Expected Utility Calculator

What does this expected utility calculator estimate?

This calculator converts two uncertain monetary outcomes into a single expected utility score. It applies a square-root utility function to each monetary value and then weights that utility by the outcome's probability. Square-root utility is concave: doubling money does not double utility. That property represents diminishing marginal utility and a risk-averse preference pattern. The score itself is expressed in utility units rather than dollars, so it is most useful for comparing scenarios that use the same utility function.

The page also reports the expected monetary value, a normalized certainty equivalent, and a risk premium. Expected monetary value is the probability-weighted average payoff. The certainty equivalent translates expected utility back into a dollar amount: it is the guaranteed payoff that produces the same utility as the uncertain scenario. The risk premium is the amount by which expected monetary value exceeds that certainty equivalent. These measures help separate the size of a possible payoff from the decision-maker's attitude toward uncertainty.

How should each input be used?

Event 1 probability and Event 2 probability

Enter each likelihood as a percentage from 0% to 100%. A higher probability increases that event's contribution to expected utility and expected monetary value. For a complete two-outcome scenario, the probabilities should total 100%. The calculator still shows a raw expected utility when the total differs, but it flags the mismatch and normalizes the dollar comparison metrics by the entered probability total. Common mistakes include typing a decimal such as 0.4 when 40% was intended, assigning overlapping events that can occur together, or omitting a possible outcome.

Event 1 monetary value and Event 2 monetary value

Enter the nonnegative payoff associated with each event in U.S. dollars. Higher values increase both utility and expected monetary value, but utility rises at a decreasing rate because the model uses the square root. For example, moving from $10,000 to $20,000 adds less utility than moving from $0 to $10,000. Use net outcomes measured on a consistent basis. Do not mix revenue for one event with profit for the other, and do not mix nominal and inflation-adjusted dollars. Negative payoffs are not accepted because the square root of a negative monetary value is not a real number under this model.

How are the results calculated?

For each event, the calculator first converts the probability percentage to a decimal. It then computes utility as the square root of the monetary payoff. Weighted utility equals probability multiplied by utility. The expected utility result is the sum of the two weighted utility contributions. With probabilities of 40% and 60% and payoffs of $10,000 and $20,000, the expected utility is approximately 124.85.

Expected monetary value is the probability-weighted average payoff. When probabilities total 100%, the certainty equivalent equals expected utility squared because the inverse of the square-root utility function is squaring. When probabilities do not total 100%, the calculator first divides expected utility by the entered probability total and then applies the inverse utility function. This normalization keeps the certainty-equivalent comparison interpretable without changing the raw expected utility formula.

How should the outputs be interpreted?

Expected utility

A larger expected utility is preferable to a smaller one when comparing alternatives under the same square-root utility function. The absolute number has no direct dollar meaning. A zero result means all weighted utilities are zero, usually because probabilities or monetary values are zero. The most important comparison is between complete scenarios built with consistent inputs.

Expected monetary value

This is the average payoff implied by the probability distribution. It does not incorporate diminishing marginal utility, so it can favor a high-variance alternative even when the certainty equivalent is lower. Expected value is useful as a financial baseline, but it should not be treated as a guaranteed result.

Certainty equivalent and risk premium

The certainty equivalent converts the utility result back to dollars. Under concave utility it will generally be less than or equal to expected monetary value. The gap is the risk premium: the theoretical amount a person with this utility function would give up to replace the uncertain scenario with a guaranteed payoff. A zero risk premium occurs when outcomes are identical, when only one certain outcome remains, or when the scenario otherwise has no dispersion relevant to the model.

Chart and calculation table

The stacked bar shows how much each event contributes to total expected utility. A large segment can come from a high probability, a high payoff, or both. The detail table exposes the exact probability, payoff, square-root utility, weighted contribution, and share for each event. Use it to verify that an apparently dominant outcome is not merely the result of a data-entry error.

What are the model's benefits and limitations?

Expected utility provides a disciplined way to compare uncertain choices while recognizing that money may have diminishing personal value. It is especially useful when alternatives have different payoff distributions. However, the square-root function is only one possible representation of risk preference. Real preferences can vary by wealth, time horizon, liquidity needs, and the nature of the decision. The model also assumes the probabilities are meaningful and that the monetary outcomes capture everything important.

For broader background, review the Stanford Encyclopedia of Philosophy discussion of utility and rational choice, the Investopedia overview of expected utility, and the Investor.gov introduction to investment risk. This calculator is an educational decision-analysis tool, not personalized financial or investment advice.