Decision trees
Draw the sequence of choices and chance events, then compute backwards from the outcomes to find the best first move.
- Time cost
- 1–2 h
- Output
- A tree with rolled-back values at every node.
- Steps
- 6
Use when
- A decision leads to another decision, with uncertainty in between.
- You want to know what a future choice is worth before making the present one.
- Several people need to agree on structure before arguing about numbers.
Do not use when
- There is one decision and one chance event. Plain expected value is enough.
- The tree would exceed about twenty end nodes. Past that it obscures more than it shows.
Inputs required
- The sequence of decisions and chance events
- Probabilities at each chance node
- Payoffs at the leaves
Procedure
- 01
Draw forward
Left to right, in time order. Squares for decisions you control, circles for chance events you do not. Keeping them visually distinct is most of the method’s value.
- 02
Fill in the leaves
The total payoff along each path, including costs incurred on the way.
- 03
Check every chance node
Branches from each circle must sum to 1. This catches missing outcomes more reliably than any other step.
- 04
Roll back
Work right to left. At a chance node, take the expected value of its branches. At a decision node, take the best branch — you control it, so you will choose the maximum.
- 05
Read the first move
The value at the root is the value of the whole decision. The branch that produced it is what to do now.
- 06
Test the thin branches
Vary the probabilities you are least sure of. If the first move does not change, you do not need better estimates. If it does, that is exactly where to spend research.
Characteristic failure mode
Worked example
A decision about whether to run a paid trial before committing to a large purchase.
- 01Decide: trial (£3k) or commit directly.
- 02Chance after trial: promising 60%, not 40%.
- 03Second decision only exists on the promising branch.
- 04Roll back: committing directly is worth £41k; trial-then-decide is worth £47k net of its cost.
Result
The trial is worth £6k more than committing, not because it improves the outcome but because it buys the right to stop. That value is invisible without the tree.
Where to go next