Approachability in Stackelberg Stochastic Games with Vector Costs

The notion of approachability was introduced by Blackwell (Pac J Math 6(1):1–8, 1956) in the context of vector-valued repeated games. The famous ‘Blackwell’s approachability theorem’ prescribes a strategy for approachability, i.e., for ‘steering’ the average vector cost of a given agent toward a given target set, irrespective of the strategies of the other agents. In this paper, motivated by the multi-objective optimization/decision-making problems in dynamically changing environments, we address the approachability problem in Stackelberg stochastic games with vector-valued cost functions. We make two main contributions. Firstly, we give a simple and computationally tractable strategy for approachability for Stackelberg stochastic games along the lines of Blackwell’s. Secondly, we give a reinforcement learning algorithm for learning the approachable strategy when the transition kernel is unknown. We also recover as a by-product Blackwell’s necessary and sufficient conditions for approachability for convex sets in this setup and thus a complete characterization. We give sufficient conditions for non-convex sets.