Emergence of Power-Law and Other Wealth Distributions in Crowd of Heterogeneous Agents

This study investigates the emergence of power-law and other concentrated distributions through a feedback loop model in crowd interactions. Agents act by their response functions to observations and external forces, while observations change by the aggregated actions of all agents, weighted by their respective influence, i.e. power or wealth. Agents wealth dynamically adjust based on the alignment between an agents actions and observation outcomes: agents gain wealth when their actions align with observed trends and lose wealth otherwise. A reward function, that describes the change of agents wealth at each time step, manifests the differences of response functions of agents to observations. When all agents responses are set to zero and feedback loop is broken, agents wealth follow a normal or lognormal distribution. Otherwise, this response-reward iterative feedback mechanism results in concentrated wealth distributions, characterized by a small number of dominant agents and the marginalization of the majority. Contrasted to past studies, such concentration is not limited only to asymptotic behavior at the upper tail for large variables, nor does it require the reward function to be linear to agents previous wealth as formulated in random growth model and network preferential attachment. Probability density functions for various distributions are more visually distinguishable for small values at the lower tail. In application of this model, key differences in income and wealth distributions in the US vs Japan are attributed to different response functions of agents in the two countries. The model applicability extends beyond social systems to other many-body systems with analogous feedback mechanisms, where power-law distributions represent a rare subset of general concentrated outcomes.