Single-dimensional Contract Design: Efficient Algorithms and Learning
Martino Bernasconi, Matteo Castiglioni, Andrea Celli
TL;DR
This paper investigates Bayesian contract design in a single-parameter setting where an agent’s type is a single cost-per-unit-of-effort, showing that additive approximation is tractable via a PTAS, but no additive FPTAS exists unless P=NP. The authors introduce a bucketing discretization approach that groups nearby types, solving a discretized problem and robustifying the solution to achieve a $(1+O(1/k))$-approximation with poly-time complexity for constant bucket count. They also show a fundamental separation between additive and multiplicative approximations, contrasted with known reductions that preserve only multiplicative guarantees. Beyond computation, the work proves that single-dimensional contracts can be learned efficiently in online settings, obtaining sublinear regret, and provides sample-complexity bounds under bounded-density assumptions, by reducing to misspecified linear bandits and discretized type spaces. Overall, the results indicate that single-dimensional contract design is computationally and informationally easier to learn than its multi-dimensional counterpart, with practical implications for designing and learning contracts under structured uncertainty. The findings open avenues for extensions to menus of contracts and to weaker assumptions on type distributions and feedback models, enhancing our understanding of the tractability frontier in contract design under uncertainty.
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.