  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 average reward. While this
goal is provably impossible for an isolated individual, we show that, in social
groups, this goal can be achieved easily with the aid of social persuasion,
i.e., communication. Specifically, we study the learning dynamics wherein an
individual sequentially decides on which arm to pull next based on not only its
private reward feedback but also the suggestions provided by randomly chosen
peers. Our learning dynamics are hard to analyze via explicit probabilistic
calculations due to the stochastic dependency induced by social interaction.
Instead, we employ the mean-field approximation method from statistical physics
and we show:
  (1) With probability $\to 1$ as the social group size $N \to \infty $, every
individual in the social group learns the best option.
  (2) Over an arbitrary finite time horizon $[0, T]$, with high probability (in
$N$), the fraction of individuals that prefer the best option grows to 1
exponentially fast as $t$ increases ($t\in [0, T]$).
  A major innovation of our mean-filed analysis is a simple yet powerful
technique to deal with absorbing states in the interchange of limits $N \to
\infty$ and $t \to \infty $. The mean-field approximation method allows us to
approximate the probabilistic sample paths of our learning dynamics by a
deterministic and smooth trajectory that corresponds to the unique solution of
a well-behaved system of ordinary differential equations (ODEs). Such an
approximation is desired because the analysis of a system of ODEs is relatively
easier than that of the original stochastic system.
