  An extension of the traditional two-armed bandit problem is considered, in
which the decision maker has access to some side information before deciding
which arm to pull. At each time t, before making a selection, the decision
maker is able to observe a random variable X_t that provides some information
on the rewards to be obtained. The focus is on finding uniformly good rules
(that minimize the growth rate of the inferior sampling time) and on
quantifying how much the additional information helps. Various settings are
considered and for each setting, lower bounds on the achievable inferior
sampling time are developed and asymptotically optimal adaptive schemes
achieving these lower bounds are constructed.
