Expected Utility: Understanding, Calculating, and Examples

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Expected utility is the probability-weighted average of the satisfaction, or utility, a decision maker gets from each possible outcome of an uncertain choice. Expected utility theory (EUT) holds that a rational person facing risk picks the option with the highest expected utility, not the one with the highest average payoff in dollars. The two can differ because an extra dollar is not worth the same to everyone, or to the same person at every level of wealth.

The Expected Utility Formula

For a choice with n possible outcomes:

EU = p₁·u(x₁) + p₂·u(x₂) + … + pₙ·u(xₙ)

where xᵢ is the outcome (usually final wealth), pᵢ is its probability (the probabilities sum to 1), and u(·) is the person’s utility function.

The shape of u carries the person’s attitude to risk. A concave function, such as the square root or the natural log of wealth, describes a risk-averse person: each additional dollar adds less utility than the one before. A linear function describes a risk-neutral person, for whom expected utility ranks options the same way expected value does. A convex function describes a risk seeker.

Utility numbers have no meaning on their own. Under the von Neumann-Morgenstern framework, a utility function can be rescaled (multiplied by a positive number and shifted) without changing any decision, so utility can even be negative: log utility, for example, gives negative values for wealth below 1. Comparing “91 utils” across two people says nothing about who is happier. Only the ranking of options for a given person matters.

Worked Example: Why People Buy Insurance

Hypothetical example. A person has wealth of $10,000 and utility u(w) = √w. There is a 10% chance of a loss of $9,900 (leaving $100) and a 90% chance of no loss.

Without insurance:

• Expected wealth: 0.9 × $10,000 + 0.1 × $100 = $9,010
• Expected utility: 0.9 × √10,000 + 0.1 × √100 = 0.9 × 100 + 0.1 × 10 = 91

With full insurance at a $1,200 premium:

• Wealth is a certain $8,800
• Utility: √8,800 ≈ 93.8

The insured position has the higher expected utility even though its expected wealth ($8,800) is lower than the uninsured $9,010. The premium exceeds the expected loss of $990, so the policy has a negative expected value, yet the person is better off by their own preferences.

The certainty equivalent shows how far this goes. The sure amount that gives utility of 91 is 91² = $8,281. This person would pay a premium of up to $10,000 − $8,281 = $1,719 to remove the risk. The gap between that and the $990 expected loss is the risk premium, the price of avoiding uncertainty under this utility function.

The St. Petersburg Paradox

Bernoulli’s motivating puzzle is a coin game. The pot starts at $2 and doubles with each head; the first tails ends the game and the player collects the pot. The payout is 2ᵏ dollars with probability (½)ᵏ, where k is the toss on which tails first appears.

Each term of the expected value is (½)ᵏ × 2ᵏ = $1, and there are infinitely many terms, so the expected payout is infinite. A player maximizing expected value should pay any finite price to play. In practice, almost no one would pay more than a modest amount.

Bernoulli’s answer was to weight utilities rather than dollars. With logarithmic utility, the expected utility of the payout is ln(2) × Σ k/2ᵏ = 2 ln(2) = ln(4), a finite number equivalent to a sure $4 (ignoring the player’s existing wealth).

How Expected Utility Is Used

Insurance and risk management. Risk aversion explains why households and firms pay to transfer large losses. The less a loss would hurt relative to wealth, the less a person will pay to avoid it.

Wealth and risk-taking. Hypothetical example. A ticket pays $1 million or nothing with equal odds, and its holder is offered a sure $500,000. With no other wealth and u(w) = √w, keeping it yields 0.5 × √1,000,000 = 500, while selling yields √500,000 ≈ 707, so selling is clearly better. For someone who already has $10 million, the outcomes are $10 million or $11 million. Under the same utility function the two choices come out within a fraction of a percent of each other, so this person is close to indifferent.

Portfolio management. Finance courses, including the CFA curriculum, often use a mean-variance utility score: U = E(r) − ½·A·σ², where E(r) is expected return, σ² is variance and A is the investor’s risk-aversion coefficient. The formula approximates expected utility for an investor who cares about return and volatility, letting portfolios be ranked by a single number.

Expected Utility vs. Related Concepts

Expected income vs. expected utility: a job paying either $40,000 or $80,000 with equal odds has expected income of $60,000. A risk-averse person would rank it below a guaranteed $60,000 salary because the gamble’s expected utility is lower.

Limitations of Expected Utility Theory

The Allais paradox (1953). Maurice Allais designed paired choices in which many people’s preferences violate the theory’s independence axiom, typically by overweighting outcomes that are certain.

Ambiguity. Daniel Ellsberg’s 1961 experiments showed people prefer bets with known probabilities over bets with unknown ones. Expected utility, which needs a probability for every outcome, does not capture that preference.

Modest stakes. Matthew Rabin argued in 1999 that if expected utility explained people turning down small, favorable bets, it would imply absurd aversion to large bets. That makes the theory a poor description of behavior when stakes are small relative to wealth.

Reference points and losses. Prospect theory (Kahneman and Tversky, 1979) proposes that people judge gains and losses relative to a reference point, feel losses more strongly than equal gains, and distort probabilities. Expected utility, defined over final wealth, has none of these features.

Inputs are subjective. The result depends entirely on the probabilities and the utility function chosen. Neither is directly observable, so two analysts can reach different conclusions from the same situation.

Expected utility theory remains the standard normative model of how a consistent decision maker should choose under risk. As a description of how people actually choose, it is a starting point that behavioral economics has repeatedly qualified.

FAQs

The Bottom Line

Expected utility matters to anyone weighing insurance, portfolio choices or other decisions where outcomes are uncertain. It explains why a policy with negative expected value can still leave a risk-averse person better off. The results depend on subjective probabilities and utility functions, and evidence from Allais, Ellsberg and prospect theory shows people often choose differently, so the model works best as a standard for consistent choice.