Decision-making & prioritisation
Decision Trees
A branching map of decisions, chance events and payoffs that turns a sequential choice under uncertainty into a tree you can price. Fold expected values back from the end branches to the root to find which first move is worth most.
Also known as Decision tree analysis, Rollback analysis. First set out by John F. Magee (populariser); Howard Raiffa and Robert Schlaifer (underlying theory) in 1964; the primary source is cited in full below.
- Format
- Mapping
- Level
- Business unit · Product · Team
- Best for
- Evaluate options · Assess risk
- Decision stage
- Explore options · Decide
- Difficulty
- Intermediate
- Time to apply
- Two to four hours for a disciplined small tree; days where probabilities need real elicitation or research.
Plate · The model
The components
Decision nodes
Points, conventionally drawn as squares, where the decision-maker chooses among alternatives. Each branch is an act you control. The root of the tree is always a decision node: the choice in front of you now.
Signals of strength
Every alternative genuinely available is drawn, including 'wait' and 'do nothing' · Later decision nodes capture real future flexibility, not a fixed plan · Each branch is an action someone could actually authorise
Chance nodes
Points, conventionally drawn as circles, where the outcome is decided by events beyond your control: demand, weather, a competitor's launch, a planning ruling. Each branch carries a probability, and the branches together must exhaust the possibilities.
Signals of strength
Probabilities sum to one at every node · Outcomes are defined sharply enough to say later which one occurred · Estimates come from data or named judgement, not committee smoothing
Outcomes and payoffs
The terminal branches of the tree, each carrying the value of ending up there: revenues less costs on a consistent basis, discounted where the horizon warrants it. Payoffs must include follow-on value, since a small payoff that preserves an option can beat a large one that closes doors.
Signals of strength
All payoffs are on the same basis, ideally net present value · Downside branches include the full cost of failure, not a rounded zero · Option value and salvage value are counted, not just first-order cash
Roll back the expected values
The solution procedure: fold values back from right to left, averaging over probabilities at chance nodes and choosing the best branch at decision nodes. Rollback converts a tangle of possible futures into a single recommended policy and a price for the decision.
Signals of strength
The arithmetic is visible and checkable, not buried in a model · Pruned branches are kept in view so the choice can be defended · The result is stated as a policy, with the sensitivity of the root value reported alongside
When it earns its keep
- You face a sequential decision under uncertainty, where today's choice opens or closes later options whose value depends on how chance events resolve.
- You can put credible numbers, even rough ones, on the probabilities and payoffs involved, and the stakes justify the modelling effort.
- You want to value flexibility explicitly, for instance the option to start small and expand if demand appears, which gut feel and single-scenario spreadsheets systematically misprice.
- A decision keeps being re-argued from intuition and you need the assumptions on the table where they can be challenged one number at a time.
And when it doesn't
- The uncertainty is genuinely unquantifiable. If nobody can defend a probability estimate even as a range, the tree will manufacture false precision; use scenario planning instead.
- The decision is a one-shot choice among options with no downstream branching. A weighted decision matrix or cost-benefit analysis gets there faster.
- Outcomes ride on what an intelligent adversary chooses rather than on chance. Trees model nature's moves, not a rival's strategy; that calls for game-theoretic thinking.
- The values at stake are not commensurable in money or a single utility measure, for instance safety against cost, without a fight over the conversion itself.
How to run it
Before starting, gather the inputs the analysis depends on:
- A clear statement of the root decision, the alternatives at each later decision point, and the time horizon.
- Probability estimates for each chance event, elicited from data or from the people best placed to judge, with their reasoning recorded.
- Payoff estimates for each terminal branch: cash flows, costs and any salvage or follow-on value, on a consistent basis such as net present value.
- The decision-maker's risk stance, since a firm that cannot survive the worst branch should not choose on expected value alone.
- 1
Frame the decision and horizon
Define the choice you face now, the later choices it may lead to, and how far ahead the tree should reach. Magee's original chemical-plant example ran a ten-year horizon; most trees mislead by stopping at the first payoff and ignoring what the first decision makes possible.
- 2
Lay out the tree
Draw the canonical branching form left to right: a square for each decision node with a branch per alternative, a circle for each chance node with a branch per outcome. Alternate them as the logic of the situation dictates until every path ends in a terminal branch.
- 3
Attach probabilities and payoffs
Assign each chance branch a probability, with the branches at each node summing to one, and each terminal branch a payoff. Record the source of every number; the tree's conclusions are only as good as its worst estimate.
- 4
Roll back the tree
Work from the terminal branches to the root. At each chance node, compute the expected value of its branches; at each decision node, keep the alternative with the best expected value and prune the rest. The value that arrives at the root prices the whole decision, and the surviving branches are the recommended policy.
- 5
Test the sensitivity
Vary the shakiest probabilities and payoffs and watch whether the recommended first move flips. A conclusion that survives wide swings in the inputs is robust; one that flips on a five-point probability change tells you where to spend your research budget.
- 6
Decide, and keep the tree
Take the first move the analysis supports and keep the tree as a living record. When a chance node resolves in reality, you re-enter the tree at that point with better information, which is exactly the situation it was built to handle.
Reading the result
A drawn tree with probabilities and payoffs on every branch, an expected value at the root, a recommended policy identifying the best choice at each decision node, and a sensitivity view showing which estimates the conclusion actually depends on.
- The recommendation is the branch kept at the root, but the policy is the whole set of kept branches: it tells you what to do later depending on how each chance event resolves.
- Read the expected value as a long-run average over many similar decisions, not a promise about this one. The firm still experiences a single branch.
- Give the sensitivity analysis equal billing. If the answer flips inside the honest range of an estimate, the tree's real output is 'go and improve that estimate'.
A worked example
A commercial solar installer prices its bid on a fixed-price contract
A UK commercial solar installation firm is bidding a fixed price to fit a 900-panel rooftop array for a distribution park. The main uncertainties are the grid operator's connection ruling, which can force expensive substation works, and steel-frame roof condition, which surveys can only partially resolve before contract. The firm can bid low to win share, bid high to cover the risks, or pay for an intrusive pre-bid survey before deciding. It builds a tree to price the three openings.
- Decision nodes
- Root decision: bid low at 610,000 pounds, bid high at 685,000, or spend 9,000 on an intrusive survey first. A second decision node follows the survey branch, since its result changes which bid is sensible. A third sits inside delivery: if the grid ruling comes back adverse, accept the substation works or negotiate a variation, each drawn as its own branch.
- Chance nodes
- Win probability is estimated at 0.7 for the low bid and 0.4 for the high bid, from the firm's log of the last thirty tenders against the same two rivals. Grid ruling adverse with probability 0.25, from the operator's published determinations for that district. Roof remediation needed with probability 0.3 unsurveyed; the survey resolves this to near-certainty either way before the bid is priced.
- Outcomes and payoffs
- Margins per terminal branch range from 92,000 pounds on a clean low-bid win to a 41,000 loss where the low bid wins and both risks land. The high bid absorbs both risks and still clears 54,000, but wins less often. Losing the tender pays zero less bid costs of 6,000. All figures are contribution after direct costs, undiscounted given the fourteen-month horizon.
- Roll back the expected values
- Rollback prices the low bid at 38,000 pounds expected, dragged down by the 0.075 joint-probability branch where both risks hit. The high bid rolls back to 18,000, mostly because it loses six tenders in ten. The survey branch rolls back to 47,000 net of its 9,000 cost: with roof condition known, the firm bids low on a clean roof and high on a bad one, and the tree prunes the loss-making branch almost entirely.
The read. The tree says pay for the survey, then let its result choose the bid. That answer was not obvious in advance; the survey looked like dead cost against a 3.5 per cent margin difference, and the firm had historically skipped it. Sensitivity testing shows the recommendation holds until the low bid's win probability falls below 0.55 or survey cost triples, but it is fragile to the grid-ruling probability, which is the firm's least-evidenced number. The honest next step is a conversation with the grid operator before the bid goes in, and the tree, not instinct, identified that.
Pitfalls
- Choosing on expected value when the firm cannot afford the worst branch. A positive-EV bet that risks insolvency is still a bad bet; adjust with a utility function or a survival constraint before trusting the rollback.
- Letting the tree sprawl. Every added chance node multiplies the branches; the craft is pruning to the two or three uncertainties that actually move the decision, and Raiffa warned from the start about trees turning into a bushy mess.
- Anchoring the analysis on the first probabilities offered. Elicit ranges, ask what evidence would change the number, and record who estimated what.
- Drawing the tree to justify a decision already made, typically by omitting the 'wait' branch or the awkward downside outcome.
- Treating the output as a forecast. Rollback recommends a policy under stated assumptions; it does not predict which branch reality will take.
- Stopping at the first payoff and ignoring follow-on value, which systematically penalises flexible, staged options, the very things trees exist to value.
What the critics say
Expected monetary value is blind to risk appetite, and people demonstrably do not choose as EV maximisers: losses loom larger than gains and small probabilities are misweighted. A tree rolled back on raw EV can recommend gambles the decision-maker is right to refuse, and the utility-function repair the theory offers is rarely applied in practice.
Kahneman, D. and Tversky, A. (1979) 'Prospect Theory: An Analysis of Decision under Risk', Econometrica, 47(2), pp. 263-291.
The whole apparatus rests on elicited probabilities, and human probability judgement is systematically biased: anchoring, availability and overconfidence distort exactly the subjective estimates trees consume. Garbage probabilities roll back into garbage recommendations with a decimal point of false authority.
Tversky, A. and Kahneman, D. (1974) 'Judgment under Uncertainty: Heuristics and Biases', Science, 185(4157), pp. 1124-1131.
Trees suffer combinatorial explosion: branches multiply geometrically with each added decision and chance node, so realistic problems either become unreadable or are pruned by judgement calls the formalism cannot audit. Raiffa acknowledged the problem from the beginning, and the practical fixes, coarse discretisation and aggressive pruning, reintroduce the informality the method promised to remove.
Raiffa, H. (1968) Decision Analysis: Introductory Lectures on Choices under Uncertainty. Reading, MA: Addison-Wesley.
Sources and further reading
- Magee, J. F. (1964) 'Decision Trees for Decision Making', Harvard Business Review, 42(4), July-August 1964. ↗
- Raiffa, H. and Schlaifer, R. (1961) Applied Statistical Decision Theory. Boston: Division of Research, Harvard Business School.
- Howard, R. A. (1966) 'Decision Analysis: Applied Decision Theory', in Proceedings of the Fourth International Conference on Operational Research. New York: Wiley-Interscience.
- Raiffa, H. (1968) Decision Analysis: Introductory Lectures on Choices under Uncertainty. Reading, MA: Addison-Wesley.