StPetersburgParadox - Daniel Bernoulli and the St....

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Unformatted text preview: Daniel Bernoulli and the St. Petersburg Paradox A L TEX file: StPetersburgParadox — Daniel A. Graham, June 19, 2005 Suppose you are offered the chance to play the following game. A fair coin will be tossed until a head appears. If a head occurs for the first time on the nth toss then you will be paid 2 n dollars. How much would you be willing to pay to play this game? The St. Petersburg Paradox is that although the expected value of the payoff is infinite, 2 1 (1/2)1 + 22 (1/2)2 + · · · = 1 + 1 + · · · = ∞, no one seems to be willing to pay anything approaching this sum to play this game. Daniel Bernoulli’s resolution of the paradox was to suppose that utility is not linear in the payoff but, instead, strictly concave, e.g., ln(x). What certain payoff, c , would an individual with such a “Bernoulli” utility function regard as equivalent to the St. Petersburg lottery? This c would have to solve ∞ (1/2)i ln(2i ) ln(c) = i=1 = ∞ i=1 i/2i ln(2) = 2 ln(2) = ln(4) and thus c = 4 — a rather modest value to place upon the infinite expectation. Page 1 of 1 ...
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This document was uploaded on 10/20/2011.

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