  We study a bandit problem where observations from each arm have an
exponential family distribution and different arms are assigned independent
conjugate priors. At each of n stages, one arm is to be selected based on past
observations. The goal is to find a strategy that maximizes the expected
discounted sum of the $n$ observations. Two structural results hold in broad
generality: (i) for a fixed prior weight, an arm becomes more desirable as its
prior mean increases; (ii) for a fixed prior mean, an arm becomes more
desirable as its prior weight decreases. These generalize and unify several
results in the literature concerning specific problems including Bernoulli and
normal bandits. The second result captures an aspect of the
exploration-exploitation dilemma in precise terms: given the same immediate
payoff, the less one knows about an arm, the more desirable it becomes because
there remains more information to be gained when selecting that arm. For
Bernoulli and normal bandits we also obtain extensions to nonconjugate priors.
