What Is Certainty Worth? | Quantitative Analysis

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What Is Certainty Worth?

Quantitative Analysis · Chapter 5: Decision Analysis with ProbabilitiesSection 5.6 · lesson 25 of 5510 min · free with a free account

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In this lesson

Calculating the expected value of perfect information - and the budget ceiling it sets for every study, pilot and expert engagement.

  • Explain what perfect information would change about a decision
  • Compute EMVPI and EVPI from a payoff table
  • Show why EVPI equals the minimum expected opportunity loss
  • Use EVPI as the price ceiling on any study or forecast

Transcript

Everything said in the video, section by section (1,586 words). Read along, or search it.

Decision Analysis · Section 5.6

What Is Certainty Worth?

Every decision in this chapter so far has been made in the dark. You choose an alternative, nature reveals a state, and the payoff follows. Now we ask a different question. What would change if you could learn the true state of nature before deciding? That question has a precise answer, and the answer is a single number. It is called the expected value of perfect information, or EVPI. EVPI is the ceiling on the price of any study, any pilot, any expert. By the end of this section you will compute it two different ways. And you will use it to reject an expensive market study in one comparison.

Perfect Information

The Clairvoyant Question

Picture a clairvoyant sitting across the table from you. Before you decide, they tell you with certainty which state of nature is about to occur. Not a probability. The actual state. What would you do with that? You would choose the best alternative for that specific state. And every cell outside that state's column becomes irrelevant. Now imagine playing that game many times over. Sometimes the clairvoyant says low demand, sometimes high, in the proportions your priors describe. Average your payoff across all of those plays. That average is the expected payoff under perfect information, written EMVPI. The gap between EMVPI and the best you can do without the clairvoyant is EVPI.

Formal Derivation

Best in Each Column, Probability-Weighted

Let us set the notation down carefully, because the formula is built from it. The payoff table has alternatives, indexed by i, and states of nature, indexed by j. P of s j is the prior probability of state j. And v of a i, s j is the payoff sitting in that cell. The expected payoff under perfect information is built column by column. For each state, scan down its column and take the largest payoff available. That is the choice the clairvoyant would let you make in that state. Weight each of those best payoffs by the probability of its state, then sum. That sum is EMVPI. Notice what it is not. It is not the value of any one alternative. It mixes decisions from different rows, so no single row can deliver it. EVPI is then EMVPI minus the best expected value reachable without information.

Three Properties

What EVPI Can and Cannot Be

Three properties fall straight out of that definition. First, EVPI is never negative. Knowing more cannot make you worse off, because at worst you ignore what you learn. If your arithmetic ever hands you a negative EVPI, the arithmetic is wrong. Second, EVPI is exactly zero in one situation. When a single alternative is best under every state of nature. The state then does not matter, the choice never changes, and information has no value. Third, EVPI is bounded by how far payoffs swing across the states. Wild swings mean a large EVPI. Tightly clustered payoffs mean a small one. And here is the part people read backwards. A zero EVPI is not a failed analysis. It says your decision is robust.

Regret View

EVPI Is the Minimum Expected Opportunity Loss

There is a second route to the same number, and it starts from regret. Recall opportunity loss from section five point three. If a state occurs and you chose a given alternative, your regret is what you left behind. It is the best payoff available in that state, minus the payoff you actually received. Weight those regrets by the state probabilities and you get expected opportunity loss. Now the elegant part. Add an alternative's expected payoff to its expected opportunity loss. By construction the two always sum to EMVPI. The same constant, for every single row of the table. So maximising expected value and minimising expected regret are the same operation. They select the same alternative, every time, and its expected regret is exactly EVPI. Do the best you can, and minimise your regret. Two philosophies, one bridge.

Budget Ceiling

The Maximum Rational Price for Information

Now the reason EVPI earns its keep in practice. It is the maximum rational price to pay for information about the state of nature. Any information, however reliable it promises to be. The logic is airtight, and it is short. Even perfect information adds only EVPI to your expected payoff. Nothing can add more, because perfect is the top of the scale. So paying more than EVPI guarantees an expected loss on the transaction. Not sometimes. In expectation, no matter how good the study turns out to be. Which gives you a screening test made of one subtraction and one comparison. This single check has saved firms from expensive but pointless research engagements many times. But note what passing it does not mean. Below EVPI is a candidate, not an approval.

Boundaries

What EVPI Is Not

EVPI is powerful, and it is also easy to over-read. So let us be precise about what it is not. It is not the value of any specific piece of information. It is the value of perfect information, which no real study provides. It also does not tell you whether to buy. A study priced below EVPI can still be a waste if it is too unreliable. Settling that question takes EVSI, in chapter six. And EVPI is purely monetary. Strategic value, reputational value, the option value of knowing early, none of it is in the number. One more trap. EVPI depends entirely on your priors. A question worth paying to answer under one set of beliefs can be worthless under another.

Toward EVSI

Real Information Is Never Perfect

In reality, almost no information is perfect. Market research surveys have non-response bias. Pilot studies have sampling error. Expert forecasts are sometimes simply wrong. So picture three regimes, side by side. With no information you decide on the priors alone, and earn the best EMV. With perfect information you earn EMVPI, and the gain over the priors is EVPI. Real studies live in between those two. The expected value of sample information measures what an imperfect study actually adds. It sits between zero and EVPI. Always. That inequality is the one relationship worth memorising here. It also gives a yardstick. Seventy per cent of EVPI extracts most of the available value. Ten per cent is barely worth the effort.

Worked Example

Three Launches, Three Demand States

Let us put real numbers on all of this. Take a firm that is choosing between three product launches. There are three demand states: low, medium and high. Low and high each carry a prior probability of thirty per cent. Medium carries forty per cent. All payoffs are in thousands of dollars. The cautious launch pays forty, sixty and eighty across the three states. Its expected value is sixty thousand. The balanced launch pays nothing in low demand, eighty in medium, one hundred and forty in high. That averages out to seventy-four thousand. The aggressive launch loses eighty thousand in low demand, makes forty in medium, three hundred and twenty in high. Its expected value is eighty-eight thousand, the largest of the three. So step one is done. With no information the firm launches aggressively.

Worked Example

The Payoff Under Perfect Information

Step two. Forget the rows for a moment and read down the columns. In low demand the best any alternative achieves is forty thousand, from the cautious launch. In medium demand the best is eighty thousand, from the balanced launch. In high demand the best is three hundred and twenty thousand, from the aggressive launch. Notice the answer changes with the state. Three states, three different launches. That is exactly what perfect information buys you. Step three. Weight those three best payoffs by their probabilities. Thirty per cent of forty, plus forty per cent of eighty, plus thirty per cent of three hundred and twenty. That comes to one hundred and forty thousand dollars. This is EMVPI. Step four. Subtract the best available without information, eighty-eight thousand. One hundred and forty minus eighty-eight leaves fifty-two thousand. That is EVPI. And no single launch could ever earn one hundred and forty on its own.

Worked Example

Step 5 - Apply the Budget Ceiling

Now the payoff for all that work. The firm is offered a market study, priced at sixty thousand dollars. EVPI is fifty-two thousand. Sixty is more than fifty-two, and that ends the conversation. The study costs more than perfect information itself is worth. Run the thought experiment to be sure of it. Suppose the study were flawless, naming the exact demand state every time. Even then it could add only fifty-two thousand to the expected payoff. Paying sixty for at most fifty-two loses money in expectation, by at least eight thousand. So reject the study at this price. And notice what you never had to do. You never examined its methodology or its accuracy. If the firm renegotiated below fifty-two thousand, the offer would clear the screen. But clearing it is necessary, not sufficient. Chapter six still has to check reliability.

Section Summary

One Number, Two Jobs

Let us close with the shape of the whole thing. EMVPI is the probability-weighted best payoff in each state of nature. It is what you would earn if the true state were revealed before you decided. EVPI is EMVPI minus the best expected value available without information. That is the analytical worth of removing all uncertainty about the state. The same number appears from the regret side, as the minimum expected opportunity loss. Two derivations, one answer. That is a good sign the framework is sound. Practically, EVPI is a ceiling on what information can be worth, at any reliability. If the price sits above it, reject, and skip the deeper analysis entirely. If the price sits below it, move on to EVSI and test whether the study is good enough. Because sample information always lands between zero and EVPI. Certainty has a price. EVPI is how you compute it before anyone quotes you one.

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FAQ

Questions students ask us

About What Is Certainty Worth?

Can the expected value of perfect information be zero or negative?
EVPI can never be negative, because knowing more cannot leave you worse off; at worst you ignore the information. If a calculation gives a negative EVPI, the arithmetic is wrong. EVPI is exactly zero when one alternative is best under every state of nature, since the forecast would never change the choice. A zero EVPI does not mean the analysis failed; it means the decision is robust.
What does a large or small expected value of perfect information tell you?
A large EVPI means payoffs swing widely across states and the best choice depends heavily on which state occurs, so the stakes are high and better information could pay. A small EVPI means current probability estimates are good enough to act on. In a launch example where the best payoffs of 40, 80 and 320 thousand dollars come from three different launches, EVPI is $52,000, so no study costing more is worth buying.
What is the difference between EVPI and the expected value of sample information?
EVPI is the value of perfect information, which no real study provides. The expected value of sample information (EVSI) measures what an imperfect source, such as a survey with non-response bias or a pilot with sampling error, actually adds. EVSI always lies between zero and EVPI. A study priced below EVPI may still be a poor buy if its EVSI is small relative to its cost.
What are the limitations of the expected value of perfect information?
EVPI is a ceiling, not a purchase decision: it values perfect information, not any specific study, and it cannot say whether an imperfect study is worth buying. It is purely monetary, so strategic, reputational and early-knowledge option value are left out. It also depends entirely on the prior probabilities, so information worth paying for under one set of beliefs can be worthless under another.

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