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The Immediate Exchange model: an analytical investigation

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  • Guy Katriel

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 not a Gamma distribution. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2015

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  • Guy Katriel, 2015. "The Immediate Exchange model: an analytical investigation," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 88(1), pages 1-6, January.
    • Handle: RePEc:spr:eurphb:v:88:y:2015:i:1:p:1-6:10.1140/epjb/e2014-50661-7
    DOI: 10.1140/epjb/e2014-50661-7
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    1. repec:cup:cbooks:9781107013445 is not listed on IDEAS
    2. Sitabhra Sinha, 2005. "The Rich Are Different!: Pareto Law from asymmetric interactions in asset exchange models," Papers physics/0504197, arXiv.org.
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    Cited by:

    1. Nicolas Lanchier & Stephanie Reed, 2022. "Distribution of money on connected graphs with multiple banks," Papers 2201.11930, arXiv.org.
    2. Su, Ran & Fang, Zhi-Ming & Hao, Qing-Yi & Sheng, Chun & Fu, Yuan-Jiao, 2024. "The evolution of cooperation affected by unidirectional acceptability mechanism on interdependent networks," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).

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