Blog 3: Diffusion in Ponzi Schemes
Diffusion, often used to denote the process of a new idea or technology being adopted within a market or social sphere, is a fascinating component of studying networks. In this course, we particularly focused on the adoption of technologies such as Social Networks, where the value of the product continues to increase with adoption, and we substantial spreads due to interconnectivity from one consumer to the next. This is a remarkable way to understand how products or phenomenons, which started off with only a few users, slowly can expand their user base via the connections of the current users, sometimes resulting in exponential grow.
In particular, we focused on modelling contagion, which makes diseases such as COVID-19 so deadly.
We used the following equation to compute the basic reproductive number, R = pk, where p = contagion probability and k = the number of neighbours that a sick person will see. Intuitively, this provides an estimate of the number of people 1 person will make sick, and acts as a prediction of the contagion’s growth rate. If R < 1 the disease will decay until it is gone, but if R is greater than 1, it will continue to spread and grow exponentially.
Ponzi Schemes
The thought occurred to me. Could we use this contagion model and the process of diffusion to try understand Ponzi schemes?
A Ponzi scheme (named after the infamous Charles Ponzi) is a long-standing form of fraud that uses funds from new investors to pay profits to earlier investors. Such illegal schemes are often used to exploit investors by offering unrealistic returns. For many, being caught in a Ponzi scheme would be much more harrowing then catching the flu, so perhaps we can borrow the terminology from our contagion model to explain a Ponzi scheme! It turns out we can quite nicely model the spread of Ponzi schemes using the growth rate R = pk, where p now is the probability an individual will invest and k is the number of people a current participant is exposed to. What we see quite clearly is that R must be greater than 1 to continue providing profit to the higher levels of the pyramid. If a large group of k is continually exposed to new scheme participants and there is a high probability p that people join, we would expect the growth rate R = pk to be very large. Such a scheme can therefore grow tremendously fast, to the inevitable dismay of many investors. As the money trickles up the pyramid, the majority of funds typically end up in the network’s earliest joining nodes (those close to patient zero, again adopting the contagion terminology)
Source: https://www.50minutes.com/title/ponzi-scheme/
In a particularly successful Ponzi scheme, such as one with R = 2 (as depicted in the drawing above) we would anticipate 1048576 investors in the bottom after 20 rounds of the pyramid (2^20 nodes). However, most rational people would agree that as the pyramid gets larger, the growth rate will probably decrease. This is the same concept in contagion models, where R and diffusion in general can decrease due to herd immunity. Here, there may be a lack of sufficient investors after a certain time of growth. Eventually, when the pyramid can no longer find enough people to join the scheme (R <= 1), or too many investors cash out, the Ponzi scheme begins to dismantle, and almost everyone (most particularly the last group of nodes) ends up losing the majority of their investment.
Indeed, there is historical precedent for the uncontrolled nature of diffusion in Ponzi networks. For instance, Bernie Madoff, arrested for securities fraud, ran the largest Ponzi scheme in existence. At the time of his arrest, Madoff’s firm had liabilities exceeding $64.8 billion.
Given the detrimental historical record, and the near certain failure of such schemes, one might ask a sensible question.
Why do people join and participate in such schemes to begin with?
We can understand the motivation of specific nodes through the Payoff models we discussed in class. When determining whether to adopt a new technology, each node asks: what benefit do I get when joining, and what pay off do I get if not joining? If some node in the network is exposed to investors in the Ponzi scheme, who are consistently making high returns in their investments, the individual pay off to them of joining may be perceived as very high. Joining the network becomes a rational decision, as they anticipate the same high returns. This is also analogous to other points in our course, where we discovered that the socially optimal solution is sometimes not the actual outcome achieved (such as Braess’s paradox).
Most interestingly, in today’s interconnected and globalised world network diffusion can happen even faster than before. A study (https://cyberleninka.org/article/n/737868) from Jinggangshan University in China noted that social media platforms can cause can cause our R naught / growth rate to be even larger. This makes sense as people are exposed to a much larger K than ever before, as social media networks are significantly larger than personal networks. Even if p is low, a large K can compensate for p and cause a large growth rate of the scheme.
So, strong adoption of social networks appears to be driving diffusion in pyramid scheme. This study fascinates me, because we have two entirely different networks: One is social in nature, connecting people through globalisation and technology. And yet the adoption of this network is causing a large R in Ponzi scheme networks. This made me appreciate the fact that networks can clearly have symbiotic relations, and have unexpected affects on totally different kinds of networks.