Approachability in Stackelberg Stochastic Games with Vector Costs

Dileep Kalathil (), Vivek S. Borkar () and Rahul Jain ()
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Dileep Kalathil: EECS at UC Berkeley
Vivek S. Borkar: IIT Bombay
Rahul Jain: University of Southern California

Dynamic Games and Applications, 2017, vol. 7, issue 3, No 5, 422-442

Abstract: Abstract 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.

Date: 2017
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DOI: 10.1007/s13235-016-0198-y

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