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Harmonic sequence paradox.

Article Excerpt
Summary. Informal evidence suggests that individuals are willing to pay only a finite and, typically, very low price for a specific lottery that converges to an infinite payment with probability one. The established decision theories (expected value, expected utility theory, cumulative prospect theory) cannot satisfactorily explain this low willingness to pay. The presented paradox strengthens the original and the super St. Petersburg paradox.

Keywords and Phrases: Expected value, EUT, Cumulative prospect theory, St. Petersburg paradox, Willingness to pay.

JEL Classification Numbers: C91, D81.

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Consider an urn that initially contains one white and one black ball. An individual draws one ball from this urn and receives one dollar (nothing) should the ball be white (black). Whatever the drawn ball happens to be, it is subsequently put back into the urn. Additionally, one more black ball is added to the urn. The individual then draws one ball again and the cycle continues ad infinitum. At each iteration, the drawn white ball pays off one dollar and the number of black balls is increased by one. What is a maximum price that a rational individual is willing to pay for participation in this lottery L?

Informal evidence suggests that an individual's willingness to pay (WTP) for lottery L is finite and, typically, very low (less than ten dollars). This finding is paradoxical because it cannot be easily accommodated within established decision theories, as demonstrated below. Let [p.sub.n](x) denote the probability of winning x [eqsilon] [0, n] dollars after n draws of a ball. Equation (1) defines the lottery distribution [p.sub.n] : [0, n] [right arrow] [0, 1] recursively. For large n, the lottery distribution pn has mean [[mu].sub.n] = 1/2 + 1/3 + ... + 1/n [congruent to]...

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