The Immediate Exchange model: an analytical investigation

Abstract

We study the Immediate Exchange model, recently introduced by Heinsalu and Patriarca [Eur. Phys. J. B 87: 170 (2014)], who showed by simulations that the wealth distribution in this model converges to a Gamma distribution with shape parameter $2$. Here we justify this conclusion analytically, in the infinite-population limit. An infinite-population version of the model is derived, describing the evolution of the wealth distribution in terms of iterations of a nonlinear operator on the space of probability densities. It is proved that the Gamma distributions with shape parameter $2$ are fixed points of this operator, and that, starting with an arbitrary wealth distribution, the process converges to one of these fixed points. We also discuss the mixed model introduced in the same paper, in which exchanges are either bidirectional or unidirectional with fixed probability. We prove that, although, as found by Heinsalu and Patriarca, the equilibrium distribution can be closely fit by Gamma distributions, the equilibrium distribution for this model is {\it{not}} a Gamma distribution.

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Cited by in corpus (7)

• Rigorous results for the distribution of money on connected graphs
• Rigorous proof of the Boltzmann-Gibbs distribution of money on connected graphs
• Rigorous results for the distribution of money on connected graphs (models with debts)
• Duality and stationary distributions of the "Immediate Exchange Model" and its generalizations
• Entropy dissipation and propagation of chaos for the uniform reshuffling model
• Econophysics Through Computation
• Thermal Equilibrium in D-dimensions: From Fluids and Polymers to Kinetic Wealth Exchange Models