  We analyze a sequential decision making model in which decision makers (or,
players) take their decisions based on their own private information as well as
the actions of previous decision makers. Such decision making processes often
lead to what is known as the \emph{information cascade} or \emph{herding}
phenomenon. Specifically, a cascade develops when it seems rational for some
players to abandon their own private information and imitate the actions of
earlier players. The risk, however, is that if the initial decisions were
wrong, then the whole cascade will be wrong. Nonetheless, information cascade
are known to be fragile: there exists a sequence of \emph{revealing}
probabilities $\{p_{\ell}\}_{\ell\geq1}$, such that if with probability
$p_{\ell}$ player $\ell$ ignores the decisions of previous players, and rely on
his private information only, then wrong cascades can be avoided. Previous
related papers which study the fragility of information cascades always assume
that the revealing probabilities are known to all players perfectly, which
might be unrealistic in practice. Accordingly, in this paper we study a
mismatch model where players believe that the revealing probabilities are
$\{q_\ell\}_{\ell\in\mathbb{N}}$ when they truly are
$\{p_\ell\}_{\ell\in\mathbb{N}}$, and study the effect of this mismatch on
information cascades. We consider both adversarial and probabilistic sequential
decision making models, and derive closed-form expressions for the optimal
learning rates at which the error probability associated with a certain
decision maker goes to zero. We prove several novel phase transitions in the
behaviour of the asymptotic learning rate.
