Decision trees and the value of information
Use a workshop booking and a van reservation to decide how much a timely forecast is worth.
Build a small decision tree and estimate the maximum worth of additional information before committing to an uncertain choice.
Before the booking deadline
A ceramics studio owner can pay $20 to see booking-page traffic before she decides whether to rent a second room for Saturday. Last lesson, she already knew traffic was heavy. Step back to the morning before she saw the count. Should she buy the report even though it might tell her to keep the room she already has?
Both businesses and all figures are fictional. Assume each decision-maker can afford either choice and wants the highest expected additional profit. Paper and a calculator are enough.
Recall Lesson 039: The room costs $240 and brings in an extra $600 before the room fee only if demand is high. At what probability of high demand does booking break even?
Draw the decisions in time order
A decision tree is a map of what you can choose, what you might learn, and what happens next. Draw a square for a choice you control and a circle for an uncertain event. At the end of each path, write the payoff: here, the extra dollars earned or lost compared with using the original room.
Start with the simplest tree. The owner can decline the extra room for $0 additional profit or book it. Only the booking branch needs a split for demand, because declining pays $0 either way.
| Choice now | Saturday outcome | Probability | Additional profit |
|---|---|---|---|
| Decline room | Either demand level | 100% | $0 |
| Book room | High demand | 20% | $600 − $240 = $360 |
| Book room | Ordinary demand | 80% | $0 − $240 = −$240 |
Read the tree forward to check timing. Solve it backward: at a chance circle, multiply each payoff by its probability and add the results. At a decision square, keep the better available choice. You assign probabilities to demand because the owner cannot choose it. You compare booking with declining because she can choose between them.
Booking gives (0.20 × $360) + (0.80 × −$240) = −$120 in expected profit. Declining gives $0. The best decision without the report is worth $0. Use that as the starting point when pricing information.
What the traffic report is worth
Add an earlier square: buy the report or skip it. On the buy branch, draw a circle for what the report says. After each possible report, draw a fresh booking square. That order matters: the owner can act on traffic, but Saturday attendance is still unknown when she books.
| Traffic report | Chance of this report | Chance of high demand given report | Expected profit if booked, before report fee | Best action |
|---|---|---|---|---|
| Heavy | 25% | 60% | 0.60 × $600 − $240 = $120 | Book |
| Not heavy | 75% | 1 in 15 | (1 ÷ 15) × $600 − $240 = −$200 | Decline for $0 |
The 60% tells you how to act after a heavy report. The 25% tells you how often you expect to have that opportunity before seeing the report. Work backward through the two booking squares first. Keep $120 on the heavy branch and $0 on the other. Then average: (0.25 × $120) + (0.75 × $0) = $30.
The value of information is the expected improvement over the best decision available without it, before paying to obtain it. Here that is $30 − $0 = $30 per workshop. At a $20 fee, buy the report: it improves expected profit by $10. Pay the fee once even if the room is declined. MIT’s decision analysis course explains why a test’s value comes from the better choices its results make possible. MIT OpenCourseWare: Value of Test.
A ceiling on the price
Suppose the owner could learn Saturday’s actual demand with certainty before noon. She would book on every high-demand date, earning $360, and decline on every ordinary date, earning $0. The expected value of perfect information is (0.20 × $360) + (0.80 × $0) − $0 = $72 per workshop.
That $72 is a ceiling for information about this same demand decision. The real report is worth only $30 because some heavy-traffic dates have ordinary demand and some quiet dates fill up. Timing matters too: a perfectly accurate report delivered after the booking deadline is worth $0 for this booking if the choice cannot be changed.
Exercise: Buy the dispatch forecast?
Allow 8–10 minutes. You manage a depot for an established distributor. By noon you can reserve an extra van for tomorrow or keep the regular fleet. Compared with keeping the fleet, reserving adds $900 profit if demand is high and loses $300 if demand is ordinary. These figures include the van fee and all other costs that change. Keeping the fleet gives $0 additional profit in either case.
High demand has a 40% chance. A forecaster offers a report for $100 before noon. It is favorable half the time. High demand occurs after 70% of favorable reports and 10% of unfavorable ones. The report does not affect demand or van availability. There are no other uses for the report or costs of obtaining it.
- Draw the no-report branch. Calculate the better choice and its expected profit.
- Draw the report branch with both results. After each result, compare reserving with keeping the fleet. What is this plan worth before the fee?
- Calculate the most you would pay for this report and for perfect information. Decide whether to pay $100, and say what you will do about the van.
- Identify one claim about the forecast you would check before using these prices in a negotiation.
Try the tree before reading the answer.
Answer
Without the report, reserving gives (0.40 × $900) + (0.60 × −$300) = $180. Keeping the fleet gives $0. Reserve the van. The $180 you can already earn in expectation belongs in both price comparisons.
After a favorable report, reserving gives (0.70 × $900) + (0.30 × −$300) = $540, so reserve. After an unfavorable report, it gives (0.10 × $900) + (0.90 × −$300) = −$180, so keep the fleet for $0. Before the fee, acting on the report gives (0.50 × $540) + (0.50 × $0) = $270.
The report is worth $270 − $180 = $90. You can also see where every dollar comes from: half the time you would still reserve; the other half you avoid a reservation that loses $180 on average. At the quoted fee the report leaves $270 − $100 = $170, which is $10 less than reserving without it. Decline the report and reserve the van.
Perfect information would let you reserve only on high-demand days. Expected profit would be (0.40 × $900) + (0.60 × $0) = $360. Subtract the $180 available without information: the ceiling is $180. The quoted $100 is below that ceiling but above what this particular forecast delivers.
Before negotiating, check timestamped forecasts against actual demand on comparable depot days, including unfavorable reports. Confirm that the forecast was available before noon. A model can give the correct price for the assumed accuracy even when those assumptions do not fit this depot.
Recall answer: $240 ÷ $600 = 40%. At that probability, expected extra earnings just cover the room fee. The traffic report has value because the heavy and quiet results fall on opposite sides of that threshold.
Applying this lesson
Keep your depot tree as part of this week’s decision tree. Beside the first square, write the action you would take without a report, the result that would change it, and the most you would pay to see the report in time. The next case uses the same reasoning when a manufacturer receives a quality alert.
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