Distributional Robustness of Linear Contracts

2026-04-27

AI Summary

This study addresses principal-agent problems under distributional ambiguity, where the principal knows only the expectation of the signal but not its full distribution, and must design a robust optimal contract against worst-case distributions. By integrating distributionally robust optimization, concave envelope analysis, and self-induced action construction, this work provides the first explanation—grounded in distributional robustness—for the pervasive optimality of linear contracts: any contract is outperformed in the worst case by some linear contract. In team production settings, non-affine contracts induce zero-effort equilibria, while closed-form approximation ratios and computational bounds are derived for homogeneous agents. The results extend to multi-principal and multi-agent environments, revealing the theoretical advantages of linear contracts under severe information constraints.

Technology Category

• Multiagent Systems: Mechanism Design
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Security and Privacy: Large-scale security measurements

Abstract

Linear contracts are ubiquitous in practice, yet optimal contract theory often prescribes complex, nonlinear structures. We provide a distributional robustness justification for linear contracts. We study a principal-agent problem where the agent exerts costly effort across multiple tasks, generating a stochastic signal upon which the principal conditions payment. The principal faces distributional ambiguity: she knows the expected signal for each effort level, but not the full distribution. She seeks a contract maximizing her worst-case payoff over all distributions consistent with this partial knowledge. Our main result shows that linear contracts are optimal for such a principal. For any contract, there exists a linear contract achieving weakly higher worst-case payoff. The proof introduces the concavification approach built around the notion of self-inducing actions; these are actions where an affine contract simultaneously induces the action as optimal and supports the concave envelope of payments from above. We show that self-inducing actions always exist as maximizers of the gap between the concave envelope and agent's cost function. We extend these results to multi-party settings. In common agency with multiple principals, we show that affine contracts improve all principals' worst-case payoffs. In team production with multiple agents, we establish a complementary necessity result: if any agent's contract is non-affine, the unique ex-post robust equilibrium is zero effort. Finally, we show that homogeneous utility and cost functions yield tractable characterizations, enabling closed-form approximation ratios and a sharp boundary between computational tractability results.

Problem

• distributional robustness
• linear contracts
• principal-agent problem
• distributional ambiguity
• worst-case payoff

Innovation

• distributional robustness
• linear contracts
• concavification
• self-inducing actions
• worst-case payoff