C04 · Lesson 2 of 22 · 16 min · 5 hands-on tasks
Value of Information
Halden won the school contract. Before choosing who drills, a ground-radar survey could say whether there's rock down there, for $15,000. Is it worth it? Work out what perfect information is worth, update the odds with Bayes' rule when the survey can be wrong, and find the most you should ever pay for a test.
🎬 Situation
Good news: Halden won the Millbrook contract. The $40,000 bid cost is spent now (it's sunk: nothing can bring it back, so it no longer affects the choice). The profits ahead are:
- Own crew: $400,000 on soft clay (70%), only $50,000 on hard rock (30%).
- Subcontract to Deepcore: a sure $250,000.
The own crew wins on expected value ($295,000). But if Halden knew there was rock, it would subcontract. A firm called GroundRadar offers a survey for $15,000. It isn't perfect: it spots rock 80% of the time when rock is there, and raises a false alarm 10% of the time when it's clay.
Ingrid asks: "Is $15,000 for a survey money well spent?"
🎮 Play
Turn test results into odds. This is Bayes' rule, done as a table (a "Bayes box"). Out of every survey, how many say rock, and of those, how many are right?
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | |||||
| 2 | |||||
| 3 | |||||
| 4 |
- D2: rock AND a rock result = prior × hit rate, =B2*C2
- D3: clay AND a (false) rock result (copy D2 down)
- D4: all rock results, =D2+D3
- E2: chance of rock after a rock result, =D2/D4
- E3: chance it's really clay after a rock result, =D3/D4
Decide whether to buy the survey. The tree includes the survey's own cost. Fold it back, then move the accuracy sliders: how good must a survey be before it's worth $15,000?
■ squares are your choices, ● circles are chance. Fold back from the right (the ends): tap a circle to work out its expected value, then pick the best move at each square, until you reach the first decision. Values: profit for Halden on the Millbrook job.
- Says rock 31%After a rock result
- Rock 77.4%$35,000
- Clay 22.6%$385,000
- Rock 77.4%
- $235,000
- Says clay 69%After a clay result
- Rock 8.7%$35,000
- Clay 91.3%$385,000
- Rock 8.7%
- $235,000
- Says rock 31%
- No survey
- Rock 30%$50,000
- Clay 70%$400,000
- Rock 30%
- $250,000
🏷️ Name it
| Term | Meaning | Millbrook |
|---|---|---|
| Sunk cost | Money already spent that no choice can recover: leave it out | The $40,000 bid |
| Prior | Your probability before new evidence | Rock 30% |
| Sensitivity (hit rate) | Chance the test says "yes" when the answer is yes | Says rock 80% of the time when it's rock |
| Specificity | Chance the test says "no" when the answer is no | Says clay 90% of the time when it's clay |
| Posterior | Your probability after the evidence (Bayes' rule) | Rock 77% after "rock", 8.7% after "clay" |
| EVPI (perfect information) | EV deciding after seeing the truth − EV deciding now | $355k − $295k = $60k |
| EVSI (sample information) | EV with an imperfect test − EV deciding now | $332.5k − $295k = $37.5k |
Bayes' rule in one line: posterior = (prior × chance of this result if true) ÷ (chance of this result overall).
Three tests before paying for information: (1) could any result change my decision? (2) is the gain bigger than the price? (3) is there a cheaper way to learn the same thing (a call to someone who drilled nearby)?
🖍️ Like you're five: peeking at the weather app before choosing park or home is only useful if a "rain" forecast would keep you home. If you'd go to the park anyway, the peek is worth nothing.
Real life: test markets, pilot stores and market research are information purchases: a retailer that would launch a product whatever the research says is wasting the research budget. In medicine, a famous puzzle shows why priors matter: when a disease is rare, most positive results from a fairly accurate screening test can be false alarms, which is why doctors confirm positives with a second test (Gigerenzer's Calculated Risks has the mammography example).
🔨 Do it: 5 tasks
0 / 5 doneTask 1 of 5 · The decision without a survey
The bid cost is now spent (sunk), so profits are: own crew $400,000 on clay (70%) or $50,000 on rock (30%); subcontractor a sure $250,000. What is the own crew's expected value?
Task 2 of 5 · What perfect information is worth
A magic test would tell Halden the truth before choosing: on clay it uses the own crew ($400,000), on rock it subcontracts ($250,000). EV with perfect information minus the $295,000 of deciding now = EVPI. What is it?
Task 3 of 5 · How often does the survey say "rock"?
The survey says rock 80% of the time when there is rock, and wrongly says rock 10% of the time when it's clay. Rock has a 30% prior chance. What % of surveys come back "rock"?
Task 4 of 5 · If the survey says rock
Given the survey says rock, what's the chance there really is rock? (In %, nearest whole number.)
Task 5 of 5 · The survey's real value
With the survey: 31% of the time it says rock and Halden subcontracts ($250,000); 69% of the time it says clay and the own crew is worth $369,565 (rock is then only 8.7% likely). What's the survey worth (EV with it, before its cost, minus $295,000)?
🧩 Case: "Buy the best test money can buy"
⭐ VIPA second firm, DeepScan, offers a seismic survey that is never wrong, for $70,000. Ingrid likes the sound of it: "Certainty is worth paying for." The ground-radar survey costs $15,000 and is right 80% of the time on rock and 90% on clay.
Should Halden buy DeepScan's perfect survey?
What would you do? VIP members make the call, see what happens next across 2 decisions, and get the expert debrief.
🔁 Remember it
A test result could never change your decision. Its value is...
EVPI is...
A test is 80% accurate on rock, prior rock 30%, and it says rock. The chance of rock is now...
Information only has value if it could change what you do. Expected value of perfect information (EVPI) = EV if you could decide after seeing the truth − EV deciding now: the ceiling on what any test is worth. A real test is imperfect: update your odds with Bayes' rule (prior × how likely the result is, then rescale), re-fold the tree for each result, and the gain is the expected value of sample information. Buy the test only if that gain beats its cost.
📅 Use it Monday
Find one piece of research, data or a report your team pays for (or wants to). Ask: what decision would it change, and how? If no realistic result would change anything, it's worth nothing for that decision. If it could, estimate the gain the way Kofi did and compare it with the cost.
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