A sharper signal on a bigger bet

Question

An asset is worth either \0 or \200, each equally likely. You can buy it for \100. Before deciding, you may buy a signal that correctly indicates the true value 90\% of the time: it says "high" with probability 0.9 when the asset is worth \200, and "low" with probability 0.9 when it is worth \0. What is the most you should pay for the signal?

Accepted Answer

Without the signal. Buying at \100 has expected value 0.5(200) + 0.5(0) - 100 = 0. The price is fair, so you break even and are indifferent. A signal only has value if it changes your decision. With the signal. By symmetry it says "high" with probability 	frac12: P(	ext{high}) = 0.5(0.9) + 0.5(0.1) = 0.5. Posterior after "high": P(V = 200 \mid 	ext{high}) = \frac{0.5 \cdot 0.9}{0.5} = 0.9, so buying is worth 0.9(200) - 100 = +\80, you buy. After "low" the posterior is 0.1 and buying is worth 0.1(200) - 100 = -\80, so you pass and take \0. Expected profit with the signal: 0.5(80) + 0.5(0) =…

A sharper signal on a bigger bet

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