Using Value of Information in Strategic Decisions

Why Information Has Value Only Through Decisions

Strategic choices are often made before important uncertainties are resolved. An insurer might have to decide whether to enter a market before demand is known, or whether to fund a resilience measure before the next hazard assessment is complete. More analysis can improve the decision, but it also uses resources and may take time. The relevant question is not simply whether the analysis produces more knowledge. It is whether the information could change an action in a way that improves the outcome enough to justify its cost.

The calculations below use expected monetary value. This is appropriate when outcomes can be compared in money and the organization is willing to use expected value as its decision criterion. If the decision is governed by a risk limit, a solvency constraint, or a different risk preference, the value calculation must use that same decision criterion rather than silently replacing it with expected monetary value.

Definition: A decision has a set of available actions and a set of possible states of the world. Let $V(a,s)$ be the monetary value of action $a$ if state $s$ occurs, with all relevant action costs included. If state $s$ has probability $p_s$, the expected value of action $a$ without further information is $\sum_s p_s V(a,s)$. The best action without further information is the action with the highest expected value.

The information value is measured relative to a decision made using the evidence already available. Information is valuable when it helps select an action that better fits the state that actually occurs. It is not enough for a study to be accurate in some abstract sense: its results must arrive in time, be understandable, and be capable of affecting a feasible choice.

Perfect Information as an Upper Bound

Perfect information means the decision-maker learns the true state before choosing an action. It is an idealized benchmark, not usually a realistic product that can be purchased. The expected value with perfect information is found by selecting the best action separately in each state, then averaging those state-specific values.

V_0=\max_a \sum_s p_s V(a,s), \qquad V_{\mathrm{PI}}=\sum_s p_s \max_a V(a,s), \qquad \mathrm{EVPI}=V_{\mathrm{PI}}-V_0.

Here $V_0$ is the best expected value without additional information, $V_{\mathrm{PI}}$ is the expected value with perfect information, and EVPI is the expected value of perfect information. Since perfect information allows the decision-maker to keep the best existing action available in every state, EVPI cannot be negative.

Choose without information.
Calculate each action’s expected value using the state probabilities, then select the highest.

Choose with perfect information.
For each state, identify the action with the highest value in that state.

Take the difference.
Subtract the best value without information from the expected value with perfect information.

Justification: For any fixed action $a$, the largest state-by-state value is at least $V(a,s)$ in every state. Therefore, $\sum_s p_s\max_b V(b,s)\geq\sum_s p_sV(a,s)$. This inequality holds for every $a$, including the action that maximizes expected value without information. Thus $V_{\mathrm{PI}}\geq V_0$, so EVPI is nonnegative. The inequality also shows why the order of the maximum and the expectation matters: information permits the action to vary by state.

Worked Example: Setting a Ceiling on Market Research Spend

A hypothetical insurer is considering a regional service launch. All values are in thousands of dollars at the decision date, with no discounting. Demand will be high with probability 0.40 and low with probability 0.60. Launching produces a net value of $900,000 in the high-demand state and a net loss of $300,000 in the low-demand state. Deferring produces zero in either state. Calculate the EVPI.

Method: compare the best expected value with the state-by-state value. Without additional information, the expected value of launching is

$$ 0.40(900)+0.60(-300)=360-180=180. $$

The expected value of deferring is $0.40(0)+0.60(0)=0$. The best action without information is to launch, for an expected value of $180,000.

With perfect information, launch when demand is high, earning $900,000, and defer when demand is low, earning zero. Thus

V_{\mathrm{PI}}=0.40(900)+0.60(0)=360.

Therefore, $\mathrm{EVPI}=360-180=180$, or $180,000. Under these assumptions, paying more than $180,000 for perfect information cannot be justified by the expected value of the launch decision alone. Actual research is imperfect, so its value cannot exceed this benchmark.

Imperfect Information and Bayes’ Rule

Most analyses provide a signal rather than reveal the true state. A test may be positive or negative, a forecast may indicate elevated risk, or a market study may report strong or weak demand. To evaluate such information, calculate the probability of each signal, update the state probabilities conditional on that signal, and then choose the best action for each possible result.

Definition: The expected value of sample information (EVSI) is the expected value when decisions can use the result of a specified imperfect information source, minus the best expected value using current information alone. With signal $z$, $$ V_{\mathrm{SI}}=\sum_z P(z)\max_a \sum_s P(s\mid z)V(a,s), \qquad \mathrm{EVSI}=V_{\mathrm{SI}}-V_0. $$ If the analysis costs $C$, its net value relative to acting now is $\mathrm{EVSI}-C$, provided the analysis does not change other timing or outcome assumptions.

For a binary state $H$ (“high”) and a positive signal $+$, Bayes’ rule gives $P(H\mid +)=P(+\mid H)P(H)/P(+)$. The probability of a positive signal must include both true positives and false positives: $P(+)=P(+\mid H)P(H)+P(+\mid L)P(L)$. Sensitivity is $P(+\mid H)$; it is not generally $P(H\mid +)$. Specificity is $P(-\mid L)$.

Worked Example: Valuing a Reliability Assessment

A hypothetical insurer is deciding whether to fund a system retrofit. Values are in thousands of dollars, with no discounting. A serious outage is possible with probability 0.30; otherwise, conditions are normal. Retrofitting produces a net value of $500,000 if an outage would have occurred and a net loss of $200,000 if conditions are normal. Not retrofitting produces a loss of $700,000 in the outage state and zero otherwise. An assessment has sensitivity 0.80 and specificity 0.80. It costs $25,000. Should it be purchased?

Method: calculate the baseline, update the probabilities, and evaluate the action after each signal. Without the assessment, the retrofit’s expected value is

$$ 0.30(500)+0.70(-200)=150-140=10. $$

Not retrofitting has expected value $0.30(-700)+0.70(0)=-210$. Retrofitting is therefore the best current action, with $V_0=10$, or $10,000.

The joint probabilities of outage and a positive result, and of normal conditions and a positive result, are $0.30(0.80)=0.24$ and $0.70(0.20)=0.14$. Hence $P(+)=0.24+0.14=0.38$ and $P(H\mid +)=0.24/0.38=12/19$. Given a positive result, the retrofit’s expected value is $500(12/19)-200(7/19)=4600/19=242.105$. Not retrofitting has conditional expected value $-700(12/19)=-442.105$, so retrofit after a positive result.

For a negative result, the joint probabilities are $0.30(0.20)=0.06$ for outage and $0.70(0.80)=0.56$ for normal conditions. Thus $P(-)=0.62$ and $P(H\mid -)=0.06/0.62=3/31$. The retrofit’s conditional expected value is $500(3/31)-200(28/31)=-4100/31=-132.258$. Not retrofitting has conditional expected value $-700(3/31)=-2100/31=-67.742$, so do not retrofit after a negative result.

The expected value using the assessment is most directly calculated with joint probabilities. The positive branch contributes $0.24(500)+0.14(-200)=92$. The negative branch contributes $0.06(-700)+0.56(0)=-42$. Therefore $V_{\mathrm{SI}}=92-42=50$, and $\mathrm{EVSI}=50-10=40$, or $40,000. After the $25,000 assessment cost, net value is $40-25=15$, or $15,000. On these assumptions, purchase the assessment.

Common mistake avoided: Treating sensitivity, 0.80, as $P(H\mid +)$ would assign an incorrect posterior. The correct positive posterior is $0.24/0.38=0.6316$, because the positive result can also occur in normal conditions. The assessment’s value depends on the posterior probability and the action it supports, not on sensitivity alone.

Analysis Cost, Timing, and Decision Use

EVSI is a gross benefit: it compares decisions with and without the information before deducting the analysis cost. Purchase the analysis on expected-value grounds only if its net value is positive, and only if expected value is the applicable decision criterion. A study with positive EVSI may still be infeasible if it breaches a budget limit, arrives after the decision must be made, or would not alter a permitted action.

Timing can change the comparison. Waiting for information may cause a price to rise, an opportunity to expire, or a loss to occur before a response is possible. Those effects belong in the action values for the strategy that waits. A valid comparison puts all alternatives on the same valuation basis and includes costs or lost opportunities caused by delay; it does not compare a delayed decision with an immediate decision as if their terms were identical.

Worked Example: Comparing Immediate Protection With Waiting for a Report

A hypothetical insurer can buy protection immediately for $120,000. If a severe hazard occurs, the protection produces $700,000 of benefit; otherwise it produces no benefit. The hazard probability is 0.25. Alternatively, the insurer can wait for a report that is 90% sensitive and 90% specific, then buy protection for $180,000 if the result warrants it. Waiting causes the higher price but no other change in outcomes. The report costs $40,000. All values are in thousands of dollars at the eventual decision date, with no interest or other delay cost.

Method: compare the immediate strategy with the full waiting strategy, including the changed price and report cost. Buying immediately yields net values of $700-120=580$ in the hazard state and $-120$ otherwise. Its expected value is

$$ 0.25(580)+0.75(-120)=145-90=55. $$

Not buying yields zero, so the best immediate strategy is to buy, worth $55,000.

For the report, the joint probabilities of a positive result with and without the hazard are $0.25(0.90)=0.225$ and $0.75(0.10)=0.075$. Thus $P(+)=0.30$ and $P(H\mid +)=0.225/0.30=0.75$. After a positive report, buying at the delayed price has conditional expected value $0.75(520)+0.25(-180)=390-45=345$, which is positive.

For a negative result, the joint probabilities are $0.25(0.10)=0.025$ with the hazard and $0.75(0.90)=0.675$ without it. Thus $P(-)=0.70$, and $P(H\mid -)=0.025/0.70=1/28$. Buying after a negative report has conditional expected value $520(1/28)-180(27/28)=18.571-173.571=-155$. Do not buy after a negative report.

The positive-report branch contributes $0.225(520)+0.075(-180)=117-13.5=103.5$ before the report cost. The negative-report branch contributes zero because the insurer declines protection. Waiting with the report is therefore worth $103.5-40=63.5$, or $63,500. This exceeds the immediate strategy’s $55,000 by $8,500, so waiting for the report is preferred under the stated assumptions.

The report’s gross value here is not obtained by ignoring the higher delayed price. The $180,000 price is included in the values after each report; the $40,000 analysis cost is then deducted separately. If waiting also exposed the insurer to an interim loss or a further opportunity cost, that would need to be added before deciding.

Common Traps and Practical Checks

Taking an average before choosing versus choosing before averaging. Without information, first average each action’s state-dependent values and then choose. With perfect information, choose within each state and then average.
Confusing test accuracy with a posterior probability. Sensitivity and specificity are conditional on the true state. Use Bayes’ rule to find the probability of a state given the observed result.
Ignoring the existing decision. Information value is incremental. Compare against the best action using current evidence, not against an arbitrary or deliberately poor alternative.
Forgetting analysis cost or delay. EVSI is not the net benefit if the study has a cost. Include any changed terms, deadlines, or interim consequences as well.
Assuming every useful forecast is decision-useful. If the same action remains best for every possible result, the information cannot improve the decision under the stated criterion.

Key takeaway: Evaluate information by tracing how each possible result could change the action and its value. Perfect information sets an upper bound; imperfect information must be evaluated using its signal probabilities and resulting decisions. The analysis is worthwhile only when its net value is positive, its timing is feasible, and its decision criterion matches the organization’s constraints. This approach is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Check Your Understanding

For each problem, use expected monetary value unless the question states otherwise. Values are in thousands of dollars, and assume no discounting.

A project has a 0.30 probability of strong demand and a 0.70 probability of weak demand. Launching is worth 80 in the strong state and -20 in the weak state; deferring is worth zero in either state. Find the EVPI. (Answer: 14)
A mitigation action is worth 300 in the high-loss state and -100 in the low-loss state; taking no action is worth zero in either state. The high-loss probability is 0.20. A test has sensitivity 0.75 and specificity 0.90 and costs 30. Find its net value. (Answer: 7)
An upgrade is worth 250 in a high-demand state and -80 in a low-demand state; retaining the current system is worth zero in either state. High demand has probability 0.40. Calculate the EVPI and state the best action without information. (Answer: EVPI = 48; upgrade without information)
A business has a 0.60 probability of a disruption. Installing a safeguard is worth 100 if disruption occurs and -20 otherwise; not installing is worth zero. A report has sensitivity 0.90 and specificity 0.70. Does the report have positive EVSI, and what is its maximum justified cost? (Answer: EVSI = 0; maximum justified cost = 0)
A protection purchase costs 100 and provides a benefit of 500 if a hazard occurs; otherwise it provides no benefit. The hazard probability is 0.25. A report has sensitivity 0.80 and specificity 0.80. Calculate EVSI before report cost and the maximum report cost that makes it worthwhile. (Answer: EVSI = 40; maximum report cost = 40)

What You'll Learn