Knapsack Contracts and the Importance of Return-on-Investment

2025-09-07

AI Summary

This paper studies the strategic stochastic knapsack contracting problem: a principal designs incentive contracts to induce an agent’s effort, thereby reducing the stochastic task completion time and optimizing resource allocation. The core challenge lies in jointly modeling task stochasticity, strategic effort exertion, and contract design. We introduce the inverse return-on-investment (IOR) ratio, denoted α, as a fundamental parameter characterizing problem hardness, and establish for the first time its decisive role in determining approximation performance. Assuming only partial distributional information and restricting to non-adaptive contracts, we propose an O(α)-approximation algorithm and prove a matching Ω(α) lower bound—demonstrating tightness of the approximation guarantee under these constraints. Our approach integrates stochastic optimization, mechanism design, and approximation analysis, yielding a novel paradigm and theoretical benchmark for strategic stochastic optimization under cost sensitivity and multiple selection constraints.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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.

Problem

• Strategic version of Stochastic Knapsack with agent incentives
• Handling stochastic processing times dependent on agent efforts
• Characterizing approximation guarantees through Return-on-Investment parameter

Innovation

• Introduces Knapsack Contracts with strategic stochastic optimization
• Uses Return-on-Investment parameter for approximation guarantees
• Develops non-adaptive O(α)-approximation algorithm for contracts