The purpose and features of decision trees

Decision trees are tools that assist businesses in making decisions under uncertainty by combining probabilities and expected pay-offs to evaluate potential outcomes. They form part of scientific decision-making processes, where costs are known but outcomes are uncertain. Managers use them to analyse options by estimating probabilities based on experience or historical data.

Key definitions in decision trees

• Probability - The likelihood of an event occurring, expressed as a decimal (e.g., 0.6 for 60% chance). The probabilities of an event happening and not happening must sum to 1.
• Expected value (EV) - The anticipated pay-off from an outcome, calculated by multiplying the probability by the pay-off and summing for all outcomes.
• Net gain - The financial benefit after subtracting initial costs from the expected value.

Symbols and elements in decision trees

• Square - Represents a decision point where choices are made.
• Lines from squares - Indicate possible courses of action, including associated costs.
• Circle - Shows points where alternative outcomes branch out.
• Lines from circles - Represent possible outcomes for each action.
• Decimals on lines - Denote probabilities of each outcome.
• Values in £s - Indicate the pay-off (e.g., revenue) for the business from each outcome.

How to construct decision trees

Constructing a decision tree involves mapping out options, outcomes, and calculations to identify the best course of action.

Steps to build a decision tree

1. Identify available courses of action and their initial costs.
2. Outline possible outcomes for each action and assign probabilities (ensuring they sum to 1 for each branch).
3. Calculate the expected value (EV) for each outcome.
4. Determine the net gain for each action by subtracting initial costs from the EV.
5. Select the action with the highest net gain.

Calculating expected value and net gain

Expected value helps quantify the average outcome if a decision were repeated many times, while net gain accounts for costs to show true benefit.

Formula for expected value (EV)

$\text{Expected value (EV)}=(\text{probability of outcome 1}\times \text{pay-off 1})+(\text{probability of outcome 2}\times \text{pay-off 2})+\dots$

Where:

• Probability = Decimal likelihood of each outcome
• Pay-off = Financial return (e.g., revenue) from each outcome

Formula for net gain

$\text{Net gain}=\text{expected value (EV)}-\text{initial costs}$

Where:

• Expected value (EV) = Sum of weighted pay-offs for all outcomes
• Initial costs = Upfront expenses for the chosen action

Worked example - Calculating expected value and net gain for a product launch

A technology company is deciding whether to launch a new app using a dedicated development team (£25,000 cost) or existing resources (£8,000 cost). For the dedicated team: success probability 0.8 (revenue £90,000), failure probability 0.2 (revenue £15,000). For existing resources: success probability 0.5 (revenue £90,000), failure probability 0.5 (revenue £15,000). Calculate the EV and net gain for each option.

Step 1: Identify the values

• Dedicated team: Cost = £25,000; Success = 0.8 (£90,000); Failure = 0.2 (£15,000)
• Existing resources: Cost = £8,000; Success = 0.5 (£90,000); Failure = 0.5 (£15,000)

Step 2: Calculate EV for dedicated team

$\text{EV}=(0.8\times \pounds \mathrm{90,000})+(0.2\times \pounds \mathrm{15,000})=\pounds \mathrm{72,000}+\pounds \mathrm{3,000}=\pounds \mathrm{75,000}$

Step 3: Calculate net gain for dedicated team

$\text{Net gain}=\pounds \mathrm{75,000}-\pounds \mathrm{25,000}=\pounds \mathrm{50,000}$

Step 4: Calculate EV for existing resources

$\text{EV}=(0.5\times \pounds \mathrm{90,000})+(0.5\times \pounds \mathrm{15,000})=\pounds \mathrm{45,000}+\pounds \mathrm{7,500}=\pounds \mathrm{52,500}$

Step 5: Calculate net gain for existing resources

$\text{Net gain}=\pounds \mathrm{52,500}-\pounds \mathrm{8,000}=\pounds \mathrm{44,500}$

Step 6: Interpretation

The dedicated team option has a higher net gain (£50,000 vs £44,500), making it the better choice despite higher initial costs.

Advantages and disadvantages of decision trees

Decision trees offer a structured approach to decision-making but have limitations, especially with uncertain or qualitative factors.

Advantages of decision trees

• Force managers to quantify probabilities and potential pay-offs, promoting thorough analysis.
• Provide a visual representation that clarifies complex options and outcomes.
• Enable quantitative and objective comparisons between different courses of action.
• Perform well in familiar scenarios where probability and pay-off estimates are reliable based on past data.

Disadvantages of decision trees

• Focus solely on quantitative data, ignoring qualitative aspects such as employee morale or ethical considerations.
• Probabilities are often subjective estimates and can be inaccurate, leading to flawed decisions.
• Pay-off estimates may not match real-world results due to unforeseen factors.
• May oversimplify outcomes by assuming binary results (e.g., success or failure), missing nuances like varying degrees of success.
• Rely on potentially incomplete or biased information, which can undermine the analysis.