A PROBABILITY MODEL OF A PYRAMID SCHEME Joseph L. Gastwirth, The George Washington University Periodically, the pyramid or "chain letter" scheme is offered to Americans under the guise of a business dealership. Recently, Glen Turner's Koscot Interplan- etary Cosmetics firm has been charged with pyramiding by the FTC, SEC and vari- ous state agencies [2]. The total loss to the public has been estimated to be 44 million dollars. The promoters offer people a dealership or sales job in which most of their remuneration comes from re- cruiting new dealers (or salespersons). The basic fraud underlying a typical pyr- amid scheme is that every participant cannot recruit enough other people to re- coup his investment, much less make a profit, since the pool of potential par- ticipants is soon exhausted. The usual method of prosecuting such schemes is to show that if the repre- sentation of the promotional brochures were valid (e.g., members could recruit two new people a month), then within a short period of time (about 18 months) the entire population of the U.S. would have to participate. Thus, the last mem- bers would have no one to recruit. Al- though this argument based on the geo- metric progression is sometimes rejected by courts as unrealistic [3], pyramid scheme operators have placed a quota (or limit) on the number of participants in a specific geographic area in order to evade this line of prosecution. This ar- ticle develops a probability model of this quota-pyramid scheme and the follow- ing results which also apply to unlimited schemes are derived: 1. The vast majority of participants have less than a 10% chance of recouping their initial investment when a small profit is achieved as soon as three people are re- cruited. 2. On the average, half of the partici- pants will recruit no one else and lose all their money. 3. On the average, about one -eighth of the participants will recruit three or more people. 4. Less than one percent of the partici- pants can expect to recruit six or more new participants. While the above results can be "approx- imately" derived by ordinary limit theo- rems, for purposes of legal cases an ab- solute statement that a probability is small is more useful than an "approxi- mate" statement. Thus, the above results are derived from a new probability bound on the sum of "small" binomial r.v.'s 65 which is related to previous work of Hodges and LeCam [4]. Description of One Pyramid Scheme - A recent legal case in Connecticut [6] illustrates the confounding of legitimate business enterprise with a pyramid oper- ation. People were offered dealerships in a "Golden Book of'Values" for a fee of $2500. In return for their investment dealers could earn money in two ways: In each geographic area dealers were to de- velop a "Book of Values" for eventual sale to the public. First, they were to sell advertisements to merchants for $195 apiece and could keep half as a commis- sion. Each advertisement offered a pro- duct or service at a discount, so that a "Book of Values" containing 50 to 100 discount offers could be sold to the pub- lic. The public was to pay $15 for the Book of Values, of which dealers were to keep $12. Second, a dealer had the right to recruit other dealers and was to re- ceive $900 for each new recruit. Since the creation of a complete "Book of Val- ues" for sale to the public takes a sub- stantial amount of time, clearly the re- cruitment of new dealers is the most lu- crative aspect of the venture. In the recruitment brochure the possi- bility of earning large sums of money was illustrated by the following example: A dealer will bring people to weekly "Op- portunity Meetings" and should be able to enroll other dealers at the rate of two per month. Thus, at the end of one year, the participant should receive $21,600 from the recruitment aspect alone. The prosecution showed that this misrepre- sents the earnings potential by asking the following question: "Suppose dealers who are enrolled can enroll two other dealers per month; as time went by, what would happen ?" Professor Margolin (of Yale) testified that there would be a tripling of the number of dealers per month and by the end of 18 months, the geometric progression would exhaust the population of the United States. Clearly the cited recruitment brochure is mis- leading as all participants cannot come close to earning the indicated amount of money. The "Golden Book of Values" pyramid system had an extra statistical nuance; i.e., there was a quota of 270 dealer- ships for the State of Connecticut. The Court noted that if each new dealer was successful in recruiting two dealers per month, only 27 would make a profit and the other 243 would lose money depending on how far down the pyramid they were.