Expected Utility

Expected Utility: Concept Architecture

Orientation and learning roadmap

Expected utility connects uncertain outcomes, their probabilities, and preferences over consequences within a coherent choice framework. The architecture below develops that framework from lotteries and utility functions through risk attitudes, health applications, estimation, interpretation, and limitations.

What expected utility evaluates

An uncertain option can lead to several mutually exclusive outcomes, each with a probability and a utility. Expected utility combines those outcome utilities into one value so that alternative lotteries can be ranked. The preferred option is the one with the highest expected utility, not necessarily the highest expected monetary value or expected clinical outcome.

For a lottery with outcomes (x_1,\ldots,x_n), probabilities (p_1,\ldots,p_n), and utility function (u(\cdot)):

[ EU=\sum_{i=1}^{n}p_i u(x_i),\qquad \sum_{i=1}^{n}p_i=1 ]

Expected utility first transforms each consequence through the utility function and then averages the transformed values. In general,

[ E[u(X)]\neq u(E[X]) ]

because the utility function may be nonlinear. That difference is central to the representation of risk attitudes.

Structure the decision before calculating

Expected-utility analysis separates choices from uncertain events and final consequences. The alternatives must be feasible, the outcomes should capture everything valued by the decision maker, and the probabilities should be conditional on the chosen action and available information. A calculation cannot repair an incomplete decision structure.

Probabilities should sum to one across mutually exclusive and collectively exhaustive outcomes for each action. Utilities should apply to final consequences rather than to labels that omit relevant cost, health, time, burden, or distributional effects.

Expected utility rests on preference axioms

The von Neumann–Morgenstern representation does not assume that every observed choice actually follows expected utility. It states that preferences over lotteries can be represented by expected utility when they satisfy specified consistency conditions. The exact formal statements vary in presentation, but completeness, transitivity, continuity, and independence are central.

• Completeness requires the decision maker to rank any two lotteries or regard them as indifferent.
• Transitivity requires consistent ordering: if A is preferred to B and B to C, then A is preferred to C.
• Continuity requires that intermediate lotteries can represent indifference between better and worse outcomes under appropriate conditions.
• Independence requires a preference between two lotteries to remain unchanged when each is mixed with the same third lottery in the same proportion.

These axioms give expected utility its normative coherence and representation result. Empirical choices may violate them through framing, probability weighting, regret, ambiguity, or other behavioural effects, so analysts should distinguish a normative decision aid from a descriptive claim about actual behaviour.

Utility scales are interval scales

A von Neumann–Morgenstern utility function represents preference ordering under risk and is unique only up to a positive affine transformation. If (u(x)) represents the preferences, then (v(x)=a+bu(x)), where (b>0), represents the same choices. Utility levels and ratios therefore have no independent meaning without anchors and scale conventions.

For any lottery (L):

[ E[v(L)]=a+bE[u(L)] ]

Because (b>0), the transformation preserves the ranking of expected utilities. Adding utilities across people or interpreting one person's utility as twice another's requires assumptions beyond individual expected-utility theory.

Risk attitudes come from curvature

When outcomes are wealth or another ordered good, the shape of (u(x)) represents attitudes toward risk. A concave utility function indicates risk aversion, a linear function indicates risk neutrality, and a convex function indicates risk seeking over the relevant range. Risk attitude is defined relative to the outcome and utility specification, not as a fixed label for every decision a person faces.

For a twice-differentiable utility function with (u'(x)>0), absolute risk aversion can be described locally by the Arrow–Pratt measure:

[ A(x)=-\frac{u''(x)}{u'(x)} ]

Relative risk aversion is:

[ R(x)=-x\frac{u''(x)}{u'(x)} ]

These measures describe local curvature and should not be applied outside the domain or outcome scale for which the utility function is credible.

Certainty equivalent and risk premium

The certainty equivalent is the sure outcome that gives the same utility as a lottery. It translates expected utility back onto the outcome scale, which makes risk attitude easier to interpret. For a strictly increasing utility function, the certainty equivalent (CE) satisfies:

[ u(CE)=E[u(X)] ]

and therefore:

[ CE=u^{-1}(E[u(X)]) ]

The risk premium for a gain lottery is commonly written as:

[ \pi=E[X]-CE ]

A risk-averse decision maker has (CE<E[X]) and a positive risk premium for a non-degenerate lottery. The premium is the maximum expected-value sacrifice the decision maker would accept to replace the risky option with certainty under the specified utility function.

A worked monetary example

Suppose a decision maker has utility (u(w)=\sqrt{w}) over wealth and chooses between a sure £100 and a lottery that pays £225 with probability 0.5 and £25 with probability 0.5. The lottery and sure option have the same expected monetary value, but their expected utilities differ because the function is concave.

[ E[X]=0.5(£225)+0.5(£25)=£125 ]

The lottery's expected utility is:

[ EU(L)=0.5\sqrt{225}+0.5\sqrt{25}=0.5(15)+0.5(5)=10 ]

Its certainty equivalent is:

[ CE=10^2=£100 ]

The risk premium is:

[ \pi=£125-£100=£25 ]

The lottery is equivalent in utility to a sure £100 even though its expected monetary value is £125. A decision maker with this utility is therefore indifferent between the lottery and £100, illustrating why expected value alone does not represent risk-averse preferences.

Apply expected utility to a treatment decision

Clinical decisions may involve uncertain survival, recovery, adverse events, disability, treatment burden, and cost. Expected-utility analysis can combine those outcomes when probabilities and utilities are defined for complete consequences. The result is decision-maker specific unless a legitimate social aggregation rule is supplied.

Suppose treatment A gives full recovery with probability 0.70, moderate disability with probability 0.25, and severe adverse harm with probability 0.05. If the corresponding utilities are 1.00, 0.60, and -0.20:

[ EU(A)=0.70(1.00)+0.25(0.60)+0.05(-0.20)=0.84 ]

Suppose treatment B gives utilities 1.00, 0.60, and -0.20 with probabilities 0.60, 0.38, and 0.02:

[ EU(B)=0.60(1.00)+0.38(0.60)+0.02(-0.20)=0.824 ]

Treatment A has the higher expected utility under these inputs. The decision remains conditional on the completeness of the outcomes, the validity of the probabilities, the source and scale of the utilities, and whether the negative utility for severe harm is defensible.

Expected utility is not expected value

Expected value averages outcomes in their natural or monetary units. Expected utility averages the utilities of those outcomes. They produce the same ranking only when utility is linear in the outcome over the relevant range or when special features of the alternatives make the distinction irrelevant.

Using a probability-weighted monetary payoff is expected monetary value, not expected utility unless money itself is the defined utility scale or a linear utility function is explicitly justified. Likewise, probability-weighted QALYs are expected QALYs; they are not automatically a complete expected-utility representation of risk preferences over lifetime health profiles.

Distinguish expected utility from health-state utility

Health economics uses the word utility in more than one way. A health-state utility value is a preference-based weight used to value time in a health state, often for QALY calculation. Expected utility is a theory for ranking lotteries over uncertain consequences under specified axioms.

• A health-state utility value can be one component of a consequence used in an expected-utility model.
• A QALY calculation usually assumes a particular way of combining quality and time and may not capture risk attitude over uncertain lifetime profiles.
• A utility elicited with a standard gamble is linked historically to choice under risk, but interpreting it as a general health-state weight requires additional assumptions.
• Population-average health-state utilities do not automatically represent one coherent social expected-utility function.

Analysts should name the object being valued: a momentary health state, a health profile over time, wealth, consumption, net benefit, or a complete consequence. Reusing the same symbol (u) for these different objects without explanation can hide important assumptions.

The standard gamble connection

The standard gamble elicits a value by finding the probability at which a respondent is indifferent between a certain health outcome and a gamble between better and worse anchor outcomes. Under expected-utility assumptions and normalized anchor utilities, the indifference probability can identify the utility assigned to the certain outcome. The elicited value is sensitive to the gamble design, anchors, risk attitude, probability understanding, and framing.

If full health has utility 1 and death has utility 0, indifference between living in health state (H) for certain and a lottery giving full health with probability (p) and death with probability (1-p) implies:

[ u(H)=p(1)+(1-p)(0)=p ]

This equality relies on the elicitation model and anchors. It should not be treated as a purely factual property of (H), independent of duration, respondent, context, and the assumptions connecting the gamble to the health-state scale.

Decisions under risk and ambiguity

Classical expected utility under risk assumes probabilities are known or treated as specified. Many health decisions involve ambiguity, where probabilities are disputed, weakly identified, or represented by competing models. Replacing ambiguity with one unexamined probability distribution can create a false sense of precision.

Bayesian analysis can use subjective probabilities and update them with evidence, while robust or ambiguity-sensitive approaches can examine sets of plausible distributions. Expected utility can still be used within a probabilistic model, but the source and interpretation of the probabilities must be explicit.

Population and policy decisions require more assumptions

Individual expected utility ranks risky prospects for one decision maker. Health policy usually affects many people, so utilities, health gains, costs, and risks must be aggregated. A policy objective such as maximizing expected QALYs or net monetary benefit is not derived solely from individual expected-utility axioms.

Distributional weights, inequality aversion, priority to severe illness, financial risk protection, and rights may matter alongside aggregate expected outcomes. Analysts should distinguish the individual's risk preference from society's attitude to distributing health and resources across people.

Implement the calculation transparently

A spreadsheet or decision tree can calculate expected utility when actions, outcomes, probabilities, and utilities are clearly separated. Each terminal consequence should appear once, probabilities should follow the correct action-specific path, and utilities should use a consistent scale. The model should expose rather than bury the assumptions about risk attitude.

For continuous outcomes with density (f(x)), expected utility becomes:

[ EU=\int u(x)f(x),dx ]

Numerical integration or simulation may be required when the distribution or utility function is complex. Monte Carlo estimates should be checked for convergence, especially when rare catastrophic outcomes materially affect expected utility.

Validate the analysis

Validation must cover probabilities, utilities, outcomes, formulas, and preference assumptions. A correct weighted sum can still answer the wrong question if consequences are incomplete or the utility function is used outside its intended domain. Small test problems and dominance checks provide strong safeguards.

1. Check probabilities. Confirm that outcome probabilities are non-negative and sum to one for each action.
2. Check consequence completeness. Ensure that each outcome includes all attributes relevant to the decision maker or explicitly states exclusions.
3. Check the utility scale. Confirm anchors, units, respondent, domain, and any transformation applied.
4. Check dominance. An option that produces outcomes at least as good in every state and better in some should not receive lower expected utility under increasing utility.
5. Check affine invariance. Verify that a positive affine transformation does not change the ranking.
6. Check manual examples. Recalculate a small decision tree independently.
7. Check sensitivity. Vary probabilities, utility curvature, anchors, and disputed outcomes.
8. Check behavioural evidence. When the model claims to predict choices, compare predicted and observed decisions rather than relying only on normative axioms.

Common violations and alternatives

Observed choices often display patterns inconsistent with standard expected utility, including certainty effects, probability weighting, loss aversion, reference dependence, and ambiguity aversion. These findings do not make expected utility mathematically incoherent; they limit its descriptive accuracy for particular contexts. Alternative models add structure but also introduce parameters and identification challenges.

Prospect theory, rank-dependent utility, regret models, and ambiguity-sensitive models may describe some choices better. Their use should be justified by the decision purpose and evidence rather than selected after seeing which model produces a preferred result.

Common errors and safeguards

Expected utility is vulnerable to category errors because utility, value, QALY weight, and net benefit are sometimes used interchangeably. The main safeguard is to state exactly what outcomes the utility function maps, whose preferences it represents, and which axioms support the calculation. The analysis should remain interpretable before numerical complexity is added.

• Do not calculate expected utility by applying utility only to the expected outcome.
• Do not assume utility differences or ratios are meaningful beyond positive affine transformations.
• Do not use a population mean utility as though it were one individual's complete preference function without justification.
• Do not describe expected QALYs as risk-adjusted expected utility unless the required assumptions are stated.
• Do not infer stable risk aversion from one gamble or transfer a utility function across unrelated outcome domains.
• Do not omit rare severe outcomes because their probability is small.
• Do not mix utilities elicited with different anchors or durations without valid rescaling.
• Do not claim that expected utility predicts actual behaviour when the analysis is being used only as a normative decision rule.

Interpreting expected utility responsibly

Expected utility provides a coherent way to rank risky alternatives when the decision structure, probabilities, outcome utilities, and preference axioms are defensible. It makes the treatment of risk explicit and allows certainty equivalents and risk premia to be calculated. Its apparent precision should not conceal incomplete outcomes, disputed probabilities, interpersonal aggregation, or empirical violations of the model.

A decision-ready report identifies the decision maker, actions, outcome domain, probabilities, utility function, elicitation source, anchors, risk attitude, uncertainty, sensitivity analyses, and any departures from expected-utility behaviour. In health economics, it should also state clearly whether utility refers to preferences over risky prospects, health-state values for QALYs, or a broader social objective.