  Modelling efforts in opinion dynamics have to a large extent ignored that
opinion exchange between individuals can also have an effect on how willing
they are to express their opinion publicly. Here, we introduce a model of
public opinion expression. Two groups of agents with different opinion on an
issue interact with each other, changing the willingness to express their
opinion according to whether they perceive themselves as part of the majority
or minority opinion. We formulate the model as a multi-group majority game and
investigate the Nash equilibria. We also provide a dynamical systems
perspective: Using the reinforcement learning algorithm of $Q$-learning, we
reduce the $N$-agent system in a mean-field approach to two dimensions which
represent the two opinion groups. This two-dimensional system is analyzed in a
comprehensive bifurcation analysis of its parameters. The model identifies
social-structural conditions for public opinion predominance of different
groups. Among other findings, we show under which circumstances a minority can
dominate public discourse.
