Generalized Principal-Agent Problem with a Learning Agent (2402.09721v5)

Abstract: Classic principal-agent problems such as Stackelberg games, contract design, and Bayesian persuasion, often assume that the agent is able to best respond 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 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.

Persuading a Learning Agent: A Formal Analysis of Generalized Principal-Agent Problems

This paper explores the dynamics of generalized principal-agent problems, specifically focusing on repeated Bayesian persuasion scenarios where the agent employs learning algorithms to respond to signals from a principal who lacks commitment power. The authors extend classical models by integrating notions of learning instead of presuming complete rationality from agents. They achieve this by transforming the repeated persuasion problem to a one-shot scenario with an agent who approximately best-responds, thus providing insights into how firms and agents strategize in dynamic environments without commitment.

Key Contributions and Results

1. Reduction to Approximate Best Response: The work demonstrates how a repeated Bayesian persuasion problem, or any generalized principal-agent problem with complete information, can be simplified into a one-shot problem where agents approximately best-respond. This crucial reduction enables a clear analysis of the utility that principals can secure under various constraints imposed by the agents' learning behaviors.
2. Analysis with Contextual No-Regret Learning: The paper establishes that if agents adopt contextual no-regret learning algorithms, the principal can achieve a utility arbitrarily close to what could be achieved in classical best-response settings with commitment power. The gap between the achievable utility with learning agents and non-learning models is bounded by the regret of the agents.
3. Contextual No-Swap-Regret Learning: When agents engage in contextual no-swap-regret learning, the principal's utility cannot significantly exceed the classic model's optimal utility. This suggests that no-swap-regret learning imposes stricter constraints on the principal's ability to enhance utility, as it ensures the agent does not systematically regret action switches.
4. Implications of Mean-Based Learning: The paper further identifies that if agents employ mean-based learning algorithms that are no-regret but not no-swap-regret, the principal can surpass the utilities achievable in models where the agent's learning is not considered. This finding highlights the exploitability of certain types of learning in strategic settings.
5. Applications to Stackelberg Games and Contract Design: The framework's application extends beyond Bayesian persuasion to cover scenarios such as Stackelberg games and contract design, showing versatility in addressing various strategic contexts where agents learn and adapt.

Numerical Findings and Model Implications

The numerical analysis in the paper underscores the susceptibility of classical models to assumptions regarding agent learning. Key results include formalized bounds on the principal's utility—displaying minimal increases over the optimal utility in certain learning contexts—thereby demonstrating the robustness of no-regret and no-swap-regret learning frameworks. This mathematical rigor reinforces the necessity for strategic models to evolve beyond static rationality assumptions in accommodating real-world learning behaviors.

Future Directions and Speculative Outlook

The research presented opens avenues for further exploration into dynamic and strategic information design where learning processes adapt over time. Future work could extend such models by incorporating private information or heterogeneous learning algorithms among agents, enriching the spectrum of outcomes and strategizing methodologies. Moreover, investigating the interactions between various forms of agent errors and learning dynamics holds potential for enhancing theoretical understanding and practical applicability in designing commitments and signals in multi-agent systems.

In conclusion, this work systematically integrates learning models with principal-agent dynamics, offering robust insights into the realms of persuasion and strategy in interactive settings. The paper challenges traditional rationality frameworks by foregrounding the empirical realities of learning agents, paving the path for more nuanced and applicable economic and computational models.