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What an oracle is worth

Expected valueHard

A trade makes $20$ with posterior chance $\frac{3}{10}$ and otherwise loses $5$ . Passing makes $0$ . An oracle would tell you the outcome before you decide. What is the most you should pay for the oracle, in expected value terms? Give a decimal.

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Hint

Compare the best blind action with what you earn when you act only on good news.

Solution

Blind, trading earns $\frac{3}{10} \times 20 - \frac{7}{10} \times 5 = 2.5$ , which beats passing.
With the oracle you trade only when it works, earning $\frac{3}{10} \times 20 = 6$ .
The oracle is worth $6 - 2.5 = 3.5$ .

Answer: $3.5$

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