Learning models on rooted regular trees with majority update policy: Convergence and phase transition

Abstract

We study a model of social learning on rooted regular trees. An agent is stationed at each vertex of 𝕋𝑚, the rooted tree in which each vertex has precisely m children, and at any time step 𝑡 ∈ℕ0, the agent is allowed to select one of two available technologies: B and R. Let the technology chosen by the agent at vertex v of 𝕋𝑚, at time step t, be 𝐶𝑡⁡(𝑣). We begin with the independent and identically distributed (i.i.d.) collection {𝐶0⁡(𝑣) : 𝑣 ∈𝕋𝑚}, where 𝐶0⁡(𝑣) =𝐵 with probability 𝜋0. During the epoch t, the agent at vertex v performs an experiment that results in success with probability 𝑝𝐵 if 𝐶𝑡⁡(𝑣) =𝐵, and with probability 𝑝𝑅 if 𝐶𝑡⁡(𝑣) =𝑅. If the children of v are denoted 𝑣1,…,𝑣𝑚, the agent at v updates their technology to 𝐶𝑡+1⁡(𝑣) =𝐵 if the number of successes among all 𝑣𝑖 (where 𝑖 ∈{1,2,…,𝑚}) with 𝐶𝑡⁡(𝑣𝑖) =𝐵 exceeds, strictly, the number of successes among all 𝑣𝑗 (where 𝑗 ∈{1,2,…,𝑚}) with 𝐶𝑡⁡(𝑣𝑗) =𝑅. If these two numbers are equal then the agent at v sets 𝐶𝑡+1⁡(𝑣) =𝐵 with probability 1/2. In all other cases, 𝐶𝑡+1⁡(𝑣) =𝑅. We show that {𝐶𝑡⁡(𝑣) : 𝑣 ∈𝕋𝑚} is i.i.d. as well, with 𝐶𝑡⁡(𝑣) =𝐵 with probability 𝜋𝑡, where the sequence {𝜋𝑡}𝑡∈ℕ0 converges to a fixed point 𝜋, in [0, 1], of a function 𝑔𝑚. We show that for 𝑚 ⩾3, there exists a 𝑝⁡(𝑚) ∈(0,1) such that 𝑔𝑚 has the unique fixed point 1/2 when 𝑝 ⩽𝑝⁡(𝑚), and three distinct fixed points, of the form 𝛼, 1/2, and 1 −𝛼, for some 𝛼 ∈[0,1/2) when 𝑝 >𝑝⁡(𝑚). When 𝑚 =3, 𝑝𝐵 =1, and 𝑝𝑅 ∈[0,1), we show that the function 𝑔3 (i) has a unique fixed point, 1, when 𝑝𝑅 <√3 −1, (ii) has two distinct fixed points, one of which is 1, when 𝑝𝑅 =√3 −1, and (iii) has three distinct fixed points, one of which is 1, when 𝑝𝑅 >√3 −1. When 𝑔𝑚 has multiple fixed points, we also specify which of these fixed points 𝜋 equals, depending on 𝜋0. Finally, for 𝑚 =2, we describe the behaviour of 𝑔2 for all values of 𝑝𝐵 and 𝑝𝑅.