Computing Stationary Equilibria in Measure-Dependent Markov Systems

2026-09-10

AI Summary

本文针对具有有限维聚合反馈的度量依赖马尔可夫系统，提出了一种计算平稳均衡的框架，并通过自洽方程和导数估计方法解决了该问题。

Abstract

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.

Problem

• Stationary Equilibria
• Measure-Dependent Markov Systems
• Self-Consistency Equation

Innovation

• measure-dependent Markov systems
• stationary equilibria
• self-consistency equation
• infinitesimal perturbation analysis
• finite-time bias