Dynamics of opinion expression

Modeling 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 multigroup 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.

Figure 1

The available pure-strategy Nash equilibria in different regimes of $\gamma$ and $\delta$. The equilibria are abbreviated by either $e$ for expression or $s$ for silence for each opinion group (the first entry is for the collective action of $G_1$, the second for the one of $G_2$). For costs $c>0$, $\gamma$ and $\delta$ below $\frac{c}{1-c}$ will lead to a situation in which the only available Nash equilibrium is one in which no one expresses her opinion publicly. An increase in the structural parameters above this threshold leads to additional Nash equilibria in which at least one of the two opinion groups speaks out. If both $\gamma$ and $\delta$ are bigger than $\frac{c+1}{1-c}$, then an additional Nash equilibrium arises in which all agents express their opinion.

Figure 2

The constant $\gamma$- and $\delta$-curves for $N=100$ agents, $q_{12}=0.2$ and $c=0.2$. They are plotted with respect to $N_1$, $N_2=N-N_1$, and $q_{11}=q_{22}$. Each blue curve (starting at $N_1$ = 100, $q_{11}=0$) stands for a combination of the number of opinion group members $N_1$ and internal connection weights $q_{11}$ that yields a constant value of the structural parameters $\gamma$, each red one (starting at $N_1=0$, $q_{11}=0$) for a combination of $N_2$ and $q_{22}$ that produces constant $\delta$. The color-coding for the different Nash equilibrium regions is analogous to Fig. 1. It is visible that the numerical minority of an opinion group cannot always be compensated by increasing $q_{11}$ (or $q_{22}$), the probability of a connection between two agents of the same opinion group. Moreover, the fixed $\gamma$- and $\delta$-curves are symmetric with respect to $N_1=N_2=50$, where they intersect. (For better readability, the dashed $\delta$ curves have not been labeled. They correspond to their $\gamma$ counterparts.)

Figure 3

Three phase portraits of the $Q_1$ ($x$ axis) and $Q_2$ values ($y$ axis) of the two-dimensional system for different configurations of $\gamma$ and $\delta$. We have $c=0.1$, $\beta =10$, and structural parameters $\gamma =\delta =0.1$ (bottom left), $\gamma =\delta =1$ (bottom right), and $\gamma =\delta =3$ (top right). The yellow (light gray, if in grayscale) and blue (dark gray) lines in the phase portraits are the isoclines of the equations for $Q_1$ and $Q_2$. The fixed points are located at their intersections.

As is visible, the $(s,s)$ fixed point disappears in the dynamical system for higher $\gamma$ and $\delta$ values. This is due to the finite exploration rate $\beta$ and the transition from the $N$-player game of Sec. 3 to the two-population game in the mean-field approximation.

Figure 4

The trajectories of the $Q$ values in simulations, averaged over 50 runs with $N\times {10}^5$ steps, for different values of $\alpha$ with a starting point close to the border (red line) of the two basins of attraction of the two stable fixed points. The starting $Q$ values were $Q_{i\in G_1}=0$, $Q_{i\in G_2}=-0.25$. There were $N=200$ agents, 100 of each opinion group, and $c=0.1$, $q_{11}=0.04$, $q_{12}=0.05$, and $q_{22}=0.15$. A relatively big $\alpha =0.1$ makes the trajectory leave the lower right basin of attraction of the two-dimensional system (black trajectory). Due to the high $\alpha$, the fixed point of the other basin of attraction is also missed by some margin. The lower $\alpha$, the closer the trajectories get to the fixed point and the more probable it is that they will stay in the basin predicted by the two-dimensional approximation. For $\alpha =0.01$ (turquoise) and $\alpha =0.001$ (light green), the trajectories run toward the predicted fixed point. The yellow (light gray) and blue (dark gray) lines are the isoclines of the equations for $Q_1$ and $Q_2$. The fixed points are located at their intersections.

Figure 5

The development of the $Q_1$ (a) and $Q_2$ value (b) of the fixed points with $\gamma$ given $\beta =10$, relatively high $\delta =2.36$, and $c=0.1$. The colors of the curves in the two plots indicate the different fixed point pairs of $Q_1$ and $Q_2$. A dashed line indicates an unstable fixed point, a continuous one a stable fixed point. It is visible in the plots that a poorly connected opinion group $G_1$ ($\gamma <0.5$) will be driven into silence by the other group [beige curve, lowest one in panel (a), highest one in panel (b)]. With increasing in-group connectivity, fixed points arise for which $G_1$ expresses their opinion in two saddle-node bifurcations [red for an an $(e,s)$-equilibrium (highest curve in panel (a), lowest in panel (b)) and blue for $(e,e)$ (in-between)]. For $\gamma >4.5$, $G_1$ is so well-connected that the equilibrium disappears in which the group is silent. The dotted grey line indicates $Q$ value 0, where the probability of expression passes 0.5.

Figure 6

The development of $Q_1$- and $Q_2$-fixed points with $\gamma$ given $\beta =10$, moderate $\delta =1.6$ and $c=0.1$. For $\gamma <0.4$, only group $G_2$ is expressive. A second fixed point arises for higher $\gamma$ in which $G_1$ is predominating public discourse. There is no fixed point in which both groups are expressive.

Figure 7

The development of the fixed points with $c$ given $\beta =10$ and $\gamma =\delta =2.1$. Panels (a) and (b) are symmetric since $c$ is the same for both and has the same impact on both groups if they also have identical structural parameters. If expression is costly, then everyone is silent; if it has negative costs, then everyone speaks out.

Figure 8

Fixed-point development with $c_1$ independent of $c_2$, given $\beta =10$, $\gamma =\delta =2.36$, $c_2=0.1$. Strongly negative $c_1$ corresponds to a strong motivational disposition (or the facilitation of opinion expression for the group) in the opinion group to express their opinion. There, only fixed points in which this opinion group is expressive exist. For decreasing motivation (or if opinion expression is impeded), fixed points arise in which the second opinion group is the only expressive one.

Figure 9

Illustration of the transitions between the game-theoretic equilibrium regions for (i) stronger internal cohesion of both opinion groups (“echo chambers”), (ii) less internal cohesion of both (heterophilious connections), and (iii) stronger internal cohesion for only one opinion group (“#metoo”).

Figure 10

The development of the fixed points with $\beta$ given $c=0.1$ and $\gamma =\delta =2.36$. Since $\gamma$ and $\delta$ are the same, the plots in panels (a) and (b) are symmetric.