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Unformatted text preview: Daniel Bernoulli and the St. Petersburg Paradox
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)
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.
- Spring '09