  People often learn from other's actions when they make decisions while doing
online shopping. This kind of observational learning may lead to information
cascades, which means agents might ignore their own signals and follow the
'trend' created collectively by the actions of their predecessors. It is
well-known that with rational agents, such a cascade model can result in either
correct or incorrect cascades. In this paper, we additionally consider the
presence of fake agents who always take fixed actions and we investigate their
influence on the outcome of these cascades. We propose an infinite Markov Chain
sequence structure and a tree structure to analyze how the fraction and the
type of such fake agents impacts behavior of the upcoming agents. We show that
an increase in the fraction of fake agents may reduce the chances of their
preferred outcome, and also there is a certain lower bound for the probability
of a wrong cascade. In particular, we discuss the probability of an agent being
fake tends to 1 and the effect of a constant portion of fake agents.
