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 [DTW03] by a variant of the celebrated Gittins Index for the classical multi-armed bandit (MAB) problem with Markovian rewards [Gittins 74, Gittins79]. However, for multi-armed bandit (MAB) with nontrivial switching cost, even if the switching cost is a constant, the classic paper by Banks and Sundaram [BS94] 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 is the first index strategy that achieves a constant approximation factor for a general 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.