Distributional Robustness of Linear Contracts

TL;DR

This paper proves that linear contracts are optimal under distributional ambiguity in principal-agent problems with multiple tasks, offering a robust justification for their widespread use.

Contribution

It introduces a distributional robustness framework showing linear contracts outperform all others in worst-case scenarios, with new proof techniques and extensions to multi-party settings.

Findings

Linear contracts maximize worst-case payoff under distributional ambiguity.

Existence of self-inducing actions supports the optimality of linear contracts.

Affine contracts improve all principals' worst-case payoffs in multi-principal settings.

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.