How much is a noisy signal worth?

Question

An asset is worth either \0 or \100, each equally likely. You can buy it for \50. Before deciding, you may purchase a signal that correctly indicates the true value 80\% of the time, that is, it says "high" with probability 0.8 when the asset is worth \100, and says "low" with probability 0.8 when it's worth \0. What is the most you should pay for the signal?

Accepted Answer

Without the signal. Buying at \50 has expected value 0.5(100) + 0.5(0) - 50 = 0. It's a fair price, so you're indifferent, expected profit \0. Information is only valuable if it changes your decision. With the signal, update, then act. By symmetry the signal says "high" with probability 	frac12 and "low" with probability 	frac12: P(	ext{high}) = 0.5(0.8) + 0.5(0.2) = 0.5. Posterior after "high": P(V=100 \mid 	ext{high}) = \frac{0.5 \cdot 0.8}{0.5} = 0.8. So if the signal says "high," buying has expected value 0.8(100) - 50 = +\30, you buy. Posterior after "low": P(V=100 \mid 	ext{low}) =…

How much is a noisy signal worth?

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