4.6 Decision Making Under Uncertainty

Expected Value and Payoff Tables

Expected value (EV) is the probability-weighted average of an alternative's possible payoffs:

EV = Σ (probability of outcome × payoff of outcome)

A payoff table arrays decision alternatives (rows) against uncertain states of nature (columns), with each cell showing the payoff. Compute EV for each row and choose the highest.

• EV(Small) = 0.40×$30,000 + 0.60×$50,000 = $12,000 + $30,000 = $42,000
• EV(Large) = 0.40×(−$10,000) + 0.60×$90,000 = −$4,000 + $54,000 = $50,000

Choose Build Large on expected value.

Decision Trees and Rollback

A decision tree diagrams sequential choices and chance events. Decision (choice) nodes are squares; chance (event) nodes are circles with branch probabilities. You evaluate by rollback (backward induction) — start at the far-right payoffs and work left.

• At a chance node, compute the expected value of its branches.
• At a decision node, pick the branch with the higher value and carry it back.

This lets you value choices made before uncertainty resolves, such as whether to run a market test before committing to full production.

Worked Example: Rollback

Launch a product (decision node). If launched, the market is a chance node: 70% success → $200,000; 30% failure → −$50,000. The alternative is do not launch → $0.

• Expected value at the chance node = 0.70 × $200,000 + 0.30 × (−$50,000) = $140,000 − $15,000 = $125,000
• At the decision node, compare Launch $125,000 vs. Do not launch $0.
• Roll back: choose Launch, valuing the decision at $125,000.

If an upfront launch cost of $40,000 applied, subtract it: net $85,000 — still launch, since it beats $0.

Expected Value of Perfect Information

Expected value of perfect information (EVPI) is the most a rational decision maker would pay to know the future state with certainty:

EVPI = (Expected payoff with perfect information) − (Expected payoff of the best action without it)

With perfect information, you would pick the best alternative for each state, then weight those best payoffs by their probabilities.

Using the earlier table: for Low demand, best = Small ($30,000); for High, best = Large ($90,000).

• EV with perfect info = 0.40×$30,000 + 0.60×$90,000 = $12,000 + $54,000 = $66,000
• Best EV without info = $50,000 (Build Large)
• EVPI = $66,000 − $50,000 = $16,000. Never pay more than $16,000 for a forecast study.

Coefficient of Variation and Risk

When alternatives have different expected values, comparing standard deviations alone is misleading. The coefficient of variation (CV) scales risk by return:

CV = Standard deviation ÷ Expected value

A lower CV means less risk per dollar of expected return. CV is the right tool when projects differ in size.

Example

Project Q has a larger absolute standard deviation but a lower CV (0.24 < 0.30), so it carries less risk per unit of expected return — generally preferable to a risk-averse manager.

Risk Attitudes and Sensitivity Analysis

Risk attitudes shape choices beyond raw expected value:

• Risk-averse: demands extra expected return to bear extra risk; prefers lower CV. Most managers and the exam default here.
• Risk-neutral: decides purely on expected value, indifferent to spread.
• Risk-seeking: accepts lower expected value for a chance at a large payoff.

Sensitivity analysis asks how the decision changes when an input changes — varying a probability, price, cost, or discount rate one at a time. It identifies the variables the decision is most sensitive to and finds break-even points (the input value at which two alternatives are equal). A conclusion that holds across a wide range of inputs is robust; one that flips with a small change demands more analysis before committing.

Expected Value of Sample Information and Conditional Probabilities

Perfect information is an upper bound; real studies give imperfect (sample) information. The expected value of sample information (EVSI) is the expected payoff with the study minus the expected payoff without it, and EVSI can never exceed EVPI.

When a forecast is imperfect, revise probabilities with Bayes' theorem: a prior probability is updated by the forecast's reliability to form a posterior probability. For example, if a market test is 80% accurate, a "favorable" result raises but does not guarantee the probability of high demand. The exam may ask only for the concept — that a paid study is worth its cost only if EVSI exceeds the study's price, and that EVSI < EVPI because the information is incomplete.

Standard Deviation, Variance, and Risk Ranking

Expected value alone ignores dispersion — the spread of possible outcomes around the mean. Two alternatives can share an expected value yet differ sharply in risk.

• Variance = Σ probability × (payoff − expected value)². Standard deviation is its square root, in the same units as the payoff.
• A larger standard deviation means a wider, riskier range of outcomes.
• The coefficient of variation (CV = standard deviation ÷ expected value) is the correct comparison when alternatives have different expected values, because it states risk per dollar of return.

Use standard deviation to compare same-sized alternatives and CV to compare differently sized ones. A risk-averse decision maker, facing two options with equal expected value, picks the one with the lower standard deviation; facing options of different scale, the one with the lower CV.