  A multi-user multi-armed bandit (MAB) framework is used to develop algorithms
for uncoordinated spectrum access. The number of users is assumed to be unknown
to each user. A stochastic setting is first considered, where the rewards on a
channel are the same for each user. In contrast to prior work, it is assumed
that the number of users can possibly exceed the number of channels, and that
rewards can be non-zero even under collisions. The proposed algorithm consists
of an estimation phase and an allocation phase. It is shown that if every user
adopts the algorithm, the system wide regret is constant with time with high
probability. The regret guarantees hold for any number of users and channels,
in particular, even when the number of users is less than the number of
channels. Next, an adversarial multi-user MAB framework is considered, where
the rewards on the channels are user-dependent. It is assumed that the number
of users is less than the number of channels, and that the users receive zero
reward on collision. The proposed algorithm combines the Exp3.P algorithm
developed in prior work for single user adversarial bandits with a collision
resolution mechanism to achieve sub-linear regret. It is shown that if every
user employs the proposed algorithm, the system wide regret is of the order
$O(T^\frac{3}{4})$ over a horizon of time $T$. The algorithms in both
stochastic and adversarial scenarios are extended to the dynamic case where the
number of users in the system evolves over time and are shown to lead to
sub-linear regret.
