We propose a tightening sequence of optimistic approximations to the Gittins index in "Optimistic Gittins Indices." We show that the use of these approximations in concert with the use of an increasing discount factor appears to offer a compelling alternative to state-of-the-art index schemes proposed for the Bayesian multiarmed bandit problem. We prove that the use of these optimistic indices constitutes a regret optimal algorithm. Perhaps more interestingly, the use of even the loosest of

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... e approximations appears to offer substantial performance improvements over state-of-the-art alternatives while incurring little to no additional computational overhead relative to the simplest of these alternatives.