Pyramid Schemes and the Unexpected Hanging Paradox

The Unexpected Hanging Paradox is an interesting paradox which is essentially described as follows:
A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.
Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the “surprise hanging” can’t be on Friday, as if he hasn’t been hanged by Thursday, there is only one day left – and so it won’t be a surprise if he’s hanged on Friday. Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.
He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.
The next week, the executioner knocks on the prisoner’s door at noon on Wednesday — which, despite all the above, was an utter surprise to him. Everything the judge said came true.” (from Wikipedia)

If you haven’t come across this paradox before, I don’t want to spoil your fun by pointing out the obvious self-contradicting assumptions involved in this paradox. It won’t take long to figure it out yourself, and in any case, a detailed analysis and exhaustive bibliography on this paradox (at least from a logical standpoint) can be found in this freely available paper by Tim Chow. 

My purpose here is to bring your attention to a real world scenario that is very similar to this paradox in many respects, but more like its inverse or reversal. The scenario I refer to is that of Pyramid schemes (a more elaborate version of Ponzi schemes that involves its victims in victimizing even more people) which is may be described as:

“a fraudulent business model that involves promising participants payment or services, primarily for enrolling other people into the scheme, rather than supplying any real investment or sale of products or services to the public” (form Wikipedia)

Now let us look at the paradox involved here.

There are a limited number of people in the world, and so the number of people who would participate in the scheme are also limited. People joining in the last level would not join because they knows there’s nobody left to enroll. If people joining in the last-but-one level know of this, they would also not join because they will not have anybody to enroll. We can similarly extend this (as in the case of the condemned prisoner in Unexpected Hanging Paradox), and that means even at the very first level, no one would want to join this Pyramid. But there are actually people who join such schemes, and hence the paradox. 

By terming this a paradox, I am not claiming that this is very difficult to explain. I am just emphasizing that human behavior at times is very surprising. A very casual analysis (as shown in the figure below borrowed from Wikimedia) is enough to convince oneself of the non-sustainability of such a scheme. However, there are still such schemes in vogue, and there are many who would gladly be part of them.

There are broadly 3 categories of people who get involved in such schemes. First, is the group of people who really don’t know what is involved, and think there is some truth or value in it because they trust their friends or acquaintances who introduce them to the. Then there are those who know what is involved, want to make money, and (perhaps due to their “moral conditioning”) also think it is ethical to engage in such activities. I have personally talked to some who said they were just doing a business like any other businessman, and what they are earning are genuine profits. But what they don’t see is that a business needs to create some value, and the value cannot be derived from a scheme where you give your money to somebody hoping somebody else (say, two others) will pay you even more and so you can have a net profit. The point they miss is that money has its value only as a measure of contribution a person makes for the society, by means of which he can claim certain privileges and services in return. So wanting to make money without contributing any value to the society is unethical, even in cases where it is legal.

Finally, there are those who know that this scheme is a fraud and that some people down the pyramid will surely lose their money, but don’t care about it as long as they are able to make money. The sad reality is that it is such people who are most likely get away with making most of the money early on and move on to another similar venture. The only consolation is that those who lose money foolishly in such frauds are often those who wanted to make quick and easy money. To resolve the Pyramid Scheme paradox through logic is easy – we just need to note that the implicit assumption that people always act rationally and ethically is unfounded. And as long as this assumption is unfounded, we’ll continue to hear about such schemes in newer names and forms, but pretty much unchanged in essence. 

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2 thoughts on “Pyramid Schemes and the Unexpected Hanging Paradox

  1. Well written. To add to this, readers, thinking that direct selling companies which sells products and services through word of mouth networks where distributors are trained and taught sales skills are also pyramid schemes, can deprive you of many possible low risk opportunities right under your nose in case you ever want to learn and become a businessman. What do you think , Mr.Raman? cheers

    1. Well, Ranjith, I would still ask my readers exercise extreme caution before being involved with direct selling companies that offer "low risk", high return "business" opportunites, because if they really want to learn and become good businessmen, there are many other genuine options available. I must reemphasize that holding on to our values is much more important than making money unethically.

      But I can't deny that there could be a few direct selling companies out there which are genuine sellers of a product through a different sales mechanism. So let me try to give a Litmus test for users to decide whether or not to entertain these products


      If you were presented the same product at the same price as just a product, and not as a "business opportunity", will you take it? If you wouldn't, don't take it even when presented as a business opportunity, because to sell it to others also you'd need to present it as a business opportunity. And such products which are passed on for a perceived value as a business opportunity more than their intrinsic value are the hallmark of a pyramid scheme.

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