Decision Tree Exam Training
Exam decision trees usually have more than one stage: you decide, something uncertain happens, then you decide again. The method stays the same. Lay out every branch, then roll back from right to left, taking expected values at chance nodes and the best option at each decision node.
Multi-stage decision trees explained simply
For a multi-stage tree, roll back from right to left: take expected values at chance nodes and the best option at each decision node. In the example, launching now is worth 60, while testing first is worth 76 after the test's cost of 10, so test first (figures in $000s).
What this decision tree exam lesson covers
- How to draw a multi-stage decision tree from a word problem
- How to roll back the tree from right to left
- How to handle a second decision after new information
- How to work out the value of a test or extra information
- Presentation tips that earn method marks
Decision tree exam formulas, explained
- pᵢ
- probability of branch i
- Valueᵢ
- value at the end of branch i
- option
- each branch you could choose at that point
How to solve a two-stage decision tree step by step (worked example)
Launching now costs 100 (in $000s): 50% chance of a 300 payoff, 50% chance of 20. Or run a test market for 10 first. Test is favorable 50% of the time (then success chance is 90%) and unfavorable 50% (success chance 10%). Ignore time value. Launch now or test first?
- Launch now: 0.5 × 300 + 0.5 × 20 − 100 = 60
- After favorable test, launch: 0.9 × 300 + 0.1 × 20 − 100 = 172
- After unfavorable test, launch would give 0.1 × 300 + 0.9 × 20 − 100 = −52, so don't launch (0)
- Test first: 0.5 × 172 + 0.5 × 0 − 10 = 76
- Compare: 76 > 60
Answer: Test first. Expected value is 76 vs 60 for launching now, so the test adds 16 (in $000s) after paying its cost of 10.
Launch now vs test first
| Launch now | Test first | |
|---|---|---|
| Up-front cost | 100 | 10 for the test, then 100 if you launch |
| What you learn first | Nothing | Whether the market looks favorable |
| Bad case | Launch anyway (−80 if it fails) | Don't launch (0) |
| Expected value | 60 | 76 |
Why plan for several futures, not one? David's take
This tree is a small version of what I call critical simulations. The usual critical path assumes one goal and one best route. I prefer branching out depending on how things go and how iterations come out, like alternate universes. Here, a cheap test (10) shows which universe you're in before the big spend, and it's worth 16 more than launching blind. The honest disclaimer: there are infinite possible branches, so you choose a few plausible ones and say why.
From David's concept: Critical Simulations. This is David's opinion, alongside the standard textbook answer above.
Harder decision tree exam questions
What is the most you should pay for the test?
Before its cost, testing is worth 0.5 × 172 = 86, against 60 for launching now. So the test is worth paying up to 26.
What would perfect information be worth here?
Knowing the outcome, you'd launch only on success: 0.5 × (300 − 100) = 100. That's 40 more than launching now, so no test is worth more than 40.
Common decision tree exam questions
How do you solve a decision tree exam question?
Draw the tree left to right with squares for decisions and circles for chance events. Write probabilities and payoffs on each branch. Then roll back from right to left, computing expected values at circles and choosing the highest value at squares.
How do you calculate the value of information in a decision tree?
Compare the best expected value with the information, before paying for it, against the best expected value without it. The difference is the most you should pay. If the test costs less than that, it is worth buying.
Do you discount cash flows in a decision tree?
If the payoffs happen in different years, yes. Convert them to present values first or use expected NPV at each node. Many exam questions simplify by giving present values or saying to ignore the time value of money.
Why must probabilities on a chance node add up to 1?
Branches from a chance node should cover every possible outcome, and exactly one of them will happen. If the probabilities do not sum to 1, an outcome is missing or a number is wrong, and the expected value will be off.