Many real decisions are not made once. A company may decide whether to test a product in one city before deciding whether to launch nationally; a contractor may decide whether to bid, and if the bid is won, whether to subcontract; a developer may drill a test well and then decide whether to build.
The second choice depends on what happened after the first, and the chance of each result may itself depend on earlier choices. Listing these as a table of options is clumsy. A decision tree shows the whole sequence on one diagram. Expected values calculated from the end of the tree back to the start show the best strategy at every point.
This topic explains how to build a tree, the two kinds of node, how to roll back the tree, how to include costs and what the answer means, with two worked examples, one of which tests whether a pilot study is worth its cost. It builds on the earlier topic on expected values and the decision rules. The next topic covers the value of perfect and imperfect information.
In 30 seconds
A decision tree is drawn from left to right, from the first decision to the final outcomes, and analysed from right to left, which is called rolling back. There are two types of node. A decision point (drawn as a square) is where the manager chooses. An outcome or chance point (drawn as a circle) is where nature decides, and the branches carry probabilities that add to 1.
At each chance point calculate the expected value of the branches. At each decision point take the best option, after deducting the cost of that option, and carry that figure back. The result at the start is the expected value of the best strategy, together with the decisions to take at each stage.
To judge a pilot study or test, compare the expected value of the strategy with the test (after its cost) and without it. Limits: the probabilities and values are estimates; the expected value is an average and ignores risk; and trees become unwieldy with many stages. Full marks come from a correct tree, the figures at each node and a clear statement of the strategy.
Building and rolling back a decision tree
The nodes and branches
- Decision point (square). The manager chooses between options. One branch leaves the square for each option. Any cost of an option is written on its branch.
- Outcome point (circle). A result that is not in the manager’s control, such as high or low demand, or a successful or failed development. One branch leaves the circle for each possible outcome, labelled with its probability; the probabilities leaving a circle add to 1.
- End of a branch. The pay-off of that path: profit, net cash flow or present value, before or after the costs on the way, as long as all are treated the same way.
A tree is built from the left: begin with the first decision, then draw what can happen after each option, then any further decision that follows each outcome, and so on until every branch ends in a pay-off. It is analysed from the right.
Rolling back
- Start at the end of each branch with the pay-offs.
- At an outcome point, multiply each pay-off by its probability and add: this is the expected value (EV) at that point. Write it above the circle.
- At a decision point, compare the EVs of the options, each after deducting the cost on its branch, and take the highest. Mark the rejected options with a double stroke. Write the best figure above the square.
- Move left and repeat until the first decision is reached.
- State the strategy: the choice at the first decision, and the choice to make at each later decision depending on what happens.
Using the tree: value of a test
A tree can show whether it is worth getting more information first. Draw the option to buy the test, with its cost, followed by the test result (favourable or unfavourable) and the decision that depends on it. Then compare the roll-back value of testing (after the cost of the test) with the value of deciding now.
If testing gives the higher EV, it is worth doing; the difference is the amount by which the test improves the decision. The next topic develops this as the value of perfect and imperfect information.
Advantages, limitations and good practice
Advantages. The tree makes the sequence of decisions and chances explicit, so nothing is missed. It breaks a complex problem into simple steps. It shows the strategy to follow whatever happens, not just a single choice. It encourages the manager to estimate the probabilities and pay-offs. And it can be used with spreadsheets to test the effect of changing the probabilities.
Limitations. The probabilities and pay-offs are estimates, and they may be little more than guesses. The EV is an average; for a one-off decision the manager will receive one of the pay-offs, not the average, so the risk is not shown and a risk-averse manager may choose differently. Trees become large and hard to read when there are many stages or many outcomes at each.
Pay-offs received at different times should be discounted to present values, which adds to the work. Qualitative factors, such as reputation, cannot be shown on the tree and must be weighed separately.
Good practice. Label every branch; check that the probabilities at each circle add to 1; deduct the cost of each option on its own branch, once; use the same basis (net or gross of cost) for all the pay-offs; and write the decision in words at the end.
Layouts for decision trees
- Draw the tree from left to right, with probabilities and pay-offs.
- Roll back from the right: EV at each circle; best option at each square, net of its cost.
- Value of the strategy = the figure at the first decision.
- Compare options (with or without a test, launch or not) and state the decisions in order.
Money is in naira and each answer is rounded once, at the end.
Worked examples
Example 1: a single-stage tree. A launch costs ₦500,000. There is a 60% chance of success, with a net inflow of ₦1,200,000, and a 40% chance of failure, with an inflow of ₦100,000. The alternative is not to launch. Required: the decision.
- Outcome point.
EV of inflows = 0.6 × 1,200,000 + 0.4 × 100,000 = 720,000 + 40,000 = ₦760,000. - Decision point.
Launch: 760,000 − 500,000 = ₦260,000.
Do not launch: ₦0. - Decision. Launch, with an expected gain of ₦260,000. The actual result will be either +₦700,000 or −₦400,000, so a very risk-averse company might still decline.
Example 2: a pilot study. A launch costs ₦600,000. Demand is high with probability 40%, giving an inflow of ₦2,000,000, or low with probability 60%, giving ₦300,000. A pilot study costs ₦100,000. It gives a positive result with probability 50%; if positive, the probability of high demand is 65%; if negative, it is 15%. After the pilot the company can launch or stop. Required: should it pilot, launch directly, or do neither?
- Launch directly.
EV of inflows 0.4 × 2,000,000 + 0.6 × 300,000 = 800,000 + 180,000 = 980,000
less cost 600,000 = ₦380,000. - After a positive pilot.
EV of inflows 0.65 × 2,000,000 + 0.35 × 300,000 = 1,300,000 + 105,000 = 1,405,000
less 600,000 = ₦805,000.
Launch. - After a negative pilot.
EV of inflows 0.15 × 2,000,000 + 0.85 × 300,000 = 300,000 + 255,000 = 555,000
less 600,000 = −₦45,000.
Do not launch: ₦0. - Pilot, rolled back.
0.5 × 805,000 + 0.5 × 0 = 402,500
less the pilot cost 100,000 = ₦302,500. - Decision. Launch directly: ₦380,000 is more than ₦302,500. The pilot improves the decision by only ₦22,500 (402,500 − 380,000), which is far less than its cost of ₦100,000, so it is not worth doing.
The probabilities are consistent: 0.5 × 0.65 + 0.5 × 0.15 = 0.40.
Expect variants: a tree with decisions at two stages, costs at several branches, a tree in which a decision to abandon is possible, and a question that asks for the maximum price to pay for a test.
Common mistakes
- Rolling forward. Always start at the right-hand end.
- Probabilities that do not add to 1 at a circle.
- Forgetting the cost on the branch, or deducting it twice.
- Choosing at a circle. The manager chooses only at squares.
- Ignoring the option to stop after an unfavourable test result.
- Stopping at a number. State the strategy in words.
Quick revision
- Square = decision; circle = chance. Build left to right; roll back right to left.
- Circle: EV. Square: best option net of cost.
- Test worth doing if EV with the test (net of its cost) exceeds EV without.
- Limits: estimates, average not outcome, size.
How the marks are earned
- Tree: structure, labels and probabilities.
- Roll-back: EV at each node, net of costs.
- Conclusion: the strategy, and the comparison with and without a test.
Practice questions
These are original questions, written in the style of the Skills examination. Work each one before you open the answer.
Recall check: multiple choice
Question 1. A square on a decision tree shows:
- A chance event
- A decision point
- A pay-off
- A probability
Show the answer
B. The manager chooses here.
Question 2. A circle on a decision tree shows:
- A decision
- A cost
- An outcome point, where an expected value is calculated
- The end of the tree
Show the answer
C. The branches carry probabilities.
Question 3. A decision tree is analysed:
- From right to left, which is called rolling back
- From left to right
- In any order
- From the middle
Show the answer
A. Later decisions must be known first.
Question 4. At a decision point you choose:
- The lowest EV
- The most likely outcome
- The first branch
- The highest EV after deducting the cost of the option
Show the answer
D. For costs, the lowest EV.
Question 5. At an outcome point you:
- Calculate the expected value of the branches
- Choose the best branch
- Ignore probabilities
- Add the costs
Show the answer
A. Pay-off × probability, summed.
Question 6. A cost written on a branch is:
- Ignored
- Added to the pay-off
- Deducted when comparing options
- Shown as a probability
Show the answer
C. Deduct it once, on its own branch.
Question 7. Decision trees are especially useful for:
- Single decisions with no uncertainty
- Multi-stage decisions where later choices depend on earlier outcomes
- Fixed budgets
- Variance analysis
Show the answer
B. The whole sequence is shown.
Question 8. A limitation of decision trees is that:
- They show no probabilities
- They are always small
- They cannot show costs
- Probabilities and pay-offs are estimates and the EV ignores risk
Show the answer
D. Large trees are also hard to read.
Question 9. A pilot study is worth doing only if:
- The improvement in expected value exceeds its cost
- It is cheap
- It is popular
- The result is favourable
Show the answer
A. Compare the EV with and without it.
Question 10. Rolling back means:
- Choosing at random
- Starting at the first decision
- Starting at the ends of the branches and working back to the first decision
- Ignoring later stages
Show the answer
C. It gives the strategy for each stage.
Section B style: written questions
Question 11 (10 marks). (a) Explain the two types of node in a decision tree. (4 marks) (b) Explain how a tree is rolled back. (3 marks) (c) State THREE limitations of decision trees. (3 marks)
Question 12 (15 marks). A project costs ₦4m. It succeeds with probability 70%, giving a present value of inflows of ₦9m, or fails with probability 30%, giving ₦2m. The company could alternatively sell the project rights for ₦1.5m.
(a) Describe the tree. (3 marks) (b) Calculate the expected value of investing. (4 marks) (c) State the decision. (3 marks) (d) Discuss the limitations of the analysis. (5 marks)
Question 13 (20 marks). A launch costs ₦8m. Demand is high with probability 35%, giving inflows of ₦26m, or low with probability 65%, giving ₦6m. A market test costs ₦1m. It is favourable with probability 50%; if favourable, the probability of high demand is 60%; if unfavourable, 10%. After the test the company may launch or stop.
(a) Calculate the expected value of launching without the test. (5 marks) (b) Roll back the tree for testing first. (9 marks) (c) State the recommendation. (2 marks) (d) Discuss the advantages and limitations of decision trees. (4 marks)
Model answers
Show the model answer to Question 11
(a) A decision point, a square, is where the manager chooses between options; one branch leaves for each option (2). An outcome point, a circle, is where an uncertain event occurs; its branches carry probabilities that add to 1, and the expected value is calculated there (2).
(b) Start at the ends of the branches; at each circle calculate the EV; at each square choose the best option after its cost; work back to the first decision (3).
(c) Any three:
- estimates of probabilities and pay-offs are uncertain
- EV is an average that ignores risk
- large trees are hard to read
- timing needs discounting
- qualitative factors are not shown (1 each)
Show the model answer to Question 12
(a) A decision point with two options: invest (cost ₦4m) or sell the rights (₦1.5m). The invest branch leads to an outcome point: success (0.7, ₦9m) or failure (0.3, ₦2m) (3).
(b)
- EV of inflows 0.7 × 9 + 0.3 × 2 = 6.3 + 0.6 = ₦6.9m
- less cost 4 = ₦2.9m (4).
(c) Invest, since ₦2.9m exceeds ₦1.5m from selling (3).
(d) Any: probabilities are estimates; EV is not the actual result, which will be +₦5m or −₦2m; risk is not shown; timing should be discounted; non-financial factors are omitted (5).
Show the model answer to Question 13
(a)
- EV of inflows 0.35 × 26 + 0.65 × 6 = 9.1 + 3.9 = ₦13.0m
- less 8 = ₦5.0m (5).
(b)
- Favourable.
0.6 × 26 + 0.4 × 6 = 15.6 + 2.4 = 18.0 - less 8 = ₦10.0m, so launch (3).
- Unfavourable.
0.1 × 26 + 0.9 × 6 = 2.6 + 5.4 = 8.0 - less 8 = ₦0, so the company is indifferent and should stop (2).
- Expected value with the test.
0.5 × 10.0 + 0.5 × 0 = 5.0 - less the cost of the test 1.0 = ₦4.0m (4).
(c) Launch without the test (₦5.0m against ₦4.0m). The test does not change the decision in a way that adds value, so it is not worth its ₦1m cost (2).
(d) Advantages: shows the whole sequence; breaks a problem into steps; gives a strategy; makes estimates explicit. Limitations: estimates are uncertain; EV ignores risk; large trees are hard to read (4).
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