Mixed Leadership Stochastic Differential Game in Feedback Information Pattern with Applications

TL;DR

This paper develops a high-dimensional stochastic differential game framework with feedback information, deriving explicit feedback equilibrium strategies for innovation and pricing in a dynamic stochastic model.

Contribution

It introduces a verification theorem for feedback Stackelberg-Nash equilibrium in high-dimensional stochastic differential games with control in the diffusion term, and applies it to a dynamic pricing and innovation problem.

Findings

Explicit feedback equilibrium strategies derived from coupled Riccati equations

Local existence and uniqueness of Riccati solutions established

Numerical analysis shows parameter effects on strategies

Abstract

This paper is devoted to a high-dimensional mixed leadership stochastic differential game on a finite horizon in feedback information mode, where the control variables enter into the diffusion term of state equation. A verification theorem for the feedback Stackelberg-Nash equilibrium is obtained by using a system of coupled and fully nonlinear parabolic partial differential equations. We apply the verification theorem to deal with a dynamic innovation and pricing decision problem where the buyer acts as the leader in the pricing decisions and the dynamic model is stochastic. Via the solutions of coupled Riccati equations, we explicitly express the feedback equilibrium strategies of innovation and pricing. And by analysis, the local existence and uniqueness of the solutions of the coupled Riccati equations is derived. We also conduct some numerical analyses to discuss the effects of model parameters on the feedback equilibrium strategies.