Contracting with a Learning Agent

Many real-life contractual relations differ completely from the clean, static model at the heart of principal-agent theory. Typically, they involve repeated strategic interactions of the principal and agent, taking place under uncertainty and over time. While appealing in theory, players seldom use complex dynamic strategies in practice, often preferring to circumvent complexity and approach uncertainty through learning. We initiate the study of repeated contracts with a learning agent, focusing on agents who achieve no-regret outcomes. Optimizing against a no-regret agent is a known open problem in general games; we achieve an optimal solution to this problem for a canonical contract setting, in which the agent's choice among multiple actions leads to success/failure. The solution has a surprisingly simple structure: for some $α> 0$, initially offer the agent a linear contract with scalar $α$, then switch to offering a linear contract with scalar $0$. This switch causes the agent to ``free-fall'' through their action space and during this time provides the principal with non-zero reward at zero cost. Despite apparent exploitation of the agent, this dynamic contract can leave \emph{both} players better off compared to the best static contract. Our results generalize beyond success/failure, to arbitrary non-linear contracts which the principal rescales dynamically. Finally, we quantify the dependence of our results on knowledge of the time horizon, and are the first to address this consideration in the study of strategizing against learning agents.

Paper

Similar papers

Peer review

Reviewer 8ua36/10 · confidence 3/52024-07-05

Summary

This theoretical paper studies repeated principal-agent contracts where the agent uses no-regret learning algorithms rather than complex strategic reasoning. The main results characterize optimal dynamic contracts against mean-based learning agents: For linear contracts (including success/failure settings), the optimal dynamic contract has a simple "free-fall" structure - offer a carefully designed contract for some fraction of time, then switch to paying nothing. This can be computed efficiently. There exist settings where both principal and agent benefit from the optimal dynamic contract compared to the best static contract. With uncertainty about the time horizon, the principal's ability to outperform static contracts degrades as the uncertainty increases.

Strengths

1. Novel and interesting problem formulation, bridging contract theory and online learning 2. Clean theoretical results with full proofs provided 3. Careful analysis of both linear and general contract settings 4. Considers practical issues like unknown time horizons 5. Results provide interesting insights, e.g. potential for "win-win" dynamic contracts

Weaknesses

Limited to mean-based learning agents; Optimal contracts for fully general (non-linear) settings not analyzed

Questions

1. Are there natural economic settings where the "win-win" dynamic contracts might arise in practice? 2. Do you expect qualitatively similar results for multiple interacting agents? What are the key challenges there? 3. How do you expect the results to change if the agent uses a more sophisticated no-regret algorithm, such as one with bounded memory or one that is aware of the principal's strategy? 4. The paper focuses on optimizing the principal's utility. How would the analysis change if we consider Pareto-optimal contracts that balance utilities between the principal and agent? 5. Are there any interesting implications of your results for the design of real-world incentive structures, such as employee compensation plans or insurance contracts? 6. Your results show that dynamic contracts can sometimes benefit both parties. Are there conditions under which this is guaranteed, or conversely, conditions under which it's impossible? 7. The paper mentions potential extensions to MDPs. How do you envision applying these ideas to more complex sequential decision-making settings?

Rating

6

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

Yes

Reviewer caMQ7/10 · confidence 3/52024-07-11

Summary

The paper studies the repeated interaction between a principal and a learning agent. In particular, the authors assume that the agent employes a mean-based learning algorithm. The goal is to design a sequence of contracts that maximizes the principal’s cumulative utility. The main result of the paper is to show that in binary outcome settings the optimal strategy for the principal is to employ a “free-fall” contract in which: the principal commits to the same contract for some rounds and than switch to the zero contract. This result generalizes to settings with more than two outcomes in which the principal restricts to use linear contracts. The second main result of the paper regards the uncertainty over the time horizon $T$. The authors show that this uncertainty hurts the effectiveness of dynamic contracts, characterizing the performance of optimal dynamic contracts.

Strengths

The paper introduces a new interesting problem and provides interesting results. The techniques are novel and non-trivial. The paper is well-written.

Weaknesses

I’ve some doubts on the realism of a model in which the agent is a no-regret minimizer, and not a swap regret minimizer. Nonetheless, this difference is clearly analyzed in the paper, and I do believe that the study of this setting is important.

Questions

None.

Rating

7

Confidence

3

Soundness

4

Presentation

4

Contribution

3

Limitations

None.

Reviewer x3aE6/10 · confidence 4/52024-07-13

Summary

This paper considers the problem of contract design against a (mean-based) no-regret agent. The papers shows several results on the optimal contract design in this dynamic setting. First, with binary outcome, dynamic linear contract, it is optimal to design a free-fall contract. Second, the paper constructs instances (with non-zero measure) where optimal dynamic contract can pareto-improve the principal and agent utility than the optimal static contract. The paper also extends some of the results to the case of non-binary outcomes and unknown time horizon.

Strengths

The paper is super well-written and easy to follow. It clearly explains the problem and the solution, as well as their relation to various lines of prior work. The examples and figures in these papers are also very carefully constructed to explain intuitions of their results. These readability optimizations help us a lot to get a quick and deep understanding of the paper.

Weaknesses

While I think the paper is well-executed from its writing to technical derivation, the problem setting and the result it derives seems to me a bit artificial. The optimality of "free-fall contract" does not make any economic sense to me, but rather an unrealistic exploitation of the agent's no-regret learning algorithm. I can be wrong here, but I cannot think of any realistic situation in practice where anything similar to the free-fall contract is implemented. Maybe something loosely related: the quality of many restaurants, hotels over time can degrade after they establish a good reputation. This is because they can exploit the reputation to attract customers and reduce the quality to cut the cost. This is a kind of "free-fall contract" in a sense, but it is not sustainable (unless time horizon T is known to be finite like your setup). In general, I would say the mean-based regret model may not be a good model to capture the real-world learning agents, which prevents the unrealistic yet optimal solution of "free-fall contract": Perhaps they are not no-regret agents but admits a strong discounting factor to the historical reward in the regret notion so that they are quickly aware of the change in the contract (due to the distribution shift in the received payment) and adapt their response to it. The last section on the case of unknown time horizons is more realistic and it indeed rules out the free-fall contract as the optimal solution, though the results are not as strong as the simpler cases. Overall, I would say the paper could have taken a much stronger stance by exploiting its economic implications.

Questions

Please address my concern in the above section if possible.

Rating

6

Confidence

4

Soundness

3

Presentation

4

Contribution

3

Limitations

n/a

Reviewer HskF7/10 · confidence 4/52024-07-13

Summary

This work studies the problem of contracting with a no-regret learning agent. They show that - In linear contracts, the optimal dynamic contract against a mean-based learning agent is a free-fall contract. - dynamic contracts can be win-win. Both principal and agent can benefit from some dynamic contractor and mean-based learning agent. - Knowing time horizon is very important. They provide a lower bound showing that there exists a problem in which no dynamic strategies outperform the optimal static linear contract.

Strengths

The problem studied by this work is very interesting --- repeated contracts with learning agents. The results are novel. There are some interesting findings in this work, including the optimality of free-fall contract, the existence of win-win scenarios, and the impact of the knowledge time horizon. The writing is clear.

Weaknesses

If I have to say some weaknesses of this work, I would say most results seem to be instance-dependent but not general.

Questions

- In line 218-226, they introduced full feedback and bandit feedback. I am a bit confused here about bandit feedback. Does this mean that the agent won't observe the chosen contract $p_t$ at each round? - In Thm 3.2, they show that there exists a problem in which optimal dynamic contracts lead to "win-win" for both principal and agent. I am curious if there are scenarios in which agents get hurt by running learning algorithms. - For Thm 4.3, in the construction for proving the theorem, is $(\epsilon, 1)$ feasible? I think they should mention this to justify that the infeasibility is indeed caused by the uncertainty of time horizon but not that the game itself is infeasible.

Rating

7

Confidence

4

Soundness

4

Presentation

4

Contribution

3

Limitations

NA

Reviewer 8ua32024-08-12

Thanks for the response

Thanks for the response. I would keep my score for accepting this paper.

Program Chairsdecision2024-09-25

Decision

Accept (poster)

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