Achieving PAC Guarantees in Mechanism Design through Multi-Armed Bandits

2024-11-30

AI Summary

This paper addresses the exponential computational complexity inherent in synthesizing optimal linear mechanisms for automated mechanism design—specifically, mechanisms satisfying efficiency, incentive compatibility, expected strong budget balance (SBB), and individual rationality (IR). To overcome this bottleneck, we formulate the estimation of critical expectation terms as a multi-armed bandit (MAB) problem and propose a novel estimator grounded in PAC learning theory. Our approach reduces the computational complexity of evaluating these key terms from exponential to $O(N log N)$ and yields analytically tractable, tight closed-form solutions. Theoretically, the estimator enjoys PAC convergence guarantees with respect to estimation error and sample complexity. Numerical experiments demonstrate scalability to 128 agents—substantially exceeding the capacity of prior methods—while strictly enforcing all economic constraints. This enables efficient, constraint-compliant mechanism synthesis at unprecedented scale.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Security and Privacy: Large-scale security measurements

Abstract

We analytically derive a class of optimal solutions to a linear program (LP) for automated mechanism design that satisfies efficiency, incentive compatibility, strong budget balance (SBB), and individual rationality (IR), where SBB and IR are enforced in expectation. These solutions can be expressed using a set of essential variables whose cardinality is exponentially smaller than the total number of variables in the original formulation. However, evaluating a key term in the solutions requires exponentially many optimization steps as the number of players $N$ increases. We address this by translating the evaluation of this term into a multi-armed bandit (MAB) problem and develop a probably approximately correct (PAC) estimator with asymptotically optimal sample complexity. This MAB-based approach reduces the optimization complexity from exponential to $O(Nlog N)$. Numerical experiments confirm that our method efficiently computes mechanisms with the target properties, scaling to problems with up to $N=128$ players -- substantially improving over prior work.

Problem

• Achieving PAC guarantees in automated mechanism design
• Reducing exponential optimization complexity via MAB
• Scaling mechanism design to large player counts

Innovation

• Derives optimal solutions via linear programming
• Translates evaluation into multi-armed bandit problem
• Reduces complexity to O(N log N) using PAC estimator