测度依赖马尔可夫系统中平稳均衡的计算

中文摘要 AI 辅助

运筹学与经济学中的许多随机系统在其长期状态分布与支配其动态的转移规律之间表现出反馈。在本文中，当这种反馈通过有限维聚合量运作时，我们为这类测度依赖马尔可夫系统中的平稳均衡开发了一个计算框架。我们证明了原始的平稳均衡问题可以简化为一个有限维自洽方程，将底层马尔可夫系统的稳态分析与均衡计算分离开来。我们利用所得自洽映射的性质来指导在不动点迭代、松弛不动点迭代以及不动点残差最小化之间的选择。最后一种方法需要自洽映射的导数，而这些导数通常无法以闭式形式获得。因此，我们为这些导数开发了有限时间无穷小扰动分析估计器，其误差界将蒙特卡洛误差与有限时间偏差分开。我们通过一个策略性$G/G/c$队列和一个意见动力学模型来说明该框架，展示不同的结构性质如何自然地导致不同的均衡计算方法。

英文摘要

Many stochastic systems in operations and economics exhibit feedback between their long-run state distribution and the transition law governing their dynamics. In this paper, we develop a computational framework for stationary equilibria in such measure-dependent Markov systems when this feedback operates through a finite-dimensional aggregate. We show that the original stationary-equilibrium problem can be reduced to a finite-dimensional self-consistency equation, separating steady-state analysis of the underlying Markov system from equilibrium computation. We use properties of the resulting self-consistency map to guide the choice among fixed-point iteration, relaxed fixed-point iteration, and minimization of the fixed-point residual. The last approach requires derivatives of the self-consistency map, which are typically unavailable in closed form. We therefore develop finite-time infinitesimal perturbation analysis estimators for these derivatives, with error bounds that separate Monte Carlo error from finite-time bias. We illustrate the framework through a strategic $G/G/c$ queue and an opinion-dynamics model, showing how different structural properties lead naturally to different equilibrium-computation methods.

发表机构

• Graduate School of Business, Columbia University（哥伦比亚大学商学院）
• Business School and Institute of Operations Research and Analytics, National University of Singapore（新加坡国立大学商学院与运筹学与分析研究所）

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