  An "Echo chamber" is the state of social learning agents whose performances
are deteriorated by excessive observation of others. We understand this to be
the collective behavior of agents in a restless multi-armed bandit. The bandit
has $M$ good levers and bad levers. A good lever changes to a bad one randomly
with probability $q_{C}$ and a new good lever appears. $N$ agents exploit ones'
lever if they know that it is a good one. Otherwise, they search for a good one
by (i) random search (success probability $q_{I}$) and (ii) observe a good
lever that is known by other agents (success probability $q_{O}$) with
probability $1-p$ and $p$, respectively. The distribution of agents in good
levers obeys the Yule distribution with power law exponent $1+\gamma$ in the
limit $N,M\to \infty$ and $\gamma=1+\frac{(1-p)q_{I}}{pq_{O}}$. The expected
value of the number of the agents with a good lever $N_{1}$ increases with $p$.
The system shows a phase transition at $p_{c}=\frac{q_{I}}{q_{I}+q_{o}}$. For
$p<p_{c}\,(>p_{c})$, the variance of $N_{1}$ per agent $\mbox{Var}(N_{1})/N$ is
finite (diverges as $\propto N^{2-\gamma}$ with $N$). There is a threshold
value $N_{s}$ for the system size that scales as $\ln N_{s} \propto
1/(\gamma-1)$. For $p>p_{c}$ and $N<N_{s}$, all agents tend to share only one
good lever. $\mbox{E}(N_{1})$ decreases to zero as $p\to 1$, which is referred
to as the "Echo chamber".

Physics and Society;Statistical Mechanics