What could a better answer be worth?

Information has decision value when it lets you choose differently. Its value is the improvement over the choice you would otherwise make, not the full proceeds of the project it helps you select.

Start with the choice without the answer

Consider an invented one-week project. Starting it requires a $400 payment that covers every project cost, including the work. At the end of the week it receives either $900 in a favorable state or $100 in an unfavorable state. Exactly one state occurs. There are no other receipts or payments in this before-tax example.

Supply a 40 percent chance of the favorable state and a 60 percent chance of the unfavorable one. These probabilities are assumptions for the exercise, not evidence about a real business.

Starting gives a net cash result of $500 or negative $300. Not starting gives zero in either state. For this comparison, choose the available action with the highest expected net cash result.

Without further information, starting has an expected result of 0.40 × $500 − 0.60 × $300 = $20. Under the stated rule, starting beats not starting. The $20 is an average across the supplied possibilities; neither state pays a net $20.

What a perfect answer would change

Suppose an answer is available before the $400 commitment. It always identifies the state correctly. The answer changes neither the state's probability nor the project's receipts or costs, and the choice to decline remains available.

You can start in the favorable state and decline in the unfavorable one. Before paying for the answer, that policy has an expected result of 0.40 × $500 + 0.60 × $0 = $200.

The improvement is $180: $200 with the answer minus the $20 available without it. Calling the answer worth $200 would forget the original alternative.

The Open University's discussion of the value of information compares what a decision can achieve with and without information. Its investment example uses net present value. This separate example uses the supplied one-week net cash results.

Put the information fee into both states

Suppose the perfect answer costs $50, paid before it is received and not refunded if you decline the project. No other information costs or delays occur.

In the favorable state, the combined result is $500 − $50 = $450. In the unfavorable state, you avoid the project but still pay $50, leaving negative $50. Their weighted result is $150. Compared with starting without the answer, the expected improvement after the fee is $130.

A $180 fee would leave the information policy with an expected result of $20, tying the original choice. A larger fee would make it worse under this model's expected-value rule. That is a limit derived from these particular alternatives and assumptions, not a general budget for research.

A useful test is not a perfect answer

An actual test may miss an unfavorable condition or discourage a project that would have worked. Its results might arrive after the decision is due. The $180 value of a perfect answer does not establish the value of that test.

For information alone, with the same actions, outcomes and decision rule, a perfect answer provides an upper bound: an imperfect signal cannot improve on always choosing correctly. To value a real test, you would need evidence about how its results relate to the states and which action each result would lead to. A claimed accuracy percentage without that relationship is not enough.

Follow the test result into the choice

Now use an imperfect test for the same project: a test that returns positive in eighty percent of favorable states and twenty percent of unfavorable states. It arrives before the project choice, changes no state or payoff, and creates no delay or other cost beyond its fee. These are supplied fictional test properties, not validated accuracy rates.

Multiplying each state's chance by the corresponding signal rate gives four joint branches. For example, a favorable state and a positive signal have a probability of 0.40 × 0.80 = 0.32. Read the table as a rule for acting on the signal: start after positive, decline after negative. The state itself remains unknown when choosing.

Same project, an imperfect test and a $50 fee paid in every branch

Scroll across to see every column. Use the arrow keys when the table is focused.

Why choose those actions? Positive signals account for 0.32 + 0.12 = 0.44 of all possibilities. Within that group, starting has an expected project result of (0.32 × $500 − 0.12 × $300) ÷ 0.44, about $281.82. That exceeds the zero from declining. Among positives, the favorable share is 0.32 ÷ 0.44, or 8/11. The supplied eighty percent used a different denominator.

For negative signals, the expected project result from starting is (0.08 × $500 − 0.48 × $300) ÷ 0.56, about negative $185.71. Declining is better under the stated expected-value rule. At either decision, the test fee has already been paid and applies whichever action is chosen.

Before its fee, the policy's expected result is 0.32 × $500 − 0.12 × $300 = $124; the declined branches add zero. Subtract the $20 best no-test result: this test's information value is $104, not the $180 value of a perfect answer.

Paying $50 in every branch leaves an expected $74. The table gives the same answer: $144 − $42 − $4 − $24 = $74. That improves on the $20 available without testing by $54. A $104 fee would leave $20 and tie that original choice.

ACCA's decision-tree explanation evaluates outcome probabilities and cash results before choosing between available actions. The test properties and figures here are original hypothetical additions. The conditional-denominator example explains why a rate among favorable states is not the favorable share among positive signals.

The original $400 payment still needs funding. Buying the $50 answer and then proceeding requires $450 before the project pays anything back. Expected value has supplied no cash for that opening payment and no guarantee against loss.

Before paying for more information, write down the decision you would make without it. Identify which result would change that choice, what the changed choice could improve and what obtaining the result would cost. The business-trial reading examines a further problem: a test may remove the very condition the final operation needs to face.