Persuading a Learning Agent

We study a repeated Bayesian persuasion problem (and more generally, any generalized principal-agent problem with complete information) where the principal does not have commitment power and the agent uses algorithms to learn to respond to the principal's signals. We reduce this problem to a one-shot generalized principal-agent problem with an approximately-best-responding agent. This reduction allows us to show that: if the agent uses contextual no-regret learning algorithms, then the principal can guarantee a utility that is arbitrarily close to the principal's optimal utility in the classic non-learning model with commitment; if the agent uses contextual no-swap-regret learning algorithms, then the principal cannot obtain any utility significantly more than the optimal utility in the non-learning model with commitment. The difference between the principal's obtainable utility in the learning model and the non-learning model is bounded by the agent's regret (swap-regret). If the agent uses mean-based learning algorithms (which can be no-regret but not no-swap-regret), then the principal can do significantly better than the non-learning model. These conclusions hold not only for Bayesian persuasion, but also for any generalized principal-agent problem with complete information, including Stackelberg games and contract design.