Ultra Efficient Contracts: Pushing the Boundaries of Tractable Contract Design

TL;DR

This paper introduces a polynomial-time algorithm for optimal contract design under extsc{Ultra} rewards, expanding tractability boundaries in combinatorial action models and bridging the gap between substitutes and complements regimes.

Contribution

It presents the first polynomial-time algorithm for optimal contracts with extsc{Ultra} rewards, a class that extends beyond gross substitutes and includes non-subadditive rewards.

Findings

Polynomial-time algorithm for extsc{Ultra} rewards.

Bridging the gap between substitutes and complements regimes.

Extension to non-additive cost functions.

Abstract

We study the optimal contract problem in the \emph{combinatorial actions} framework of D\"utting et al.~[FOCS'21], where a principal delegates a project to an agent who chooses a subset of hidden, costly actions, and the resulting reward is given by a monotone set function over the actions. The principal offers a contract that specifies the fraction of the reward the agent receives, and the goal is to compute a contract that maximizes the principal's expected utility. Prior work established polynomial-time algorithms for \emph{gross substitutes} rewards, while showing NP-hardness for general submodular rewards; subsequent work extended tractability to \emph{supermodular} rewards, demonstrating that tractable cases exist in both the substitutes and complements regimes. This left open the precise boundary of tractability for the optimal contract problem. Our main result is a polynomial-time algorithm for the optimal contract problem under \Ultra\ rewards, a class that strictly contains gross substitutes but is not confined to subadditive rewards, thereby bridging the substitutes and complements regimes. We further extend our results beyond additive costs, establishing a polynomial-time algorithm for \Ultra\ rewards and cost functions that are the sum of additive and symmetric functions. To the best of our knowledge, this is the first application of \Ultra\ functions in a prominent economic setting.