  From large-scale organizations to decentralized political systems,
hierarchical strategic decision making is commonplace. We introduce a novel
class of structured hierarchical games (SHGs) that formally capture such
hierarchical strategic interactions. In an SHG, each player is a node in a
tree, and strategic choices of players are sequenced from root to leaves, with
root moving first, followed by its children, then followed by their children,
and so on until the leaves. A player's utility in an SHG depends on its own
decision, and on the choices of its parent and all the tree leaves. SHGs thus
generalize simultaneous-move games, as well as Stackelberg games with many
followers. We leverage the structure of both the sequence of player moves as
well as payoff dependence to develop a gradient-based back propagation-style
algorithm, which we call Differential Backward Induction (DBI), for
approximating equilibria of SHGs. We provide a sufficient condition for
convergence of DBI and demonstrate its efficacy in finding approximate
equilibrium solutions to several SHG models of hierarchical policy-making
problems.
