Engine · Chapter 1, Observation, Beliefs, and Decision

What a signal is worth

When does a risk signal change a decision, and how much is it worth at each prior?

The decision

Prior ρ (%) 10 probability that the consignment carries contraband

Inspection cost k 1

Loss L if released 20 cost of releasing contraband

The signal

β = P(flag | contraband) % 80

α = P(flag | clean) % 20

compare a second signal

β₂ % 95

α₂ % 40

The book's scenarios

Value of the signal

\frac{17}{50}

≈ 0.3400

expected loss $- 1$ without the signal (inspect everything), $- \frac{33}{50}$ with it.

The rule

After a flag	posterior $\frac{4}{13}$	inspect
After a clear	posterior $\frac{1}{37}$	release

Inspect when the posterior exceeds $k / L = \frac{1}{20}$ .

D The book's base case: value $- 1$ without the signal, $- \frac{33}{50}$ with it, worth $\frac{17}{50}$ . Agreement, exactly.

signal (β, α) = (4/5, 1/5); breakpoints at ρ = 0, 1/77, 1/20, 4/23, 1
second signal (19/20, 2/5); breakpoints at ρ = 0, 8/369, 1/20, 12/31, 1
The curve is piecewise linear; the dots are its exact breakpoints, where a decision switches. Where the two curves cross, neither signal is better for every decision.

What the rule does, per consignment

Inspected: $\frac{13}{50}$
Contraband found: $\frac{2}{25}$
Contraband missed: $\frac{1}{50}$
Clean consignments inspected: $\frac{9}{50}$

The arithmetic

$V_{0} = max (- k, - ρ L) = - 1$

$P (flag) = ρβ + (1 - ρ) α = \frac{13}{50}, ρ_{flag} = \frac{ρβ}{P ( flag )} = \frac{4}{13}$

$P (clear) = \frac{37}{50}, ρ_{clear} = \frac{1}{37}$

$V_{1} = P (flag) max (- k, - ρ_{flag} L) + P (clear) max (- k, - ρ_{clear} L) = - \frac{33}{50}, VoI = V_{1} - V_{0} = \frac{17}{50}$

The model it runs

Model O1 · p. 54

What is the signal worth, and when is one signal better than another for every decision?

1Decision-makers: one
2Time: one decision
3Information: a binary experiment $(β, α)$ about the hidden state; prior $ρ$
4Commitment: none (one decision-maker)
5Payoffs: costs $c_{I}$ (inspect), $c_{M}$ (contraband released)
6Strategies: decision rules: signal $\mapsto$ \{inspect, release\}, possibly randomised

Analysis contract

(i) concept: Bayesian expected-utility maximisation; value of information $VoI =$ value with the signal minus value without
(ii) deviations: all four deterministic rules (randomised rules are mixtures of them)
(iii) beliefs: Bayes' rule from the stated prior
(iv) selection: ties broken towards inspection; values do not depend on it
(v) attainment: attained; exact closed form, checked against enumeration of all rules