How this instrument works
Expected utility reshapes a payoff through a curve before averaging it, instead of averaging the raw dollars. Here the curve is a square root: it takes each outcome's payoff, pulls its square root, multiplies by the outcome's probability, and adds the two branches together. Because a square root rises fast at small numbers and flattens as numbers grow, a $10,000 payoff does not count twice as much toward the total as a $2,500 one — the formula treats extra dollars as worth less once you already have some.
Daniel Bernoulli proposed this kind of curve in 1738 to resolve the St. Petersburg paradox, a game whose plain average payout is infinite yet that almost nobody would pay much to enter. His insight — later formalized by von Neumann and Morgenstern — was that people do not price a gamble by its raw expected dollars; they price it by the expected value of some concave function of those dollars, which is exactly what this sheet computes for a two-outcome wager with a square-root curve standing in for that function.
A square root is a convenient, teachable stand-in for diminishing marginal utility, not a universal law. Actuaries, options theorists, and behavioral economists reach for it (or close cousins like logarithmic or exponential curves) to explain why a household buys insurance at a price above its expected claims, or why an investor accepts a lower average return for a steadier one. The curve is a modeling choice — pick a flatter or steeper one and the same two payoffs produce a different expected utility, though the ranking between two options tested under the identical curve stays meaningful.
- Enter Probability of outcome 1 (0-1) and Payoff of outcome 1, $ for the first branch of the wager.
- Enter Probability of outcome 2 (0-1) and Payoff of outcome 2, $ for the second branch.
- Confirm the two probabilities together cover the whole gamble — the formula will not flag it if they do not.
- Read Expected utility (√ utility function) and square it to get the certainty-equivalent dollar figure.
- Compare that reading against a second gamble's expected utility, never against a raw expected-value number.
Worked example — a fair $10,000 coin flip
Take a fair coin flip that pays $10,000 on heads and $0 on tails: p1 = 0.5, payoff1 = $10,000, p2 = 0.5, payoff2 = $0, entered exactly as the wager is framed. The square root of $10,000 is 100 and the square root of $0 is 0, so the instrument computes EU = 0.5 times 100 plus 0.5 times 0, which equals 50.0.
That 50.0 sits well below the utility of receiving the coin flip's own $5,000 average outright, since the square root of $5,000 is about 70.7. The gap between 50.0 and 70.7 is the arithmetic reason a risk-averse decision-maker under this curve would trade the coin flip for a guaranteed $2,500 (50.0 squared) or even somewhat more, despite both the flip and the $5,000 having the identical expected dollar value.
Questions
How is expected utility different from expected monetary value?
Expected monetary value multiplies each payoff by its probability and adds the raw dollar results; expected utility runs each payoff through a curve — here a square root — before doing the same weighted sum. The curve bends large payoffs down relative to small ones, so a $10,000 outcome no longer contributes exactly four times a $2,500 outcome the way it would in plain dollars. That bend is the entire source of the gap between the two totals for the same gamble.
Why use a square root instead of the dollar amount itself?
Because money has diminishing marginal value for most people: the jump from $0 to $10,000 tends to matter more than the jump from $90,000 to $100,000, even though both are $10,000. A square root is a simple curve with exactly that shape — steep near zero, flat further out — which is why it has stood in for risk aversion in economics since Bernoulli's 1738 essay on the St. Petersburg paradox.
What does the certainty equivalent tell me that expected utility alone does not?
Squaring the expected utility answer converts it back into dollars — the guaranteed payment that feels exactly as good as the gamble under this curve. An expected utility of 50.0 squares to $2,500, well under the gamble's own $5,000 average payout; that $2,500 gap is effectively the price of bearing the risk rather than taking a sure thing.
Can either payoff be negative?
No — a square root has no real value for a negative number, so this formula only accepts payoffs of zero or more. If your decision includes a loss branch, a square-root curve cannot price it; a utility function built for both gains and losses, such as one from prospect theory, would be needed instead.
Who actually uses expected utility outside a textbook?
Insurance actuaries lean on it to explain why a household willingly pays a premium above its own expected claims and still comes out ahead once risk is priced in; options and portfolio theorists use related curves to model why investors accept a lower average return in exchange for a steadier one. This two-outcome sheet is the simplest version of that reasoning — real pricing work usually chains many outcomes against a curve fitted to a specific client's risk tolerance.
Does a higher expected utility always mean the better choice?
Only when comparing options scored with the same curve and the same decision-maker's risk tolerance. Swap the square root for a flatter or steeper curve and the ranking between two gambles can flip, since each curve prices diminishing returns differently. Expected utility numbers from two different curves are not comparable to each other — only to other results from the identical formula.
References
- MIT OpenCourseWare — Uncertainty, Principles of Microeconomics
- Stanford Encyclopedia of Philosophy — Expected Utility
Read this first: This instrument shows arithmetic, not advice. Real offers add fees, taxes and terms that vary by lender and place — verify the figures against your actual paperwork before deciding anything.