From Ponzi Schemes to Benign Investment Dynamics: modelling Collapse, Stability, and a Path to Sustainability

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.