Linear Contracts in Multitasking: Robustness, Uniformity, and Learning (2405.20642v2)

Abstract: In this work, we study the multitasking principal-agent problem. The agent performs several task for the principal, and the principal posts a contract incentivizing the agent to exert effort. The principal can observe a signal for each task, and the contract is a mapping from the space of possible signals to a payment. We study the special class of linear contracts from three perspectives: robustness, uniformity, and learning. Firstly, we show a robustness result: in an ambiguous setting when only first moment information is known, there is a linear contract maximizing the principal's payoff in a worst-case scenario. Secondly, we show a uniformity result: when the agent's cost function is homogeneous to a certain degree and the the principal's utility takes a linear form across tasks, then the optimal contract depends on the agent's cost function only through its homogeneuity degree. Thirdly, we study the problem of learning an optimal linear contract through observational data. We identify this as an measurement error model, and propose instrumental regression methods to estimate the optimal contract parameters in an offline setting, or to learn the optimal contract in an online setting.

• The paper establishes that under conditions like tasks being perfect substitutes and homogeneous cost functions, the optimal contract remains uniform across agents.
• It introduces learning algorithms using instrumental regression and a GMM estimator to estimate unknown task importance from observational data.
• Numerical experiments validate that repeated observations and agent diversity significantly reduce cumulative utility loss, achieving faster convergence rates.

Uniformity and Learning in Multitask Principal-Agent Problems

This work addresses the multitask principal-agent problem, focusing on the uniformity of optimal contracts and efficient learning mechanisms via instrumental regression. It presents conditions under which uniform contracts are optimal and introduces a GMM framework for learning optimal contracts from observational data, particularly when task importance is unknown.

Uniformity of Optimal Contracts

The paper establishes a "uniformity" result, demonstrating that under specific conditions, the optimal contract is consistent across different agents. This result hinges on two key assumptions: tasks are perfect substitutes, and the agent's cost function is homogeneous. Specifically, the optimal contract depends solely on the marginal utility of each task and the degree of homogeneity of the agent's cost function. This finding suggests that principals need not discriminate among agents to maximize their utility, promoting fairness and simplifying contract design, particularly in scenarios like franchising or e-commerce platforms. The paper highlights that when agents have a quadratic cost function ($k=2$), the 50-50 split rule emerges as the optimal linear contract.

Learning Optimal Contracts via Instrumental Regression

The work addresses the scenario where the marginal utility of each task is unknown, necessitating learning from observational data. The principal's challenge is to estimate the "importance" of each task to design an optimal contract. The problem is framed as a linear regression with measurement errors, where the total revenue is the response variable and the observed signals of each task are the covariates. The paper identifies the contract itself as a valid instrumental variable and proposes estimation and learning algorithms based on the GMM estimator. When repeated observations are available, the GMM estimator achieves faster convergence, especially when agents exhibit diversity.

Figure 1: Causal relationship between variables, illustrating the principal-agent interaction.

The authors use the framework of Stackelberg games from revealed preferences to paper similar problem settings. However, this work diverges by identifying and exploiting a "uniformity" result, enabling the application of instrumental regression with the GMM method. In contrast, prior work solves this problem via a two-stage optimization algorithm.

Algorithms for Learning Optimal Contracts

The paper introduces algorithms for both offline and online settings. In the offline setting, the GMM estimator is used to estimate the unknown parameters from a given dataset of contracts, signals, and total revenue. In the online setting, the principal interacts with agents sequentially and must balance exploration and exploitation to minimize cumulative utility loss. The work shows that a simple explore-then-commit algorithm can achieve $O(d\sqrt{T})$ cumulative utility loss, where $d$ is the number of tasks and $T$ is the number of rounds.

Exploiting Repeated Observations and Diversity

To improve learning rates, the paper considers scenarios with repeated observations and agent diversity. It demonstrates that under certain diversity conditions, a "pure exploitation" algorithm can achieve logarithmic regret. The analysis focuses on agents with diagonal quadratic cost functions and assumes that agent cost functions are drawn from an unknown distribution with a lower-bounded minimum eigenvalue. This diversity, combined with repeated observations, allows the second set of observations to serve as an instrumental variable, leading to faster convergence.

Numerical Experiments

Figure 2: Simulation results demonstrating the total loss using different instrumental variable approaches.

Numerical simulations validate the performance of the proposed learning algorithms. The experiments demonstrate that using the contract as an instrumental variable with small random perturbations achieves a cumulative loss on the order of $O(d\sqrt{T})$. Furthermore, when repeated observations are used with diverse agents, the cumulative loss is significantly reduced.

Conclusion

This work provides valuable insights into the multitask principal-agent problem by justifying the use of uniform contracts and proposing efficient learning mechanisms based on instrumental regression and the GMM estimator. The results highlight the importance of agent diversity and repeated observations in accelerating learning rates. Future research could extend the diversity condition to more general cost functions and explore alternative notions of diversity related to agent abilities in each task.