  One of two independent stochastic processes (arms) are to be selected at each
of n stages. The selection is sequential and depends on past observations as
well as the prior information. Observations from arm i are independent given a
distribution P_i, and, following Clayton and Berry (1985), P_i's have
independent Dirichlet process priors. The objective is to maximize the expected
future-discounted sum of the n observations. We study structural properties of
the bandit, in particular how the maximum expected payoff and the optimal
strategy vary with the Dirichlet process priors. The main results are (i) for a
particular arm and a fixed prior weight, the maximum expected payoff increases
as the mean of the Dirichlet process prior becomes larger in the increasing
convex order; (ii) for a fixed prior mean, the maximum expected payoff
decreases as the prior weight increases. Specializing to the one-armed bandit,
the second result captures the intuition that, given the same immediate payoff,
the more is known about an arm, the less desirable it becomes because there is
less to learn when selecting that arm. This extends some results of Gittins and
Wang (1992) on Bernoulli bandits and settles a conjecture of Clayton and Berry
(1985).
