C04 · Lesson 1 of 22 · 15 min · 5 hands-on tasks
Decision Trees
Kofi's first week at Halden Heat: should the company spend $40,000 bidding for a school heating contract, and if it wins, drill with its own crew or pay a subcontractor? Draw the choice as a tree, fold it back, and find the probabilities that would flip the answer.
🎬 Situation
Meet Kofi, the new decision analyst at Halden Heat. Halden installs ground-source heat pumps: it drills a borehole about 120 metres deep next to a house, runs a loop of pipe down it, and uses the steady warmth of the ground to heat the home. It does about 600 homes a year, at around $24,000 each.
On Kofi's first morning, the CEO, Ingrid, drops a folder on the desk: "Millbrook schools want three buildings heated. 40 boreholes. Fixed price. Do we bid?"
Here's what Kofi learns by lunch:
- Preparing a serious bid (site survey, design) costs $40,000, win or lose.
- Halden would bid $1.2M. Dana, head of sales, thinks the chance of winning is 50%.
- If Halden wins, it can drill with its own crew: cost $800,000 in soft clay, but $1,150,000 if the crew hits hard rock. The geology report says 30% chance of rock.
- Or it can subcontract to Deepcore Drilling for a fixed $950,000, rock or not.
Three people, three opinions. Kofi draws a tree.
🎮 Play
Fold the tree back. Start at the right: tap the circle to work out the own-crew expected value, pick the best way to drill, then work out the bid circle and make the first decision. Then move the sliders and watch when the best plan flips.
■ 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, after the $40,000 bid cost.
- Win 50%How to drill?
- Soft clay 70%$360,000 · 1.2M − 800k − 40k
- Hard rock 30%$10,000 · 1.2M − 1.15M − 40k
- Soft clay 70%
- $210,000 · 1.2M − 950k − 40k, sure
- Lose 50%−$40,000 · the bid cost is gone
- Win 50%
- $0
Play the bid 1,000 times. The tree says bidding is worth $107,500. But any single bid ends at one leaf. Run it and watch the average settle.
Nothing yet. Press a button to run some bids.
🏷️ Name it
| Term | Meaning | At Halden |
|---|---|---|
| Decision node ■ | A point where you choose | Bid or not; own crew or subcontract |
| Chance node ● | A point where the world chooses, with probabilities that add up to 1 | Win or lose; clay or rock |
| Payoff | What you end with at the end of a path | $360,000 for win + clay |
| Expected value (EV) | Probability × payoff, added up over a chance node | Own crew: 0.7 × 360k + 0.3 × 10k = $255k |
| Folding back (rolling back) | Working from the ends to the start: average at circles, pick the best at squares | Bid is worth $107,500 |
| Sensitivity analysis | Moving one input to see where the decision flips | Rock above 43% → subcontract |
How to draw a good tree: choices in time order (you decide, then learn, then decide again); every circle's branches cover everything that can happen and their chances add up to 100%; payoffs written at the ends as the total for that whole path (here, after the $40,000 bid cost).
🖍️ Like you're five: you can play in the park or stay in. If you go, it might rain. Draw the choices as a fork in the road, the weather as a coin flip, and a smiley face at every end. Then ask: which road gives the most smiles on average?
Real life: decision trees spread through business after Harvard's Howard Raiffa and Robert Schlaifer developed them in the 1960s; one of the first big uses was oil and gas drilling, studied by C. Jackson Grayson in Decisions Under Uncertainty: Drilling Decisions by Oil and Gas Operators (1960). Drug companies still value research projects this way: fewer than one in ten drugs that enter human trials is approved, so each trial stage is a chance node with a price tag.
🔨 Do it: 5 tasks
0 / 5 doneTask 1 of 5 · The own-crew circle
If Halden drills with its own crew, there's a 70% chance of soft clay (profit $360,000) and a 30% chance of hard rock (profit $10,000). What is the expected value of using the own crew?
Task 2 of 5 · Is bidding worth it?
Winning (50%) is worth $255,000 (the best move after a win). Losing (50%) costs the $40,000 spent on the bid. What is the expected value of bidding?
Task 3 of 5 · When does rock flip the choice?
Own-crew value = 360,000 − 350,000 × (chance of rock). The subcontractor is a sure $210,000. Above what chance of rock (in %, nearest whole number) should Halden subcontract?
Task 4 of 5 · The lowest win chance worth bidding for
Bidding is worth p × 255,000 − (1 − p) × 40,000. Below what chance of winning (in %, nearest whole number) is bidding not worth it?
Task 5 of 5 · What the $107,500 means
The tree says bidding is worth $107,500. What will Halden actually get from this one bid?
🧩 Case: "Bid low, win more"
⭐ VIPDana, head of sales, wants to cut the bid from $1.2M to $1.05M: "We'd win 75% of the time instead of 50%." At $1.05M, own-crew profit is $210,000 on clay and −$140,000 on rock; the subcontractor gives $60,000. The bid still costs $40,000 to prepare (already inside those figures).
Fold back the low-bid tree. What is bidding $1.05M worth?
What would you do? VIP members make the call, see what happens next across 2 decisions, and get the expert debrief.
🔁 Remember it
At a chance node (circle) you...
Why do we fold the tree back from the right?
A probability in your tree sits close to the value that flips the decision. You should...
A decision tree has squares (your choices) and circles (chance). Write the payoff at every end, then fold back from the right: at each circle take the expected value (probability × payoff, added up), at each square keep the best branch. The value at the far left is what the whole plan is worth on average. Then ask which probability would flip the decision: that's the number worth checking.
📅 Use it Monday
Pick one real decision on your desk this week. Draw it as a tree on paper: your options (squares), the 2–3 things that could happen (circles, with your best-guess chances), and a rough payoff at each end. Fold it back. Then find the one probability that would flip your choice, and ask: how sure am I about that number?
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