即时交换模式：一项分析研究
The Immediate Exchange model: an analytical investigation
摘要：
我们研究的即时交换模型，最近介绍了Heinsalu和Patriarca [Eur.第87章：一个女人170（2014）]，他通过模拟表明，该模型中的财富分布收敛于具有形状参数$2$的Gamma分布。在这里，我们证明这一结论分析，在无限人口的限制。一个无限的人口版本的模型推导出，描述的财富分布的概率密度空间上的非线性算子的迭代的演变。证明了带形状参数$2$的Gamma分布是该算子的不动点，并且从任意财富分布开始，该过程收敛于其中一个不动点.我们还讨论了在同一篇文章中介绍的混合模型，其中交换是双向的或单向的固定概率。我们证明了，虽然Heinsalu和Patriarca发现，平衡分布可以紧密地拟合的Gamma分布，该模型的平衡分布是{ \it{not} }的Gamma分布。
英文摘要：
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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