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 $U^{\ast }-\hat{I}˜\left(\sqrt{\frac{\mathrm{Reg}(T)}{T}}\right)$, 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.

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