Single-dimensional Contract Design: Efficient Algorithms and Learning

TL;DR

This paper introduces efficient algorithms for single-dimensional Bayesian contract design, providing an additive PTAS and demonstrating that learning optimal contracts is computationally easier than in multi-dimensional settings.

Contribution

It offers the first additive PTAS for single-dimensional contract design and shows that learning these contracts is more efficient than in multi-dimensional cases.

Findings

An additive PTAS exists for single-dimensional contract design.

No additive FPTAS can exist for these problems.

Optimal contracts can be learned efficiently with favorable regret and sample complexity.

Abstract

We study a Bayesian contract design problem in which a principal interacts with an unknown agent. We consider the single-parameter uncertainty model introduced by Alon et al. [2021], in which the agent's type is described by a single parameter, i.e., the cost per unit-of-effort. Despite its simplicity, several works have shown that single-dimensional contract design is not necessarily easier than its multi-dimensional counterpart in many respects. Perhaps the most surprising result is the reduction by Castiglioni et al . [2025] from multi- to single-dimensional contract design. However, their reduction preserves only multiplicative approximations, leaving open the question of whether additive approximations are easier to obtain than multiplicative ones. In this paper, we answer this question -- to some extent -- positively. In particular, we provide an additive PTAS for these problems while also ruling out the existence of an additive FPTAS. This, in turn, implies that no reduction from multi- to single-dimensional contracts can preserve additive approximations. Moreover, we show that single-dimensional contract design is fundamentally easier than its multi-dimensional counterpart from a learning perspective. Under mild assumptions, we show that optimal contracts can be learned efficiently, providing results on both regret and sample complexity.