Generalized Principal-Agent Problem with a Learning Agent

Abstract

In classic principal-agent problems such as Stackelberg games, contract design, and Bayesian persuasion, the agent best responds to the principal's committed strategy. We study repeated generalized principal-agent problems under the assumption that the principal does not have commitment power and the agent uses algorithms to learn to respond to the principal. We reduce this problem to a one-shot problem where the agent approximately best responds, and prove that: (1) If the agent uses contextual no-regret learning algorithms with regret $\mathrm{Reg}(T)$, then the principal can guarantee utility at least $U^{\ast }-\Theta \left(\sqrt{\frac{\mathrm{Reg}(T)}{T}}\right)$, where $U^{\ast }$ is the principal's optimal utility in the classic model with a best-responding agent. (2) If the agent uses contextual no-swap-regret learning algorithms with swap-regret $\mathrm{SReg}(T)$, then the principal cannot obtain utility more than $U^{\ast }+O(\frac{\mathrm{SReg}(\mathrm{T})}{T})$. (3) In addition, if the agent uses mean-based learning algorithms (which can be no-regret but not no-swap-regret), then the principal can sometimes do significantly better than $U^{\ast }$. These results not only refine previous works on Stackelberg games and contract design, but also lead to new results for Bayesian persuasion with a learning agent and all generalized principal-agent problems where the agent does not have private information.

arXiv commentA short version of this work appeared on ICLR 2025 (spotlight). This full version has been accepted by Quantitative Economics