Strategy A pays 6 or 14 with equal chances. Strategy B pays 2, 10 or 18 with chances 1/4, 1/2 and 1/4. For a payoff X, let S_X(t) = E[max(t − X, 0)]. What is S_B(12) − S_A(12)? Give a fraction in lowest terms.

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Shortfall of a wider bet

Expected valueMedium

Strategy $A$ pays $6$ or $14$ with equal chances. Strategy $B$ pays $2$ , $10$ or $18$ with chances $\frac{1}{4}$ , $\frac{1}{2}$ and $\frac{1}{4}$ . For a payoff $X$ , let $S_{X} (t) = E [max (t - X, 0)]$ . What is $S_{B} (12) - S_{A} (12)$ ? Give a fraction in lowest terms.

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Hint

Only payoffs below the target contribute to the shortfall.

Solution

For $A$ , only the payoff $6$ is below the target, with shortfall $12 - 6 = 6$ , so $S_{A} (12) = \frac{1}{2} \times 6 = 3$ .
For $B$ , the payoffs $2$ and $10$ are below it, with shortfalls $10$ and $2$ , so $S_{B} (12) = \frac{1}{4} \times 10 + \frac{1}{2} \times 2 = \frac{7}{2}$ .
The difference is $\frac{7}{2} - 3 = \frac{1}{2}$ .

Answer: $\frac{1}{2}$

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