Expected Utility Calculator
With the utility function U(x)=√x, compute the expected utility of two uncertain outcomes: EU = Σ probability × √amount.
Input Data
Results
At a glance:Expected Utility is the probability-weighted sum of the utilities of uncertain outcomes. This calculator uses the concave U(x)=√x, reflecting risk aversion and diminishing marginal utility: EU = Σ probability × √amount. A concave function 'penalises' dispersion (risk), so a certain amount can beat a risky gamble with the same expected value.
Formula
EU = probability₁ × √amount₁ + probability₂ × √amount₂.
U(x)=√x is concave, showing risk aversion and diminishing marginal utility.
Probabilities in decimal (40%=0.4); the two usually sum to 100%.
$$EU = \sum_i p_i \, U(x_i) = p_1\sqrt{x_1} + p_2\sqrt{x_2}$$How to Use
- Enter the first outcome's probability and amount.
- Enter the second outcome's probability and amount.
- View the expected utility under the √ utility function.
Expected utility examples (U(x)=√x)
| Case | Outcome 1 (p×amt) | Outcome 2 (p×amt) | EU | EV |
|---|---|---|---|---|
| Risky gamble | 40% × 10,000 | 60% × 20,000 | 124.85 | 16,000 |
| Fair gamble | 50% × 0 | 50% × 20,000 | 70.71 | 10,000 |
| Certain 16,000 | 100% × 16,000 | 0% × 0 | 126.49 | 16,000 |
EU = Σ p×√amount. Compare 'risky gamble' vs 'certain 16,000': both EV = 16,000, but the certain option's EU (126.49) exceeds the gamble's (124.85) — that is risk aversion.
Case Studies
Case 1: Expected utility of a risky gamble
An investment with two outcomes: 40% to get 10,000, 60% to get 20,000. U(x)=√x.
EU = 0.4×√10,000 + 0.6×√20,000 = 0.4×100 + 0.6×141.42 = 40 + 84.85 ≈ 124.85.
By contrast the gamble's EV = 0.4×10,000 + 0.6×20,000 = 16,000. EU 124.85 and EV 16,000 are different scales: EV measures average money; EU measures the decision-maker's satisfaction from the uncertain gain, already embedding risk attitude via the concave function.
Case 2: Risk aversion and certainty equivalent
From above, gamble EU = 124.85. Invert to amount (certainty equivalent CE): CE = EU² = 124.85² ≈ 15,588.
Meaning: to this U(x)=√x decision-maker, a '16,000-EV risky gamble' gives the same satisfaction as 'certainly getting 15,588'. He would give up 16,000 − 15,588 = 412 (risk premium) to gain certainty and avoid risk.
More extreme: a fair gamble '50% get 0, 50% get 20,000', EV also 10,000, but EU = 0.5×0 + 0.5×√20,000 ≈ 70.71, CE = 70.71² ≈ 5,000 — far below 10,000. The concave utility weighs downside pain more than upside joy — the mathematical form of risk aversion, and why insurance and diversification have value.
FAQ
Expected utility vs expected value?
Expected value weights the amounts by probability directly; expected utility first converts each amount via a utility function into 'satisfaction' then weights. Because the utility is usually concave, expected utility reflects risk aversion rather than merely chasing the highest average amount.
Why use the square root as the utility function?
U(x)=√x is one of the most common concave utilities — simple and it captures diminishing marginal utility: the more wealth, the less extra satisfaction each extra dollar brings. In practice log or power functions are also used.
Must the two probabilities sum to 100%?
With two mutually exclusive outcomes, they should sum to 100%. If you have more outcomes, sum each (probability × √amount); this calculator focuses on the common two-outcome case.
How do expected utility and expected value differ?
Expected Value (EV) and Expected Utility (EU) are easily confused but fundamentally different: EV weights the amounts by probability, answering 'on average how much money'; EU first converts each amount via U(x) into 'utility (satisfaction)' then weights by probability, answering 'how much satisfaction this uncertain gain brings on average'. The key is that U is usually concave (here U(x)=√x), reflecting diminishing marginal utility — an extra dollar when rich pleases far less than when poor. So the same gamble evaluated two ways can differ: EV only sees average money and is blind to risk; EU, because of concavity, penalises uncertainty — the more dispersed (riskier), the lower the EU. That explains a key phenomenon: a gamble with positive or high EV may still be rejected by a rational risk-averse person (its EU is below the status quo). Decision theory holds that people maximise expected utility, not expected value — that is why people buy insurance (negative EV yet willing), diversify, and prefer caution on big bets.
Why does a concave utility mean risk aversion?
The equivalence 'concave utility ⇔ risk aversion' is the core insight; the key is that a concave function 'rises ever more slowly'. As amount x grows, utility U keeps rising but each extra dollar's increment (marginal utility) shrinks — the satisfaction from 0 to 10,000 far exceeds that from 10,000 to 20,000. This diminishing marginal utility directly causes risk aversion. Imagine a fair gamble: 50% to gain W, 50% to lose the same W (EV=0). For a concave utility, the utility gained from winning W is smaller than the utility lost from losing W (loss hits the steeper part of the curve). Weighted, this fair gamble's EU is negative — even at zero expected value, bearing the risk lowers satisfaction, so the rational person rejects it and keeps the status quo. Graphically, the straight line joining the two outcome points (EU) lies below the utility curve (certain amount's utility); the gap is the utility loss from risk, and its money equivalent is the risk premium (what you pay to avoid risk). Thus: a straight-line utility (fixed marginal utility) = risk neutral (EV only); convex = risk seeking (loves gambles); the curve's bend and degree precisely depict one's risk attitude.
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References
Content review: Calculatorism Finance Team. Results are for reference only; please refer to the relevant authorities for the official figures.