Expected Utility
Weighted average of the utility of outcomes, not just their cash value.
Formula
Variables
Example
Diminishing utility explains why people buy insurance despite negative EV.
How it works
Expected utility is the average value of a choice when each possible outcome is weighted by how likely it is and by how much the decision-maker cares about that outcome. It comes from the idea that people often do not evaluate choices by money alone: the same $100 can matter differently depending on wealth, risk attitude, or the context of the decision. By converting outcomes into utility first and then averaging, the formula captures rational choice under uncertainty for someone whose preferences are consistent.
When to use it
Use expected utility when comparing risky choices, especially in economics, game theory, finance, and decision analysis, where outcomes have both probabilities and subjective value. It is especially useful whenever payoffs are not linear in money, such as with risk-averse or risk-seeking preferences.
Worked example, step by step
A manager must choose a project with three possible profits: - $0 with probability 0.2 - $50,000 with probability 0.5 - $200,000 with probability 0.3 Assume utility is u(x) = sqrt(x), with x measured in dollars. Step 1: Compute utility of each outcome u(0) = sqrt(0) = 0 u(50,000) = sqrt(50,000) ≈ 223.61 u(200,000) = sqrt(200,000) ≈ 447.21 Step 2: Weight each utility by its probability 0.2 × 0 = 0 0.5 × 223.61 = 111.805 0.3 × 447.21 = 134.163 Step 3: Add them up EU = 0 + 111.805 + 134.163 = 245.968 Final answer: EU ≈ 245.97 utility units.
Common mistakes to avoid
- Averaging the money outcomes directly instead of averaging utility values
- Using probabilities that do not add up to 1
- Forgetting that the utility function can change the ranking of risky options even when expected cash value is lower
Did You Know?
Expected utility, formalised by von Neumann and Morgenstern in 1944, explains risk-averse behaviour.
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