Value of information: a check pays only when it changes an action

Type the two states a test could tell apart, what each action pays in each, the price and the wait. Where both results name the same action the check is worth nothing, at any accuracy.

Live result

At 80.0% declared accuracy the check is worth 20.00, more than its price and waiting.

Price and waiting7.00
Left over13.00

A Good result names Action A and a Bad result names Action B.

Three amounts against the ceiling

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Declared accuracy80.0percent

The accuracy you are stating for this check. The page holds no record of how any check has performed.

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The two states

State 1

The two states the check could tell apart. Up to 24 characters each, so both fit their own column.

State 2

Probability on the first state60.0percent

Your own probability on the first state before any check. The second state carries the rest.

Exact probability on the first statepercent

Four decimal places, so a stated probability can be typed exactly.

Actions and payoffs

Action 1

Action name

Up to 24 characters, so the name fits its own row.

Pays in ⁦Good⁩

Use one currency for all money values. Every payoff must already include the consequences you want compared, over one period.

Pays in ⁦Bad⁩

Action 2

Action name

Up to 24 characters, so the name fits its own row.

Pays in ⁦Good⁩

Pays in ⁦Bad⁩

Two actions are required because one action leaves no decision to change.

Price of the check

What the check itself costs, on the same basis as the payoffs.

Waitingdays

Whole days between commissioning the check and having its result.

Cost of a day of waiting

What one day of waiting costs you, on the same basis as the payoffs.

Exact declared accuracypercent

Four decimal places, so an exact crossing can be typed.

Every action

Action A

Pays in Good: 100.00
Pays in Bad: 0.00
Expected now: 60.00
Named by a Good result: yes
Named by a Bad result: no

Action B

Pays in Good: 0.00
Pays in Bad: 100.00
Expected now: 40.00
Named by a Good result: no
Named by a Bad result: yes

Every action
Action	Pays in Good	Pays in Bad	Expected now	Named by a Good result	Named by a Bad result
Action A	100.00	0.00	60.00	yes	no
Action B	0.00	100.00	40.00	no	yes

What the check is worth

Expected payoff of the baseline action

Value: 60.00

Ceiling at certainty

Value: 40.00

Worth at your accuracy

Value: 20.00

Price

Value: 5.00

Cost of waiting

Value: 2.00

Left over

Value: 13.00

What the check is worth
What the check is worth	Value
Expected payoff of the baseline action	60.00
Ceiling at certainty	40.00
Worth at your accuracy	20.00
Price	5.00
Cost of waiting	2.00
Left over	13.00

The accuracy you are declaring for the check

percent

Frompercent

Topercent

In steps ofpercent

It does not check the values in between.

Across the values sampled, whether the check clears its price and its waiting stays Clears its cost from 70.0 percent to 100.0 percent, the highest value sampled. The nearest value where it changes is 65.0 percent, where it becomes Does not clear its cost.

The answer	Clears its cost
Values sampled	50.0 percent to 100.0 percent, 11 values, step 5.0 percent
Holds from	70.0 percent
Holds to	100.0 percent, highest sampled
Nearest value that changes it	65.0 percent
The answer there	Does not clear its cost

What an unchanged region does and does not prove is set out in a short lesson.

The sweep samples only the values you entered. Between samples the answer is not checked, so read the boundary as observed, not proven.

Here is the maths

The formula, in words

Each action's expected payoff now is its payoff in each state weighted by the probability of that state, and the baseline is the largest of them. With perfect knowledge you would take the best action in each state, and the ceiling is that figure less the baseline. A check of a declared accuracy $q$ reports each state correctly with that probability $q$ and the other state with the rest. Its worth is the expected payoff when the action may be chosen after the result, less the baseline. Price and the cost of waiting come out afterwards. The worth can never exceed the ceiling, and where the same action is named under both results the worth is zero.

A worked example

Action $A$ now $= 0.60 \times 100 + 0.40 \times 0 = 60.00$ . Action $B$ now $= 0.60 \times 0 + 0.40 \times 100 = 40.00$ , so the baseline is Action $A$ at 60.00.
Perfect knowledge $= 0.60 \times \max(100, 0) + 0.40 \times \max(0, 100) = 100.00$ . The ceiling is $100.00 - 60.00 = 40.00$ .
A Good result gives Action $A$ a weighted score of 48.00 and Action $B$ a score of 8.00, so it names Action $A$ .
A Bad result gives Action $A$ a weighted score of 12.00 and Action $B$ a score of 32.00, so it names Action $B$ .
The informed payoff is $48.00 + 32.00 = 80.00$ . The check is worth $80.00 - 60.00 = 20.00$ .
Price and waiting $= 5.00 + 2 \times 1.00 = 7.00$ . Left over $= 20.00 - 7.00 = 13.00$ .

What this does not model

It prices only the decision you described. It reads no accuracy from anywhere, verifies no check, and knows nothing about whether a check performs as declared. It assumes both payoffs are on one currency and one period and already include the consequences you wanted compared. It assumes a symmetric accuracy, meaning the check is equally likely to report either state correctly. It compares two states and two or three actions, because those are the calculation's own limits. A third action that is never best under either result changes no number on this page.

Check it yourself

Set the sweep to run from 66.5 to 67.5 in steps of 0.1. Eleven values are checked. The answer holds through 67.0 and changes at 67.1. Then set it to run from 59 to 61 in steps of 1. The middle value has no answer at all, because at exactly 60 percent a Bad result leaves the two actions exactly equal.

Where the ceiling is stronger than the rail

The rail reports only the values it checked. The ceiling does not depend on accuracy at all, so a ceiling of zero settles every accuracy, including the values between the samples.

The calculation domain

The calculation accepts two states, two or three actions, payoffs to two decimals from $-1{,}000{,}000{,}000$ to a billion, probability and accuracy to four decimals, and waiting to 3,650 whole days. This page offers a narrower payoff range so the money labels fit the chart.

Boundary

It prices the decision you described and nothing else. A record you owe to somebody may be required whether or not it changes an action, and this page has no opinion about that. It takes your declared accuracy as declared, and it verifies no check.

Source anchor

Ronald A. Howard, Information Value Theory, 1966. The published paper sets out the decision analysis structure this page follows. Every number on this page comes from your own typing.

Calculation version

1.0.0

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This page will never advise whether to meet, avoid or reinterpret an obligation to hold a record. It describes the economic choice inside the decision model you typed. It does not determine legal, regulatory, technical, financing, assurance or evidentiary sufficiency, and it does not tell you to commission anything.

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