02 Emergence-Power-Law-Wealth-Distributions 2024
The paper develops a general feedback‑loop framework for a crowd of heterogeneous agents whose actions (decisions) respond to external forces and to observations of the system’s state. Each agent i has a response coefficient to external shocks (Ci) and to the observed change (Bi). Agents possess wealth/power Ai that weights their influence on the observation. The model specifies how observations evolve as the weighted sum of agents’ actions and how agents’ wealth updates via a reward function that depends on the alignment between their action and the resulting observation. When response functions are non‑zero, the iterative response‑reward loop drives the system toward concentrated wealth distributions, ranging from exponential‑like to power‑law forms, depending on the shape of Bi and boundary conditions. The paper contrasts this mechanism with random‑growth and preferential‑attachment models, showing that linear‑ish reward functions plus minimum‑wealth constraints can generate power‑law tails, but power‑law is a special case of a broader class of “general concentrated distributions.” Analytical derivations link response functions to the emergence of specific PDFs, CDFs, and rank‑size laws. Empirical illustrations compare US and Japan income and wealth data, demonstrating steeper power‑law slopes in the US, which the author attributes to higher average B_i values. Simulations illustrate wealth concentration over time, and the discussion extends the framework to city sizes, market returns, firm sizes, citation counts, online content popularity, collective intelligence, and the inverse problem of designing response functions for desired macro‑outcomes. The conclusions emphasize the central role of the feedback loop and heterogeneous response functions in generating inequality, the diagnostic value of lower‑tail PDFs, and the potential for policy or AI‑agent design to shape system‑level distributions.
Topics
Feedback‑Loop Model Linking Agent Responses to Wealth Concentration
Introduces the core dynamic where agents' decisions (dS_i) respond to external forces (dE) and observations (dO) via coefficients C_i and B_i, and where observations aggregate weighted actions of agents (A_i), creating a closed loop that updates both observations and agents' wealth.
Mathematical Derivation of Response and Reward Functions
Presents the formal equations for decision updates, observation aggregation, and wealth reward (dA_i), including special cases (binary outcomes, linear reward, boundary conditions) and the definition of the aggregate response D(t)=∑A_iB_i.