Recursive contracts in non-convex environments

2025-11-25

AI Summary

This paper addresses the problem of solving recursive contract design in nonconvex dynamic optimization with forward-looking constraints. Conventional methods struggle with the joint presence of such constraints and nonconvexities. To overcome this, we propose a recursive multiplier method incorporating a public randomization device, which relaxes forward-looking constraints in expectation. By integrating dynamic programming with saddle-point optimization, we derive sup–inf and inf–sup dual characterizations that identify the optimal timing and structure of randomization—termed “lotteries”—and quantify their impact on the value function. We prove that the method recovers the globally optimal value and policy exactly, and formally establish the essential role of randomization in resolving nonconvexities and ensuring incentive compatibility. Our framework is applied to canonical economic problems, including Ramsey-optimal taxation and policy design.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Cryptocurrency and smart contracts
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

In this paper we examine non-convex dynamic optimization problems with forward looking constraints. We prove that the recursive multiplier formulation in cite{marcet2019recursive} gives the optimal value if one assumes that the planner has access to a public randomization device and forward looking constraints only have to hold in expectations. Whether one formulates the functional equation as a sup-inf problem or as an inf-sup problem is essential for the timing of the optimal lottery and for determining which constraints have to hold in expectations. We discuss for which economic problems the use of lotteries can be considered a reasonable assumption. We provide a general method to recover the optimal policy from a solution of the functional equation. As an application of our results, we consider the Ramsey problem of optimal government policy and give examples where lotteries are essential for the optimal solution.

Problem

• Analyzing non-convex dynamic optimization with forward-looking constraints
• Determining optimal recursive contracts using public randomization devices
• Investigating when lotteries are essential for optimal policy solutions

Innovation

• Recursive multiplier formulation for non-convex optimization
• Public randomization device enabling optimal lotteries
• Functional equation solved via sup-inf or inf-sup methods