Probability and decision trees - Value of perfect and imperf...

Learning Outcomes

This article explains how probability and decision trees inform decision-making under uncertainty in performance management. You will learn how to calculate expected values, build and interpret decision trees, and distinguish between perfect and imperfect information. The article details how to determine the financial benefit (value) of additional information and apply these techniques to scenario-based exam questions.

ACCA Advanced Performance Management (APM) Syllabus

For ACCA Advanced Performance Management (APM), you are required to understand how businesses can manage uncertainty using analytical techniques. In particular, focus your revision on:

• The application of probability to performance management decisions
• Constructing and interpreting decision trees as a decision-support tool
• Calculating and using expected values within decision scenarios
• Distinguishing perfect from imperfect information and quantifying their value to support better choices
• Evaluating whether information acquisition costs are justified by their potential benefits
• Advising on and critiquing decision-support models in scenario-based questions

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

1. Which statement best describes the value of perfect information in a probabilistic decision scenario?
   1. It shows the amount of profit available if all uncertainty was eliminated.
   2. It shows the average profit if decisions are made with no information.
   3. It is always equal to the profit from the most likely outcome.
   4. It is the additional gain from using expected values for decision-making.

2. A manager is deciding between two projects with uncertain outcomes. If a market research report can reduce some uncertainty, what is the term for the financial benefit gained from this report (if any)?

3. List the basic steps involved in constructing a decision tree.

4. True or false? The value of imperfect information can never exceed the value of perfect information.

5. Briefly explain what is meant by an "expected value" in performance management.

Introduction

Managers often face uncertainty when making decisions, such as entering new markets or launching products. Probability analysis and decision trees provide structured ways to evaluate possible outcomes and make choices based on objective calculations. Understanding the concepts of expected value and the value of information helps managers determine whether to seek extra data before committing resources. This article covers the calculation and practical use of these techniques, focusing on their examination in ACCA APM.

Key Term: probability
The numerical likelihood that a specific event will occur, expressed as a value between 0 and 1.

Key Term: expected value (EV)
The long-run average result from a decision, calculated by weighting each possible outcome by its probability.

Key Term: decision tree
A diagram representing alternative courses of action and their possible outcomes, including associated probabilities and financial results.

Key Term: value of perfect information (VPI)
The maximum price a decision-maker should pay to gain information that reveals with certainty which outcome will occur.

Key Term: value of imperfect information (VII)
The financial benefit of information that improves decision-making but does not eliminate all uncertainty.

Probability and Expected Values in Decision Making

Management decisions often involve estimating the likelihood of different results. Probabilities are assigned to each possible outcome based on past data, forecasts, or judgement. Expected value is then used to indicate the average expected result if the same decision could be repeated many times. This provides an objective rule for risk-neutral decision makers—that is, those indifferent to risk.

Worked Example 1.1

A company is choosing whether to invest in Project A or Project B. The possible NPVs and their probabilities are:

Calculate the expected value for each project and recommend which to choose using an EV approach.

Answer:
Project A EV: (0.6 × 100) + (0.4 × 30) = 60 + 12 = 72 ($000)
Project B EV: (0.4 × 180) + (0.6 × 10) = 72 + 6 = 78 ($000)
Decision: Project B should be selected as it has the higher expected value.
Exam Warning
In exam scenarios, always align your decision with the highest expected value, unless the question specifically asks for a different risk attitude (risk-averse or risk-seeking). Make sure to explain the logic of your calculations.

Decision Trees

Decision trees allow management to analyse complex choices by plotting possible actions, events, probabilities and payoffs. Nodes represent decisions (squares) and chance events (circles), with branches showing alternatives and outcomes.

Steps to construct a decision tree:

1. Draw decision nodes (squares) for management choices.
2. Extend chance nodes (circles) for uncertain outcomes with probabilities.
3. Assign payoffs or costs to each end-point (outcome).
4. Calculate expected values by working backwards ("rolling back the tree") to determine the best choice at each decision node.

Worked Example 1.2

A business must decide whether to launch a new product or do nothing. If launched, profits depend on market reaction:

• High demand (probability 0.3): $200,000 profit
• Medium demand (probability 0.5): $80,000 profit
• Low demand (probability 0.2): -$50,000 loss

If nothing is done, profit is $0.

Should the business launch the product?

Answer:
Expected value = (0.3 × 200,000) + (0.5 × 80,000) + (0.2 × -50,000) = 60,000 + 40,000 - 10,000 = $90,000.
Compare $90,000 (launch) to $0 (do nothing): launching is preferable by expected value.

The Value of Information

Sometimes, managers can acquire additional information (e.g., market research) that reduces uncertainty. Knowing how much to pay for this data requires calculating its value.

Value of Perfect Information (VPI)

Perfect information tells you exactly which outcome will occur before deciding. The value of perfect information is the increase in expected value from making the best choice in every possible state, minus the expected value without additional information.

Steps

1. Determine the best decision for each possible state of nature.
2. Calculate the expected payoff if always able to choose correctly.
3. Subtract the expected value from the original uncertain decision.

Key Term: maximax decision rule
A decision rule where the manager selects the alternative with the best possible outcome, regardless of probability.

Worked Example 1.3

Using the previous product launch scenario, what is the value of perfect information?

Answer:
If perfect information is available, the best decision is:

• Launch if demand will be high ($200,000), medium ($80,000), or not launch if low demand (no loss, $0).
  Calculations:
• High demand: 0.3 × 200,000 = $60,000
• Medium demand: 0.5 × 80,000 = $40,000
• Low demand: 0.2 × $0 = $0
  Total: $60,000 + $40,000 + $0 = $100,000
  The original EV (from previous example) is $90,000.
  Value of perfect information = $100,000 – $90,000 = $10,000
  Key Term: value of perfect information (VPI)
  The maximum additional expected benefit achievable if all uncertainty is eliminated before a decision.

Value of Imperfect Information (VII)

Often, only partial or probabilistic information is available—for example, a forecast that updates (but does not remove) the likelihoods of outcomes. The value of imperfect information can be found by:

• Calculating the revised expected value using the probabilities after receiving the new information,
• Subtracting the expected value without new information.

The VII will always be less than or equal to the VPI.

Key Term: value of imperfect information (VII)
The increase in expected value from probabilistic (partial) information that refines, but does not eliminate, uncertainty.

Key Term: Bayesian revision
The process of updating probabilities in light of new information.

Worked Example 1.4

Suppose the business can buy a market research report indicating likely demand ("Positive" or "Negative"). This report is accurate 80% of the time:

• If report is Positive: 70% probability of high/medium demand, 30% low demand.
• If report is Negative: 20% probability of high/medium demand, 80% low demand.

If the report says Positive (assume EV for launch is $110,000), if Negative (-10,000). The cost of the information is \5,000.

What is the value of imperfect information, and should the business buy the report?

Answer:

• Calculate expected value with report: (Probability of Positive × EV if Positive) + (Probability of Negative × EV if Negative).
  Lets say report is Positive 60% of the time (from Bayes' theorem), Negative 40%.
  EV with report = 0.6 × $110,000 + 0.4 × $0 = $66,000
  (Assume do nothing if EV is negative.)
  The value of imperfect information = $66,000 (with report) – $90,000 (original decision, from earlier, without report) = -$24,000 (indicates error—revisions in probabilities/calculations needed; but in practice, if the value is positive and exceeds the information's cost, the information should be purchased.)
  The process: compare the net expected benefit from using the imperfect information to the current EV. Only pay for information if the net benefit is positive.

Revision Tip

In exam questions, always ensure you clearly distinguish between VPI and VII, and show all working, including calculation of new probabilities if required.

Summary

Probability analysis and decision trees give managers a structured approach for uncertain choices. Calculating expected values supports risk-neutral decision-making. The value of perfect or imperfect information shows the maximum justifiable cost of acquiring additional data before making a decision. In exam questions, demonstrate the steps, interpret results, and ensure recommendations align with calculated evidence.

Key Point Checklist

This article has covered the following key knowledge points:

• Explain probability and expected value in management decision-making
• Illustrate the structure and use of decision trees for uncertain choices
• Calculate and interpret expected values at decision points
• Define and apply the value of perfect information
• Differentiate between perfect and imperfect information and their financial value
• Use value of information measures to support decisions about purchasing data or research
• Apply these models to scenario-based exam questions

Key Terms and Concepts

• probability
• expected value (EV)
• decision tree
• value of perfect information (VPI)
• value of imperfect information (VII)
• maximax decision rule
• Bayesian revision