A model of social learning on rooted regular trees

Sunday, October 27, 2024, 10:10 -- 10:40

Let $\mathbb{T}_{m}$ be the rooted regular tree in which each vertex has $m$ children. An agent is stationed at each vertex of $\mathbb{T}_{m}$, and is allowed, at any point in time, to select one of two available technologies: $B$ and $R$. Let $C_{t}(v)$ denote the technology chosen by the agent at vertex $v$ at time-step $t$, for $t \in \mathbb{N}_{0}$. During the epoch $t$, the agent at $v$ performs an experiment that results in success with probability $p_{B}$ if $C_{t}(v)=B$, and with probability $p_{R}$ if $C_{t}(v)=R$. Letting $v_{1}, v_{2}, \ldots, v_{m}$ denote the children of $v$, the agent at $v$ updates their technology to $C_{t+1}(v)=B$ if the number of successes among all $v_{j}$, $1 \leqslant j \leqslant m$, with $C_{t}(v_{j})=B$, strictly exceeds the number of successes among all $v_{j}$, $1 \leqslant j \leqslant m$, with $C_{t}(v_{j})=R$. If these two numbers are equal, the agent at $v$ sets $C_{t+1}(v)=B$ with probability $1/2$. In all other cases, $C_{t+1}(v)=R$. We study the limit, as $t \rightarrow \infty$, of the process $\{C_{t}(v): v \in \mathbb{T}_{m}\}$, when \begin{align*} (1) \qquad & p_{B}, p_{R} \in [0,1] \text{ and } m=2; \\ (2) \qquad & p_{B}=p_{R}=p \text{ and } m \text{ is an arbitrary positive integer with } m \geqslant 3; \\ (3) \qquad & p_{B}=1, p_{R} \in [0,1) \text{ and } m=3. \end{align*} Phase transition phenomena are shown to take place in the latter two scenarios. This work can also be interpreted as a model for the diffusion of technologies throughout a population of agents.