A signal that cries "high" too often

Question

An asset is worth \0 or \100, each equally likely, and its fair price is \50. You may buy a signal with asymmetric accuracy: when the asset is truly worth \100 it says "high" 90\% of the time, but when the asset is worth \0 it still says "high" 40\% of the time (a false alarm). What is the most you should pay for this signal?

Accepted Answer

Without the signal. Buying at \50 has expected value 0.5(100) - 50 = 0: fair, so profit \0. Signal probabilities. The false-alarm rate inflates how often "high" appears: P(	ext{high}) = 0.5(0.9) + 0.5(0.4) = 0.45 + 0.20 = 0.65, \qquad P(	ext{low}) = 0.35. Posterior after "high": P(V = 100 \mid 	ext{high}) = \frac{0.45}{0.65} \approx 0.692, so buying is worth 0.692(100) - 50 \approx +\19, you buy. Posterior after "low": P(V = 100 \mid 	ext{low}) = \frac{0.5(0.1)}{0.35} = \frac{0.05}{0.35} \approx 0.143, worth 0.143(100) - 50 \approx -\36, so you pass. Expected profit with the signal. Only …

A signal that cries "high" too often

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