The Central Idea
Information is valuable only when it can change what we do or improve the result of what we do.
The basic comparison is:
\[ \boxed{ \text{Value of Information} = \text{Value with Information} - \text{Value without Information} }. \]If information does not improve the decision, its decision value is zero.
Information Changes Decisions by Reducing Uncertainty
Before obtaining information, a decision maker chooses using current probabilities, estimates, or scenarios.
After obtaining information, the decision maker may:
- update probabilities;
- eliminate implausible states;
- choose a different action;
- avoid a bad outcome;
- take advantage of a favorable condition.
Information Has Decision Value, Not Merely Descriptive Value
A report can be interesting without being decision-relevant.
Value-of-information analysis asks whether the new information changes the expected quality of the decision.
If the same action remains optimal no matter what the information says, the information may have little or no decision value.
The Baseline: Best Decision Without New Information
Begin by finding the best available decision using current information.
Suppose two decisions have expected values
\[ E[A]=42 \]and
\[ E[B]=38. \]Without additional information, choose A and obtain baseline expected value
\[ \boxed{42}. \]Perfect Information
Perfect information reveals the true state before the decision is made.
It is an ideal benchmark. Real information is rarely perfect, but perfect information tells us the maximum possible value that any information about the uncertain state could provide.
Expected Value with Perfect Information
With perfect information, the decision maker selects the best action for each state after learning which state will occur.
The expected value with perfect information is
\[ \boxed{ EV_{\text{wPI}} = \sum_j p_j \left( \max_i V_{ij} \right) }. \]Here \(V_{ij}\) is the value of decision \(i\) in state \(j\).
Expected Value of Perfect Information
The expected value of perfect information, or EVPI, is
\[ \boxed{ EVPI = EV_{\text{wPI}} - EV_{\text{woPI}} }. \]The second term is the expected value of the best decision without perfect information.
Example: Decision Table
| Decision | Low Demand \(0.4\) | High Demand \(0.6\) |
|---|---|---|
| Small Run | $20,000 | $35,000 |
| Large Run | -$5,000 | $60,000 |
Best Decision Without Perfect Information
For the small run,
\[ E[\text{small}] =0.4(20{,}000)+0.6(35{,}000) \] \[ =29{,}000. \]For the large run,
\[ E[\text{large}] =0.4(-5{,}000)+0.6(60{,}000) \] \[ =34{,}000. \]Therefore,
\[ \boxed{EV_{\text{woPI}}=34{,}000}. \]Best Decision with Perfect Information
If demand is known to be low, choose the small run because
\[ 20{,}000>-5{,}000. \]If demand is known to be high, choose the large run because
\[ 60{,}000>35{,}000. \]Therefore,
\[ EV_{\text{wPI}} =0.4(20{,}000)+0.6(60{,}000). \] \[ =8{,}000+36{,}000. \] \[ \boxed{EV_{\text{wPI}}=44{,}000}. \]Compute EVPI
\[ EVPI =44{,}000-34{,}000. \] \[ \boxed{EVPI=10{,}000}. \]Perfect information about demand is worth at most $10,000 in expected monetary value.
EVPI Is an Upper Bound
No imperfect information source about the same uncertainty should be worth more than perfect information under the same model.
Therefore,
\[ \boxed{EVSI\le EVPI} \]for ordinary sample-information settings.
Information Cannot Have Negative Gross Value
If information can be ignored, access to it cannot make the optimal decision worse before considering its cost.
Therefore the gross expected value of information satisfies
\[ \boxed{VOI\ge0}. \]Net value can be negative after information costs are included.
Sample Information
Sample information is imperfect information obtained from data, testing, surveys, inspections, experiments, or other observations.
Unlike perfect information, sample information may sometimes point in the wrong direction.
Expected Value of Sample Information
The expected value of sample information, or EVSI, is
\[ \boxed{ EVSI = EV_{\text{wSI}} - EV_{\text{woSI}} }. \]Here \(EV_{\text{wSI}}\) is the expected value when the decision maker can observe the sample information before choosing an action.
Sample Information Usually Changes Probabilities
Suppose a test can return either:
- favorable;
- unfavorable.
After observing the result, the decision maker updates probabilities such as
\[ P(\text{high demand}\mid\text{favorable}) \]and
\[ P(\text{high demand}\mid\text{unfavorable}). \]Bayes' Theorem Connects Information to Decisions
When a test result is informative about an uncertain state, Bayes' theorem can be used to update probabilities.
\[ \boxed{ P(H\mid F) = \frac{P(F\mid H)P(H)}{P(F)} }. \]The updated probability then enters the decision calculation.
Example: Prior Probability
Suppose the prior probability of high demand is
\[ P(H)=0.6. \]Therefore,
\[ P(L)=0.4. \]A market test can return favorable \(F\) or unfavorable \(U\).
Test Accuracy
Suppose
\[ P(F\mid H)=0.8 \]and
\[ P(F\mid L)=0.3. \]The test is more likely to be favorable when demand is actually high.
Probability of a Favorable Test
By total probability,
\[ P(F) = P(F\mid H)P(H) + P(F\mid L)P(L). \] \[ =0.8(0.6)+0.3(0.4). \] \[ =0.48+0.12. \] \[ \boxed{P(F)=0.60}. \]Posterior Probability after a Favorable Test
\[ P(H\mid F) = \frac{0.8(0.6)}{0.60}. \] \[ =\frac{0.48}{0.60}. \] \[ \boxed{P(H\mid F)=0.80}. \]The favorable test raises the probability of high demand from \(0.60\) to \(0.80\).
Probability of an Unfavorable Test
\[ P(U)=1-P(F)=0.40. \]Also,
\[ P(U\mid H)=0.2 \]and
\[ P(U\mid L)=0.7. \]Posterior Probability after an Unfavorable Test
\[ P(H\mid U) = \frac{P(U\mid H)P(H)}{P(U)}. \] \[ =\frac{0.2(0.6)}{0.40}. \] \[ =\frac{0.12}{0.40}. \] \[ \boxed{P(H\mid U)=0.30}. \]Use the Posterior Probability to Choose Again
Recall the payoffs:
| Decision | Low Demand | High Demand |
|---|---|---|
| Small Run | $20,000 | $35,000 |
| Large Run | -$5,000 | $60,000 |
After each test result, recompute the expected values using the updated probabilities.
Decision after a Favorable Test
Given \(F\),
\[ P(H\mid F)=0.8,\qquad P(L\mid F)=0.2. \]Small run:
\[ E[\text{small}\mid F] =0.2(20{,}000)+0.8(35{,}000) \] \[ =4{,}000+28{,}000 \] \[ =32{,}000. \]Large run:
\[ E[\text{large}\mid F] =0.2(-5{,}000)+0.8(60{,}000) \] \[ =-1{,}000+48{,}000 \] \[ =47{,}000. \]Choose the large run.
Decision after an Unfavorable Test
Given \(U\),
\[ P(H\mid U)=0.3,\qquad P(L\mid U)=0.7. \]Small run:
\[ E[\text{small}\mid U] =0.7(20{,}000)+0.3(35{,}000) \] \[ =14{,}000+10{,}500 \] \[ =24{,}500. \]Large run:
\[ E[\text{large}\mid U] =0.7(-5{,}000)+0.3(60{,}000) \] \[ =-3{,}500+18{,}000 \] \[ =14{,}500. \]Choose the small run.
Expected Value with Sample Information
The favorable test occurs with probability \(0.60\), and the unfavorable test occurs with probability \(0.40\).
Therefore,
\[ EV_{\text{wSI}} = 0.60(47{,}000)+0.40(24{,}500). \] \[ =28{,}200+9{,}800. \] \[ \boxed{EV_{\text{wSI}}=38{,}000}. \]Compute EVSI
Without sample information, the best expected value was
\[ 34{,}000. \]Therefore,
\[ EVSI = 38{,}000-34{,}000 \] \[ \boxed{EVSI=4{,}000}. \]Compare EVSI with EVPI
We found
\[ EVPI=10{,}000 \]and
\[ EVSI=4{,}000. \]As expected,
\[ \boxed{EVSI<EVPI}. \]The test is useful, but it does not reveal demand perfectly.
Cost of Information
Information is not free.
Possible costs include:
- survey expense;
- testing expense;
- consulting fees;
- data acquisition;
- staff time;
- delay;
- opportunity cost.
Net Value of Information
If information costs \(C\), then
\[ \boxed{ \text{Net Value of Information} = \text{Gross Value of Information} - C }. \]Obtain the information when its expected improvement is worth more than its full decision-relevant cost.
Example: Is the Market Test Worth Buying?
Suppose the test costs
\[ 3{,}000. \]Since
\[ EVSI=4{,}000, \]the net value is
\[ 4{,}000-3{,}000 \] \[ \boxed{1{,}000}. \]Under this model, purchasing the test is worthwhile.
Maximum Worthwhile Information Cost
The gross value of information gives the maximum price the decision maker should be willing to pay under the chosen criterion.
If
\[ EVSI=4{,}000, \]then paying more than $4,000 would make the net expected value negative.
Information Has No Value When It Cannot Change the Action
Suppose Decision A is best under every possible test result.
Then the information does not alter the optimal action.
In that case,
\[ \boxed{EVSI=0} \]even if the information is statistically interesting.
Information Is Most Valuable Near Decision Boundaries
Sensitivity analysis often reveals threshold values where the preferred decision changes.
When current beliefs lie near such a threshold, better information may have substantial value because a modest update can reverse the decision.
Information May Be Low Value When the Decision Is Robust
If one action remains best across a wide range of plausible probabilities or assumptions, reducing uncertainty may not change the recommendation.
In such cases, information can have low decision value even when uncertainty itself is large.
Value of Information Depends on the Available Actions
Information is useful only if the decision maker can act on it.
If there is only one feasible action, learning the state may improve understanding without improving the decision.
Value of Information Depends on Timing
Information received after an irreversible decision may have little value for that decision.
The same information received before commitment can be highly valuable.
Timing is therefore part of the information problem.
Delayed Decisions Can Create Option Value
Sometimes waiting allows information to arrive before a major commitment is made.
The ability to delay, stage, or reverse a decision can create option value.
Flexibility can therefore be valuable even before the information itself is obtained.
Perfect Information and Expected Loss
Value-of-information ideas can also be written in terms of loss.
Without information, choose the action with minimum expected loss.
With information, choose the minimum-loss action for each information state.
The value of information is then the expected reduction in loss.
Expected Reduction in Loss
If
\[ R_{\text{without}} \]is the minimum expected loss without information and
\[ R_{\text{with}} \]is the minimum expected loss with information, then
\[ \boxed{ VOI=R_{\text{without}}-R_{\text{with}} }. \]Value of Information and Expected Utility
If decisions are evaluated by utility instead of money, compare expected utilities.
Information is valuable when it increases the maximum expected utility achievable before the decision is made.
This allows value-of-information analysis to incorporate risk preferences.
More Accurate Information Is Not Automatically More Valuable
Accuracy matters only through its effect on the decision.
A highly accurate signal can have low value if the same action is chosen regardless of the result.
A moderately accurate signal can have high value if it frequently moves the decision across an important threshold.
Information Quality
Useful information can be evaluated along several dimensions:
- accuracy;
- reliability;
- timeliness;
- relevance;
- resolution;
- cost;
- ability to change the decision.
False Positives and False Negatives Affect Information Value
A test may make two kinds of mistakes.
Their effect on value depends on:
- how often each error occurs;
- which decision each result triggers;
- the loss caused by choosing incorrectly.
Information quality must therefore be evaluated in the context of the decision.
Information Gathering as a Decision
Whether to gather information is itself a decision.
The decision tree may begin with:
- buy the information;
- do not buy the information.
The information branch then contains chance nodes for possible signals and later decision nodes for actions taken after observing those signals.
Rollback Analysis for Information Decisions
Evaluate information decisions from right to left.
- Find the best action after each possible signal.
- Compute the expected value across signal outcomes.
- Subtract the information cost.
- Compare with the best decision available without information.
Sequential Information
Information can arrive in stages.
A decision maker might:
- run a low-cost screening test;
- perform an expensive test only if the first result is ambiguous;
- make the final decision after the available evidence is assembled.
Decision trees are particularly useful for these staged information strategies.
Partial Information
Information need not identify the exact state to be useful.
It may narrow the possibilities or divide them into groups.
Even partial reduction of uncertainty can improve a decision.
Information and Learning
A new observation can change beliefs even when it does not immediately change the current decision.
If the updated knowledge affects future decisions, the information may have broader long-term value.
The scope of the analysis should match the scope of the decisions being evaluated.
Common Error: Comparing Information Value with the Wrong Baseline
Value of information is measured relative to the best decision available without the information.
Comparing against an inferior baseline exaggerates the value of information.
Common Error: Forgetting to Reoptimize after the Signal
The purpose of information is to allow a better decision after the signal is observed.
Each information result should therefore be followed by a fresh comparison of the available actions.
Common Error: Subtracting Information Cost Too Early
First compute the gross expected value produced by the information.
Then subtract the information cost.
Keeping these quantities separate makes the analysis easier to interpret and audit.
Common Error: Assuming Perfect Information Is Realistically Available
Perfect information is usually a benchmark, not a practical option.
Its main purpose is to establish an upper bound on how valuable better information could possibly be under the model.
Common Error: Paying More Than EVPI
Under a consistent expected-value model, no information about the uncertain state should be worth more than perfect information about that state.
If a proposed information source costs more than EVPI, it cannot be justified by decision improvement alone within that model.
Common Error: Ignoring Delay and Opportunity Cost
Information can have hidden costs if obtaining it delays action.
The total cost may therefore include:
- direct research expense;
- time cost;
- lost market opportunity;
- expiration of alternatives;
- cost of maintaining flexibility.
A Reliable Procedure for Perfect Information
- Construct the decision table.
- Compute the best expected value without information.
- For each state, identify the best available action.
- Weight those state-specific best values by their probabilities.
- Add to obtain expected value with perfect information.
- Subtract the baseline expected value.
- Interpret the result as the maximum worth of perfect information under the model.
A Reliable Procedure for Sample Information
- Find the best decision without new information.
- List the possible information signals.
- Find the probability of each signal.
- Update state probabilities after each signal.
- Find the best decision after each signal.
- Weight those optimized conditional values by signal probabilities.
- Add to obtain expected value with sample information.
- Subtract the no-information baseline.
- Subtract information cost to obtain net value.
Practice 1: Baseline Expected Value
Decision A has expected value \(35\). Decision B has expected value \(42\).
What is the expected value without additional information?
Practice 2: Expected Value with Perfect Information
There are two states:
- State 1 has probability \(0.3\) and best available payoff \(50\);
- State 2 has probability \(0.7\) and best available payoff \(80\).
Find \(EV_{\text{wPI}}\).
Practice 3: EVPI
Suppose
\[ EV_{\text{wPI}}=71 \]and
\[ EV_{\text{woPI}}=62. \]Find EVPI.
Practice 4: Information Cost
A test has EVSI of $6,000 and costs $4,500.
Find its net value.
Practice 5: Maximum Worthwhile Price
A survey has EVSI of $2,800.
What is the maximum price that can be justified by expected decision improvement alone?
Practice 6: Total Probability
Suppose
\[ P(H)=0.5,\qquad P(L)=0.5, \] \[ P(F\mid H)=0.9, \]and
\[ P(F\mid L)=0.2. \]Find \(P(F)\).
Practice 7: Posterior Probability
Using Practice 6, find
\[ P(H\mid F). \]Practice 8: Zero Information Value
A test changes the estimated probability of an event, but Decision A remains optimal after every possible test result.
What is the gross decision value of the test?
Practice 9: Compare EVSI and EVPI
A model gives
\[ EVPI=12 \]and
\[ EVSI=15. \]What should you conclude?
Practice 10: Information near a Threshold
A sensitivity analysis shows the decision changes when a probability crosses \(0.50\). The current estimate is \(0.51\).
Would additional information about that probability potentially have high or low decision value? Explain.
Practice Answers
Practice 1
Choose the larger baseline expected value:
\[ \boxed{42}. \]Practice 2
\[ EV_{\text{wPI}} =0.3(50)+0.7(80) \] \[ =15+56 \] \[ \boxed{71}. \]Practice 3
\[ EVPI=71-62 \] \[ \boxed{9}. \]Practice 4
\[ 6{,}000-4{,}500 \] \[ \boxed{1{,}500}. \]Practice 5
The maximum justified price is the gross value of the information:
\[ \boxed{\$2{,}800}. \]Practice 6
\[ P(F) = 0.9(0.5)+0.2(0.5) \] \[ =0.45+0.10 \] \[ \boxed{0.55}. \]Practice 7
\[ P(H\mid F) = \frac{0.9(0.5)}{0.55} \] \[ =\frac{0.45}{0.55} \] \[ \boxed{\frac{9}{11}\approx0.818}. \]Practice 8
If the same action remains optimal after every signal, the information cannot improve the decision.
\[ \boxed{EVSI=0}. \]Practice 9
The result is inconsistent with the ordinary value-of-information framework because sample information should not exceed perfect information.
\[ \boxed{EVSI\le EVPI}. \]The calculations or assumptions should be checked.
Practice 10
The information could have relatively high decision value because the current estimate is close to the threshold. A small update could reverse the preferred action.
What to Remember
Value-of-information analysis asks whether reducing uncertainty is worth the cost.
- Begin with the best decision available without new information.
- Perfect information reveals the true state before the decision.
- EVPI is the improvement from perfect information.
- Sample information is imperfect but may still improve decisions.
- EVSI measures the expected improvement produced by sample information.
- Bayesian updating connects signals to revised probabilities.
- Information cost must be subtracted to obtain net value.
- EVPI provides an upper bound on the value of imperfect information.
- Information has little value when it cannot change the action.
- Information is often most valuable near important decision thresholds.
The most important principle is: information is valuable because of the decisions it improves, not merely because it reduces uncertainty.
Where We Go Next
Value-of-information analysis treats information as something that can improve decisions by reducing uncertainty. The next lesson studies a broader mathematical concept that measures uncertainty itself.
The next lesson, Entropy and Information, develops quantitative measures of uncertainty and information content and connects probability to coding, prediction, and communication.