Combinatorial Contract Design: Recent Progress and Emerging Frontiers

Contract theory studies how a principal can incentivize agents to exert costly, unobservable effort through performance-based payments. While classical economic models provide elegant characterizations of optimal solutions, modern applications, ranging from online labor markets and healthcare to AI delegation and blockchain protocols, call for an algorithmic perspective. The challenge is no longer only which contracts induce desired behavior, but whether such contracts can be computed efficiently. This viewpoint has given rise to \emph{algorithmic contract design}, paralleling the rise of algorithmic mechanism design two decades ago. This article focuses on \emph{combinatorial contracts}, an emerging frontier within algorithmic contract design, where agents may choose among exponentially many combinations of actions, or where multiple agents must work together as a team, and the challenge lies in selecting the right composition. These models capture a wide variety of real-world contracting environments, from hospitals coordinating physicians across treatment protocols to firms hiring teams of engineers for interdependent tasks. We review three combinatorial settings: (i) a single agent choosing multiple actions, (ii) multiple agents with binary actions, and (iii) multiple agents each selecting multiple actions. For each, we highlight structural insights, algorithmic techniques, and complexity barriers. Results include tractable cases such as gross substitutes reward functions, hardness results, and approximation guarantees under value- and demand-oracle access. By charting these advances, the article maps the emerging landscape of combinatorial contract design, and highlights fundamental open questions and promising directions for future work.