[论文解读] From Ponzi Schemes to Benign Investment Dynamics: modelling Collapse, Stability, and a Path to Sustainability

Bernhard R. Parodi|arXiv (Cornell University)|Feb 21, 2026

Economic theories and models被引用 0

一句话总结

本文提出三种离散时间投资模型，具有闭式解（几何、准逻辑斯蒂增长和基于 SIR 的模型），在共同的资本预算等式下统一庞氏式崩溃与良性、有限期投资动态。

ABSTRACT

The population and capital dynamics of three stylized investment systems are mathematically described using discrete-time difference equations with closed-form solutions. The models share a common capital budget equation but differ in their demographic laws, which are geometric, quasi-logistic, or epidemiologic (SIR-based). The quasi-logistic model is designed as an analytically tractable non-Ponzi investment system: it generalizes the geometric model (and, in the limit of a constant growth rate, reproduces classical Ponzi dynamics) while closely mirroring the behaviour of an SIR-based model with decreasing effective growth. In all cases, promised returns are modeled as fixed per-period payouts on initial investment with principal repaid upon exit, so that aggregate liabilities depend only on the current number of active investors. Within this unified framework, classical Ponzi schemes arise as special cases that inevitably collapse, while suitable parameter choices in the quasi-logistic and SIR-based versions generate finite-horizon, legally benign "no-Ponzi game" investment schemes with analytically transparent conditions for collapse, stability, and sustained operation.

研究动机与目标

• 为可能同时表现出崩溃与可持续性的投资体系建立统一的数学框架的动机。
• 发展三种人口学模型（几何增长、准逻辑斯蒂增长、基于 SIR 的增长）并给出闭式资本动态。
• 识别决定崩溃与可行性的参数各自的区间，并勾勒良性运作的条件。
• 提供分析性洞见，说明锁定期和有限投资者池如何影响长期结果。

提出的方法

• 定义具有固定每期支出和本金偿还的共同资本预算恒等式。
• 推导三种人口学法则下的闭式资本动态：几何增长、准逻辑斯蒂增长、以及基于 SIR 的增长。
• 引入一种新颖的准逻辑斯蒂人口学，产生有限-Horizon 的非庞氏 regimes。
• 给出离散时间模型的精确/显式解，并与 SIR 变体进行比较。
• 分析初始资本、锁定时间和人口学速率等参数如何影响稳定性与崩溃。

Figure 1: Growth rates $n_{t}$ for the quasi-logistic growth model and for a couple of SIR-models. (For better visualization, only the interpolation lines connecting the discrete data points will be shown in this figure and in all other figures.) Top panel: Sigmoidally decreasing growth rates $n_{t}

实验结果

研究问题

• RQ1 在何种参数条件下，庞氏式计划崩溃、持续或变为良性？
• RQ2 不同的投资-人口学法则（几何、准逻辑斯蒂、SIR 基于）如何影响资本动态与可持续性？
• RQ3 在统一模型中，准逻辑或基于 SIR 的框架是否能产生有限 Horizon、法律上良性的投资计划？
• RQ4 锁定期和有限投资者池在崩溃与可行性之间的过渡中起到怎样的作用？

主要发现

• 三个离散时间模型具有闭式解描述在不同人口学下的资本动态。
• 在不利参数下，庞氏型崩溃会内生出现，而准逻辑斯蒂和基于 SIR 的模型可以产生有限 Horizon 的良性方案。
• 准逻辑模型将几何增长嵌入其中作为特例，并映射出具有佐吉罗型投资者基础的 SIR 类下降增长。
• 统一框架通过共同的预算恒等式将经典庞氏骗局、脆弱系统和良性集体收入产品联系起来。
• 关键参数区间划分了崩溃与稳定的边界，并指示通向可持续投资动态的路径。
• 本文提出一种在受控条件下可行的新型临时性工具。

Figure 2: Demographic development for the quasi-logistic model. Developments are shown under different lock-up period conditions: without exits from the system (lock-up period $T=200$ , representing $T\rightarrow\infty$ , dashed lines) and with exits after $T$ = 30 periods of participation (solid li