Decision Trees

Spec mapping: AQA 7132 Section 3.2 — Managers, leadership and decision making (refer to the official AQA specification document for exact wording). This lesson develops decision trees at A-Level depth — how to construct and interpret a decision tree, how to calculate the expected value (EV) at a chance node, how to derive the net gain of each option, how to select the option with the highest net gain, two fully worked numerical examples with every arithmetic step shown, and the advantages, limitations and evaluative framework an examiner expects on a calculation-plus-evaluation question.

Connects to:

Section 3.2 (The Decision-Making Process) — the decision tree is the canonical scientific decision-making tool; it sits inside the scientific-vs-intuitive frame established in the previous lesson.
Section 3.2 (Influences on Decision Making) — risk, reward and uncertainty are exactly what a decision tree attempts to quantify; the model's reliability depends on the quality of the probability estimates.
Section 3.5 (Finance) — net gain is a close cousin of investment-appraisal logic; both compare an expected return against an up-front cost, and both rest on opportunity cost.
Section 3.4 (Operations) — capacity, outsourcing and launch decisions are the operational choices decision trees most often model.
Synoptic across all three 7132 papers — calculation items recur in data-response questions (Papers 2 and 3), and the evaluative limitations of the technique are tested in higher-tariff questions.

What a Decision Tree Is

Definition: A decision tree is a diagram that maps the alternative courses of action available to a decision-maker, the possible outcomes of each, and the probability and financial value of each outcome — allowing the options to be compared on an expected-value basis.

A decision tree is the visual, quantitative expression of scientific decision-making. It forces the manager to (a) lay out the options explicitly, (b) assign a probability to each possible outcome, and (c) attach a financial value to each outcome — then to compute which option has the highest expected payoff net of its cost. Its great virtue is discipline: it converts a vague "I think the national launch is riskier" into a defensible number.

The Components of a Decision Tree

A decision tree is read left to right and evaluated right to left. Three symbols matter.

Symbol	Name	Meaning
Square (▢)	Decision node	A point where the manager chooses between alternatives
Circle (◯)	Chance node	A point where the outcome is determined by chance (probabilities apply here)
Triangle / line end	Outcome (terminal)	The final payoff of a particular path

The structural rules:

Branches from a decision node are the manager's choices (Option A, Option B …). The manager controls which branch is taken.
Branches from a chance node are the possible states of the world (high demand, low demand …). The manager does not control which occurs; instead each is assigned a probability.
The probabilities on the branches leaving any one chance node must sum to 1 (e.g. 0.6 + 0.4 = 1). If they do not, the tree is mis-specified — this is one of the most common errors and an easy way to lose marks.

The Two Formulae You Must Know

There are only two calculations in a decision tree, and they are the load-bearing maths of this lesson.

1. Expected value at a chance node is the probability-weighted sum of the outcomes flowing from that node:

$EV = \sum (\text{probability} \times \text{payoff})$

For a node with two outcomes:

$EV = (p_1 \times \text{payoff}_1) + (p_2 \times \text{payoff}_2)$

2. Net gain of an option is its expected value minus the cost of taking that option:

$\text{Net gain} = EV - \text{cost of the decision}$

The decision rule: calculate the net gain of every option, then choose the option with the highest net gain.

Why subtract the cost? The expected value is the probability-weighted revenue (or benefit) the option is expected to generate. The cost is what the manager must spend to pursue that option. Net gain is therefore the expected surplus — the figure that is comparable across options with different up-front costs. Comparing EVs alone would unfairly favour expensive options; net gain is the like-for-like comparison.

Worked Example 1 — A Capacity Decision

Scenario. A hypothetical manufacturer must choose how to add capacity to meet forecast demand. Option A is to install a new automated production line at a cost of £450,000. Option B is to outsource production to a contract manufacturer at a cost of £150,000. Demand could be strong or weak.

Option A (automated line, cost £450,000): 0.55 probability of strong demand → revenue £1,100,000; 0.45 probability of weak demand → revenue £400,000.

Option B (outsource, cost £150,000): 0.70 probability of strong demand → revenue £520,000; 0.30 probability of weak demand → revenue £240,000.

Figures are fabricated for illustrative purposes.

Step 1 — Draw the tree

graph LR
  D[["Decision ▢"]] --> A["Option A: automated line<br/>cost £450,000"]
  D --> B["Option B: outsource<br/>cost £150,000"]
  A --> AC(("Chance ◯"))
  B --> BC(("Chance ◯"))
  AC --> A1["Strong demand p = 0.55<br/>→ £1,100,000"]
  AC --> A2["Weak demand p = 0.45<br/>→ £400,000"]
  BC --> B1["Strong demand p = 0.70<br/>→ £520,000"]
  BC --> B2["Weak demand p = 0.30<br/>→ £240,000"]

  style D fill:#1d4ed8,color:#fff

Check the probabilities first: Option A's chance node, $0.55 + 0.45 = 1.00$ ; Option B's chance node, $0.70 + 0.30 = 1.00$ . Both sum to 1, so the tree is well-specified.

Step 2 — Expected value at each chance node

Option A:

$EV_A = (0.55 \times 1{,}100{,}000) + (0.45 \times 400{,}000)$

Term by term: $0.55 \times 1{,}100{,}000 = 605{,}000$ ; and $0.45 \times 400{,}000 = 180{,}000$ .

$EV_A = 605{,}000 + 180{,}000 = \pounds785{,}000$

Option B:

$EV_B = (0.70 \times 520{,}000) + (0.30 \times 240{,}000)$

Term by term: $0.70 \times 520{,}000 = 364{,}000$ ; and $0.30 \times 240{,}000 = 72{,}000$ .

$EV_B = 364{,}000 + 72{,}000 = \pounds436{,}000$

Step 3 — Net gain of each option

$\text{Net gain}_A = EV_A - \text{cost}_A = 785{,}000 - 450{,}000 = \pounds335{,}000$

$\text{Net gain}_B = EV_B - \text{cost}_B = 436{,}000 - 150{,}000 = \pounds286{,}000$

Step 4 — Apply the decision rule

Option	Expected value	Cost	Net gain
A — automated line	£785,000	£450,000	£335,000
B — outsource	£436,000	£150,000	£286,000

Net gain is higher for Option A (£335,000 > £286,000), so on a purely quantitative basis the manufacturer should install the automated line. Note that Option A has both the higher cost and the higher net gain — which is exactly why you must compare net gain, not cost or EV alone. (Notice, too, that Option B is less risky: its worst-case payoff of £240,000 still exceeds its £150,000 cost, whereas Option A's worst case of £400,000 against a £450,000 cost is an outright loss. The numbers favour A; risk appetite is the qualitative judgement that sits on top — see the evaluation below.)

Worked Example 2 — An Expansion Decision

Scenario. A hypothetical café chain must choose between opening a flagship city-centre store and refitting its existing stores. Option X is to open the flagship store at a cost of £280,000. Option Y is to refit existing stores at a cost of £120,000.

Option X (flagship store, cost £280,000): 0.60 probability of success → revenue £700,000; 0.40 probability of underperformance → revenue £180,000.

Option Y (refit, cost £120,000): 0.75 probability of success → revenue £380,000; 0.25 probability of underperformance → revenue £140,000.

Figures are fabricated for illustrative purposes.

Probability check: Option X, $0.60 + 0.40 = 1.00$ ; Option Y, $0.75 + 0.25 = 1.00$ . Both valid.

Expected values

Option X:

$EV_X = (0.60 \times 700{,}000) + (0.40 \times 180{,}000)$

$EV_X = 420{,}000 + 72{,}000 = \pounds492{,}000$

Option Y:

$EV_Y = (0.75 \times 380{,}000) + (0.25 \times 140{,}000)$

$EV_Y = 285{,}000 + 35{,}000 = \pounds320{,}000$

Net gains

$\text{Net gain}_X = 492{,}000 - 280{,}000 = \pounds212{,}000$

$\text{Net gain}_Y = 320{,}000 - 120{,}000 = \pounds200{,}000$

Decision

Option	Expected value	Cost	Net gain
X — flagship store	£492,000	£280,000	£212,000
Y — refit existing	£320,000	£120,000	£200,000

The flagship store (Option X) has the higher net gain (£212,000 vs £200,000), so the quantitative recommendation is to open the flagship store. But the margin is narrow — only £12,000, or about 6% of the flagship's net gain. This is the analytically important feature of the second example: when net gains are close, the decision is sensitive to the probability estimates. If the success probability of the flagship were 0.55 rather than 0.60, its EV would fall to $(0.55 \times 700{,}000) + (0.45 \times 180{,}000) = 385{,}000 + 81{,}000 = \pounds466{,}000$ and its net gain to $466{,}000 - 280{,}000 = \pounds186{,}000$ — below the refit option. A 5-percentage-point error in a single probability flips the recommendation. This is the case for treating decision-tree outputs as one input to judgement, not as a mechanical verdict.

Worked Example 3 — A Sequential (Multi-Stage) Decision

Decision trees come into their own when a decision unfolds in stages — when the outcome of a first decision determines what second decision the manager then faces. These are evaluated by rolling back from the right: the EV of a later decision becomes the payoff that feeds the earlier chance node.

Scenario. A hypothetical firm must decide whether to run a £40,000 market test before committing to a £300,000 full launch.

Skip the test and launch immediately (cost £300,000): 0.50 success → £700,000; 0.50 failure → £120,000.

Run the test first (cost £40,000), then launch (further cost £300,000) only if the test is favourable. The test is favourable with probability 0.6. If favourable, the launch then succeeds with probability 0.8 → £700,000, or fails with probability 0.2 → £120,000. If the test is unfavourable (probability 0.4), the firm abandons the launch (payoff £0, having spent only the £40,000 test).

Figures are fabricated for illustrative purposes.

Launch immediately:

$EV = (0.50 \times 700{,}000) + (0.50 \times 120{,}000) = 350{,}000 + 60{,}000 = \pounds410{,}000$

$\text{Net gain} = 410{,}000 - 300{,}000 = \pounds110{,}000$

Test first — roll back from the launch decision: if the test is favourable, the conditional launch has

$EV_{\text{launch}} = (0.8 \times 700{,}000) + (0.2 \times 120{,}000) = 560{,}000 + 24{,}000 = \pounds584{,}000$

and a net gain (after the £300,000 launch cost) of $584{,}000 - 300{,}000 = \pounds284{,}000$ . Because £284,000 is positive, the firm would launch after a favourable test. If the test is unfavourable, the firm abandons (net £0 from that branch). Rolling these back to the test decision, and then subtracting the £40,000 test cost:

$EV_{\text{test branch}} = (0.6 \times 284{,}000) + (0.4 \times 0) = \pounds170{,}400$

$\text{Net gain (test first)} = 170{,}400 - 40{,}000 = \pounds130{,}400$

Decision: testing first yields the higher net gain (£130,400 vs £110,000), so the firm should run the test. The £40,000 test buys the option to abandon the launch when the signal is bad — the value of information. This is the most sophisticated use of decision trees and a strong synoptic link to the value of market research in marketing.

Interpreting the Result — Reading the Tree

Once the net gains are computed, the tree is interpreted right-to-left: at each chance node you have the EV, and at the decision node you select the branch with the highest net gain. The non-selected branches are conventionally marked with a double line (║) through them to show they have been rejected — a small but examinable convention. In a multi-stage tree (Example 3) the roll-back proceeds in the same direction: the net gain of each later decision becomes the payoff fed into the chance node immediately to its left.

The key interpretive points:

The recommended option is the highest net gain, not the highest EV and not the lowest cost.
A negative net gain means the option is expected to lose money and should be rejected outright.
The size of the gap between options matters. A large gap (Example 1) is a robust recommendation; a narrow gap (Example 2) is sensitive to the probability estimates and warrants caution.

Advantages of Decision Trees

Advantage	Explanation
Visual and structured	Presents complex, multi-outcome decisions in a clear, logical format that is easy to communicate
Quantitative discipline	Forces managers to assign explicit probabilities and values, replacing vague hunches with comparable numbers
Like-for-like comparison	Net gain compares options with different costs on a common basis
Incorporates uncertainty	Explicitly weights outcomes by probability rather than assuming a single certain result
Supports justification	Provides defensible evidence to support decisions to stakeholders, lenders and boards
Handles sequential decisions	Can model multi-stage choices (e.g. "if we launch and demand is high, should we then expand?")

Limitations of Decision Trees

Limitation	Explanation
Probabilities are estimates	The output is only as reliable as the probability estimates, which are often subjective — "garbage in, garbage out"
False precision	A single-pound net-gain figure can imply a certainty the underlying estimates do not support
Ignores qualitative factors	Brand fit, ethics, staff morale, strategic fit and risk appetite are not captured in the numbers
Assumes risk-neutrality	EV treats a 50/50 gamble for £200k the same as a certain £100k; a risk-averse owner may rationally prefer the certain outcome
Static snapshot	The tree is built at a moment in time; conditions, probabilities and payoffs can shift before the outcome resolves
Bias in construction	Optimistic managers tend to inflate success probabilities and payoffs, skewing the result toward their preferred option

Decision Trees

Decision Trees

What a Decision Tree Is

The Components of a Decision Tree

The Two Formulae You Must Know

Worked Example 1 — A Capacity Decision

Step 1 — Draw the tree

Step 2 — Expected value at each chance node

Step 3 — Net gain of each option

Step 4 — Apply the decision rule

Worked Example 2 — An Expansion Decision

Expected values

Net gains

Decision

Worked Example 3 — A Sequential (Multi-Stage) Decision

Interpreting the Result — Reading the Tree

Advantages of Decision Trees

Limitations of Decision Trees

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