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  Theory of Expected Utility

Theory of Expected Utility:

St. Petersburg Paradox (Nicholas Bernoulli):

Suppose you had the opportunity to pay $100 and then play of the following gambles, each of which is fair bet.

• You get back $100.

• I toss a fair coin You receive $200 if heads.
Or 0 if tails.

• I roll a fair die You receive
$400 if 1, $70 if 2, $55 if 3, $25 if 4, $40 if 5, and $10 if 6.

All the gambles depicted above have expected value of $100 but would you be equally willing to play each one? For one thing the variances are dissimilar:

• 0

• 1/2 (200 −100)2 + 1/2 (0 −100)2 = 10,000

•1/6(3002+ 302 + 452 + 752 + 602 + 902)= 18,375

You might be more eager to play the gamble with the lower variance than the one with the higher variance.

This point is exemplified by what is called the St. Petersburg paradox. This was illustrious by Bernoulli a Swiss mathematician of the 18th century. He proposed a difference of the following gamble. Suppose a fair coin is tossed until it comes up heads. Your payoff depends on the number of tosses before heads appears for the first time. Recognizing that tosses of a fair coin are independent and that probabilities get multiplied together on successive tosses, your payoffs in Bernoulli’s game are constructed as follows:

$2 if heads comes up first on the first try ( p = 1/ 2 )
$4 if heads comes up first on the second try ( p = 1/ 4 )
$8 if heads comes up first on the third try ( p = 1/ 8 )
.
.
.
$ 2n if heads comes up first on the n-th try (p = 1/ (2)n )
.
.
.

The expected value of the gamble set out above is:


(1/2)2 + (1/4)4 + (1/8)8 + ....(1/2n)2n +.... = ∑ [(1/2n])2n = 1 + 1 + 1 + .... = ∞

However no one would pay an infinite amount to play this gamble. In fact, few would play much more than a few dollars. One reason might be that the discrepancy of this gamble is as well infinite as well as most people prefer lower variance (less uncertainty) to more.

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