  An individual's decisions are often guided by those of his or her peers,
i.e., neighbors in a social network. Presumably, being privy to the experiences
of others aids in learning and decision making, but how much advantage does an
individual gain by observing her neighbors? Such problems make appearances in
sociology and economics and, in this paper, we present a novel model to capture
such decision-making processes and appeal to the classical multi-armed bandit
framework to analyze it. Each individual, in addition to her own actions, can
observe the actions and rewards obtained by her neighbors, and can use all of
this information in order to minimize her own regret. We provide algorithms for
this setting, both for stochastic and adversarial bandits, and show that their
regret smoothly interpolates between the regret in the classical bandit setting
and that of the full-information setting as a function of the neighbors'
exploration. In the stochastic setting the additional information must simply
be incorporated into the usual estimation of the rewards, while in the
adversarial setting this is attained by constructing a new unbiased estimator
for the rewards and appropriately bounding the amount of additional information
provided by the neighbors. These algorithms are optimal up to log factors;
despite the fact that the agents act independently and selfishly, this implies
that it is an approximate Nash equilibria for all agents to use our algorithms.
Further, we show via empirical simulations that our algorithms, often
significantly, outperform existing algorithms that one could apply to this
setting.
