The Value of Perfect and Imperfect Information in Decision Making

A property developer in Abuja is about to commit ₦200 million to an estate. A consultant offers a demand forecast for a fee. The developer should not ask only whether the forecast is good, but how much it is worth. If the forecast will change the decision, it has value; if the developer would build anyway, it is worth nothing.

If it is sometimes wrong, its value is smaller than if it were always right. The value of information turns this reasoning into numbers: the extra expected profit that the information makes possible, which is the most it is worth paying for it.

This topic explains how to calculate the value of perfect information, which tells the decision maker exactly what will happen, and of imperfect information, which is a forecast that is right only part of the time, using probability tables and revised probabilities. It builds on the earlier topic on decision trees, where a pilot study was tested in the same way.

The next topic covers budgetary systems and the main types of budget.

Perfect information

What it means

Perfect information is a forecast that is always right: it tells the decision maker which state will occur before the decision is made. Few real forecasts are perfect, but the value of perfect information is useful because it sets an upper limit on what any information could be worth. If a forecast is not worth the price even if it were perfect, it is not worth buying at all.

How to calculate it

1. EV without information. Calculate the EV of each option with the existing probabilities and take the highest.
2. EV with perfect information. For each state, decide which option would be chosen if the state were known, and take its pay-off. Multiply by the probability of the state and add.
3. Value of perfect information = (2) − (1). This is never negative.
4. Decision. If the fee for the information is less than the value, buy it; the net gain is the value less the fee.

The method works because, with perfect information, the decision maker avoids the cost of making the wrong choice in each state, and the value of the information is the expected size of those avoided losses, or foregone gains.

Imperfect information

Real forecasts are right with some probability, for example 80% of the time. If the forecast says the market will be good, it may still be poor. The value of such a forecast is less than that of perfect information, because it sometimes leads to the wrong decision. Calculate it in four steps.

1. Joint probabilities. For each state and each forecast, multiply the prior probability of the state by the probability of the forecast given the state (the accuracy, or its complement for a wrong forecast). Put them in a table whose rows are the forecasts and whose columns are the states.
2. Probability of each forecast. Add each row of the table. The row totals add to 1.
3. Revised probabilities. The probability of a state given a forecast is the joint probability divided by the row total (the probability of the forecast).
4. Decision after each forecast, and the EV. With the revised probabilities, calculate the EV of each option after each forecast and take the best, which may be to do nothing. Weight the best EV after each forecast by the probability of that forecast, and add. Subtract the EV without information to get the value of imperfect information.

Two checks help. The probability of each state from the joint table must equal its prior. And the value of imperfect information must not exceed the value of perfect information. Equal values mean that the forecast, although in principle fallible, is as good as perfect for this decision.

Uses and limitations

The value of information tells management how much to spend on market research, a consultant, a test market or a better forecasting system.

Its limits:

• the probabilities, both of the states and of the accuracy of the forecast, are estimates
• the value is an expected value, not a guaranteed gain
• the cost of delay while information is obtained is not included
• information may bring other benefits, such as better understanding of the market, that are not in the pay-offs

The method also assumes a risk-neutral decision maker who chooses by EV.

Layouts for the value of information

1. EV without information: highest EV of the options.
2. Perfect information: Σ (probability of state × best pay-off in the state) − (1).
3. Imperfect information: joint probability table; row totals; revised probabilities; best option after each forecast; weighted EV − (1).
4. Decision: buy if the fee is below the value.

Money is in naira and each answer is rounded once, at the end.

Worked examples

1. EV without information.
   A 0.3 × 100 + 0.5 × 300 + 0.2 × 700 = 320
   B = 280
   C = 295.
   Best: A, 320.
2. EV with perfect information.
   Weak: choose B (200) × 0.3 = 60.
   Normal: choose A (300) × 0.5 = 150.
   Strong: choose C (850) × 0.2 = 170.
   Total 380.
3. Value.
   380 − 320 = 60, that is ₦60,000.
   The company should pay no more than this, and a fee of ₦50,000 would leave a net gain of ₦10,000.

1. EV without information.
   Invest: 0.6 × 500,000 − 0.4 × 300,000 = 300,000 − 120,000 = ₦180,000.
   Invest.
2. Perfect information.
   Invest only if good: 0.6 × 500,000 = ₦300,000.
   Value of perfect information = 300,000 − 180,000 = ₦120,000.
3. Joint probabilities.
   Says good and market good: 0.6 × 0.8 = 0.48.
   Says good and market poor: 0.4 × 0.2 = 0.08.
   Says poor and market good: 0.6 × 0.2 = 0.12.
   Says poor and market poor: 0.4 × 0.8 = 0.32.
   Check: the total is 1.00, and market good is 0.48 + 0.12 = 0.60.
4. Probability of each forecast.
   Says good: 0.48 + 0.08 = 0.56.
   Says poor: 0.12 + 0.32 = 0.44.
5. After a “good” forecast.
   Probability of good market = 0.48 ÷ 0.56 = 0.857.
   Invest: 0.857 × 500,000 − 0.143 × 300,000 = ₦385,714.
   Invest.
6. After a “poor” forecast.
   Probability of good market = 0.12 ÷ 0.44 = 0.273.
   Invest: 0.273 × 500,000 − 0.727 × 300,000 = −₦81,818.
   Do not invest: ₦0.
7. EV with imperfect information.
   0.56 × 385,714 + 0.44 × 0 = ₦216,000 (equivalently 0.48 × 500,000 − 0.08 × 300,000 = 216,000).
8. Value.
   216,000 − 180,000 = ₦36,000.
   This is the most worth paying; it is much less than the ₦120,000 value of perfect information, because an 80% accurate forecast is wrong one time in five.

Expect variants: a decision tree that includes the cost of a test, three states, forecasts with different accuracy for each state, and a question that asks only for the most to pay for perfect information.

Common mistakes

• Using the wrong EV for “without”. It is the best EV with the existing probabilities, not the average of all.
• Weighting the best option by the wrong probabilities in the perfect case: use the probability of each state.
• Using the accuracy as the revised probability. Revise by dividing the joint probability by the forecast probability.
• Forgetting the option to do nothing after an unfavourable forecast.
• A value of imperfect information above the perfect value. Re-check the table.
• Ignoring the fee. The net gain is the value less the fee.

Quick revision

• VPI = EV with perfect information − EV without; the best option in each state × probability of the state.
• Imperfect: joint probabilities, forecast probabilities, revised probabilities, best option after each forecast.
• Value of imperfect ≤ value of perfect. Pay only if the fee is below the value.

How the marks are earned

• EV without and with information, each shown.
• Probability table for imperfect information.
• Conclusion on the most to pay and whether to buy.

Practice questions

These are original questions, written in the style of the Skills examination. Work each one before you open the answer.

Recall check: multiple choice

Section B style: written questions

Model answers

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