⚖️ Expected Utility Calculator

A gamble can have positive expected dollars and still be irrational to take. Expected utility theory explains why: people value the first dollars more than the last, so a 50/50 chance to win $1,000 or lose $500 hurts more in utility than it helps, even though the dollar expectation is +$250. This calculator finds the certainty equivalent, the guaranteed amount worth exactly as much as the gamble to a risk-averse person.

How to Use This Calculator

The Formula

Log utility is the classic model of risk aversion: each extra dollar adds less utility than the last. The risk premium, expected value minus certainty equivalent, prices your fear. For pure dollar expectation without risk aversion, see our EMV calculator.

Worked Example

With $10,000 of wealth, a 50/50 gamble to gain $1,000 or lose $500 has an expected dollar value of $10,250. But expected utility = 0.5 × ln(11,000) + 0.5 × ln(9,500) ≈ 9.2323, giving a certainty equivalent of e^9.2323 ≈ $10,222.52. The risk premium is $27.48: a guaranteed $10,223 beats the gamble for this investor, even though the gamble's average is $10,250. Risk aversion has a price, and here it is twenty-seven dollars.

Tips for Applying It

Scale matters enormously: the same gamble is trivial at $10 million of wealth and terrifying at $1,000, because utility curves with wealth. Use it for one-off, unrepeatable decisions like job offers or legal settlements, where you cannot rely on the law of large numbers. For repeatable small bets, expected value rules; insurance companies profit precisely because customers rationally pay the risk premium. And remember the model assumes rationality: real humans add loss aversion on top, rejecting even more gambles than log utility predicts.

Common Mistakes

Applying utility to repeatable bets is the classic error; if you face the gamble a thousand times, take every positive-EV one and let averages work. Using wealth instead of total lifetime resources understates risk tolerance for young earners with high human capital. Forgetting that the gamble must be evaluated at the margin: a bet already taken changes wealth and therefore the next bet's utility. And treating the certainty equivalent as a market price: it is personal, varying with each individual's wealth and curvature.

Why Insurance Exists

Insurance is negative-expected-value by design, yet rational people buy it: the certainty equivalent of living uninsured sits below wealth minus the premium. The insurer aggregates thousands of independent risks, converting your scary gamble into their boring average, and pockets the difference between the premium and expected claims. Both sides win in utility terms while the insurer wins in dollars. Expected utility is the only framework where this is not a paradox.

Prospect Theory vs Expected Utility

Kahneman and Tversky showed real humans deviate from expected utility systematically: losses hurt about twice as much as gains please, people overweight tiny probabilities, and framing changes choices. A gamble this calculator approves may still feel terrible, because loss aversion adds a kink the smooth log curve lacks.

Neither theory is "right"; they describe different things. Expected utility is normative, how rational agents should decide; prospect theory is descriptive, how humans actually decide. The wise move is computing the rational answer here, then noticing your emotional resistance as data about your own loss aversion rather than as a veto.

Design around the bias: frame decisions as gains where possible, pre-commit to rational rules, and separate the decision from the outcome. You can know the gamble is correct and still feel sick taking it; the theory prices the risk, experience teaches the stomach.

Utility in Insurance Pricing

Insurers set premiums in the gap between expected claims and customers' certainty equivalents. If expected annual claims are $800 but risk-averse homeowners would pay $1,200 for certainty, any premium between $800 and $1,200 leaves both sides better off in utility terms. Competition squeezes the premium toward cost plus a loading for expenses and profit.

Deductibles are utility engineering: the insurer sheds small claims where administrative cost exceeds value, and the customer retains risk they can bear. Raising your deductible from $500 to $2,500 typically cuts premiums far more than the expected extra out-of-pocket, because you are selling the insurer risk they price highly and you bear cheaply.

This logic inverts for the insurer's own reinsurance: they pay to shed catastrophic tails they cannot bear, at premiums above expected loss. Everyone in the chain buys certainty from someone more diversified, each paying a risk premium that shrinks with scale. Utility theory is the quiet engine of the entire industry.

Estimates for planning only, not financial advice.