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

Abstract

Abstract We study a model of social learning on rooted regular trees. An agent is stationed at each vertex of double struck upper T Subscript m <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> </mml:math> $\mathbb{T}_{m}$ , the rooted tree in which each vertex has precisely m children, and at any time step t element of double struck upper N 0 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> $t \in \mathbb{N}_{0}$ , 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 double struck upper T Subscript m <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> </mml:math> $\mathbb{T}_{m}$ , at time step t , be upper C Subscript t Baseline left parenthesis v right parenthesis <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> $C_{t}(v)$ . We begin with the independent and identically distributed (i.i.d.) collection StartSet upper C 0 left parenthesis v right parenthesis colon v element of double struck upper T Subscript m Baseline EndSet <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo stretchy="false" fence="false">{</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mo>:</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>v</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false" fence="false">}</mml:mo> </mml:math> $\{C_{0}(v)\,:\, v \in \mathbb{T}_{m}\}$ , where upper C 0 left parenthesis v right parenthesis equals upper B <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo>