Figure 2.

Belief dynamics and actions of a single agent in response to a sequence of Hashtag observations from two fictive neighbours. Shown are the history of Bernoulli parameters defining three marginal posterior beliefs of the focal agent: the belief about the truth value of Idea 1 ($Q(s_t^{\mathbf{Idea}}=\mathbf{Idea}\mathbf{1})$, in red), and its beliefs about the beliefs of its two neighbours regarding Idea 1 ($Q(s_t^{\mathbf{MB}1}=\mathbf{Idea}\mathbf{1})$ and $Q(s_t^{\mathbf{MB}2}=\mathbf{Idea}\mathbf{1})$, shown in two shades of blue). Through its generative model, the focal agent believes that its Hashtag observations are caused by two neighbour ‘meta-belief’ states. The focal agent is exposed to a sequence of Hashtag observations for 100 timesteps, where in case of attending to the first neighbour ($u_t^{\mathbf{Who}}=0$), the agent receives observation $o_t^{\mathbf{NT}1}=$Hashtag 1, $o_t^{\mathbf{NT}2}=$Null, and in case of sampling the other neighbour ($u_t^{\mathbf{Who}}=0$), the agent receives observation $o_t^{\mathbf{NT}1}=$Null, $o_t^{\mathbf{NT}2}=$Hashtag 2. Due to the ‘Hashtag semantics’ matrix in its generative model, these two Hashtags, respectively, lend evidence for the two levels of $s^{\mathbf{Idea}}$. At each timestep the focal agent performs inferences with respect to hidden states $Q\left({\mathbf{s}}_t\right)$ as well as policies (control states) $Q({\mathbf{u}}_t$), and then samples a Neighbour Attendance action from the posterior over control states $Q(u^{\mathbf{Who}}=0,u^{\mathbf{Who}}=1)$. Below each subplot is a heatmap showing the temporal evolution of the probability of sampling Neighbour 1 vs. Neighbour 2 over time. Subfigure (a) shows an agent with low $\gamma$ (3.0) and low ${\omega }^{\mathbf{Soc}}$ (3.0). The agent’s beliefs about both of their neighbors does not lead it to converge on an idea being true or not. Subfigure (b) shows an agent shows an agent with low $\gamma$ (3.0) and high ${\omega }^{\mathbf{Soc}}$ (9.0). The agent will be more certain about the beliefs of their neighbors, attend less often to their neighbors, quickly converging to neighbour 2. Subfigure (c) shows an agent shows an agent with high $\gamma$ (9.0) and low ${\omega }^{\mathbf{Soc}}$ (3.0). This agent believes in high volatility and will be driven to continue sampling their neighbors, which will lead them to take longer to converge towards an idea. However, given their $\gamma$, the agent does converge towards the first sampled idea. Subfigure (d) shows an agent shows an agent with high $\gamma$ (9.0) and high ${\omega }^{\mathbf{Soc}}$ (9.0). This agent believes in low volatility and will be driven to sample the same neighbor very quickly, which will lead them to converge towards an idea quickly.