Linear-Quadratic Stochastic Differential Games on Random Directed Networks

TL;DR

This paper extends the analysis of linear-quadratic stochastic differential games to random directed networks, explicitly solving for Nash equilibria and characterizing the resulting Gaussian processes on complex network structures.

Contribution

It introduces a random directed chain model with explicit Nash equilibrium solutions and extends the framework to two-sided chains and directed trees.

Findings

Explicit Nash equilibrium for random directed chains.

Equilibrium dynamics form an infinite-dimensional Gaussian process.

Extension to random two-sided chains and directed trees.

Abstract

The study of linear-quadratic stochastic differential games on directed networks was initiated in Feng, Fouque \& Ichiba \cite{fengFouqueIchiba2020linearquadratic}. In that work, the game on a directed chain with finite or infinite players was defined as well as the game on a deterministic directed tree, and their Nash equilibria were computed. The current work continues the analysis by first developing a random directed chain structure by assuming the interaction between every two neighbors is random. We solve explicitly for an open-loop Nash equilibrium for the system and we find that the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain introduced in \cite{fengFouqueIchiba2020linearquadratic}. The discussion about stochastic differential games is extended to a random two-sided directed chain and a random directed tree structure.