Boltzmann social learning with heterogeneous rationality

This paper analyzes a novel social learning model in which, at each discrete time step, agents with private preferences repeatedly select actions via a softmax (Boltzmann) rule, and update their preferences based on public observations of others' choices. This work addresses a critical gap by introducing rational heterogeneity through agent-specific rationality parameters gamma-i. Unlike previous models, our approach accounts for the diverse ways individuals process social information by using a discrete-time deterministic mean-field approximation map. We establish fundamental equilibrium properties that were previously unexplored. In particular, we prove the existence of fixed points and show that, on complete graphs, every mean-field equilibrium is a consensus state, where all agents share identical preferences. We further derive sufficient conditions for the uniqueness of this equilibrium and its local asymptotic stability. Numerical simulations validate our theoretical findings and illustrate how rational heterogeneity and network structure interact to shape collective behavior in social learning systems.