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

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.

Figures (8)

• 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 $n_t$ for the quasi-logistic model (according to equation \ref{['ntQL']}). Adopting a fraction $N_0/N=0.01$, the examples shown start with initial values $n$ as indicated in the legend, pass turning points at times $t_{TP}$ (equation \ref{['tTP']}) and at heights $\frac{1}{2}n$, and finally approach zero. The lower the value of $n$, the larger $t_{TP}$, and hence the slower the dropping of the growth rate. Bottom panel: Decreasing effective growth rates for two types of SIR-models, adopting the same parameter values $\beta=0.3, ..., 0.01$ and $\gamma=0.02$: the non-standard SIR-model (equation \ref{['ntnsSIR']}, dashed to solid lines) and, for comparison, the standard SIR-model (equation \ref{['ntSIR']}, dotted lines). For the purpose of illustration, equal values $\beta=n$ are chosen, hence in the case of additionally setting $\gamma=0$ the curves would coincide with those in the top panel.
• 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 lines). Sizes are normalized by $N=N_{t\,pool}+N_{t}+N_{t\,exit}$, where $N_{t\,pool}$, $N_{t}$, and $N_{t\,exit}=\sum_{k=0}^t \Delta N_k^{out}$ are the respective numbers of potential, current, and former investors in the system at time $t$.
• Figure 3: Traffic light scenarios: The evolution of capital and its associated contributions within a rudimentary investment framework is examined. The numerical dynamics can be traced sequentially. The interaction among the initial capital, compounded interest, the number of investors, and the particular lock-up period $T = 1$ yields qualitatively distinct outcomes. The accompanying graphs depict the temporal evolution of the capital and its aggregated contributions. The red light scenario corresponds to a final collapse (top panel), the yellow light scenario encounters profiting investors but a promoter with some losses (middle panel), and the green light scenario only sees agents with gains at termination (bottom panel).
• Figure 4: Capital dynamics with geometric growth ($n$ = constant). Parameter values used: constant rate of return on investment $r = 10\%$ and growth rate $n = r$ in the upper panel, but $n = 9.95\% <r$ in the lower panel. Market-based rate of return $i$ and duration $T$ of investments as indicated in the legends. Initial values $K_{0}^{pro}= 100$, $I_{0} = 3$, $N_{0}= 10$.
• Figure 5: Quasi-logistic investment dynamics: capital evolution for a Ponzi scheme-like model with a decreasing demographic growth rate $n_t$. Initial values are $K_{0}^{pro}= 100$, $I_{0} = 3$, $N_{0}= 10$, and fixed rates $n=10\%$, $r=5.2\%$, and $i=3\%$. Examples for schemes with quasi-logistic growth of member numbers (using formulas \ref{['QLKttsT']} and \ref{['QLKttgeT']}). The investment timespan (or lock-up period) $T$ varies as indicated, with the case $T=100$ periods representing here continued participation. Ultimately collapsing systems are shown in black ($T\ge 7$), while red lines represent surviving systems ($T\le 6$).