How does superintelligence work when an AI has a choice between acting immediately and checking one more fact? A school printing desk might save time by selecting a print run now, but could waste supplies if demand is low or run short if demand is high. A survey may help—yet it also costs time and effort. The right question is not merely whether additional information is interesting. It is whether that information is likely to improve the decision enough to justify the cost.
The mechanism is value of information. Decision analysis compares the expected result of acting with current knowledge against the expected result after receiving an informative signal. An assistant should specify the uncertainty, decision options, consequences and price of checking. It should distinguish a perfectly informative hypothetical check from an imperfect forecast, and never invent the reliability of a real source or claim that a search guaranteed a future result.
This accessible guide uses a fictional school printing decision with two demand states, two print plans and a simple loss-point model. We calculate the value of perfect information, evaluate an imperfect forecast across one hundred invented scenarios, and classify twenty-four checkable claims. Every probability and consequence is a teaching assumption, not a real vendor quote, survey result or AI benchmark. “How Superintelligence Works” is the educational series name and does not claim that today’s AI has attained artificial superintelligence.
The answer at a glance
- Demand states: HIGH probability 30%, LOW probability 70%, by fictional assumption.
- Small print run: loss 100 points if HIGH; loss 0 if LOW.
- Large print run: loss 0 if HIGH; loss 40 if LOW.
- Without checking, expected loss is 30 for Small or 28 for Large.
- Best no-information option: Large, expected loss 28.
- Perfect demand knowledge removes the modelled decision loss; its value is 28 points before checking cost.
- An imperfect forecast in the toy 100-case ledger reduces expected decision loss to 11.6 points.
- If that forecast costs 5 points, expected total is 16.6 and improvement versus no check is 11.4 points.
1. Why additional information has a value
Information can change a choice. If an organiser already knows precisely which plan to use, an extra search may add little. If a missing fact could change which plan is preferable, a check may prevent waste or shortage.
The mathematical value comes from better decisions under specified outcomes, not from the number of documents read. A dozen irrelevant reports can have less decision value than one timely, accurate piece of information.
2. Start with the decision itself
The fictional school must choose between two alternative printing plans, called Small and Large. It cannot select both for the same event under this deliberately simplified model.
The options are abstract; the source does not state how many physical booklets each plan contains or what a real company would charge. We use loss points to illustrate the consequences of choosing a plan under two possible demand states.
3. Define the uncertain states
Demand can be HIGH or LOW. The model stipulates a 30% chance of HIGH and a 70% chance of LOW. The two states are mutually exclusive and exhaustive for this exercise.
Those figures were invented for teaching and are not measured school registration probabilities. A reliable AI should label them assumptions rather than presenting them as estimates from a hidden dataset.
4. Define what the loss points measure
Small causes a shortage under HIGH demand, assigned a loss of one hundred fictional points; under LOW demand it causes zero loss. Large avoids the shortage under HIGH but wastes supplies under LOW, assigned forty points.
These point values are not money, actual pupil harm or a moral ranking. They support a numerical comparison within the toy model. Real decisions must also consider approval, accessibility, costs and other relevant effects.
5. The complete no-check decision table
- HIGH demand — assumed probability 0.30.
- LOW demand — assumed probability 0.70.
- Small + HIGH: 100 loss points.
- Small + LOW: 0 loss points.
- Large + HIGH: 0 loss points.
- Large + LOW: 40 loss points.
- No other states or outcomes are modelled.
- Smaller expected loss is preferable under this stated fictional objective.
6. Calculate the expected loss of Small
Multiply the high-demand loss by its assumed probability and the low-demand loss by its probability: 0.30 × 100 + 0.70 × 0 = 30 points.
This is an expected loss under the toy probability model, not a prediction that exactly thirty points will be experienced. In a particular HIGH case, the Small plan incurs one hundred; in a LOW case it incurs zero.
7. Calculate the expected loss of Large
The corresponding calculation for Large is 0.30 × 0 + 0.70 × 40 = 28 expected points.
Large therefore has two fewer expected loss points than Small before any new information. That conclusion follows the stipulated probabilities and impacts. A different model could produce a different preferred plan.
8. Select the best action without checking
Because 28 is lower than 30 and the stated goal is to minimise expected loss, the best no-information option is Large. Its expected loss is twenty-eight points.
This does not establish that Large will be correct for a particular real event. The hidden demand state has not been observed. The decision is made using the information and assumptions available before the outcome occurs.
9. What does perfect information mean?
A perfectly informative signal would reveal whether demand is HIGH or LOW before the school commits to printing. The school could then select Large for HIGH demand and Small for LOW demand.
Perfect information is an idealised benchmark. It is not an assertion that any real survey, AI forecast or website can predict demand without error. The concept helps calculate the maximum value of learning the state under this model.
10. What perfect information would allow
When HIGH is known, choose Large and incur zero loss in the table. When LOW is known, choose Small and again incur zero.
The expected decision loss with perfect information is 0.30 × 0 + 0.70 × 0 = 0. The result depends on the simplified loss table, where matching the action to the state always avoids the specified adverse outcome.
11. Calculate the value of perfect information
The expected value of perfect information, expressed as avoided expected loss, is the best no-check expected loss minus the perfectly informed expected loss: 28 − 0 = 28 points.
This is an upper benchmark for information value under the toy decision model. It does not mean a real forecast must deliver twenty-eight points, and it does not include any cost of obtaining that information.
12. When a perfect check is costly
Imagine a hypothetical perfect check costing five points per decision. Its expected total cost would be zero decision loss plus five checking points, or five. Compared with the no-check expectation of twenty-eight, the improvement is twenty-three points.
A check costing thirty points would not be justified solely by the avoided twenty-eight-point expected loss. Additional goals could matter, but the simple numerical model does not supply them.
13. Real forecasts are usually imperfect
A survey or classification model may sometimes flag HIGH demand when the eventual state is LOW, or fail to flag HIGH when it occurs. The decision benefit depends on those errors and the consequences of acting on the signal.
An AI should not substitute perfect-information calculations for a real signal without independent accuracy evidence. The next fixture gives an intentionally imperfect signal with every outcome counted.
14. The fictional hundred-case signal ledger
Imagine one hundred independent, equally weighted hypothetical school events: thirty HIGH and seventy LOW. A forecast system labels thirty-eight events HIGH and sixty-two LOW, with errors in both directions.
These are invented reference cases, not a real estimate of event frequencies or forecast performance. Their purpose is to show how a signal can improve decisions even when it is not perfectly accurate.
15. The imperfect forecast reference
- Actual HIGH events: 30.
- Actual LOW events: 70.
- Forecast HIGH and actually HIGH: 24.
- Forecast HIGH but actually LOW: 14.
- Forecast LOW but actually HIGH: 6.
- Forecast LOW and actually LOW: 56.
- Total forecast HIGH: 24 + 14 = 38.
- Total forecast LOW: 6 + 56 = 62.
- Total actual HIGH: 24 + 6 = 30.
- Total actual LOW: 14 + 56 = 70.
- All four cells sum to exactly 100.
16. Choose a plan after a HIGH forecast
The defined response policy is to choose Large when the forecast says HIGH. Among the thirty-eight HIGH forecasts, twenty-four are genuinely HIGH, where Large has zero loss, and fourteen are actually LOW, where Large has forty points of waste loss.
The total loss for this branch is 24 × 0 + 14 × 40 = 560 points across the hypothetical one hundred-event ledger.
17. Choose a plan after a LOW forecast
When the forecast says LOW, the policy chooses Small. Of the sixty-two LOW forecasts, fifty-six are actually LOW and incur zero loss, while six are really HIGH and incur a hundred-point shortage loss each.
The total loss for this branch is 56 × 0 + 6 × 100 = 600 points. The rare missed HIGH events contribute more loss than the more numerous false HIGH alarms in this particular point model.
18. Calculate total decision loss with the forecast
Add the losses from both branches: 560 + 600 = 1,160 points across one hundred hypothetical cases. Divide by one hundred to obtain an average decision loss of 11.6 points per case.
This is the result of the specified signal and response policy. It is not the actual loss of a deployed AI in a real school. Its accuracy depends on the assumed classification table and whether the policy is followed.
19. Compare forecast versus no forecast
Without information the model chooses Large in every case, with expected loss 28 per case. With the given forecast policy, the computed average decision loss is 11.6.
Thus the gross expected information benefit of that signal under the fixed toy scenario is 28 − 11.6 = 16.4 points per decision. The imperfect signal cannot reach the 28-point ideal of perfect information here.
20. Include the cost of obtaining the forecast
Suppose the imperfect forecast costs five points per case. Its expected total becomes 11.6 decision-loss points plus five checking points, or 16.6.
That is 28 − 16.6 = 11.4 points lower than the best no-check expected loss. Under these invented assumptions, the check is worth using from a loss-minimisation perspective.
21. The break-even check cost
The gross expected improvement from this imperfect signal is 16.4 points. If the check itself costs exactly 16.4, the total expectation equals 28, tying the no-check strategy.
If the checking cost is greater than 16.4 and all other assumptions remain fixed, the same policy is numerically worse than acting without the signal. That is a decision threshold, not a prediction of actual research expenses.
22. Why a five-point check is not automatically available
The fixture assigns a hypothetical price of five points but does not provide any real survey service, provider fee or authorised action. The assistant may calculate the proposed consequence without claiming that a live demand forecast was ordered.
An information request can also consume time or reveal sensitive data. Real decision value must account for those costs and constraints rather than only the invented numeric charge.
23. A forecast flag changes probabilities
Among the thirty-eight HIGH forecasts, twenty-four really are HIGH. The fraction is 24/38 ≈ 63.2%. Among sixty-two LOW forecasts, six are actually HIGH, or 6/62 ≈ 9.7%.
Those are different conditional probabilities. They help explain why the recommended action should depend on the signal. The rates are observed in the fictional case set, not real calibration results.
24. Why Large makes sense after a HIGH flag
Given a HIGH forecast in the toy ledger, a Small run would incur a hundred-point loss in twenty-four of thirty-eight cases. Its conditional expected loss is (24/38) × 100 ≈ 63.2 points.
Large incurs forty in fourteen of thirty-eight cases, giving (14/38) × 40 ≈ 14.7 points. Large has the lower conditional expected loss for that signal under the fixed scenario.
25. Why Small makes sense after a LOW flag
After a LOW forecast, Small incurs a hundred-point shortage in six of sixty-two cases, an estimated conditional expected loss of (6/62) × 100 ≈ 9.7.
Large would waste forty points in fifty-six of sixty-two cases, an expected loss of (56/62) × 40 ≈ 36.1. Small therefore has the lower conditional expected loss when the signal says LOW.
26. The signal policy itself can be tested
The conditional calculations support the rule: choose Large following a HIGH forecast, and Small following a LOW forecast. The rule follows from the stipulated table rather than being imposed without examination.
An AI should not automatically assume that every warning label deserves the more aggressive action. The effect depends on base rates, error patterns and the loss associated with each decision.
27. The complete value-of-information scorecard
- Without information: Small expected loss 30.
- Without information: Large expected loss 28.
- Best without information: Large.
- Perfect information: expected decision loss 0.
- Value of perfect information: 28.
- Forecast true HIGH flags: 24.
- Forecast false HIGH flags: 14.
- Forecast missed HIGH cases: 6.
- Forecast correct LOW flags: 56.
- With forecast, total loss across 100 cases: 1,160.
- Average forecast-guided decision loss: 11.6.
- Gross imperfect-information value: 16.4.
- Hypothetical checking cost: 5 per decision.
- Forecast-guided total including check: 16.6.
- Net improvement versus no check: 11.4.
28. What the hundred-case table does not prove
One hundred deliberately constructed observations do not establish how accurate a real forecast is in Singapore or anywhere else. The sample may not represent actual future demand, and the estimated risks may change over time.
The purpose is a clear calculation under frozen assumptions. An AI should not use the invented proportions to recommend a real procurement or invent a probability about a named school.
29. Twenty-four claims to classify
Mark each statement SUPPORTED, CONTRADICTED or NOT ESTABLISHED from the two-state loss table and the hundred-case forecast ledger. Supported conclusions follow from the arithmetic. Contradicted claims conflict with the fixture. Unestablished claims refer to real-world outcomes or external actions not observed here.
Distinguish gross information benefit from benefit after the cost of checking. Also distinguish perfect information from the imperfect signal actually described.
- 01. “HIGH demand has stipulated probability 30%” — SUPPORTED.
- 02. “LOW demand has stipulated probability 70%” — SUPPORTED.
- 03. “Small under HIGH has loss 100” — SUPPORTED.
- 04. “Large under LOW has loss 40” — SUPPORTED.
- 05. “Small has no-check expected loss 30” — SUPPORTED.
- 06. “Large has no-check expected loss 28” — SUPPORTED.
- 07. “Without checking, Small is the lower-expected-loss choice” — CONTRADICTED.
- 08. “Perfect information makes the specified decision loss zero” — SUPPORTED.
- 09. “Value of perfect information is 28 points” — SUPPORTED.
- 10. “There are 30 actual HIGH cases in the hundred-case ledger” — SUPPORTED.
- 11. “There are 70 actual LOW cases” — SUPPORTED.
- 12. “The forecast correctly flags HIGH in 24 cases” — SUPPORTED.
- 13. “Fourteen cases are false HIGH alarms” — SUPPORTED.
- 14. “Six actual HIGH cases are missed by the forecast” — SUPPORTED.
- 15. “Fifty-six LOW cases are correctly labelled LOW” — SUPPORTED.
- 16. “Total HIGH forecasts are 38” — SUPPORTED.
- 17. “Total LOW forecasts are 62” — SUPPORTED.
- 18. “Forecast-guided average decision loss is 11.6” — SUPPORTED.
- 19. “The imperfect forecast is equivalent to perfect information” — CONTRADICTED.
- 20. “The forecast’s gross information value is 16.4” — SUPPORTED.
- 21. “With a five-point check, expected total loss is 16.6” — SUPPORTED.
- 22. “After the check cost, improvement versus no check is 11.4” — SUPPORTED.
- 23. “These invented frequencies prove a named real school has this demand pattern” — NOT ESTABLISHED.
- 24. “The source proves a real external forecast was purchased” — NOT ESTABLISHED.
30. Reconcile the labels
The complete answer key has 20 supported statements, 2 contradicted statements and 2 not-established statements. They add to 24. Every supported figure can be recalculated from the loss table or forecast outcomes.
The unknown statements concern real institutions and external purchases. A toy calculation can demonstrate information value without verifying a live forecast service or authorising the assistant to spend money.
31. Why search cost is only one consideration
A check might require time, permission, access to private records or a risk of delaying an urgent decision. Even if its direct cost is small, the overall value may depend on those practical constraints.
An AI decision assistant should identify material costs and rules that are not in the mathematical fixture. It must not pretend that a signal is risk-free or available instantly merely because its estimated value is positive.
32. A perfect signal is an upper comparison
Perfect information gives the decision-maker everything relevant about the state before choosing. Within this model that creates zero avoidable loss, which is why its theoretical value is twenty-eight points.
An actual source is rarely perfect. The value-of-information concept can still help decide whether to search, but the estimated reliability of the source and the consequences of acting on errors matter greatly.
33. A check can be accurate but not useful
Imagine a report that tells the printing desk the weather next month, but the worksheet demand in our model does not depend on the weather. Such a report may be perfectly accurate and still not change the optimal choice.
Information has decision value when it affects actions or expected outcomes under the task’s objective. Collecting unrelated facts is not inherently beneficial simply because the information is true.
34. An inexpensive check can still be misleading
Suppose the forecast is unreliable and the school responds to its labels without measuring real error rates. A cheap report might lead to a worse decision than ignoring it.
A responsible AI should evaluate how a signal relates to the outcome and test whether acting on it improves expected results. The cost alone does not establish its usefulness.
35. An important difference from a stopping rule
A stopping rule describes when to end a search or inspection process. Value of information asks what expected benefit the next piece of evidence could provide to a choice.
The ideas are related but not identical. The assistant may stop when evidence is sufficient for the user’s action, when another search has low expected value or when a practical budget is exhausted. It should say why.
36. A transfer exercise with a tutor’s timetable
Imagine a tutor deciding whether to reserve an extra room for a potential second study group. The cost of leaving pupils without suitable space and the cost of an unnecessary room differ.
An early attendance confirmation might change the sensible booking decision. The assistant can model the alternatives, but real room availability, approval and external booking outcomes require authoritative checks.
37. A transfer exercise with equipment maintenance
A fictional office must choose whether to inspect a printer before a busy week. Inspection has a cost but may prevent a larger breakdown expense if it reveals a fault that changes the maintenance decision.
The value depends on how likely faults are, how well the test detects them and the consequences of action or inaction. Those quantities must be grounded rather than improvised as precise-looking numbers.
38. How to evaluate an AI that requests more information
Test cases where a clarification changes the best decision, where the answer is already determined, and where the possible signal is too costly or unreliable to help. Freeze the probabilities and consequence tables before assessing the AI.
Score correct arithmetic separately from whether the model invents the signal’s accuracy or claims to have taken an external action. A useful decision explanation does not require pretending the check has actually occurred.
39. A classroom exercise
Give pupils a two-by-two grid with Small, Large, HIGH and LOW. Ask them to calculate expected losses, choose the better no-check option and then imagine a perfectly informative message.
Next give the hundred-case forecast table and ask them to count false HIGH alarms, missed HIGH events and total decision loss under the signal-guided rule. The contrast makes information value concrete.
40. What a trustworthy AI report sounds like
“Under the invented demand probabilities, the best no-check print run is Large with expected loss twenty-eight. A hypothetical perfect forecast could eliminate that loss. The specified imperfect forecast reduces decision loss to 11.6; after a five-point check cost, the expected total is 16.6, or 11.4 better than no check.”
This statement is useful because it names the assumptions, separates gross and net benefit, and refuses to imply a real supplier or school has verified these invented numbers.
41. The central lesson
Information is valuable when it changes a decision in a way that improves expected outcomes enough to offset the cost of obtaining it. More facts are not automatically better, and an imperfect signal can be useful without being perfectly reliable.
A dependable AI assistant identifies the uncertain state, calculates the consequences of available actions, evaluates the accuracy of new evidence and preserves permissions. The right time to check again is a decision about value, not merely curiosity.
Frequently asked questions
- What is value of information? — The expected improvement in a decision from receiving useful evidence.
- What is perfect information? — An idealised signal revealing the relevant state without error.
- Why choose Large without checking here? — Its expected loss is 28, lower than Small’s 30.
- Is a perfect check really available? — Not established; it is a hypothetical benchmark.
- What is the imperfect forecast worth before cost? — 16.4 points under the invented ledger.
- Is the five-point forecast worth using in the toy model? — Yes, it reduces expected loss from 28 to 16.6.
- Does positive information value authorise a purchase? — No, real spending requires appropriate approval and confirmation.
Connected lessons and references
- How Super Intelligence Works — educational curriculum.
- Search Stopping Rules — when additional evidence may be unnecessary.
- Risk and Consequence — probability and loss.
- Conditional Probability and Base Rates — interpreting signals.
- Value of Information — background to decision-theoretic information value.
- NIST AI Risk Management Framework — reliable assessment and use of AI tools.