  We study the effect of probabilistic distortions to the local majority rules
used in the Galam model of opinion dynamics and bottom-up hierarchical voting.
A different probability for a flip against the local majority within the
discussion group is associated with each ratio of majority / minority. The
cases of groups of sizes 3 and 5 are investigated in detail. For hierarchical
voting, the local flip corresponds to a `faithless elector', a representative
who decides to vote against the choice of their electing group. Depending on
the flip probabilities, the model exhibits a rich variety of patterns for the
dynamics, which include novel features in the topology of the landscape. In
particular, for size 5, we uncover for the first time an interplay between five
fixed points, which split into either three attractors and two tipping points
or two attractors and three tipping points, depending on the flip
probabilities. Larger groups are also analysed. These features were absent in
the former versions of the Galam model, which has at maximum three fixed points
for any group size. The results shed a new light on a series of social
phenomena triggered by one single individual who acts against the local
majority.
