Expected Utility Calculator

The Expected Utility Calculator compares two uncertain nonnegative monetary outcomes. Enter a probability and value for outcome 1, then a probability and value for outcome 2. The calculator takes the square root of each value, weights each utility by its probability divided by 100, and adds the weighted utilities. It also reports expected monetary value, the utility from each outcome, and the total probability entered.

Expected utility is useful when the average dollar payoff is not enough. A person, household, or company may prefer a steadier result over a volatile one even when both choices have the same expected monetary value. This calculator applies the explicit scenario assumption U(x) = √x to each entered nonnegative monetary outcome; it does not estimate a user’s utility curve or preference. For time-value analysis, pair the result with the present value annuity calculator or compound interest calculator. For the cash-flow consequences of a risky decision, use the budget calculator. For return comparison, see the roi calculator.

What the calculator assumes

The model has two outcomes only. Each outcome has a probability entered as a percent and a nonnegative monetary value. The calculator rejects probabilities below 0% or above 100%, and it rejects negative values. It does not require the two probabilities to add to 100%. Instead, it displays the probability total and marks it positively only when the total is effectively 100%. That design allows a user to inspect incomplete scenarios, but for a normal two-outcome lottery the probabilities should sum to 100%.

The utility function is the square root of the monetary value. Square-root utility is concave: it increases as value increases, but it increases more slowly at higher values. In plain terms, the first dollars matter more than later dollars. On nonnegative x, √x is increasing and concave; here it is a chosen teaching assumption, not an empirically fitted risk preference. It is not a universal description of every decision maker. A risk-neutral model would use dollars directly, while a risk-seeking model would reward upside volatility more heavily.

Formula used by the calculator

Utility from outcome 1 is:

$\text{utility one} = \sqrt{\text{value one}}$

Utility from outcome 2 is:

$\text{utility two} = \sqrt{\text{value two}}$

Expected utility is:

$\text{expected utility} = \frac{\text{probability one}}{100} \times \text{utility one} + \frac{\text{probability two}}{100} \times \text{utility two}$

Expected monetary value is:

$\text{expected monetary value} = \frac{\text{probability one}}{100} \times \text{value one} + \frac{\text{probability two}}{100} \times \text{value two}$

The probability total shown in the result is:

$\text{probability total} = \text{probability one} + \text{probability two}$

Example: comparing two uncertain outcomes

Use the default inputs: outcome 1 has a 60% probability and $10,000 value. Outcome 2 has a 40% probability and $2,500 value. The square root of $10,000 is 100.00, so utility from outcome 1 is 100.00. The square root of $2,500 is 50.00, so utility from outcome 2 is 50.00.

Expected utility equals 0.60 multiplied by 100.00 plus 0.40 multiplied by 50.00. That is 60.00 plus 20.00, or 80.00. Expected monetary value equals 0.60 multiplied by $10,000 plus 0.40 multiplied by $2,500. That is $6,000 plus $1,000, or $7,000. The probability total is 60% plus 40%, or 100.0%, so the calculator treats the total as complete.

Now compare a safer scenario with the same expected monetary value: 100% probability of $7,000 and 0% probability of $0. Expected monetary value is still $7,000. Expected utility is the square root of $7,000, or about 83.67. Under square-root utility, the certain $7,000 has higher utility than the 60/40 gamble, even though the expected dollars are identical. That is the risk-aversion message the calculator is designed to show.

How to use expected utility in decisions

Expected utility is most helpful when outcomes are uncertain and the pain of downside differs from the pleasure of upside. Insurance deductibles, settlement offers, investment payoffs, startup equity, bonus plans, warranties, and project bids can all have this shape. A higher expected monetary value may not be attractive if the bad outcome would force borrowing, missed bills, or unacceptable stress.

The result is a score, not a dollar amount. The expected utility value of 80.00 in the default example is in square-root utility units. It should be compared with other scores calculated using the same utility function and value scale. Do not compare it with dollars, interest rates, or utility scores from a different formula.

Caveats and model limits

This calculator has only two outcomes. Real decisions may have many outcomes, dependencies, taxes, fees, timing differences, and nonfinancial effects. If a project can lose money, the current square-root model is not enough because negative values are invalid. If an investment pays over time, discount the cash flows first or analyze present values before entering outcomes. If probabilities are subjective, document the source and test optimistic and pessimistic cases.

The square-root curve also assumes a particular type of risk aversion. It may be too cautious for a well-capitalized organization and not cautious enough for someone facing a critical bill. Treat the calculator as a structured way to make assumptions visible. The best decision is not always the highest score; it is the choice that fits the decision maker’s constraints, obligations, and preferences.

Sources

• Stanford Encyclopedia of Philosophy, Normative theories of rational choice: expected utility — philosophical and decision-theory background.
• OpenStax, Confronting objections to the economic approach — economic reasoning about preferences, utility, and choices under constraints.