makes 20 copies and distributes them. When Jane gets a copy of a chain letter in the mail she habitually glances at it and throws it away without reading the body of text, figuring it wants her to send money. When she gets a copy of V she glances at the title and at the bottom of the letter where one might look to see who the sender was. There is no sender listed, but the words "Do not send money!" appear prominently. This communicates at once that this letter does not ask for money. Jane reads the full text and, since she is waiting to hear if she was selected for a part in play, she is persuading not to take a chance on bad luck and complies with the demand for 20 copies. Some others may react as Jane did: say the new postscript induces just one additional person per hundred to fully and effectively comply to the demand for 20 copies. So now for every 100 V letters received about 120 receipts are generated in turn. This is a weekly rate of growth of p = 120/100 = 1.2, and as long as this is maintained the circulation of this variation will undergo exponential growth. Using equation (3) for this example: y0=20, P=7, p=1.2, and the active population y after t days is y = 20*(1.2)t/7. With this growth rate, the population will double every month since y(t+30) / y(t) = (1.2)30/7 = 2.18 > 2. Starting with 20 copies, two years of such doubling would produce a quarter billion receipts per week - more than the number of adults in the United States. The scenario employed in this example is realistic: in fact the postscript "Do not send money!" did appear on a mainline luck chain letter around 1939 and rapidly expanded its numbers (> Section 4.6). Computer simulations by John Burkhardt provide additional evidence for the "great advantage of a small advantage." For twenty letters initially launched, and (in effect) a weekly rate of growth of p = 1., of 150 simulated launchings not one produced a lineage of over 1000 letters. But when the weekly rate of growth was increased slightly to 1.02 then 85 of the 150 simulating launchings still continued after 1000 generations {Meditations on the Chain Letter - link no longer available}. Immunization. In the example above, the model of exponential growth produced a doubling of circulation every month. Obviously such growth cannot be sustained for years. The number of possible recipients is limited, and there is an immunization effect whereby receiving more than one chain letter of the same category makes one less likely to comply with copy demands. If one variation of a luck chain is abundant, another variation may be deprived of the attention and resources required for making and distributing copies. Eventually the abundant variation will foul its own nest by the same process. Thus for population booms, the exponential growth model applies only at the onset. More sophisticated mathematical models of growth are available but we will not pursue that approach. Exponential decline. The exponential growth model may also be applied to a declining chain letter variation. In the John Doe example above, consider the fate of the unimproved mainline letters (call them variation "U"), which were just hanging on before the V letter appeared. For U,
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