A gambler values a payout of x dollars as √x. A lottery pays 64 dollars with probability 3/4 and 4 dollars with probability 1/4. What sure amount, in dollars, is worth exactly as much to the gambler as the lottery?

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A gambler values a payout of $x$ dollars as $x$ . A lottery pays $64$ dollars with probability $\frac{3}{4}$ and $4$ dollars with probability $\frac{1}{4}$ . What sure amount, in dollars, is worth exactly as much to the gambler as the lottery?

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Hint

Find the expected utility, then invert the utility.

Solution

The expected utility is $\frac{3}{4} \times 8 + \frac{1}{4} \times 2 = \frac{13}{2}$ .
The sure amount $c$ satisfies $c = \frac{13}{2}$ .
So $c = (\frac{13}{2})^{2} = \frac{169}{4}$ .

Answer: $\frac{169}{4}$

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