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Decision Analysis with Probabilities
Quantitative Analysis · Chapter 5: Decision Analysis with ProbabilitiesSection 5.4 · lesson 23 of 5511 min · free with a free account
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In this lesson
Applying decision analysis when likelihood data is available - EMV, EOL, and the value of perfect information.
- Compute EMV systematically for multi-alternative, multi-state decisions
- Build an opportunity-loss table and compute EOL
- Show that the EOL-minimizing alternative equals the EMV-maximizing one
- Calculate EMV under certainty, derive EVPI, and screen an information offer
Transcript
Everything said in the video, section by section (1,666 words). Read along, or search it.
Decision Analysis · Section 5.4
Probability in Action
Welcome to section five point four, Decision Analysis with Probabilities. In the previous section, the manager was working blind, with no idea how likely each state was. So she fell back on rules of thumb. Maximin, maximax, equally likely. Now change one thing, and the whole picture changes. Suppose probabilities can be attached to those states. They might come from historical data, from market research, or from expert judgement. With those numbers in hand, decision analysis becomes considerably more powerful. This section builds three closely related tools. Expected monetary value, expected opportunity loss, and the expected value of perfect information. Together they do two jobs, and the second one is easy to overlook. They pick the best alternative, and they tell you whether to buy better information first.
From Rules Of Thumb
What Changes When Probabilities Are Available
Start with what probabilities actually buy you. Without them, every criterion is really a posture toward risk, not a use of likelihood data. Maximin is pessimism. Maximax is optimism. Equally likely is a shrug. Add probabilities, and the analysis becomes arithmetic rather than temperament. The work runs in four stages, and each one is built from the one before it. First, lay out the payoff table. Alternatives down the side, states across the top. Second, compute expected monetary value for every alternative, and take the largest. Third, rebuild the same table as regret, and compute expected opportunity loss. Fourth, ask what a perfect forecast would have been worth. That last stage is the one managers most often skip. It sets the most you should ever pay for information.
Expected Monetary Value
EMV - The Probability-Weighted Payoff
Expected monetary value is the probability weighted average of an alternative's payoffs. Weighted across every state of nature that could occur. The formula is compact. E M V of A equals the sum of P sub i times V sub i. P sub i is the probability that state i occurs. V sub i is the payoff that alternative A produces in that state. Multiply each payoff by its probability, then add the products. The decision rule is as simple as it gets. Choose the alternative with the highest expected monetary value. Now be careful about what that average actually means. It is what the alternative would earn on average, if the same decision were faced many times under the same probabilities. It is a long run figure. It is not a promise about the one decision in front of you.
Expected Opportunity Loss
EOL - Measuring Regret Instead of Gain
Expected monetary value asks a single question. What do we gain on average? There is an opposite question, and it is just as useful. What do we lose on average, by not having chosen perfectly? That loss is called opportunity loss, or more plainly, regret. For each state, take the best payoff any alternative could have achieved. Then subtract the payoff your chosen alternative actually produced. The maximum is taken across alternatives, within a single state. So you are reading down one column of the payoff table, not across a row. Weight those regrets by the state probabilities, and you have expected opportunity loss. The rule now flips direction. Choose the alternative with the smallest E O L. It is the option you will regret least, on average.
The Equivalence
Why EOL and EMV Never Disagree
Here is a theorem worth knowing, because it settles a worry before you have it. The alternative that minimises E O L is always the alternative that maximises E M V. The two criteria never disagree. Not usually. Never. The proof takes one line of algebra. Expected opportunity loss is the weighted sum of the best payoff minus your payoff. Split it in two. The first piece weights the best payoff in each state. Call that constant K. The second piece is simply the expected monetary value of your alternative. So E O L of A equals K minus E M V of A, and K does not depend on A. Subtracting a larger number from a fixed constant leaves a smaller result. So minimising regret and maximising expected value are the same operation. Then why compute both? Because the minimum E O L is the value of perfect information. Building the regret table is the cleanest path to that number.
Perfect Information
EMV Under Certainty, and the EVPI Formula
Now the third tool, and the one with the most commercial bite. How much would it be worth to know, with certainty, which state will occur? Imagine a perfectly accurate forecaster who tells you before you decide. You would then always pick whichever alternative pays best in that state. Here is the subtle part, and it trips a lot of students. Before the forecast arrives, each state still occurs with its original probability. So expected value under certainty is the probability weighted sum of the column maxima. Take the best payoff in each column, weight it, and add. The value of perfect information is the gap between that figure and the best expected value under risk. It is what perfect knowledge would add to a decision you can already make. And it equals the minimum expected opportunity loss, by the algebra we just did.
An Upper Bound
EVPI Is a Ceiling, Not a Price Tag
One warning now, and it is the most practical idea in the section. Perfect information is never actually available. No market study, no consultant, no forecast can tell you exactly which state will occur. Every real source is imperfect. It shifts your probabilities, but it does not eliminate the uncertainty. So this number is an upper bound on what any information could possibly be worth. That gives you a clean rule for screening offers. If a consultant charges more than the value of perfect information, reject the offer on principle. It does not matter how good the consultant is. The price is above the ceiling. If the price sits below it, the offer may be worth taking. Whether it actually is depends on how good their information is. That is the expected value of sample information, and it is the next section's subject.
Exercise 5.4
The Plant-Building Decision
Let us put all three tools on one worked example. A firm is deciding whether to build a plant, and how big. Three alternatives. Build a large plant, build a small plant, or do not build at all. Three states of demand, each with a probability attached. High demand, forty per cent. Medium, thirty-five. Low, twenty-five. All payoffs are in thousands of dollars. The large plant earns two hundred in high demand, and one hundred in medium. But in low demand it loses one hundred and twenty. The small plant earns ninety, sixty, and twenty across the three states. It never loses money, but it never earns much either. Doing nothing pays zero in every state. That is the baseline the other two must beat. The firm has also been offered a demand study, priced at eighty thousand dollars.
Step One
Computing EMV for Every Alternative
Start with expected monetary value, one row at a time. For the large plant, multiply each payoff by its probability. Point four zero times two hundred is eighty. Point three five times one hundred is thirty-five. Point two five times minus one hundred and twenty is minus thirty. Eighty plus thirty-five minus thirty gives eighty-five thousand dollars. Now the small plant. Thirty-six, plus twenty-one, plus five. That comes to sixty-two thousand dollars. Doing nothing pays zero in every state, so its expected value is zero. Eighty-five beats sixty-two, and both beat zero, so the firm builds the large plant. Notice what that choice accepts. A one in four chance of losing a hundred and twenty thousand. It takes that risk to reach a forty per cent chance of two hundred thousand.
Step Two
The Regret Table and EOL
Now build the regret table, and watch it agree with what we just found. First, find the best payoff available in each state. In high demand the best is two hundred, from the large plant. In medium demand the best is one hundred, again the large plant. In low demand the best is twenty, from the small plant. Those three numbers are the column maxima. Now subtract each alternative's payoff from the maximum in its own column. The large plant regrets nothing in high or medium demand, because it is the best there. But in low demand it regrets one hundred and forty, the twenty it forgoes plus the hundred and twenty it loses. Weight that regret by twenty-five per cent, and its expected opportunity loss is thirty-five thousand. The small plant comes to fifty-eight thousand, and doing nothing to one hundred and twenty thousand. The smallest regret belongs to the large plant. The same answer, reached from the opposite direction.
Step Three
Computing EVPI - What a Perfect Forecast Is Worth
One number left. The expected value of perfect information. Take the column maxima as though they were a row of their own. Two hundred, one hundred, and twenty. That row is not an alternative anyone can choose. It is what a perfect forecaster would let the firm choose, state by state. Weight those three numbers by the same probabilities. Eighty, plus thirty-five, plus five, which is one hundred and twenty thousand dollars. The best the firm can do without a forecast is eighty-five thousand. So perfect information is worth the gap. Thirty-five thousand dollars. And there is the check we were promised. Thirty-five thousand is exactly the minimum expected opportunity loss from the regret table.
The Verdict
Should the Firm Pay $80,000 for the Study?
So, should the firm pay eighty thousand dollars for the demand study? The arithmetic answers that before you know anything about the study itself. Even perfect, certainty providing information is worth at most thirty-five thousand dollars. The study is priced at eighty thousand. More than double that ceiling. And a demand study cannot possibly deliver perfect information anyway. The decision is unambiguous. Reject the study, and build the large plant on the probabilities already in hand. That is what makes this measure so practical. Before you investigate how good a study or a consultant is, check the price. If the price sits above the ceiling, no further investigation is needed. The number is also a diagnostic about the decision itself. A high value signals high stakes, and rewards careful information gathering. A low value, as here, says your current probability estimates are good enough to act on.
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