Generalized Principal-Agent Problem with a Learning Agent

Tao Lin, Yiling Chen

Keywords: principal-agent problems, Bayesian persuasion, no-regret learning, no-swap-regret

TL;DR: We study generalized principal-agent problem with a learning agent, which not only sharpens previous works on contract design and Stackelberg games but also leads to new results for Bayesian persuasion.

Abstract:

Generalized principal-agent problems, including Stackelberg games, contract design, and Bayesian persuasion, are a class of economic problems where an agent best responds to a 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 generalized principal-agent problem where the agent approximately best responds. Using this reduction, we show that: (1) if the agent uses contextual no-regret learning algorithms with regret $\mathrm{Reg}(T)$, then the principal can guarantee utility at least $U^* - \Theta\big(\sqrt{\tfrac{\mathrm{Reg}(T)}{T}}\big)$, where $U^*$ 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 $U^* + O(\frac{\mathrm{SReg(T)}}{T})$.
But (3) 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 $U^*$.
These results not only refine previous results in 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.

Primary Area: learning theory

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Submission Number: 7667