Why a picture helps
Many game choices are not a single step. You decide whether to attack, and if the attack works you decide whether to press on. Keeping the probabilities and payoffs of every stage in your head is hard. A decision tree puts the whole problem on paper so that you can check each part separately.
A tree has two kinds of points, called nodes.
- Decision nodes are where you choose. Draw them as squares, with one branch for each option.
- Chance nodes are where luck decides. Draw them as circles, with one branch per possible outcome, each labelled with its probability. The probabilities leaving a chance node must add up to 1.
At the end of each path you write the final payoff. Costs you pay along the way are subtracted on the branch where you pay them.
Building a tree: an invented example
Suppose you are playing a board game with points as the score. On your turn you can choose between two actions.
- Route A (safe): you gain 10 points for certain.
- Route B (risky): you pay 4 points to start. There is a 0.5 chance it works, in which case you reach a second decision. If it fails, you gain nothing further.
If Route B works, you then choose again:
- Take the sure thing: gain 20 points.
- Push on: a 0.4 chance of gaining 60 points, otherwise 0.
These numbers are made up for practice and are not from any real game.
Folding back: solve from the right
The key technique is to start at the end of the tree and work backwards, replacing each node with its value. This is often called folding back.
Step 1, the second decision. Compare the two options.
- Sure thing: 20.
- Push on: 0.4 x 60 + 0.6 x 0 = 24.
Pushing on has the higher average, 24 against 20, so at that square you would choose it. The square’s value is 24.
Step 2, the first chance node on Route B. There is a 0.5 chance of reaching the 24-value square and a 0.5 chance of ending with 0.
0.5 x 24 + 0.5 x 0 = 12.
Step 3, subtract the cost. You paid 4 to start, so Route B is worth 12 - 4 = 8.
Step 4, the first decision. Route A is worth 10, and Route B is worth 8. On average Route A is better.
The tree has told you something that is easy to miss by intuition alone: the exciting 60-point prize sits behind two uncertain steps and an entry fee, and by the time those are accounted for it is not worth more than the plain 10.
Testing how sensitive the answer is
Probabilities in real games are usually estimates, so it is worth asking how much the answer depends on them. Change the chance that Route B works from 0.5 to 0.6. Then:
- 0.6 x 24 + 0.4 x 0 = 14.4
- 14.4 - 4 = 10.4
Now Route B is slightly ahead of Route A, 10.4 against 10. A move of 0.1 in one estimate flipped the ranking. That tells you the two routes are close, so other factors such as your position in the game may reasonably decide it. A close call is itself useful information.
Common mistakes
Probabilities that do not add to 1. If the branches from a circle sum to something else, an outcome is missing or a number is wrong. Check this first.
Forgetting costs on the way. A fee, a lost turn or a spent resource belongs on the branch where it occurs, even if it feels like part of the setup.
Treating later choices as fixed. The point of folding back is that you assume you will choose well at each later square. If you ignore the option to stop or switch, you undervalue early choices that keep options open.
Dependent branches. When what happens at one step changes the odds at the next, as in drawing cards without replacement, the probabilities on later branches must be recalculated for each path. Probability trees in counting problems work the same way, and the tree here is simply that idea with decisions added.
Limits of the method
A tree uses expected value, so it ranks options by their average. It does not say anything about the spread of outcomes, so two routes with equal averages can feel very different. If a bad result would knock you out of the game, you may sensibly prefer the lower-variance route even when its average is a little lower. Trees also grow quickly: with many options and stages, you will have to prune unlikely branches or simplify.
None of this makes a risky game safe. If every branch includes a negative expectation, as it does in casino games with a built-in house edge, no tree will turn it into a positive one. The tree can only show you which choice loses least, or confirm that the real decision is whether to play at all.
A habit worth keeping
Draw the tree before you commit, even roughly, on scrap paper. Write the probabilities you are least sure about with a question mark, then try a higher and lower value for each. If the answer holds across your range, you can act with more confidence. If it flips, accept that the choice is close and decide on other grounds.