How to Calculate Expected Utility
Expected utility equals the sum of each outcome's probability times the utility of that outcome, so payoffs are turned into utility first and averaged second.
The Expected Utility formula
Calculator
Enter the utility of each outcome and its probability to get expected utility, then rank it against a sure thing.
The utility number the problem gives for the best state, not the dollar payoff.
How likely that state is. All three probabilities should total 100%.
Utility in the middle state.
Chance of the middle state.
Utility in the worst state. Set its probability to zero for a two-outcome gamble.
Chance of the worst state.
The utility of the option with no risk, so the two can be ranked.
The gamble is worth 57.5 utils, which is a ranking number rather than an amount of money.
- Probabilities entered
- 100%
- Weighted utility from outcome 1
- 22.5
- Weighted utility from outcome 2
- 30
- Weighted utility from outcome 3
- 5
- Choice against the sure option
- Take the gamble
The probabilities total 100%, so every possible outcome is on the list.
Outcome 1 contributes 22.5 utils once its utility is scaled by how likely it is.
Outcome 2 contributes 30 utils.
Outcome 3 contributes 5 utils.
Expected utility of 57.5 against 55 for the certain option decides it, and the size of the gap carries no extra meaning.
How to calculate Expected Utility, step by step
- 1Turn every outcome into utility. Apply the utility function to each possible payoff, or read the utility straight off the table the question gives you.
- 2Attach a probability to each outcome. Write percentages as decimals and check they add to 1, because a set that falls short means an outcome has been left off the list.
- 3Weight each utility. Multiply the utility of every outcome by the probability of that outcome.
- 4Add the weighted utilities. The total is expected utility. It is measured in utils, so it ranks options and does not price them.
- 5Rank against the alternative. Work out the same number for the safe option and pick whichever scores higher.
Worked example: Expected Utility
A risky job is worth a utility of 90 in a boom, 60 in normal conditions and 20 in a bust, with probabilities of 25%, 50% and 25%. EU = 0.25 × 90 + 0.5 × 60 + 0.25 × 20 = 22.5 + 30 + 5 = 57.5 utils. A salaried offer worth a certain 55 utils scores lower, so the risky job wins.
Expected Utility questions
Do the probabilities have to add up to 1?
Yes. Every state the world can end in has to appear exactly once, so the probabilities must total 1, or 100% if you are working in percentages. A set that adds to less than 1 means an outcome is missing, and the expected utility you get back will be too low.
What if the question gives dollar payoffs instead of utilities?
Run each payoff through the utility function before you weight anything. With the square root of wealth, a $400 payoff becomes 20 utils and a $1,600 payoff becomes 40. Weighting the dollars first and converting afterwards gives a different answer and erases risk aversion from it.
Why is expected utility not the same as the utility of the expected value?
Because a risk-averse person's utility curve bends. Taking utility first and averaging second lands below the utility of the average payoff whenever the curve is concave, and that gap is exactly what makes a sure thing attractive.
What does the size of an expected utility number mean?
On its own, nothing. Utils have no natural scale, so 57.5 is only useful next to the score of another option. Convert it to a certainty equivalent if you need an answer in dollars.
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