In the Multi-Armed Bandit (MAB) problem, there are a given set of arms with
unknown reward distributions. At each time, a player selects one arm to play,
aiming to maximize the total expected reward over a horizon of length T. The
essence of the problem is the tradeoff between exploration and exploitation:
playing a less explored arm to learn its reward statistics for future benefit
or playing the arm with the largest sample mean at the current time.

We consider decentralized restless multi-armed bandit problems with unknown
dynamics and multiple players. The reward state of each arm transits according
to an unknown Markovian rule when it is played and evolves according to an
arbitrary unknown random process when it is passive. Players activating the
same arm at the same time collide and suffer from reward loss. The objective is
to maximize the long-term reward by designing a decentralized arm selection
policy to address unknown reward models and collisions among players.

In the classic Bayesian restless multi-armed bandit (RMAB) problem, there are
$N$ arms, with rewards on all arms evolving at each time as Markov chains with
known parameters. A player seeks to activate $K \geq 1$ arms at each time in
order to maximize the expected total reward obtained over multiple plays. RMAB
is a challenging problem that is known to be PSPACE-hard in general. We
consider in this work the even harder non-Bayesian RMAB, in which the
parameters of the Markov chain are assumed to be unknown \emph{a priori}.

We consider multi-armed bandit with distributed players, where each player
independently samples one of N stochastic processes with unknown parameters and
accrues reward in each slot without information exchange. Users choosing the
same arm collide, and none or only one receives reward depending on the
collision model. This problem can be formulated as a decentralized multi-armed
bandit problem.

In this paper we investigate pointed Hopf algebras via quiver methods. We
classify all possible Hopf structures arising from minimal Hopf quivers, namely
basic cycles and the linear chain. This provides full local structure
information for general pointed Hopf algebras.