A Lagrangian Approach to Optimal Lotteries in Non-Convex Economies

2025-04-22

AI Summary

This paper addresses the design of optimal lottery allocation mechanisms in non-convex economic models. It tackles settings involving moral hazard and hidden-type taxation under private information. The proposed method leverages Lagrangian duality to develop an efficient algorithm: it establishes, for the first time, that the saddle-point value of the non-convex lottery problem equals the optimal value of its deterministic dual—enabling iterative reconstruction of the optimal lottery by solving a sequence of tractable deterministic subproblems, thereby circumventing the exponential complexity of standard linear programming approaches. The framework integrates Lagrangian duality theory, subgradient descent, and non-convex optimization analysis. Experiments demonstrate substantial gains in computational efficiency and scalability; notably, it solves, for the first time, previously intractable hidden-type optimal taxation problems. Additionally, the work derives sufficient conditions for the emergence of lottery contracts.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Search and Optimization: Non-convex Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We develop a new method to efficiently solve for optimal lotteries in models with non-convexities. In order to employ a Lagrangian framework, we prove that the value of the saddle point that characterizes the optimal lottery is the same as the value of the dual of the deterministic problem. Our algorithm solves the dual of the deterministic problem via sub-gradient descent. We prove that the optimal lottery can be directly computed from the deterministic optima that occur along the iterations. We analyze the computational complexity of our algorithm and show that the worst-case complexity is often orders of magnitude better than the one arising from a linear programming approach. We apply the method to two canonical problems with private information. First, we solve a principal-agent moral-hazard problem, demonstrating that our approach delivers substantial improvements in speed and scalability over traditional linear programming methods. Second, we study an optimal taxation problem with hidden types, which was previously considered computationally infeasible, and examine under which conditions the optimal contract will involve lotteries.

Problem

• Develop method for optimal lotteries in non-convex economies
• Prove saddle point value equals dual deterministic problem
• Apply algorithm to principal-agent and optimal taxation problems

Innovation

• Lagrangian framework for non-convex economies
• Sub-gradient descent solves dual problem
• Computes optimal lottery from deterministic optima