In this paper, we study N -player finite-horizon discrete-time dynamic stochastic games where the uncertainty is described by an event tree. We consider linear-state dynamics, with a one-period lag structure, and quadratic costs. We derive necessary and sufficient conditions for the existence of S -adapted Nash equilibria with open-loop and closed-loop (no-memory) information structures. We observe that the existence of these equilibria is related to the solvability of a generalized backward Riccati recursion defined on the event tree. Next, we consider these games with (node-specific) linear constraints. We show that the S -adapted Nash equilibria (both open-loop and closed-loop) can be obtained by solving a parametric linear-complementarity problem defined on the entire event tree.