Multi-token Markov Game with Switching Costs

We study a general Markov game with metric switching costs: in each round, the player adaptively chooses one of several Markov chains to advance with the objective of minimizing the expected cost for at least k chains to reach their target states. If the player decides to play a different chain, an additional switching cost is incurred. The special case in which there is no switching cost was solved optimally by Dumitriu, Tetali, and Winkler~citeDTW03 by a variant of the celebrated Gittins Index for the classical multi-armed bandit (MAB) problem with Markovian rewards citeGit74,Git79. However, for Markovian multi-armed bandit with nontrivial switching cost, even if the switching cost is a constant, the classic paper by Banks and Sundaram citeBS94 showed that no index strategy can be optimal. In this paper, we complement their result and show there is a simple index strategy that achieves a constant approximation factor if the switching cost is constant and k=1. To the best of our knowledge, this index strategy is the first strategy that achieves a constant approximation factor for a general Markovian MAB variant with switching costs. For the general metric, we propose a more involved constant-factor approximation algorithm, via a nontrivial reduction to the stochastic k-TSP problem, in which a Markov chain is approximated by a random variable. Our analysis makes extensive use of various interesting properties of the Gittins index.