On the (In)approximability of Combinatorial Contracts

We study two recent combinatorial contract design models, which highlight different sources of complexity that may arise in contract design, where a principal delegates the execution of a costly project to others. In both settings, the principal cannot observe the choices of the agent(s), only the project s outcome (success or failure), and incentivizes the agent(s) using a contract, a payment scheme that specifies the payment to the agent(s) upon a project s success. We present results that resolve open problems and advance our understanding of the computational complexity of both settings. In the multi-Agent setting, the project is delegated to a team of agents, where each agent chooses whether or not to exert effort. A success probability function maps any subset of agents who exert effort to a probability of the project s success. For the family of submodular success probability functions, Dötting et al. [2023] established a poly-Time constant factor approximation to the optimal contract, and left open whether this problem admits a PTAS. We answer this question on the negative, by showing that no poly-Time algorithm guarantees a better than 0.7-Approximation to the optimal contract. For XOS functions, they give a poly-Time constant approximation with value and demand queries. We show that with value queries only, one cannot get any constant approximation. In the multi-Action setting, the project is delegated to a single agent, who can take any subset of a given set of actions. Here, a success probability function maps any subset of actions to a probability of the project s success. Dötting et al. [2021a] showed a poly-Time algorithm for computing an optimal contract for gross substitutes success probability functions, and showed that the problem is NP-hard for submodular functions. We further strengthen this hardness result by showing that this problem does not admit any constant factor approximation. Furthermore, for the broader class of XOS functions, we establish the hardness of obtaining a n^1/2+-Approximation for any < 0.