The main features of decision trees

Decision trees provide a structured method for evaluating business choices by mapping out possible options and their potential results. They help managers assess risks and select the most profitable path based on likely outcomes and financial returns.

Key elements in decision trees

• Options available - All possible choices a manager can make in a given situation.
• Possible outcomes - The different results that could follow from each option.
• Probabilities of outcomes - Numerical estimates of the likelihood that each outcome will happen, often based on forecasts or historical data.
• Economic returns - The expected financial gains or losses associated with each outcome, helping to identify the option with the best potential profitability.

Decision trees minimise risks by combining these elements to compare options logically, focusing on those with the highest overall value.

How to construct decision trees

Decision trees are built in a visual format that starts from the initial choice and branches out to show consequences. They use specific symbols to represent decisions and uncertainties, making complex choices easier to analyse.

Symbols used in decision trees

• Decision nodes - Shown as squares, indicating points where a choice must be made between options.
• Chance nodes - Represented by circles, marking where multiple outcomes are possible from a decision, each with its own probability.
• Branches - Lines connecting nodes, with probabilities (e.g., 0.6 for 60% chance) and economic returns (e.g., revenue or costs) labelled alongside.

Trees can become more complex with sequential decisions, where one choice leads to further options and outcomes.

How to calculate expected values

Expected values quantify the potential worth of each option in a decision tree by weighting financial outcomes by their probabilities. This helps identify the choice with the highest likely return.

Formula for expected value

$\text{Expected value}=\sum (\text{probability of outcome}\times \text{economic return of outcome})$

Where:

• Probability of outcome = Likelihood of each result (must sum to 1 for all outcomes from a chance node)
• Economic return of outcome = Financial gain or loss for that result (£)

Calculations proceed from right to left: first compute expected values at chance nodes, then compare options at decision nodes, selecting the highest value path.

Worked example - Calculating expected values for venue choice

A festival organiser is deciding between an indoor hall costing £1,500 or an outdoor marquee costing £2,500. Weather forecasts predict a 70% chance of fine weather and 30% chance of rain. In fine weather, the hall generates £4,000 revenue and the marquee £8,000. In rain, the hall generates £6,000 and the marquee £3,000. Calculate the net expected value for each option.

Step 1: Identify the values

• Indoor hall cost = £1,500
• Outdoor marquee cost = £2,500
• Probability of fine weather = 0.7
• Probability of rain = 0.3
• Revenues: Hall (fine) = £4,000; Hall (rain) = £6,000; Marquee (fine) = £8,000; Marquee (rain) = £3,000

Step 2: Calculate expected value for indoor hall

$\text{Expected value}=(0.7\times \pounds \mathrm{4,000})+(0.3\times \pounds \mathrm{6,000})=\pounds \mathrm{2,800}+\pounds \mathrm{1,800}=\pounds \mathrm{4,600}$

Step 3: Calculate net return for indoor hall

Net return = £4,600 - £1,500 = £3,100

Step 4: Calculate expected value for outdoor marquee

$\text{Expected value}=(0.7\times \pounds \mathrm{8,000})+(0.3\times \pounds \mathrm{3,000})=\pounds \mathrm{5,600}+\pounds 900=\pounds \mathrm{6,500}$

Step 5: Calculate net return for outdoor marquee

Net return = £6,500 - £2,500 = £4,000

Step 6: Interpretation

The outdoor marquee has a higher net expected value (£4,000 vs £3,100), making it the better choice despite higher costs.

Advantages and limitations of decision trees

Decision trees offer a clear framework for analysing choices but have drawbacks related to data reliability and scope.

Advantages of using decision trees

• Comprehensive analysis - They require managers to identify all possible options, outcomes, and variables, ensuring nothing is overlooked.
• Visual clarity - Information is presented in a simple diagram, making it easier to understand and discuss numerical factors like probabilities and returns.
• Logical decision-making - They promote structured thinking, helping teams evaluate risks and select options with the highest expected value.

Limitations of decision trees

• Data accuracy issues - Probabilities and economic returns are often estimates or based on past data, which may not account for future changes or unexpected events.
• Ignores qualitative factors - They focus on numerical values but overlook non-financial aspects, such as environmental effects, employee morale, or ethical considerations.
• Averages, not guarantees - Expected values represent long-term averages, assuming repeated decisions; for one-off choices, actual results may differ significantly.
• Does not eliminate risk - While they quantify risks, they cannot remove uncertainties, and complex trees can become time-consuming to create.

Worked example - Calculating expected values in a complex decision tree

A tech firm is deciding whether to sell its current app for £700,000 immediately or invest £350,000 to upgrade it. After upgrading, it can sell the full app or invest another £180,000 to develop add-ons. Market competition has a 70% chance of being high and 30% low. If sold full after upgrade: high competition yields £1,100,000, low yields £1,900,000. If add-ons developed: high competition yields £1,400,000, low yields £2,600,000. Calculate the best path and net expected values.

Step 1: Identify the values

• Immediate sale = £700,000 (no costs)
• Upgrade cost = £350,000
• Add-ons cost = £180,000
• Probability of high competition = 0.7
• Probability of low competition = 0.3
• Revenues: Full app (high) = £1,100,000; Full app (low) = £1,900,000; Add-ons (high) = £1,400,000; Add-ons (low) = £2,600,000

Step 2: Calculate expected value for add-ons option

$\text{Expected value}=(0.7\times \pounds \mathrm{1,400,000})+(0.3\times \pounds \mathrm{2,600,000})=\pounds \mathrm{980,000}+\pounds \mathrm{780,000}=\pounds \mathrm{1,760,000}$ Net after add-ons cost = £1,760,000 - £180,000 = £1,580,000

Step 3: Calculate expected value for full app sale (no add-ons)

$\text{Expected value}=(0.7\times \pounds \mathrm{1,100,000})+(0.3\times \pounds \mathrm{1,900,000})=\pounds \mathrm{770,000}+\pounds \mathrm{570,000}=\pounds \mathrm{1,340,000}$

Step 4: Choose best path after upgrade and calculate net

Best after upgrade = Add-ons (£1,580,000 > £1,340,000) Net after upgrade cost = £1,580,000 - £350,000 = £1,230,000

Step 5: Compare initial options

Upgrade path net = £1,230,000 Immediate sale = £700,000 The upgrade path has higher net expected value.