  We consider multi-armed bandit problems in social groups wherein each
individual has bounded memory and shares the common goal of learning the best
arm/option. We say an individual learns the best option if eventually (as $t\to
\infty$) it pulls only the arm with the highest expected reward. While this
goal is provably impossible for an isolated individual due to bounded memory,
we show that, in social groups, this goal can be achieved easily with the aid
of social persuasion (i.e., communication) as long as the communication
networks/graphs satisfy some mild conditions. To deal with the interplay
between the randomness in the rewards and in the social interaction, we employ
the {\em mean-field approximation} method. Considering the possibility that the
individuals in the networks may not be exchangeable when the communication
networks are not cliques, we go beyond the classic mean-field techniques and
apply a refined version of mean-field approximation:
  (1) Using coupling we show that, if the communication graph is connected and
is either regular or has doubly-stochastic degree-weighted adjacency matrix,
with probability $\to 1$ as the social group size $N \to \infty$, every
individual in the social group learns the best option.
  (2) If the minimum degree of the graph diverges as $N \to \infty$, over an
arbitrary but given finite time horizon, the sample paths describing the
opinion evolutions of the individuals are asymptotically independent. In
addition, the proportions of the population with different opinions converge to
the unique solution of a system of ODEs. In the solution of the obtained ODEs,
the proportion of the population holding the correct opinion converges to $1$
exponentially fast in time.
  Notably, our results hold even if the communication graphs are highly sparse.
