We analyze and solve a game in which a player chooses which of several Markov chains to advance, with the object of minimizing the expected time (or cost) for one of the chains to reach a target state. The solution entails computing (in polynomial time) a function γ-a variety of "Gittins index"-on the states of the individual chains, the minimization of which produces an optimal strategy. It turns out that γ is a useful cousin of the expected hitting time of a Markov chain but is defined, for

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... ample, even for random walks on infinite graphs. We derive the basic properties of γ and consider its values in some natural situations.