  We consider a decentralized stochastic multi-armed bandit problem with
multiple players. Each player aims to maximize his/her own reward by pulling an
arm. The arms give rewards based on i.i.d. stochastic Bernoulli distributions.
Players are not aware about the probability distributions of the arms. At the
end of each turn, the players inform their neighbors about the arm he/she
pulled and the reward he/she got. Neighbors of players are determined according
to an Erd{\H{o}}s-R{\'e}nyi graph with connectivity $\alpha$. This graph is
reproduced in the beginning of every turn with the same connectivity. When more
than one player choose the same arm in a turn, we assume that only one of the
players who is randomly chosen gets the reward where the others get nothing. We
first start by assuming players are not aware of the collision model and offer
an asymptotically optimal algorithm for $\alpha = 1$ case. Then, we extend our
prior work and offer an asymptotically optimal algorithm for any connectivity
but zero, assuming players aware of the collision model. We also study the
effect of $\alpha$, the degree of communication between players, empirically on
the cumulative regret by comparing them with traditional multi-armed bandit
algorithms.
