Many real-world systems possess a hierarchical structure where a strategic
plan is forwarded and implemented in a top-down manner. Examples include
business activities in large companies or policy making for reducing the spread
during pandemics. We introduce a novel class of games that we call structured
hierarchical games (SHGs) to capture these strategic interactions. In an SHG,
each player is represented as a vertex in a multi-layer decision tree and
controls a real-valued action vector reacting to orders from its predecessors
and influencing its descendants' behaviors strategically based on its own
subjective utility. SHGs generalize extensive form games as well as Stackelberg
games. For general SHGs with (possibly) nonconvex payoffs and high-dimensional
action spaces, we propose a new solution concept which we call local subgame
perfect equilibrium. By exploiting the hierarchical structure and strategic
dependencies in payoffs, we derive a back propagation-style gradient-based
algorithm which we call Differential Backward Induction to compute an
equilibrium. We theoretically characterize the convergence properties of DBI
and empirically demonstrate a large overlap between the stable points reached
by DBI and equilibrium solutions. Finally, we demonstrate the effectiveness of
our algorithm in finding \emph{globally} stable solutions and its scalability
for a recently introduced class of SHGs for pandemic policy making