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Risk-seeking comparison

Expected valueHard

A risk-seeking gambler values a payout of $x$ dollars as $x^{2}$ . Lottery A pays the larger of $2$ fair $6$ -sided dice. Lottery B pays $4.5$ dollars for sure. By how much does the expected utility of A exceed that of B?

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Hint

The probability that the larger die equals k is (2k − $1$ )/36.

Solution

The larger die equals $k$ with probability $\frac{2 k - 1}{36}$ , so $E [max^{2}] = \sum \frac{k ^{2} ( 2 k - 1 )}{36} = \frac{2 \times 441 - 91}{36} = \frac{791}{36}$ .
Lottery B has utility $4. 5^{2} = \frac{81}{4} = \frac{729}{36}$ .
The difference is $\frac{791}{36} - \frac{729}{36} = \frac{62}{36}$ , which equals $\frac{31}{18}$ .

Answer: $\frac{31}{18}$

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