  The fidelity bandits problem is a variant of the $K$-armed bandit problem in
which the reward of each arm is augmented by a fidelity reward that provides
the player with an additional payoff depending on how 'loyal' the player has
been to that arm in the past. We propose two models for fidelity. In the
loyalty-points model the amount of extra reward depends on the number of times
the arm has previously been played. In the subscription model the additional
reward depends on the current number of consecutive draws of the arm. We
consider both stochastic and adversarial problems. Since single-arm strategies
are not always optimal in stochastic problems, the notion of regret in the
adversarial setting needs careful adjustment. We introduce three possible
notions of regret and investigate which can be bounded sublinearly. We study in
detail the special cases of increasing, decreasing and coupon (where the player
gets an additional reward after every $m$ plays of an arm) fidelity rewards.
For the models which do not necessarily enjoy sublinear regret, we provide a
worst case lower bound. For those models which exhibit sublinear regret, we
provide algorithms and bound their regret.
