eduKateSG · Judgement · The Gold Standard of Education
Compare risky choices without pretending that expected value captures every consequence
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Full chapter index · Independent practice · Judgement subject guide
Expected value averages outcomes using their probabilities. Expected utility first translates each consequence through a utility function, then takes the probability-weighted average. The distinction matters when an extra unit of outcome does not have the same value at every level or when a decision-maker is willing to trade some expected value for less exposure to loss.
Certainty equivalents and risk premiums turn that preference into an interpretable comparison. They are useful decision tools, not universal measurements of personality. A utility model belongs to a defined stakeholder, consequence scale, time and decision.
The worked examples use fictional points and simple functions. They explain the structure; they do not prescribe financial, medical or safety choices.
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Full chapter index
See the mechanism · Chapters 1–2
Build the worked model · Chapters 3–5
Test what changes · Chapters 6–8
Check evidence and limits · Chapters 9–10
Practise and continue · Chapters 11–12
CHAPTER 1 OF 12 · See the mechanism
1. Expected value is a benchmark, not the whole decision
For outcomes x_i with probabilities p_i, expected value is the sum of p_i x_i. A 50–50 lottery paying 0 or 100 points has expected value 50. If repeated many independent times under stable conditions, the average tends toward that value.
A one-off choice can still feel different from receiving 50 for certain. The low outcome may threaten a requirement, while the high outcome has diminishing value. Expected value does not represent those preferences by itself.
Keep probabilities and consequences separate. A vague label such as “high risk” conceals both. Build the outcome table before selecting a rule.
Evidence check: for Expected value is a benchmark, not the whole decision, preserve the observation before the interpretation. Record population, binary action, outcome and time horizon. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
A utility function assigns numbers so that choices consistent with the expected-utility model can be compared by their expected utility. The utility numbers are not dollars, happiness units or moral worth.
Positive affine transformations preserve rankings: replacing u(x) with a+bu(x), where b is positive, changes the scale but not the preferred option. Zero and one can therefore be anchored to convenient reference consequences.
The model needs a single consequence measure or a defensible multiattribute utility construction. Do not hide safety, fairness or irreversible harm inside a convenient monetary total.
Evidence check: for Utility represents ranked tradeoffs, preserve the observation before the interpretation. Record whose tradeoff the threshold represents. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
CHAPTER 3 OF 12 · Build the worked model
3. Risk-neutral, risk-averse and risk-seeking curvature
Linear utility treats expected value and expected utility as equivalent: this is risk neutrality over the modelled range. Concave utility represents risk aversion; the utility of the expected outcome exceeds the expected utility of the lottery.
Convex utility represents risk seeking over that range. Curvature can change with wealth, stakes or consequence domain. Calling a person globally “risk-averse” from one small choice overstates the evidence.
Draw the function and state its domain. A logarithmic utility is undefined for non-positive wealth unless the model is shifted and justified.
Evidence check: for Risk-neutral, risk-averse and risk-seeking curvature, preserve the observation before the interpretation. Record the formula, denominator and threshold weight. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Consider a 50–50 lottery between 0 and 100 points with u(x)=sqrt(x). Expected utility is 0.5×0 + 0.5×10 = 5. The certainty equivalent CE solves sqrt(CE)=5, so CE=25.
The expected value is 50, while the risk premium is EV−CE=25. Under this utility model, receiving 25 for certain is indifferent to the lottery. More than 25 certain is preferred; less than 25 is rejected.
The result comes from the chosen function and range. A linear utility gives CE=50. The attached table compares the assumptions rather than presenting one answer as universal.
| Utility model | Expected utility | Certainty equivalent | Risk premium from EV 50 |
|---|---|---|---|
| Linear u(x)=x | 50 | 50 | 0 |
| Square-root u(x)=√x | 5 | 25 | 25 |
Evidence check: for A worked certainty equivalent, preserve the observation before the interpretation. Record TP and FP counts recalculated at that threshold. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
One method asks for the certain amount that makes a person indifferent to a lottery between defined best and worst outcomes. Repeat at several points to estimate curvature and check consistency.
Choices can be noisy, strategically reported or poorly understood. Use clear probabilities, practice questions and consequence descriptions. Repeat or bracket indifference values instead of forcing false precision.
Observed decisions also include framing, liquidity, regret and ambiguity. A fitted utility curve may absorb effects that belong in separate model components.
Evidence check: for Elicit utility from choices, preserve the observation before the interpretation. Record treat-all, treat-none and current-practice comparators. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
The certainty equivalent puts a lottery on the consequence scale. For a risk-averse model, CE is below expected value and the difference is a positive risk premium. It can be interpreted as the most expected value the model would surrender to remove the risk.
Risk premium is not an insurance market price. Real insurance includes administration, capital, adverse selection, taxes and contract limits. Compare the modelled premium with these real features separately.
For non-monetary consequences, a certainty equivalent may be expressed in the same performance unit only if the utility mapping is meaningful and ethically acceptable.
Evidence check: for Certainty equivalents and risk premiums, preserve the observation before the interpretation. Record the relevant range and uncertainty around differences. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Option A yields 40 for certain. Option B yields 0 or 100 with equal probability. Expected values are 40 and 50, so expected value favours B. Under square-root utility, utilities are about 6.32 for A and 5 for B, so expected utility favours A.
The disagreement is useful: it identifies the curvature assumption that changes the choice. Do not average the rankings. Ask whether the utility function represents the stakeholder and stakes.
Test nearby probabilities and outcomes. A choice supported only by one fragile parameter deserves review or a staged option.
| Option | Outcomes | Expected value | Expected utility with √x |
|---|---|---|---|
| A | 40 certain | 40 | 6.32 |
| B | 0 or 100 equally likely | 50 | 5.00 |
| Preferred | — | B | A |
Evidence check: for Compare two risky options, preserve the observation before the interpretation. Record calibration in the intended validation population. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
CHAPTER 8 OF 12 · Test what changes
8. Wealth, reference points and repeated decisions
Utility for money is often modelled over final wealth rather than isolated gains. The same 100-point risk can be small for one decision-maker and consequential for another.
Repeated independent risks can diversify, but only when exposures and constraints permit. A business facing bankruptcy cannot rely on long-run averages it may never survive to observe.
State the time horizon, starting resources and whether outcomes are additive. Correlated losses defeat simple diversification stories.
Evidence check: for Wealth, reference points and repeated decisions, preserve the observation before the interpretation. Record target prevalence and sampling weights. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
If one option is at least as good in every state and better in some, it dominates another under increasing utility. Remove dominated options before debating fine differences in curvature.
First-order stochastic dominance provides a broader check across outcome distributions. A model that prefers a stochastically dominated option may have a coding, probability or utility error.
Inspect units, probability totals and impossible outcomes. Decision analysis gains credibility from simple checks before advanced calculations.
Evidence check: for Dominance and model checks, preserve the observation before the interpretation. Record resampling boundaries and external validation. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Expected utility is a normative and descriptive model under specific axioms. Real choices can violate those axioms through framing, probability weighting, loss aversion, ambiguity or inconsistent preferences.
That does not make the model useless. It makes assumptions visible and creates a baseline for diagnosing why observed choice differs. Do not rename every deviation irrationality.
For public or group decisions, one person’s utility function cannot automatically stand for everyone. Distribution, rights and governance require explicit treatment.
Separate model disagreement from value disagreement. Two analysts may share the same utility curve but estimate different probabilities; or agree on probabilities and disagree about consequences. Run those cases separately. A sensitivity table that changes probability, utility and outcome together cannot show which assumption drives the decision.
Keep the original consequence table beside every transformed utility table. Reviewers should be able to see the real outcomes and probabilities before accepting the compressed ranking.
Utility elicitation is not a one-time personality test. Repeat important questions with equivalent probability frames, reverse the option order and check whether small wording changes create large shifts. When they do, retain an uncertainty band over the utility parameters. A decision that remains preferred across that band is more defensible than one chosen from a single precise-looking curve.
Record who approved the consequence scale and which outcomes were excluded. A mathematically consistent calculation can still answer the wrong question if the decision boundary, stakeholder or feasible alternatives changed after the model was built. Reopen the structure before adjusting decimals.
Evidence check: for Limits and behavioural evidence, preserve the observation before the interpretation. Record capacity, downstream harms and missing utilities. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Practice one: compare 30 for certain with a 50–50 lottery of 0 or 100 under linear and square-root utility. Show both rankings and the switching certainty equivalent.
Practice two: a fitted utility function changes sharply when questions are framed as losses. Record the reference point and analyse gain and loss domains separately rather than averaging them.
Practice three: two options have equal expected utility, but one can breach a safety threshold. Add the constraint before ranking; do not let compensating utility override a non-negotiable requirement.
Transfer the framework to time, reliability or learning outcomes only after defining a consequence scale and stakeholder. The formula travels more easily than its meaning.
Evidence check: for Independent practice and transfer, preserve the observation before the interpretation. Record which changed context reverses the preferred rule. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Is expected utility better than expected value?
It answers a different question by representing consequence preferences. Its quality depends on probabilities, outcomes and a defensible utility model.
Is the certainty equivalent guaranteed to be unique?
With a continuous strictly increasing utility function over the relevant range, it is. Flat or discontinuous preferences can complicate the result.
Does concavity prove someone is always risk-averse?
It represents risk aversion over the modelled consequence range. Preferences can change with context and stakes.
Can expected utility decide ethical tradeoffs?
Not by itself. Rights, constraints, distribution and stakeholder governance may not be compensable on one utility scale.
Authoritative starting points include John Pratt’s original paper Risk Aversion in the Small and in the Large, NASA’s Risk-Informed Decision Making Handbook, and the NASA systems-engineering guidance on decision analysis. Use the responsible professional framework for real financial, safety or policy decisions.
Continue with How Expected Value Works, robust decisions and switching assumptions, and decision curves and net benefit. The next useful move is to calculate one certainty equivalent under two plausible utility functions and identify which assumption changes the preferred option.
Evidence check: for FAQ, sources and the next useful move, preserve the observation before the interpretation. Record the decision sentence that the curve is meant to support. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.