  We study a stochastic model of anonymous influence with conformist and
anti-conformist individuals. Each agent with a `yes' or `no' initial opinion on
a certain issue can change his opinion due to social influence. We consider
anonymous influence, which depends on the number of agents having a certain
opinion, but not on their identity. An individual is conformist/anti-conformist
if his probability of saying `yes' increases/decreases with the number of
`yes'-agents. We focus on three classes of aggregation rules (pure conformism,
pure anti-conformism, and mixed aggregation rules) and examine two types of
society (without, and with mixed agents). For both types we provide a complete
qualitative analysis of convergence, i.e., identify all absorbing classes and
conditions for their occurrence. Also the pure case with infinitely many
individuals is studied. We show that, as expected, the presence of
anti-conformists in a society brings polarization and instability: polarization
in two groups, fuzzy polarization (i.e., with blurred frontiers), cycles,
periodic classes, as well as more or less chaotic situations where at any time
step the set of `yes'-agents can be any subset of the society. Surprisingly,
the presence of anti-conformists may also lead to opinion reversal: a majority
group of conformists with a stable opinion can evolve by a cascade phenomenon
towards the opposite opinion, and remains in this state.
