Leader-Follower semi-Markov Decision Problems: Theoretical Framework and Approximate Solution

Leader-follower problems are hierarchical decision problems in which a leader uses incentives to induce certain desired behavior among a set of self-interested followers. Dynamic leader-follower problems extend this structure to multi-period decision situations. In this work we propose a Markov decision process (MDP) framework for a class of dynamic leader-follower problems that have important applications and discuss their approximate solution using reinforcement learning (RL). In these problems, the leader makes incentive decisions intermittently while the followers make their decisions in every period. Our theoretical framework and computational approach are based on the observation that such dynamic problems can be thought of as consisting of two coupled sequential decision processes, that of the leader and of the followers. In our formulation, the leader´s decision problem that has the structure of a single-agent semi-Markov decision process (SMDP), and the followers´ sequential decision problem structured as a stochastic game (multiagent competitive MDP) operate over the same state space. We call this MDP framework a leader-follower semi-Markov decision process (LFSMDP). We consider approximate solution of these problems using RL and demonstrate the solution approach in the special case where the followers´ stochastic game is a repeated game.