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

TL;DR

This paper investigates discrete-time stochastic two-player non-zero-sum LQ games with random coefficients, deriving conditions for Nash equilibria and explicit strategies using advanced stochastic difference equations.

Contribution

It introduces a novel approach with coupled FBSΔE and stochastic Riccati equations to explicitly characterize open-loop Nash equilibria in games with fully random coefficients.

Findings

Derived necessary and sufficient conditions for Nash equilibria.

Provided explicit feedback strategies for players.

Extended Riccati equations to fully nonlinear stochastic difference equations.

Abstract

This paper presents a pioneering investigation into discrete-time two-person non-zero-sum linear quadratic (LQ) stochastic games with random coefficients. We derive necessary and sufficient conditions for the existence of open-loop Nash equilibria using convex variational calculus. To obtain explicit expressions for the Nash equilibria, we introduce fully coupled forward-backward stochastic difference equations (FBS$\Delta$E, for short), which provide a dual characterization of these Nash equilibria. Additionally, we develop non-symmetric stochastic Riccati equations that decouple the stochastic Hamiltonian system for each player, enabling the derivation of closed-loop feedback forms for open-loop Nash equilibrium strategies. A notable aspect of this research is the complete randomness of the coefficients, which results in the corresponding Riccati equations becoming fully nonlinear higher-order backward stochastic difference equations. It distinguishes our non-zero-sum difference game from the deterministic case, where the Riccati equations reduce to algebraic forms.