Set up the decision
Every decision analysis has the same parts: the choices you control (alternatives), the events you do not control (states of nature), the probability of each state and the payoff for each combination. Write them in a payoff table before you calculate anything.
Product launch (hypothetical, $ thousands)
A firm decides whether to launch a new product. If the product succeeds (probability 0.6) it earns 500. If it fails (probability 0.4) it loses 300. Not launching earns 0.
| Alternative | Success (0.6) | Failure (0.4) | Expected monetary value |
|---|---|---|---|
| Launch | 500 | -300 | 0.6 x 500 + 0.4 x (-300) = 300 - 120 = 180 |
| Do not launch | 0 | 0 | 0 |
The expected monetary value (EMV) of launching is $180,000, which beats 0, so launch.
Break-even probability and sensitivity
Probabilities are often guesses, so ask how wrong they can be before the decision changes. The break-even probability p solves: p x 500 + (1 - p) x (-300) = 0, so 500p - 300 + 300p = 0, giving 800p = 300 and p = 0.375.
If you believe the chance of success is above 37.5 percent, launching has a positive expected value. The estimate of 60 percent has room to be wrong by more than 20 points before the choice changes. Reporting this tells a manager how much to trust the recommendation.
| Probability of success | EMV of launch ($ thousands) | Decision |
|---|---|---|
| 0.30 | 0.30 x 500 - 0.70 x 300 = 150 - 210 = -60 | Do not launch |
| 0.375 | 0 | Indifferent |
| 0.50 | 250 - 150 = 100 | Launch |
| 0.60 | 180 | Launch |
| 0.80 | 400 - 60 = 340 | Launch |
The value of perfect information
How much should you pay to remove the uncertainty? The expected value with perfect information (EVwPI) assumes you would learn the state before choosing: launch if success, skip if failure. EVwPI = 0.6 x 500 + 0.4 x 0 = 300. The expected value of perfect information (EVPI) is the difference between that and the best EMV without information: EVPI = 300 - 180 = 120 ($120,000).
No information source can be worth more than $120,000 here, and any real source, which is imperfect, is worth less. EVPI is an upper bound for deciding whether research is worth considering.
Imperfect information and Bayes' theorem
A market test is informative but not perfect. Suppose it gives a favorable result 80 percent of the time when the product would succeed and 30 percent of the time when it would fail. Use Bayes' theorem to update the probabilities.
Updating after a test (hypothetical)
Probability of a favorable result: 0.6 x 0.8 + 0.4 x 0.3 = 0.48 + 0.12 = 0.60. Probability of unfavorable: 0.40.
After a favorable result: P(success) = 0.48 / 0.60 = 0.80. EMV of launch = 0.8 x 500 - 0.2 x 300 = 400 - 60 = 340. Launch.
After an unfavorable result: P(success) = 0.6 x 0.2 / 0.40 = 0.12 / 0.40 = 0.30. EMV of launch = 0.3 x 500 - 0.7 x 300 = 150 - 210 = -60. Do not launch (EMV 0).
Expected value with the test: 0.60 x 340 + 0.40 x 0 = 204.
The expected value of sample information (EVSI) = 204 - 180 = 24 ($24,000). If the test costs $15,000, doing it adds $9,000 of expected value. If it costs $30,000, it is not worth running.
Note a general lesson: information is valuable only if it can change your decision. If the test results never led you to choose differently, EVSI would be zero, whatever its accuracy.
| Quantity | Value ($ thousands) | Meaning |
|---|---|---|
| Best EMV with no information | 180 | Launch now |
| EVSI (market test) | 24 | Most you would pay for the test |
| EVPI | 120 | Most you would pay for any information |
Beyond expected value
EMV treats a 50 percent chance of winning $1 million as equal to a certain $500,000, but most organizations and people do not feel indifferent. Risk attitudes matter when the stakes are large relative to the firm's resources. Ways to include them are listed below.
- Utility functions: convert dollars into utility to reflect risk aversion, so that a large loss weighs more than an equal gain.
- Minimax regret: choose the option that minimizes the maximum regret, useful when probabilities are unknown.
- Maximin: choose the best of the worst cases, a cautious approach.
- Scenario analysis and simulation: test results under several possible futures, or run thousands of random trials to see the distribution of outcomes.
For the launch decision, you might note that a $300,000 loss is survivable for a large firm but not for a start-up, so the same EMV could lead to different choices.
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Get an instant quoteA second worked decision: bid high or low
A contractor can bid high (probability of winning 0.3, profit $200,000 if won) or bid low (probability 0.6, profit $80,000 if won). Losing earns nothing.
| Option | Win probability | Profit if won ($ thousands) | EMV ($ thousands) |
|---|---|---|---|
| Bid high | 0.3 | 200 | 0.3 x 200 = 60 |
| Bid low | 0.6 | 80 | 0.6 x 80 = 48 |
Bidding high has the higher EMV (60 against 48), but a risk-averse firm that needs work to keep its crew busy might prefer to bid low, since it wins twice as often. The difference of 12 is the price of that preference. Decision analysis does not tell the manager what to prefer; it makes the trade-off explicit.
Risk and the spread of outcomes
Two options with the same EMV can differ greatly in risk. Report the spread as well as the average.
Scenario analysis (hypothetical, profit in $ thousands)
| Scenario | Probability | Profit | Probability x profit |
|---|---|---|---|
| Weak demand | 0.25 | 40 | 10.0 |
| Expected demand | 0.50 | 100 | 50.0 |
| Strong demand | 0.25 | 180 | 45.0 |
| Expected value | 105.0 |
Variance = 0.25 x (40 - 105) squared + 0.50 x (100 - 105) squared + 0.25 x (180 - 105) squared = 0.25 x 4,225 + 0.50 x 25 + 0.25 x 5,625 = 1,056.25 + 12.5 + 1,406.25 = 2,475. Standard deviation = square root of 2,475 = about 49.8.
Say the expected profit is $105,000 with a standard deviation of about $50,000, so outcomes range widely. If an alternative has the same mean and a standard deviation of $15,000, many managers would prefer it. Monte Carlo simulation extends this idea: draw many random values for each uncertain input, compute profit each time and read off the chance of losing money.
Certainty equivalents and risk premium
Suppose a firm is offered a venture with a 50 percent chance of $1 million and a 50 percent chance of nothing. The EMV is $500,000. If the firm would swap the venture for a certain $350,000 but no less, the certainty equivalent is $350,000, and the risk premium is 500,000 - 350,000 = $150,000, the amount of expected value given up for certainty.
Use this idea in written answers to explain why identical EMVs can lead to different choices in a large corporation and in a start-up, and why corporations diversify the projects they hold: they can take many positive-EMV risks and be close to risk neutral across the portfolio.
Writing the analysis
- Show the payoff table Alternatives, states, probabilities, payoffs.
- Calculate and interpret State the EMV and what it means in a sentence.
- Test sensitivity Give break-even probabilities and the range where the decision holds.
- Value information EVPI as an upper bound, EVSI for real tests.
- Add judgment Risk tolerance, non-financial factors and ethics.
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