Discrete-Time LQ Stochastic Two Person Nonzero Sum Difference Games With Random Coefficients:~Closed-Loop Nash Equilibrium

2025/07/22 by Qingxin Meng, Yiwei Wu, Meng, Qingxin +1
Decision Sciences · Economics, Econometrics and Finance · #Economic theories and models #FOS: Mathematics #Game Theory and Applications #Game Theory and Voting Systems #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2507.16412

openalex publication_date 2025/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper investigates closed-loop Nash equilibria for discrete-time linear-quadratic (LQ) stochastic nonzero-sum difference games with random coefficients. Unlike existing works, we consider randomness in both state dynamics and cost functionals, leading to a complex structure of fully coupled cross-coupled stochastic Riccati equations (CCREs). The key contributions lie in characterizing the equilibrium via state-feedback strategies derived by decoupling stochastic Hamiltonian systems governed by two symmetric CCREs-these random coefficients induce a higher-order nonlinear backward stochastic difference equation (BS\triangleE) system, fundamentally differing from deterministic counterparts. Under minimal regularity conditions, we establish necessary and sufficient conditions for closed-loop Nash equilibrium existence, contingent on the regular solvability of CCREs without requiring strong assumptions. Solutions are constructed using a dynamic programming principle (DPP), linking equilibrium strategies to coupled Lyapunov-type equations. Our analysis resolves critical challenges in modeling inherent randomness and provides a unified framework for dynamic decision-making under uncertainty.

Citations

• Discrete-Time LQ Stochastic Two-Person Nonzero-Sum Difference Games with Random Coefficients:~Open-Loop Nash Equilibrium
• Mean-Field Linear-Quadratic Stochastic Differential Games
• Two-Person Zero-Sum Stochastic Linear-Quadratic Differential Games
• Time-Inconsistent Mean-Field Stochastic LQ Problem: Open-Loop Time-Consistent Control
• Linear Quadratic Stochastic Two-Person Nonzero-Sum Differential Games: Open-Loop and Closed-Loop Nash Equilibria
• Mean-Field Stochastic Linear Quadratic Optimal Control Problems: Closed-Loop Solvability
• Linear Quadratic Stochastic Two-Person Zero-Sum Differential Games in an Infinite Horizon
• Linear Quadratic Stochastic Differential Games: Open-Loop and Closed-Loop Saddle Points

Related

• Discrete-Time LQ Stochastic Two-Person Nonzero-Sum Difference Games with Random Coefficients:~Open-Loop Nash Equilibrium
• Linear Quadratic Stochastic Two-Person Nonzero-Sum Differential Games: Open-Loop and Closed-Loop Nash Equilibria
• Open-loop and Closed-loop Local and Remote Stochastic Nonzero-sum Game with Inconsistent Information Structure
• Two-player nonZero–sum stopping games in discrete time
• Stochastic linear-quadratic differential game with Markovian jumps in an infinite horizon