Solving Structured Hierarchical Games Using Differential Backward Induction

Zun Li, Feiran Jia, Aditya Mate, Shahin Jabbari, Mithun Chakraborty, Milind Tambe, Yevgeniy Vorobeychik

Keywords: game theory, dynamical systems, graphical models, hierarchical decision making

Abstract: From large-scale organizations to decentralized political systems, hierarchical strategic decision making is commonplace. We introduce a novel class of \emph{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 \emph{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 \emph{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.