  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.
