing the target action as soon as possible, or as often as possible over a fixed time period, it is optimal for a principal with a per-period budget to assign the budget to the target action and wait for the agent to want to make that choice. But with an across-period budget, no algorithm can provide good performance on all instances without knowledge of the agent's update process, except in the particular case in which the goal is to induce the agent to select the target action once. We demonstrate ways to overcome this strong negative result with knowledge about the agent's beliefs, by providing a tractable algorithm for solving the offline problem when the principal has perfect knowledge, and an analytical solution for an instance of the problem in which partial knowledge is available.