Knapsack Contracts and the Importance of Return-on-Investment

We formulate the Knapsack Contracts problem -- a strategic version of the classic Stochastic Knapsack problem, which builds upon the inherent randomness shared by stochastic optimization and contract design. In this problem, the principal incentivizes agents to perform jobs with stochastic processing times, the realization of which depends on the agents' efforts. Algorithmically, we show that Knapsack Contracts can be viewed as Stochastic Knapsack with costs and multi-choice, features that introduce significant new challenges. We identify a crucial and economically meaningful parameter -- the Return on Investment (ROI) value. We show that the Inverse of ROI (or IOR for short) precisely characterizes the extent to which the approximation guarantees for Stochastic Knapsack extend to its strategic counterpart. For IOR of $α$, we develop an algorithm that finds an $O(α)$-approximation policy that does not rely on adaptivity. We establish matching $Ω(α)$ lower bounds, both on the adaptivity gap, and on what can be achieved without full distributional knowledge of the processing times. Taken together, our results show that IOR is fundamental to understanding the complexity and approximability of Knapsack Contracts, and bounding it is both necessary and sufficient for achieving non-trivial approximation guarantees. Our results highlight the computational challenges arising when stochasticity in optimization problems is controlled by strategic effort.